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.. default-domain:: dynare
.. |br| raw:: html
<br>
.. _model-file:
##############
The model file
##############
.. _conv:
Conventions
===========
A model file contains a list of commands and of blocks. Each command
and each element of a block is terminated by a semicolon (;). Blocks
are terminated by ``end;``.
If Dynare encounters an unknown expression at the beginning of a line
or after a semicolon, it will parse the rest of that line as native
MATLAB code, even if there are more statements separated by semicolons
present. To prevent cryptic error messages, it is strongly recommended
to always only put one statement/command into each line and start a
new line after each semicolon. [#feol]_
Lines of codes can be commented out line by line or as a block. Single-line
comments begin with ``//`` and stop at the end of the line. Multiline comments
are introduced by ``/*`` and terminated by ``*/``.
*Examples*
::
// This is a single line comment
::
var x; // This is a comment about x
::
/* This is another inline comment about alpha */ alpha = 0.3;
::
/*
This comment is spanning
two lines.
*/
Note that these comment marks should not be used in native MATLAB code regions
where the `%` should be preferred instead to introduce a comment. In a
``verbatim`` block, see :ref:`verbatim`, this would result in a crash since
``//`` is not a valid MATLAB statement).
Most Dynare commands have arguments and several accept options,
indicated in parentheses after the command keyword. Several options
are separated by commas.
In the description of Dynare commands, the following conventions are
observed:
* Optional arguments or options are indicated between square brackets:
‘[]’;
* Repeated arguments are indicated by ellipses: “...”;
* Mutually exclusive arguments are separated by vertical bars: ‘|’;
* INTEGER indicates an integer number;
* INTEGER_VECTOR indicates a vector of integer numbers separated by
spaces, enclosed by square brackets;
* DOUBLE indicates a double precision number. The following syntaxes
are valid: ``1.1e3``, ``1.1E3``, ``1.1d3``, ``1.1D3``. In some
places, infinite Values ``Inf`` and ``-Inf`` are also allowed;
* NUMERICAL_VECTOR indicates a vector of numbers separated by spaces,
enclosed by square brackets;
* EXPRESSION indicates a mathematical expression valid outside the
model description (see :ref:`expr`);
* MODEL_EXPRESSION (sometimes MODEL_EXP) indicates a mathematical
expression valid in the model description (see :ref:`expr` and
:ref:`model-decl`);
* MACRO_EXPRESSION designates an expression of the macro processor
(see :ref:`macro-exp`);
* VARIABLE_NAME (sometimes VAR_NAME) indicates a variable name
starting with an alphabetical character and can’t contain:
‘()+-\*/^=!;:@#.’ or accentuated characters;
* PARAMETER_NAME (sometimes PARAM_NAME) indicates a parameter name
starting with an alphabetical character and can’t contain:
‘()+-\*/^=!;:@#.’ or accentuated characters;
* LATEX_NAME (sometimes TEX_NAME) indicates a valid
LaTeX expression in math mode (not including the
dollar signs);
* FUNCTION_NAME indicates a valid MATLAB function name;
* FILENAME indicates a filename valid in the underlying operating
system; it is necessary to put it between quotes when specifying the
extension or if the filename contains a non-alphanumeric character;
* QUOTED_STRING indicates an arbitrary string enclosed between (single)
quotes.
.. _var-decl:
Variable declarations
=====================
While Dynare allows the user to choose their own variable names, there
are some restrictions to be kept in mind. First, variables and
parameters must not have the same name as Dynare commands or built-in
functions. In this respect, Dynare is not case-sensitive. For example,
do not use ``Ln`` or ``shocks`` to name your variable. Not conforming
to this rule might yield hard-to-debug error messages or
crashes. Second, when employing user-defined steady state files it is
recommended to avoid using the name of MATLAB functions as this may cause
conflicts. In particular, when working with user-defined steady state files, do not
use correctly-spelled greek names like ``alpha``, because there are
MATLAB functions of the same name. Rather go for ``alppha`` or
``alph``. Lastly, please do not name a variable or parameter
``i``. This may interfere with the imaginary number i and the index in
many loops. Rather, name investment ``invest``. Using ``inv`` is also
not recommended as it already denotes the inverse operator. Commands
for declaring variables and parameters are described below.
.. command:: var VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)]...;
var(log) VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)]...;
var(deflator=MODEL_EXPR) VAR_NAME (... same options apply)
var(log, deflator=MODEL_EXPR) VAR_NAME (... same options apply)
var(log_deflator=MODEL_EXPR) VAR_NAME (... same options apply)
|br| This required command declares the endogenous variables in
the model. See :ref:`conv` for the syntax of *VAR_NAME* and
*MODEL_EXPR*. Optionally it is possible to give a
LaTeX name to the variable or, if it is
nonstationary, provide information regarding its deflator. The variables in
the list can be separated by spaces or by commas. ``var``
commands can appear several times in the file and Dynare will
concatenate them. Dynare stores the list of declared parameters,
in the order of declaration, in a column cell array
``M_.endo_names``.
If the model is nonstationary and is to be written as such in the
``model`` block, Dynare will need the trend deflator for the
appropriate endogenous variables in order to stationarize the
model. The trend deflator must be provided alongside the variables
that follow this trend.
*Options*
.. option:: log
In addition to the endogenous variable(s) thus declared, this option
also triggers the creation of auxiliary variable(s) equal to the log of
the corresponding endogenous variable(s). For example, given a
``var(log) y`` statement, two endogenous will be created (``y`` and
``LOG_y``), and an auxiliary equation linking the two will also be
added (equal to ``LOG_y = log(y)``). Moreover, every occurence of ``y``
in the model will be replaced by ``exp(LOG_y)``. This option is for
example useful when one wants to perform a loglinear approximation of
some variable(s) in the context of a first-order stochastic
approximation; or when one wants to ensure the variable(s) stay(s) in
the definition domain of the function defining the steady state or the
dynamic residuals when the nonlinear solver is used.
.. option:: deflator = MODEL_EXPR
The expression used to detrend an endogenous variable. All
trend variables, endogenous variables and parameters
referenced in MODEL_EXPR must already have been declared by
the ``trend_var, log_trend_var, var`` and ``parameters``
commands. The deflator is assumed to be multiplicative; for an
additive deflator, use ``log_deflator``.
This option can be used together with the ``log`` option (the latter
must come first).
.. option:: log_deflator = MODEL_EXPR
Same as ``deflator``, except that the deflator is assumed to
be additive instead of multiplicative (or, to put it
otherwise, the declared variable is equal to the log of a
variable with a multiplicative trend).
This option cannot be used together with the ``log`` option, because it
would not make much sense from an economic point of view (the
corresponding auxiliary variable would correspond to the log taken
two times on a variable with a multiplicative trend).
.. _long-name:
.. option:: long_name = QUOTED_STRING
This is the long version of the variable name. Its value is
stored in ``M_.endo_names_long`` (a column cell array, in the
same order as ``M_.endo_names``). In case multiple
``long_name`` options are provided, the last one will be
used. Default: ``VAR_NAME``.
.. _partitioning:
.. option:: NAME = QUOTED_STRING
This is used to create a partitioning of variables. It results
in the direct output in the ``.m`` file analogous to:
``M_.endo_partitions.NAME = QUOTED_STRING``;.
*Example (variable partitioning)*
::
var c gnp cva (country='US', state='VA')
cca (country='US', state='CA', long_name='Consumption CA');
var(deflator=A) i b;
var c $C$ (long_name=`Consumption');
.. command :: varexo VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];
|br| This optional command declares the exogenous variables in the
model. See :ref:`conv` for the syntax of ``VAR_NAME``. Optionally
it is possible to give a LaTeX name to the
variable. Exogenous variables are required if the user wants to be
able to apply shocks to her model. The variables in
the list can be separated by spaces or by commas. ``varexo`` commands can appear
several times in the file and Dynare will concatenate them.
*Options*
.. option:: long_name = QUOTED_STRING
Like :ref:`long_name <long-name>` but value stored in ``M_.exo_names_long``.
.. option:: NAME = QUOTED_STRING
Like :ref:`partitioning <partitioning>` but QUOTED_STRING
stored in ``M_.exo_partitions.NAME``.
*Example*
::
varexo m gov;
*Remarks*
An exogenous variable is an innovation, in the sense
that this variable cannot be predicted from the knowledge of the
current state of the economy. For instance, if logged TFP is a first
order autoregressive process:
.. math::
a_t = \rho a_{t-1} + \varepsilon_t
then logged TFP :math:`a_t` is an endogenous variable to be
declared with ``var``, its best prediction is :math:`\rho
a_{t-1}`, while the innovation :math:`\varepsilon_t` is to be
declared with ``varexo``.
.. command:: varexo_det VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];
|br| This optional command declares exogenous deterministic
variables in a stochastic model. See :ref:`conv` for the syntax of
VARIABLE_NAME. Optionally it is possible to give a LaTeX
name to the variable. The variables in
the list can be separated by spaces or by commas. ``varexo_det`` commands can appear several
times in the file and Dynare will concatenate them.
It is possible to mix deterministic and stochastic shocks to build
models where agents know from the start of the simulation about
future exogenous changes. In that case ``stoch_simul`` will
compute the rational expectation solution adding future
information to the state space (nothing is shown in the output of
``stoch_simul``) and forecast will compute a simulation
conditional on initial conditions and future information.
Note that exogenous deterministic variables cannot appear with a lead or a
lag in the model.
*Options*
.. option:: long_name = QUOTED_STRING
Like :ref:`long_name <long-name>` but value stored in
``M_.exo_det_names_long``.
.. option:: NAME = QUOTED_STRING
Like :ref:`partitioning <partitioning>` but QUOTED_STRING stored
in ``M_.exo_det_partitions.NAME``.
*Example*
::
varexo m gov;
varexo_det tau;
.. command :: parameters PARAM_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];
|br| This command declares parameters used in the model, in variable
initialization or in shocks declarations. See :ref:`conv` for the
syntax of ``PARAM_NAME``. Optionally it is possible to give a
LaTeX name to the parameter.
The parameters must subsequently be assigned values (see :ref:`param-init`).
The parameters in the list can be separated by spaces or by commas.
``parameters`` commands can appear several times in the file and Dynare
will concatenate them.
*Options*
.. option:: long_name = QUOTED_STRING
Like :ref:`long_name <long-name>` but value stored in ``M_.param_names_long``.
.. option:: NAME = QUOTED_STRING
Like :ref:`partitioning <partitioning>` but QUOTED_STRING stored in ``M_.param_partitions.NAME``.
*Example*
::
parameters alpha, bet;
.. command :: change_type (var|varexo|varexo_det|parameters) VAR_NAME | PARAM_NAME...;
Changes the types of the specified variables/parameters to another
type: endogenous, exogenous, exogenous deterministic or
parameter. It is important to understand that this command has a
global effect on the ``.mod`` file: the type change is effective
after, but also before, the ``change_type`` command. This command
is typically used when flipping some variables for steady state
calibration: typically a separate model file is used for
calibration, which includes the list of variable declarations with
the macro processor, and flips some variable.
*Example*
::
var y, w;
parameters alpha, beta;
...
change_type(var) alpha, beta;
change_type(parameters) y, w;
Here, in the whole model file, ``alpha`` and ``beta`` will be
endogenous and ``y`` and ``w`` will be parameters.
.. command:: var_remove VAR_NAME | PARAM_NAME...;
Removes the listed variables (or parameters) from the model. Removing a
variable that has already been used in a model equation or elsewhere will
lead to an error.
.. command:: predetermined_variables VAR_NAME...;
|br| In Dynare, the default convention is that the timing of a variable
reflects when this variable is decided. The typical example is for
capital stock: since the capital stock used at current period is
actually decided at the previous period, then the capital stock
entering the production function is ``k(-1)``, and the law of
motion of capital must be written::
k = i + (1-delta)*k(-1)
Put another way, for stock variables, the default in Dynare is to
use a “stock at the end of the period” concept, instead of a
“stock at the beginning of the period” convention.
The ``predetermined_variables`` is used to change that
convention. The endogenous variables declared as predetermined
variables are supposed to be decided one period ahead of all other
endogenous variables. For stock variables, they are supposed to
follow a “stock at the beginning of the period” convention.
Note that Dynare internally always uses the “stock at the end of
the period” concept, even when the model has been entered using
the ``predetermined_variables`` command. Thus, when plotting,
computing or simulating variables, Dynare will follow the
convention to use variables that are decided in the current
period. For example, when generating impulse response functions
for capital, Dynare will plot ``k``, which is the capital stock
decided upon by investment today (and which will be used in
tomorrow’s production function). This is the reason that capital
is shown to be moving on impact, because it is ``k`` and not the
predetermined ``k(-1)`` that is displayed. It is important to
remember that this also affects simulated time series and output
from smoother routines for predetermined variables. Compared to
non-predetermined variables they might otherwise appear to be
falsely shifted to the future by one period.
*Example*
The following two program snippets are strictly equivalent.
Using default Dynare timing convention::
var y, k, i;
...
model;
y = k(-1)^alpha;
k = i + (1-delta)*k(-1);
...
end;
Using the alternative timing convention::
var y, k, i;
predetermined_variables k;
...
model;
y = k^alpha;
k(+1) = i + (1-delta)*k;
...
end;
.. command:: trend_var (growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;
|br| This optional command declares the trend variables in the
model. See :ref:`conv` for the syntax of MODEL_EXPR and
VAR_NAME. Optionally it is possible to give a
LaTeX name to the variable.
The variable is assumed to have a multiplicative growth trend. For
an additive growth trend, use ``log_trend_var`` instead.
Trend variables are required if the user wants to be able to write
a nonstationary model in the ``model`` block. The ``trend_var``
command must appear before the var command that references the
trend variable.
``trend_var`` commands can appear several times in the file and
Dynare will concatenate them.
If the model is nonstationary and is to be written as such in the
``model`` block, Dynare will need the growth factor of every trend
variable in order to stationarize the model. The growth factor
must be provided within the declaration of the trend variable,
using the ``growth_factor`` keyword. All endogenous variables and
parameters referenced in MODEL_EXPR must already have been
declared by the var and parameters commands.
*Example*
::
trend_var (growth_factor=gA) A;
.. command :: log_trend_var (log_growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;
|br| Same as ``trend_var``, except that the variable is supposed to
have an additive trend (or, to put it otherwise, to be equal to
the log of a variable with a multiplicative trend).
.. command:: model_local_variable VARIABLE_NAME [LATEX_NAME]... ;
|br| This optional command declares a model local variable. See
:ref:`conv` for the syntax of VARIABLE_NAME. As you can create
model local variables on the fly in the model block (see
:ref:`model-decl`), the interest of this command is primarily to
assign a LATEX_NAME to the model local variable.
*Example*
::
model_local_variable GDP_US $GDPUS$;
.. _on-the-fly-declaration:
On-the-fly Model Variable Declaration
-------------------------------------
Endogenous variables, exogenous variables, and parameters can also be declared
inside the model block. You can do this in two different ways: either via the
equation tag (only for endogenous variables) or directly in an equation (for
endogenous, exogenous or parameters).
To declare an endogenous variable on-the-fly in an equation tag, simply write
``endogenous`` followed by an equal sign and the variable name in single
quotes. Hence, to declare a variable ``c`` as endogenous in an equation tag,
you can type ``[endogenous='c']``.
To perform on-the-fly variable declaration in an equation, simply follow the
symbol name with a vertical line (``|``, pipe character) and either an ``e``
(for endogenous), an ``x`` (for exogenous), or a ``p`` (for parameter). For
example, to declare a parameter named ``alphaa`` in the model block, you could
write ``alphaa|p`` directly in an equation where it appears. Similarly, to
declare an endogenous variable ``c`` in the model block you could write
``c|e``. Note that in-equation on-the-fly variable declarations must be made on
contemporaneous variables.
On-the-fly variable declarations do not have to appear in the first place where
this variable is encountered.
*Example*
The following two snippets are equivalent:
::
model;
[endogenous='k',name='law of motion of capital']
k(+1) = i|e + (1-delta|p)*k;
y|e = k^alpha|p;
...
end;
delta = 0.025;
alpha = 0.36;
::
var k, i, y;
parameters delta, alpha;
delta = 0.025;
alpha = 0.36;
...
model;
[name='law of motion of capital']
k(1) = i|e + (1-delta|p)*k;
y|e = k|e^alpha|p;
...
end;
.. _expr:
Expressions
===========
Dynare distinguishes between two types of mathematical expressions:
those that are used to describe the model, and those that are used
outside the model block (e.g. for initializing parameters or
variables, or as command options). In this manual, those two types of
expressions are respectively denoted by MODEL_EXPRESSION and
EXPRESSION.
Unlike MATLAB or Octave expressions, Dynare expressions are
necessarily scalar ones: they cannot contain matrices or evaluate to
matrices. [#f1]_
Expressions can be constructed using integers (INTEGER), floating
point numbers (DOUBLE), parameter names (PARAMETER_NAME), variable
names (VARIABLE_NAME), operators and functions.
The following special constants are also accepted in some contexts:
.. constant:: inf
Represents infinity.
.. constant:: nan
“Not a number”: represents an undefined or unrepresentable value.
Parameters and variables
------------------------
Parameters and variables can be introduced in expressions by simply
typing their names. The semantics of parameters and variables is quite
different whether they are used inside or outside the model block.
Inside the model
^^^^^^^^^^^^^^^^
Parameters used inside the model refer to the value given through
parameter initialization (see :ref:`param-init`) or ``homotopy_setup``
when doing a simulation, or are the estimated variables when doing an
estimation.
Variables used in a MODEL_EXPRESSION denote current period values when
neither a lead or a lag is given. A lead or a lag can be given by
enclosing an integer between parenthesis just after the variable name:
a positive integer means a lead, a negative one means a lag. Leads or
lags of more than one period are allowed. For example, if ``c`` is an
endogenous variable, then ``c(+1)`` is the variable one period ahead,
and ``c(-2)`` is the variable two periods before.
When specifying the leads and lags of endogenous variables, it is
important to respect the following convention: in Dynare, the timing
of a variable reflects when that variable is decided. A control
variable — which by definition is decided in the current period — must
have no lead. A predetermined variable — which by definition has been
decided in a previous period — must have a lag. A consequence of this
is that all stock variables must use the “stock at the end of the
period” convention.
Leads and lags are primarily used for endogenous variables, but can be
used for exogenous variables. They have no effect on parameters and
are forbidden for local model variables (see Model declaration).
Outside the model
^^^^^^^^^^^^^^^^^
When used in an expression outside the model block, a parameter or a
variable simply refers to the last value given to that variable. More
precisely, for a parameter it refers to the value given in the
corresponding parameter initialization (see :ref:`param-init`); for an
endogenous or exogenous variable, it refers to the value given in the
most recent ``initval`` or ``endval`` block.
Operators
---------
The following operators are allowed in both MODEL_EXPRESSION and
EXPRESSION:
* Binary arithmetic operators: ``+``, ``-``, ``*``, ``/``, ``^``
* Unary arithmetic operators: ``+``, ``-``
* Binary comparison operators (which evaluate to either 0 or 1): ``<``,
``>``, ``<=``, ``>=``, ``==``, ``!=``
Note the binary comparison operators are differentiable everywhere except on a
line of the 2-dimensional real plane. However for facilitating
convergence of Newton-type methods, Dynare assumes that, at the points
of non-differentiability, the partial derivatives of these operators
with respect to both arguments is equal to 0 (since this is the value
of the partial derivatives everywhere else).
The following special operators are accepted in MODEL_EXPRESSION (but
not in EXPRESSION):
.. operator:: STEADY_STATE (MODEL_EXPRESSION)
This operator is used to take the value of the enclosed expression
at the steady state. A typical usage is in the Taylor rule, where
you may want to use the value of GDP at steady state to compute
the output gap.
Exogenous and exogenous deterministic variables may not appear in
MODEL_EXPRESSION.
.. warning::
The concept of a steady state is ambiguous in a perfect foresight
context with permament and potentially anticipated shocks occuring.
Dynare will use the contents of ``oo_.steady_state`` as its reference
for calls to the ``STEADY_STATE()`` operator. In the presence of
``endval``, this implies that the terminal state provided by the
user is used. This may be a steady state computed by Dynare (if ``endval``
is followed by ``steady``) or simply the terminal state provided by the
user (if ``endval`` is not followed by ``steady``). Put differently,
Dynare will not automatically compute the steady state conditional on
the specificed value of the exogenous variables in the respective periods.
.. operator:: EXPECTATION (INTEGER) (MODEL_EXPRESSION)
This operator is used to take the expectation of some expression
using a different information set than the information available
at current period. For example, ``EXPECTATION(-1)(x(+1))`` is
equal to the expected value of variable x at next period, using
the information set available at the previous period. See
:ref:`aux-variables` for an explanation of how this operator is
handled internally and how this affects the output.
Functions
---------
Built-in functions
^^^^^^^^^^^^^^^^^^
The following standard functions are supported internally for both
MODEL_EXPRESSION and EXPRESSION:
.. function:: exp(x)
Natural exponential.
.. function:: log(x)
.. function:: ln(x)
Natural logarithm.
.. function:: log10(x)
Base 10 logarithm.
.. function:: sqrt(x)
Square root.
.. function:: cbrt(x)
Cube root.
.. function:: sign(x)
Signum function, defined as:
.. math::
\textrm{sign}(x) =
\begin{cases}
-1 &\quad\text{if }x<0\\
0 &\quad\text{if }x=0\\
1 &\quad\text{if }x>0
\end{cases}
Note that this function is not continuous, hence not differentiable, at
:math:`x=0`. However, for facilitating convergence of Newton-type
methods, Dynare assumes that the derivative at :math:`x=0` is
equal to :math:`0`. This assumption comes from the observation
that both the right- and left-derivatives at this point exist and
are equal to :math:`0`, so we can remove the singularity by
postulating that the derivative at :math:`x=0` is :math:`0`.
.. function:: abs(x)
Absolute value.
Note that this continuous function is not differentiable at
:math:`x=0`. However, for facilitating convergence of Newton-type
methods, Dynare assumes that the derivative at :math:`x=0` is
equal to :math:`0` (even if the derivative does not exist). The
rational for this mathematically unfounded definition, rely on the
observation that the derivative of :math:`\mathrm{abs}(x)` is equal to
:math:`\mathrm{sign}(x)` for any :math:`x\neq 0` in :math:`\mathbb R` and
from the convention for the value of :math:`\mathrm{sign}(x)` at
:math:`x=0`).
.. function:: sin(x)
.. function:: cos(x)
.. function:: tan(x)
.. function:: asin(x)
.. function:: acos(x)
.. function:: atan(x)
Trigonometric functions.
.. function:: sinh(x)
.. function:: cosh(x)
.. function:: tanh(x)
.. function:: asinh(x)
.. function:: acosh(x)
.. function:: atanh(x)
Hyperbolic functions.
.. function:: max(a, b)
.. function:: min(a, b)
Maximum and minimum of two reals.
Note that these functions are differentiable everywhere except on
a line of the 2-dimensional real plane defined by
:math:`a=b`. However for facilitating convergence of Newton-type
methods, Dynare assumes that, at the points of
non-differentiability, the partial derivative of these functions
with respect to the first (resp. the second) argument is equal to
:math:`1` (resp. to :math:`0`) (i.e. the derivatives at the kink
are equal to the derivatives observed on the half-plane where the
function is equal to its first argument).
.. function:: normcdf(x)
normcdf(x, mu, sigma)
Gaussian cumulative density function, with mean *mu* and standard
deviation *sigma*. Note that ``normcdf(x)`` is equivalent to
``normcdf(x,0,1)``.
.. function:: normpdf(x)
normpdf(x, mu, sigma)
Gaussian probability density function, with mean *mu* and standard
deviation *sigma*. Note that ``normpdf(x)`` is equivalent to
``normpdf(x,0,1)``.
.. function:: erf(x)
Gauss error function.
.. function:: erfc(x)
Complementary error function, *i.e.*
:math:`\mathrm{erfc}(x) = 1-\mathrm{erf}(x)`.
External functions
^^^^^^^^^^^^^^^^^^
Any other user-defined (or built-in) MATLAB or Octave function may be
used in both a MODEL_EXPRESSION and an EXPRESSION, provided that this
function has a scalar argument as a return value.
To use an external function in a MODEL_EXPRESSION, one must declare
the function using the ``external_function`` statement. This is not
required for external functions used in an EXPRESSION outside of a
``model`` block or ``steady_state_model`` block.
.. command:: external_function (OPTIONS...);
This command declares the external functions used in the model
block. It is required for every unique function used in the model
block.
``external_function`` commands can appear several times in the
file and must come before the model block.
*Options*
.. option:: name = NAME
The name of the function, which must also be the name of the
M-/MEX file implementing it. This option is mandatory.
.. option:: nargs = INTEGER
The number of arguments of the function. If this option is not
provided, Dynare assumes ``nargs = 1``.
.. option:: first_deriv_provided [= NAME]
If NAME is provided, this tells Dynare that the Jacobian is
provided as the only output of the M-/MEX file given as the
option argument. If NAME is not provided, this tells Dynare
that the M-/MEX file specified by the argument passed to NAME
returns the Jacobian as its second output argument. When this option is
not provided, Dynare will use finite difference approximations for
computing the derivatives of the function, whenever needed.
.. option:: second_deriv_provided [= NAME]
If NAME is provided, this tells Dynare that the Hessian is
provided as the only output of the M-/MEX file given as the
option argument. If NAME is not provided, this tells Dynare
that the M-/MEX file specified by the argument passed to NAME
returns the Hessian as its third output argument. NB: This
option can only be used if the ``first_deriv_provided`` option
is used in the same ``external_function`` command. When this option is
not provided, Dynare will use finite difference approximations for
computing the Hessian derivatives of the function, whenever needed.
*Example*
::
external_function(name = funcname);
external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided);
external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv);
A few words of warning in stochastic context
--------------------------------------------
The use of the following functions and operators is strongly
discouraged in a stochastic context: ``max``, ``min``, ``abs``,
``sign``, ``<``, ``>``, ``<=``, ``>=``, ``==``, ``!=``.
The reason is that the local approximation used by ``stoch_simul`` or
``estimation`` will by nature ignore the non-linearities introduced by
these functions if the steady state is away from the kink. And, if the
steady state is exactly at the kink, then the approximation will be
bogus because the derivative of these functions at the kink is bogus
(as explained in the respective documentations of these functions and
operators).
Note that ``extended_path`` is not affected by this problem, because
it does not rely on a local approximation of the mode.
.. _param-init:
Parameter initialization
========================
When using Dynare for computing simulations, it is necessary to
calibrate the parameters of the model. This is done through parameter
initialization.
The syntax is the following::
PARAMETER_NAME = EXPRESSION;
Here is an example of calibration::
parameters alpha, beta;
beta = 0.99;
alpha = 0.36;
A = 1-alpha*beta;
Internally, the parameter values are stored in ``M_.params``:
.. matvar:: M_.params
Contains the values of model parameters. The parameters are in the
order that was used in the ``parameters`` command, hence ordered as
in ``M_.param_names``.
The parameter names are stored in ``M_.param_names``:
.. matvar:: M_.param_names
Cell array containing the names of the model parameters.
.. matcomm:: get_param_by_name ('PARAMETER_NAME');
Given the name of a parameter, returns its calibrated value as it is
stored in ``M_.params``.
.. matcomm:: set_param_value ('PARAMETER_NAME', MATLAB_EXPRESSION);
Sets the calibrated value of a parameter to the provided expression.
This does essentially the same as the parameter initialization syntax
described above, except that it accepts arbitrary MATLAB/Octave expressions,
and that it works from MATLAB/Octave scripts.
.. _model-decl:
Model declaration
=================
The model is declared inside a ``model`` block:
.. block:: model ;
model (OPTIONS...);
|br| The equations of the model are written in a block delimited by
``model`` and ``end`` keywords.
There must be as many equations as there are endogenous variables
in the model, except when computing the unconstrained optimal
policy with ``ramsey_model``, ``ramsey_policy`` or
``discretionary_policy``.
The syntax of equations must follow the conventions for
MODEL_EXPRESSION as described in :ref:`expr`. Each equation
must be terminated by a semicolon (‘;’). A normal equation looks
like:
MODEL_EXPRESSION = MODEL_EXPRESSION;
|br| When the equations are written in homogenous form, it is possible
to omit the ‘=0’ part and write only the left hand side of the
equation. A homogenous equation looks like:
MODEL_EXPRESSION;
|br| Inside the model block, Dynare allows the creation of
*model-local variables*, which constitute a simple way to share a
common expression between several equations. The syntax consists
of a pound sign (#) followed by the name of the new model local
variable (which must **not** be declared as in :ref:`var-decl`,
but may have been declared by :comm:`model_local_variable`), an
equal sign, and the expression for which this new variable will
stand. Later on, every time this variable appears in the model,
Dynare will substitute it by the expression assigned to the
variable. Note that the scope of this variable is restricted to
the model block; it cannot be used outside. To assign a LaTeX name
to the model local variable, use the declaration syntax outlined
by :comm:`model_local_variable`. A model local variable declaration
looks like:
#VARIABLE_NAME = MODEL_EXPRESSION;
|br| It is possible to tag equations written in the model block. A tag
can serve different purposes by allowing the user to attach
arbitrary informations to each equation and to recover them at
runtime. For instance, it is possible to name the equations with a
``name`` tag, using a syntax like::
model;
[name = 'Budget constraint'];
c + k = k^theta*A;
end;
Here, ``name`` is the keyword indicating that the tag names the
equation. If an equation of the model is tagged with a name, the
``resid`` command will display the name of the equations (which
may be more informative than the equation numbers) in addition to
the equation number. Several tags for one equation can be
separated using a comma::
model;
[name='Taylor rule',mcp = 'r > -1.94478']
r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
end;
More information on tags is available at `<https://git.dynare.org/Dynare/dynare/-/wikis/Equations-Tags>`__.
There can be several ``model`` blocks, in which case they are simply
concatenated. The set of effective options is also the concatenation of the
options declared in all the blocks, but in that case you may rather want to
use the :comm:`model_options` command.
*Options*
.. option:: linear
Declares the model as being linear. It spares oneself from
having to declare initial values for computing the steady
state of a stationary linear model. This option can’t be used
with non-linear models, it will NOT trigger linearization of
the model.
.. option:: use_dll
Instructs the preprocessor to create dynamic loadable
libraries (DLL) containing the model equations and
derivatives, instead of writing those in M-files. You need a
working compilation environment, i.e. a working ``mex``
command (see :ref:`compil-install` for more details).
Using this option can result in
faster simulations or estimations, at the expense of some
initial compilation time. Alternatively, this option can be
given to the ``dynare`` command (see :ref:`dyn-invoc`). [#f2]_
.. option:: block
Perform the block decomposition of the model, and exploit it
in computations (steady-state, deterministic simulation,
stochastic simulation with first order approximation and
estimation). See `<https://archives.dynare.org/DynareWiki/FastDeterministicSimulationAndSteadyStateComputation>`__ for details on the algorithms
used in deterministic simulation and steady-state computation.
.. option:: bytecode
Instead of M-files, use a bytecode representation of the
model, i.e. a binary file containing a compact representation
of all the equations.
.. option:: cutoff = DOUBLE
Threshold under which a jacobian element is considered as null
during the model normalization. Only available with option
``block``. Default: ``1e-15``
.. option:: mfs = INTEGER
Controls the handling of minimum feedback set of endogenous
variables. Only available with option ``block``. Possible
values:
``0``
All the endogenous variables are considered as feedback
variables.
``1``
The endogenous variables assigned to equation naturally
normalized (i.e. of the form :math:`x=f(Y)` where
:math:`x` does not appear in :math:`Y`) are potentially
recursive variables. All the other variables are forced to
belong to the set of feedback variables. (Default)
``2``
In addition of variables with ``mfs = 1`` the endogenous
variables related to linear equations which could be
normalized are potential recursive variables. All the
other variables are forced to belong to the set of
feedback variables.
``3``
In addition of variables with ``mfs = 2`` the endogenous
variables related to non-linear equations which could be
normalized are potential recursive variables. All the
other variables are forced to belong to the set of
feedback variables.
.. option:: no_static
Don’t create the static model file. This can be useful for
models which don’t have a steady state.
.. option:: differentiate_forward_vars
differentiate_forward_vars = ( VARIABLE_NAME [VARIABLE_NAME ...] )
Tells Dynare to create a new auxiliary variable for each
endogenous variable that appears with a lead, such that the
new variable is the time differentiate of the original
one. More precisely, if the model contains ``x(+1)``, then a
variable ``AUX_DIFF_VAR`` will be created such that
``AUX_DIFF_VAR=x-x(-1)``, and ``x(+1)`` will be replaced with
``x+AUX_DIFF_VAR(+1)``.
The transformation is applied to all endogenous variables with
a lead if the option is given without a list of variables. If
there is a list, the transformation is restricted to
endogenous with a lead that also appear in the list.
This option can useful for some deterministic simulations
where convergence is hard to obtain. Bad values for terminal
conditions in the case of very persistent dynamics or
permanent shocks can hinder correct solutions or any
convergence. The new differentiated variables have obvious
zero terminal conditions (if the terminal condition is a
steady state) and this in many cases helps convergence of
simulations.
.. option:: parallel_local_files = ( FILENAME [, FILENAME]... )
Declares a list of extra files that should be transferred to
follower nodes when doing a parallel computation (see
:ref:`paral-conf`).
.. option:: balanced_growth_test_tol = DOUBLE
Tolerance used for determining whether cross-derivatives are zero in
the test for balanced growth path (the latter is documented on
`<https://archives.dynare.org/DynareWiki/RemovingTrends>`__). Default:
``1e-6``
*Example* (Elementary RBC model)
::
var c k;
varexo x;
parameters aa alph bet delt gam;
model;
c = - k + aa*x*k(-1)^alph + (1-delt)*k(-1);
c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet);
end;
*Example* (Use of model local variables)
The following program::
model;
# gamma = 1 - 1/sigma;
u1 = c1^gamma/gamma;
u2 = c2^gamma/gamma;
end;
...is formally equivalent to::
model;
u1 = c1^(1-1/sigma)/(1-1/sigma);
u2 = c2^(1-1/sigma)/(1-1/sigma);
end;
*Example* (A linear model)
::
model(linear);
x = a*x(-1)+b*y(+1)+e_x;
y = d*y(-1)+e_y;
end;
.. command:: model_options (OPTIONS...);
This command accepts the same options as the :bck:`model` block.
The purpose of this statement is to specify the options that apply to the
whole model, when there are several ``model`` blocks, so as to restore the
symmetry between those blocks (since otherwise one ``model`` block would
typically bear the options, while the other ones would typically have no
option).
.. command:: model_remove (TAGS...);
This command removes equations that appeared in a previous :bck:`model`
block.
The equations must be specified by a list of tag values, separated by
commas. Each element of the list is either a simple quoted string, in which
case it designates an equation by its ``name`` tag; or a tag name (without
quotes), followed by an equal sign, then by the tag value (within quotes).
Each removed equation must either have an ``endogenous`` tag, or have a
left hand side containing a single endogenous variable. The corresponding
endogenous variable will be either turned into an exogenous (if it is still
used in somewhere in the model at that point), otherwise it will be removed
from the model.
*Example*
::
var c k dummy1 dummy2;
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1) + dummy1;
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
[ name = 'eq:dummy1', endogenous = 'dummy1' ]
c*k = dummy1;
[ foo = 'eq:dummy2' ]
log(dummy2) = k + 2;
end;
model_remove('eq:dummy1', foo = 'eq:dummy2');
In the above example, the last two equations will be removed,
``dummy1`` will be turned into an exogenous, and ``dummy2`` will be
removed.
.. block:: model_replace (TAGS...);
This block replaces several equations in the model. It removes the
equations given by the tags list (with the same syntax as in
:comm:`model_remove`), and it adds equations given within the block (with
the same syntax as :bck:`model`).
No variable is removed or has its type changed in the process.
*Example*
::
var c k;
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
[ name = 'dummy' ]
c*k = 1;
end;
model_replace('dummy');
c^(-gam) = (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;
In the above example, the dummy equation is replaced by a proper
Euler equation.
Dynare has the ability to output the original list of model equations
to a LaTeX file, using the ``write_latex_original_model``
command, the list of transformed model equations using the
``write_latex_dynamic_model command``, and the list of static model
equations using the ``write_latex_static_model`` command.
.. command:: write_latex_original_model (OPTIONS);
|br| This command creates two LaTeX files: one
containing the model as defined in the model block and one
containing the LaTeX document header information.
If your ``.mod`` file is ``FILENAME.mod``, then Dynare will create
a file called ``FILENAME/latex/original.tex``, which includes a file
called ``FILENAME/latex/original_content.tex`` (also created by Dynare)
containing the list of all the original model equations.
If LaTeX names were given for variables and parameters
(see :ref:`var-decl`), then those will be used; otherwise, the
plain text names will be used.
Time subscripts (``t``, ``t+1``, ``t-1``, ...) will be appended to
the variable names, as LaTeX subscripts.
Compiling the TeX file requires the following LaTeX
packages: ``geometry, fullpage, breqn``.
*Options*
.. option:: write_equation_tags
Write the equation tags in the LaTeX output. The
equation tags will be interpreted with LaTeX markups.
.. command:: write_latex_dynamic_model ;
write_latex_dynamic_model (OPTIONS);
|br| This command creates two LaTeX files: one containing
the dynamic model and one containing the LaTeX document
header information.
If your ``.mod`` file is ``FILENAME.mod``, then Dynare will create
a file called ``FILENAME/latex/dynamic.tex``, which includes a file
called ``FILENAME/latex/dynamic_content.tex`` (also created by Dynare)
containing the list of all the dynamic model equations.
If LaTeX names were given for variables and parameters
(see :ref:`var-decl`), then those will be used; otherwise, the
plain text names will be used.
Time subscripts (``t``, ``t+1``, ``t-1``, ...) will be appended to
the variable names, as LaTeX subscripts.
Note that the model written in the TeX file will differ from the
model declared by the user in the following dimensions:
* The timing convention of predetermined variables (see
:comm:`predetermined_variables`) will have been changed to
the default Dynare timing convention; in other words,
variables declared as predetermined will be lagged on period
back,
* The ``EXPECTATION`` operators will have been removed, replaced by
auxiliary variables and new equations (as explained in the
documentation of
:op:`EXPECTATION <EXPECTATION (INTEGER) (MODEL_EXPRESSION)>`),
* Endogenous variables with leads or lags greater or equal
than two will have been removed, replaced by new auxiliary
variables and equations,
* Exogenous variables with leads or lags will also have been replaced
by new auxiliary variables and equations.
For the required LaTeX packages, see
:comm:`write_latex_original_model`.
*Options*
.. option:: write_equation_tags
See :opt:`write_equation_tags`
.. command:: write_latex_static_model (OPTIONS);
|br| This command creates two LaTeX files: one
containing the static model and one containing the LaTeX
document header information.
If your ``.mod`` file is ``FILENAME.mod``, then Dynare will create
a file called ``FILENAME/latex/static.tex``, which includes a file
called ``FILENAME/latex/static_content.tex`` (also created by Dynare)
containing the list of all the steady state model equations.
If LaTeX names were given for variables and parameters
(see :ref:`var-decl`), then those will be used; otherwise, the
plain text names will be used.
Note that the model written in the TeX file will differ from the
model declared by the user in the some dimensions (see
:comm:`write_latex_dynamic_model` for details).
Also note that this command will not output the contents of the
optional ``steady_state_model`` block (see
:bck:`steady_state_model`); it will rather output a static version
(i.e. without leads and lags) of the dynamic ``model`` declared in
the model block. To write the LaTeX contents of the
``steady_state_model`` see :comm:`write_latex_steady_state_model`.
For the required LaTeX packages, see
:comm:`write_latex_original_model`.
*Options*
.. option:: write_equation_tags
See :opt:`write_equation_tags`.
.. command:: write_latex_steady_state_model
|br| This command creates two LaTeX files: one containing the steady
state model and one containing the LaTeX document header
information.
If your ``.mod`` file is ``FILENAME.mod``, then Dynare
will create a file called ``FILENAME/latex/steady_state.tex``,
which includes a file called
``FILENAME/latex/steady_state_content.tex`` (also created by
Dynare) containing the list of all the steady state model
equations.
If LaTeX names were given for variables and parameters
(see :ref:`var-decl`), then those will be used;
otherwise, the plain text names will be used.
Note that the model written in the ``.tex`` file will differ from
the model declared by the user in some dimensions
(see :comm:`write_latex_dynamic_model` for details).
For the required LaTeX packages, see :comm:`write_latex_original_model`.
.. _aux-variables:
Auxiliary variables
===================
The model which is solved internally by Dynare is not exactly the
model declared by the user. In some cases, Dynare will introduce
auxiliary endogenous variables—along with corresponding auxiliary
equations—which will appear in the final output.
The main transformation concerns leads and lags. Dynare will perform a
transformation of the model so that there is only one lead and one lag on
endogenous variables and no leads/lags on exogenous variables.
This transformation is achieved by the creation of auxiliary variables
and corresponding equations. For example, if ``x(+2)`` exists in the
model, Dynare will create one auxiliary variable ``AUX_ENDO_LEAD =
x(+1)``, and replace ``x(+2)`` by ``AUX_ENDO_LEAD(+1)``.
A similar transformation is done for lags greater than 2 on endogenous
(auxiliary variables will have a name beginning with
``AUX_ENDO_LAG``), and for exogenous with leads and lags (auxiliary
variables will have a name beginning with ``AUX_EXO_LEAD`` or
``AUX_EXO_LAG`` respectively).
Another transformation is done for the ``EXPECTATION`` operator. For
each occurrence of this operator, Dynare creates an auxiliary variable
defined by a new equation, and replaces the expectation operator by a
reference to the new auxiliary variable. For example, the expression
``EXPECTATION(-1)(x(+1))`` is replaced by ``AUX_EXPECT_LAG_1(-1)``,
and the new auxiliary variable is declared as ``AUX_EXPECT_LAG_1 =
x(+2)``.
Auxiliary variables are also introduced by the preprocessor for the
``ramsey_model`` and ``ramsey_policy`` commands. In this case, they
are used to represent the Lagrange multipliers when first order
conditions of the Ramsey problem are computed. The new variables take
the form ``MULT_i``, where *i* represents the constraint with which
the multiplier is associated (counted from the order of declaration in
the model block).
Auxiliary variables are also introduced by the
``differentiate_forward_vars`` option of the model block. The new
variables take the form ``AUX_DIFF_FWRD_i``, and are equal to
``x-x(-1)`` for some endogenous variable ``x``.
Finally, auxiliary variables will arise in the context of employing the
``diff`` operator.
Once created, all auxiliary variables are included in the set of
endogenous variables. The output of decision rules (see below) is such
that auxiliary variable names are replaced by the original variables
they refer to.
The number of endogenous variables before the creation of auxiliary
variables is stored in ``M_.orig_endo_nbr``, and the number of
endogenous variables after the creation of auxiliary variables is
stored in ``M_.endo_nbr``.
See `<https://git.dynare.org/Dynare/dynare/-/wikis/Auxiliary-variables>`__ for more technical details on auxiliary variables.
.. _init-term-cond:
Initial and terminal conditions
===============================
For most simulation exercises, it is necessary to provide initial (and
possibly terminal) conditions. It is also necessary to provide initial
guess values for non-linear solvers. This section describes the
statements used for those purposes.
In many contexts (deterministic or stochastic), it is necessary to
compute the steady state of a non-linear model: ``initval`` then
specifies numerical initial values for the non-linear solver. The
command ``resid`` can be used to compute the equation residuals for
the given initial values.
Used in perfect foresight mode, the types of forward-looking models
for which Dynare was designed require both initial and terminal
conditions. Most often these initial and terminal conditions are
static equilibria, but not necessarily.
One typical application is to consider an economy at the equilibrium
at time 0, trigger a shock in first period, and study the trajectory
of return to the initial equilibrium. To do that, one needs
``initval`` and ``shocks`` (see :ref:`shocks-exo`).
Another one is to study how an economy, starting from arbitrary
initial conditions at time 0 converges towards equilibrium. In this
case models, the command ``histval`` permits to specify different
historical initial values for variables with lags for the periods
before the beginning of the simulation. Due to the design of Dynare,
in this case ``initval`` is used to specify the terminal conditions.
.. block:: initval ;
initval(OPTIONS...);
|br| The ``initval`` block has two main purposes: providing guess
values for non-linear solvers in the context of perfect foresight
simulations and providing guess values for steady state
computations in both perfect foresight and stochastic
simulations. Depending on the presence of ``histval`` and
``endval`` blocks it is also used for declaring the initial and
terminal conditions in a perfect foresight simulation
exercise. Because of this interaction of the meaning of an
``initval`` block with the presence of ``histval`` and ``endval``
blocks in perfect foresight simulations, it is strongly
recommended to check that the constructed ``oo_.endo_simul`` and
``oo_.exo_simul`` variables contain the desired values after
running ``perfect_foresight_setup`` and before running
``perfect_foresight_solver``. In the presence of leads and lags,
these subfields of the results structure will store the historical
values for the lags in the first column/row and the terminal
values for the leads in the last column/row.
The ``initval`` block is terminated by ``end;`` and contains lines
of the form:
VARIABLE_NAME = EXPRESSION;
|br| *In a deterministic (i.e. perfect foresight) model*
First, both the ``oo_.endo_simul`` and ``oo_.exo_simul`` variables
storing the endogenous and exogenous variables will be filled with
the values provided by this block. If there are no
other blocks present, it will therefore provide the initial and
terminal conditions for all the endogenous and exogenous
variables, because it will also fill the last column/row of these
matrices. For the intermediate simulation periods it thereby
provides the starting values for the solver. In the presence of a
``histval`` block (and therefore absence of an ``endval`` block),
this ``histval`` block will provide/overwrite the historical
values for the state variables (lags) by setting the first
column/row of ``oo_.endo_simul`` and ``oo_.exo_simul``. This
implies that the ``initval`` block in the presence of ``histval``
only sets the terminal values for the variables with leads and
provides initial values for the perfect foresight solver.
Because of these various functions of ``initval`` it is often
necessary to provide values for all the endogenous variables in an
``initval`` block. Initial and terminal conditions are strictly
necessary for lagged/leaded variables, while feasible starting
values are required for the solver. It is important to be aware
that if some variables, endogenous or exogenous, are not mentioned
in the ``initval`` block, a zero value is assumed. It is
particularly important to keep this in mind when specifying
exogenous variables using ``varexo`` that are not allowed to take
on the value of zero, like e.g. TFP.
Note that if the ``initval`` block is immediately followed by a
``steady`` command, its semantics are slightly changed. The
``steady`` command will compute the steady state of the model for
all the endogenous variables, assuming that exogenous variables
are kept constant at the value declared in the ``initval``
block. These steady state values conditional on the declared
exogenous variables are then written into ``oo_.endo_simul`` and
take up the potential roles as historical and terminal conditions
as well as starting values for the solver. An ``initval`` block
followed by ``steady`` is therefore formally equivalent to an
``initval`` block with the specified values for the exogenous
variables, and the endogenous variables set to the associated
steady state values conditional on the exogenous variables.
|br| *In a stochastic model*
The main purpose of ``initval`` is to provide initial guess values
for the non-linear solver in the steady state computation. Note
that if the ``initval`` block is not followed by ``steady``, the
steady state computation will still be triggered by subsequent
commands (``stoch_simul``, ``estimation``...).
As such, ``initval`` allows specifying the initial instrument value for
steady state finding when providing an analytical conditional steady state
file for ``ramsey_model``-computations.
It is not necessary to declare 0 as initial value for exogenous
stochastic variables, since it is the only possible value.
The subsequently computed steady state (not the initial values,
use histval for this) will be used as the initial condition at all
the periods preceeding the first simulation period for the three
possible types of simulations in stochastic mode:
* :comm:`stoch_simul`, if the ``periods`` option is specified.
* :comm:`forecast` as the initial point at which the forecasts
are computed.
* :comm:`conditional_forecast` as the initial point at which
the conditional forecasts are computed.
To start simulations at a particular set of starting values that
are not a computed steady state, use :bck:`histval`.
*Options*
.. option:: all_values_required
Issues an error and stops processing the .mod file if there is
at least one endogenous or exogenous variable that has not
been set in the initval block.
*Example*
::
initval;
c = 1.2;
k = 12;
x = 1;
end;
steady;
.. block:: endval ;
endval (OPTIONS...);
|br| This block is terminated by ``end;`` and contains lines of the form:
VARIABLE_NAME = EXPRESSION;
|br| The ``endval`` block makes only sense in a deterministic model and
cannot be used together with ``histval``. Similar to the
``initval`` command, it will fill both the ``oo_.endo_simul`` and
``oo_.exo_simul`` variables storing the endogenous and exogenous
variables with the values provided by this block. If no
``initval`` block is present, it will fill the whole matrices,
therefore providing the initial and terminal conditions for all
the endogenous and exogenous variables, because it will also fill
the first and last column/row of these matrices. Due to also
filling the intermediate simulation periods it will provide the
starting values for the solver as well.
If an ``initval`` block is present, ``initval`` will provide the
historical values for the variables (if there are states/lags),
while ``endval`` will fill the remainder of the matrices, thereby
still providing *i*) the terminal conditions for variables
entering the model with a lead and *ii*) the initial guess values
for all endogenous variables at all the simulation dates for the
perfect foresight solver.
Note that if some variables, endogenous or exogenous, are NOT
mentioned in the ``endval`` block, the value assumed is that of
the last ``initval`` block or ``steady`` command (if
present). Therefore, in contrast to ``initval``, omitted variables
are not automatically assumed to be 0 in this case. Again, it is
strongly recommended to check the constructed ``oo_.endo_simul``
and ``oo_.exo_simul`` variables after running
``perfect_foresight_setup`` and before running
``perfect_foresight_solver`` to see whether the desired outcome
has been achieved.
Like ``initval``, if the ``endval`` block is immediately followed
by a ``steady`` command, its semantics are slightly changed. The
``steady`` command will compute the steady state of the model for
all the endogenous variables, assuming that exogenous variables
are kept constant to the value declared in the ``endval``
block. These steady state values conditional on the declared
exogenous variables are then written into ``oo_.endo_simul`` and
therefore take up the potential roles as historical and terminal
conditions as well as starting values for the solver. An
``endval`` block followed by ``steady`` is therefore formally
equivalent to an ``endval`` block with the specified values for
the exogenous variables, and the endogenous variables set to the
associated steady state values.
*Options*
.. option:: all_values_required
See :opt:`all_values_required`.
*Example*
::
var c k;
varexo x;
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;
initval;
c = 1.2;
k = 12;
x = 1;
end;
steady;
endval;
c = 2;
k = 20;
x = 2;
end;
steady;
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
In this example, the problem is finding the optimal path for
consumption and capital for the periods :math:`t=1` to
:math:`T=200`, given the path of the exogenous technology
level ``x``. ``c`` is a forward-looking variable and the
exogenous variable ``x`` appears with a lead in the expected
return of physical capital, while ``k`` is a purely backward-looking
(state) variable.
The initial equilibrium is computed by ``steady`` conditional
on ``x=1``, and the terminal one conditional on ``x=2``. The
``initval`` block sets the initial condition for ``k`` (since it is the
only backward-looking variable), while
the ``endval`` block sets the terminal condition for
``c`` (since it is the only forward-looking endogenous variable).
The starting values for the perfect foresight solver
are given by the ``endval`` block. See below for more details.
*Example*
::
var c k;
varexo x;
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;
initval;
k = 12;
end;
endval;
c = 2;
x = 1.1;
end;
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
In this example, there is no `steady` command, hence the
conditions are exactly those specified in the `initval` and `endval` blocks.
We need terminal conditions for
``c`` and ``x``, since both appear with a lead, and an initial
condition for ``k``, since it appears with a lag.
Setting ``x=1.1`` in the ``endval`` block without a ``shocks``
block implies that technology is at :math:`1.1` in :math:`t=1`
and stays there forever, because ``endval`` is filling all
entries of ``oo_.endo_simul`` and ``oo_.exo_simul`` except for
the very first one, which stores the initial conditions and
was set to :math:`0` by the ``initval`` block when not
explicitly specifying a value for it.
Because the law of motion for capital is backward-looking, we
need an initial condition for ``k`` at time :math:`0`. Due to
the presence of ``endval``, this cannot be done via a
``histval`` block, but rather must be specified in the
``initval`` block. Similarly, because the Euler equation is
forward-looking, we need a terminal condition for ``c`` at
:math:`t=201`, which is specified in the ``endval`` block.
As can be seen, it is not necessary to specify ``c`` and ``x``
in the ``initval`` block and ``k`` in the ``endval`` block,
because they have no impact on the results. Due to the
optimization problem in the first period being to choose
``c,k`` at :math:`t=1` given the predetermined capital stock
``k`` inherited from :math:`t=0` as well as the current and
future values for technology ``x``, the values for ``c`` and
``x`` at time :math:`t=0` play no role. The same applies to
the choice of ``c,k`` at time :math:`t=200`, which does not
depend on ``k`` at :math:`t=201`. As the Euler equation shows,
that choice only depends on current capital as well as future
consumption ``c`` and technology ``x``, but not on future
capital ``k``. The intuitive reason is that those variables
are the consequence of optimization problems taking place in
at periods :math:`t=0` and :math:`t=201`, respectively, which
are not modeled here.
*Example*
::
initval;
c = 1.2;
k = 12;
x = 1;
end;
endval;
c = 2;
k = 20;
x = 1.1;
end;
In this example, initial conditions for the forward-looking
variables ``x`` and ``c`` are provided, together with a
terminal condition for the backward-looking variable ``k``. As
shown in the previous example, these values will not affect
the simulation results. Dynare simply takes them as given and
basically assumes that there were realizations of exogenous
variables and states that make those choices equilibrium
values (basically initial/terminal conditions at the
unspecified time periods :math:`t<0` and :math:`t>201`).
The above example suggests another way of looking at the use
of ``steady`` after ``initval`` and ``endval``. Instead of
saying that the implicit unspecified conditions before and
after the simulation range have to fit the initial/terminal
conditions of the endogenous variables in those blocks, steady
specifies that those conditions at :math:`t<0` and
:math:`t>201` are equal to being at the steady state given the
exogenous variables in the ``initval`` and ``endval``
blocks. The endogenous variables at :math:`t=0` and
:math:`t=201` are then set to the corresponding steady state
equilibrium values.
The fact that ``c`` at :math:`t=0` and ``k`` at :math:`t=201`
specified in ``initval`` and ``endval`` are taken as given
has an important implication for plotting the simulated vector
for the endogenous variables, i.e. the rows of
``oo_.endo_simul``: this vector will also contain the initial
and terminal conditions and thus is 202 periods long in the
example. When you specify arbitrary values for the initial and
terminal conditions for forward- and backward-looking
variables, respectively, these values can be very far away
from the endogenously determined values at :math:`t=1` and
:math:`t=200`. While the values at :math:`t=0` and
:math:`t=201` are unrelated to the dynamics for
:math:`0<t<201`, they may result in strange-looking large
jumps. In the example above, consumption will display a large
jump from :math:`t=0` to :math:`t=1` and capital will jump
from :math:`t=200` to :math:`t=201` when using :comm:`rplot`
or manually plotting ``oo_.endo_val``.
.. block:: histval ;
histval (OPTIONS...);
|br| *In a deterministic perfect foresight context*
In models with lags on more than one period, the ``histval`` block
permits to specify different historical initial values for
different periods of the state variables. In this case, the
``initval`` block takes over the role of specifying terminal
conditions and starting values for the solver. Note that the
``histval`` block does not take non-state variables.
This block is terminated by ``end;`` and contains lines of the form:
VARIABLE_NAME(INTEGER) = EXPRESSION;
|br| EXPRESSION is any valid expression returning a numerical value
and can contain already initialized variable names.
By convention in Dynare, period 1 is the first period of the
simulation. Going backward in time, the first period before the
start of the simulation is period 0, then period -1, and so on.
State variables not initialized in the ``histval`` block are
assumed to have a value of zero at period 0 and before. Note that
``histval`` cannot be followed by ``steady``.
*Example*
::
model;
x=1.5*x(-1)-0.6*x(-2)+epsilon;
log(c)=0.5*x+0.5*log(c(+1));
end;
histval;
x(0)=-1;
x(-1)=0.2;
end;
initval;
c=1;
x=1;
end;
In this example, ``histval`` is used to set the historical
conditions for the two lags of the endogenous variable ``x``,
stored in the first column of ``oo_.endo_simul``. The
``initval`` block is used to set the terminal condition for
the forward looking variable ``c``, stored in the last column
of ``oo_.endo_simul``. Moreover, the ``initval`` block defines
the starting values for the perfect foresight solver for both
endogenous variables ``c`` and ``x``.
*In a stochastic simulation context*
In the context of stochastic simulations, ``histval`` allows
setting the starting point of those simulations in the state
space. As for the case of perfect foresight simulations, all not
explicitly specified variables are set to 0. Moreover, as only
states enter the recursive policy functions, all values specified
for control variables will be ignored. This can be used
* In :comm:`stoch_simul`, if the ``periods`` option is
specified. Note that this only affects the starting point
for the simulation, but not for the impulse response
functions. When using the :ref:`loglinear <logl>` option,
the ``histval`` block nevertheless takes the unlogged
starting values.
* In :comm:`forecast` as the initial point at which the
forecasts are computed. When using the :ref:`loglinear
<logl>` option, the ``histval`` block nevertheless takes the
unlogged starting values.
* In :comm:`conditional_forecast` for a calibrated model as
the initial point at which the conditional forecasts are
computed. When using the :ref:`loglinear <logl>` option, the
``histval`` block nevertheless takes the unlogged starting
values.
* In :comm:`Ramsey policy <ramsey_model>`, where it also
specifies the values of the endogenous states (including
lagged exogenous) at which the objective function of the
planner is computed. Note that the
initial values of the Lagrange multipliers associated with
the planner’s problem cannot be set (see
:comm:`evaluate_planner_objective`).
*Options*
.. option:: all_values_required
See :opt:`all_values_required`.
*Example*
::
var x y;
varexo e;
model;
x = y(-1)^alpha*y(-2)^(1-alpha)+e;
end;
initval;
x = 1;
y = 1;
e = 0.5;
end;
steady;
histval;
y(0) = 1.1;
y(-1) = 0.9;
end;
stoch_simul(periods=100);
.. command:: resid ;
|br| This command will display the residuals of the static
equations of the model, using the values given for the endogenous
in the last ``initval`` or ``endval`` block (or the steady state
file if you provided one, see :ref:`st-st`).
*Options*
.. option:: non_zero
Only display non-zero residuals.
.. command:: initval_file (OPTIONS...);
|br| In a deterministic setup, this command is used to specify a
path for all endogenous and exogenous variables. The length of
these paths must be equal to the number of simulation periods,
plus the number of leads and the number of lags of the model (for
example, with 50 simulation periods, in a model with 2 lags and 1
lead, the paths must have a length of 53). Note that these paths
cover two different things:
* The constraints of the problem, which are given by the path
for exogenous and the initial and terminal values for
endogenous
* The initial guess for the non-linear solver, which is given
by the path for endogenous variables for the simulation
periods (excluding initial and terminal conditions)
In perfect foresight and stochastic contexts, ``steady`` uses the
first observation loaded by ``initval_file`` as guess value to
solve for the steady state of the model. This first observation is
determined by the ``first_obs`` option when it is used.
Don’t mix ``initval_file`` with ``initval`` statements. However,
after ``initval_file``, you can modify the historical initial
values with ``histval`` or ``histval_file`` statement.
There can be several ``initval_file`` statements in a model
file. Each statement resets ``oo_.initval_series``.
*Options*
.. option:: datafile = FILENAME
filename = FILENAME (deprecated)
The name of the file containing the data. It must be included in quotes if the filename
contains a path or an extension. The command accepts the following file formats:
* M-file (extension ``.m``): for each endogenous and exogenous
variable, the file must contain a row or column vector of
the same name.
* MAT-file (extension ``.mat``): same as for M-files.
* Excel file (extension ``.xls`` or ``.xlsx``): for each
endogenous and exogenous variable, the file must contain a
column of the same name. NB: Octave only supports the
``.xlsx`` file extension and must have the `io`_ package
installed (easily done via octave by typing ‘``pkg
install -forge io``’). The first column may contain the date
of each observation.
* CSV files (extension ``.csv``): for each endogenous and
exogenous variable, the file must contain a column of the
same name. The first column may contain the date of each
observation.
.. option:: first_obs = {INTEGER | DATE}
The observation number or the date (see
:ref:`dates-members`) of the first observation to be used in the file
.. option:: first_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see
:ref:`dates <dates-members>`) at which the simulation (or the forecast) is
starting. This option avoids to have to compute the maximum
number of lags in the model. The observation corresponding to
the first period of simulation doesn’t need to exist in the
file as the only dates necessary for initialization are before
that date.
.. option:: last_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see
:ref:`dates <dates-members>`) at which the simulation (or the forecast) is
ending. This option avoids to have to compute the maximum
number of leads in the model.
.. option:: last_obs = {INTEGER | DATE}
The observaton number or the date (see
:ref:`dates-members`) of the last observation to be used in
the file.
.. option:: nobs = INTEGER
The number of observations to be used in the file (starting
with first of ``first_obs`` observation).
.. option:: series = DSERIES NAME
The name of a DSERIES containing the data (see :ref:`dseries-members`)
*Example 1*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from file
``mydata.csv`` (nothing guarantees that these vales are the
steady state of the model). The guess value for the
trajectories are also taken from the file. The file must
contain at least 203 observations of variables ``c``, ``x``
and ``e``. If there are more than 203 observations available
in the file, the first 203 are used by
``perfect_foresight_setup(periods=200)``.
Note that the values for the auxiliary variable corresponding
to ``x(-2)`` are automatically computed by ``initval_file``.
*Example 2*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_obs=10);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from file
``mydata.csv`` starting with the 10th observation in the
file. There must be at least 212 observations in the file.
*Example 3*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
ds = dseries(mydata.csv);
lds = log(ds);
initval_file(series=lds,
first_obs=2010Q1);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from dseries
``lds``. All observations are loaded starting with the 1st quarter of
2010 until the end of the file. There must be data available
at least until 2050Q3.
*Example 4*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_simulation_period=2010Q1);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from file
``mydata.csv``. The observations in the file must have
dates. All observations are loaded from the 3rd quarter of
2009 until the end of the file. There must be data available
in the file at least until 2050Q1.
*Example 5*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
last_obs = 212);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from file
``mydata.csv``. The first 212 observations are loaded and the
first 203 observations will be used by
``perfect_foresight_setup(periods=200)``.
*Example 6*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_obs = 10,
nobs = 203);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
The initial and terminal values are taken from file
``mydata.csv``. Observations 10 to 212 are loaded.
*Example 7*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_obs = 10);
steady;
The values of the 10th observation of ``mydata.csv`` are used
as guess value to compute the steady state. The exogenous
variables are set to values found in the file or zero if these
variables aren't present.
.. command:: histval_file (OPTIONS...);
|br| This command is equivalent to ``histval``, except that it
reads its input from a file, and is typically used in conjunction
with ``smoother2histval``.
*Options*
.. option:: datafile = FILENAME
filename = FILENAME (deprecated)
The name of the file containing the data. The command accepts
the following file formats:
* M-file (extension ``.m``): for each endogenous and exogenous
variable, the file must contain a row or column vector of
the same name.
* MAT-file (extension ``.mat``): same as for M-files.
* Excel file (extension ``.xls`` or ``.xlsx``): for each
endogenous and exogenous variable, the file must contain a
column of the same name. NB: Octave only supports the
``.xlsx`` file extension and must have the `io`_ package
installed (easily done via octave by typing ‘``pkg
install -forge io``’). The first column may contain the
date of each observation.
* CSV files (extension ``.csv``): for each endogenous and
exogenous variable, the file must contain a column of the
same name. The first column may contain the date of each
observation.
.. option:: first_obs = {INTEGER | DATE}
The observation number or the date (see :ref:`dates-members`) of
the first observation to be used in the file
.. option:: first_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see
:ref:`dates-members`) at which the simulation (or the forecast) is
starting. This option avoids to have to compute the maximum
number of lags in the model. The observation corresponding to
the first period of simulation doesn’t need to exist in the
file as the only dates necessary for initialization are before
that date.
.. option:: last_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see
:ref:`dates <dates-members>`) at which the simulation (or the forecast) is
ending. This option avoids to have to compute the maximum
number of leads in the model.
.. option:: last_obs = {INTEGER | DATE}
The observation number or the date (see :ref:`dates-members`) of the
last observation to be used in the file.
.. option:: nobs = INTEGER
The number of observations to be used in the file (starting
with first of ``first_obs`` observation).
.. option:: series = DSERIES NAME
The name of a DSERIES containing the data (see :ref:`dseries-members`)
*Example 1*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
steady_state_model;
x = 0;
c = exp(c*x/(1 - d));
end;
histval_file(datafile=mydata.csv);
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from the two first rows of file ``mydata.csv``.
*Example 2*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
histval_file(datafile=mydata.csv,
first_obs=10);
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from rows 10 and 11 of file ``mydata.csv``.
*Example 3*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
histval_file(datafile=mydata.csv,
first_obs=2010Q1);
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from observations 2010Q1 and 2010Q2 of file ``mydata.csv``.
*Example 4*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
histval_file(datafile=mydata.csv,
first_simulation_period=2010Q1)
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from observations 2009Q3 and 2009Q4 of file ``mydata.csv``.
*Example 5*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
histval_file(datafile=mydata.csv,
last_obs = 4);
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from the two first rows of file ``mydata.csv``.
*Example 6*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_obs = 10,
nobs = 4);
stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken
from rows 10 and 11 of file ``mydata.csv``.
*Example 7*
::
var c x;
varexo e;
parameters a b c d;
a = 1.5;
b = -0,6;
c = 0.5;
d = 0.5;
model;
x = a*x(-1) + b*x(-2) + e;
log(c) = c*x + d*log(c(+1));
end;
initval_file(datafile=mydata.csv,
first_obs=10);
histval_file(datafile=myotherdata.csv);
perfect_foresight_setup(periods=200);
perfect_foresight_solver;
Historical initial values for the simulation are taken from
the two first rows of file ``myotherdata.csv``.
Terminal values and guess values for the simulation are taken
from file ``mydata.csv`` starting with the 12th observation in
the file. There must be at least 212 observations in the file.
.. _shocks-exo:
Shocks on exogenous variables
=============================
In a deterministic context, when one wants to study the transition of
one equilibrium position to another, it is equivalent to analyze the
consequences of a permanent shock and this in done in Dynare through
the proper use of ``initval`` and ``endval``.
Another typical experiment is to study the effects of a temporary
shock after which the system goes back to the original equilibrium (if
the model is stable...). A temporary shock is a temporary change of
value of one or several exogenous variables in the model. Temporary
shocks are specified with the command ``shocks``.
In a stochastic framework, the exogenous variables take random values
in each period. In Dynare, these random values follow a normal
distribution with zero mean, but it belongs to the user to specify the
variability of these shocks. The non-zero elements of the matrix of
variance-covariance of the shocks can be entered with the ``shocks``
command.
If the variance of an exogenous variable is set to zero, this variable
will appear in the report on policy and transition functions, but
isn’t used in the computation of moments and of Impulse Response
Functions. Setting a variance to zero is an easy way of removing an
exogenous shock.
Note that, by default, if there are several ``shocks`` or ``mshocks``
blocks in the same ``.mod`` file, then they are cumulative: all the
shocks declared in all the blocks are considered; however, if a
``shocks`` or ``mshocks`` block is declared with the ``overwrite``
option, then it replaces all the previous ``shocks`` and ``mshocks``
blocks.
.. block:: shocks ;
shocks(overwrite);
|br| See above for the meaning of the ``overwrite`` option.
*In deterministic context*
For deterministic simulations, the ``shocks`` block specifies
temporary changes in the value of exogenous variables. For
permanent shocks, use an ``endval`` block.
The block should contain one or more occurrences of the following
group of three lines::
var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
It is possible to specify shocks which last several periods and
which can vary over time. The ``periods`` keyword accepts a list
of several dates or date ranges, which must be matched by as many
shock values in the ``values`` keyword. Note that a range in the
``periods`` keyword can be matched by only one value in the
``values`` keyword. If ``values`` represents a scalar, the same
value applies to the whole range. If ``values`` represents a
vector, it must have as many elements as there are periods in the
range.
Note that shock values are not restricted to numerical constants:
arbitrary expressions are also allowed, but you have to enclose
them inside parentheses.
The feasible range of ``periods`` is from 0 to the number of ``periods``
specified in ``perfect_foresight_setup``.
.. warning:: Note that the first endogenous simulation period is period 1.
Thus, a shock value specified for the initial period 0 may conflict with
(i.e. may overwrite or be overwritten by) values for the
initial period specified with ``initval`` or ``endval`` (depending on
the exact context). Users should always verify the correct setting
of ``oo_.exo_simul`` after ``perfect_foresight_setup``.
*Example* (with scalar values)
::
shocks;
var e;
periods 1;
values 0.5;
var u;
periods 4:5;
values 0;
var v;
periods 4:5 6 7:9;
values 1 1.1 0.9;
var w;
periods 1 2;
values (1+p) (exp(z));
end;
*Example* (with vector values)
::
xx = [1.2; 1.3; 1];
shocks;
var e;
periods 1:3;
values (xx);
end;
|br| *In stochastic context*
For stochastic simulations, the ``shocks`` block specifies the non
zero elements of the covariance matrix of the shocks of exogenous
variables.
You can use the following types of entries in the block:
* Specification of the standard error of an exogenous variable.
::
var VARIABLE_NAME; stderr EXPRESSION;
* Specification of the variance of an exogenous variable.
::
var VARIABLE_NAME = EXPRESSION;
* Specification the covariance of two exogenous variables.
::
var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;
* Specification of the correlation of two exogenous variables.
::
corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;
In an estimation context, it is also possible to specify variances
and covariances on endogenous variables: in that case, these
values are interpreted as the calibration of the measurement
errors on these variables. This requires the ``varobs`` command to
be specified before the ``shocks`` block.
*Example*
::
shocks;
var e = 0.000081;
var u; stderr 0.009;
corr e, u = 0.8;
var v, w = 2;
end;
|br| *In stochastic optimal policy context*
When computing conditional welfare in a ``ramsey_model`` or ``discretionary_policy``
context, welfare is conditional on the state values inherited by planner when making
choices in the first period. The information set of the first period includes the
respective exogenous shock realizations. Thus, their known value can be specified
using the perfect foresight syntax. Note that i) all other values specified for
periods than period 1 will be ignored and ii) the value of lagged shocks (e.g.
in the case of news shocks) is specified with ``histval``.
*Example*
::
shocks;
var u; stderr 0.008;
var u;
periods 1;
values 1;
end;
*Mixing deterministic and stochastic shocks*
It is possible to mix deterministic and stochastic shocks to build
models where agents know from the start of the simulation about
future exogenous changes. In that case ``stoch_simul`` will
compute the rational expectation solution adding future
information to the state space (nothing is shown in the output of
``stoch_simul``) and ``forecast`` will compute a simulation
conditional on initial conditions and future information.
*Example*
::
varexo_det tau;
varexo e;
...
shocks;
var e; stderr 0.01;
var tau;
periods 1:9;
values -0.15;
end;
stoch_simul(irf=0);
forecast;
.. block:: mshocks ;
mshocks(overwrite);
|br| The purpose of this block is similar to that of the
``shocks`` block for deterministic shocks, except that the numeric
values given will be interpreted in a multiplicative way. For
example, if a value of ``1.05`` is given as shock value for some
exogenous at some date, it means 5% above its steady state value
(as given by the last ``initval`` or ``endval`` block).
The syntax is the same as ``shocks`` in a deterministic context.
This command is only meaningful in two situations:
* on exogenous variables with a non-zero steady state, in a
deterministic setup,
* on deterministic exogenous variables with a non-zero steady
state, in a stochastic setup.
See above for the meaning of the ``overwrite`` option.
.. block:: heteroskedastic_shocks ;
heteroskedastic_shocks(overwrite);
|br| In *estimation context*, it implements heteroskedastic filters, where the standard error of shocks may unexpectedly change in every period.
The standard deviation of shocks may be either provided directly or set/modified in each observed period by a scale factor.
If ``std0`` is the usual standard error for ``shock1``, then:
* using a scale factor in period ``t`` implies: ``std(shock1|t)=std0(shock1)*scale(t)``
* using a provided value in period ``t`` implies: ``std(shock1|t)=value(t)``.
The block has a similar syntax as the ``shocks`` block in a perfect foresight context.
It should contain one or more occurrences of the following
group of three lines (for setting values)::
var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
OR (for setting scale factors)::
var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
scales DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
NOTE: ``scales`` and ``values`` cannot be simultaneously set for the same shock in the same period, but it is
possible to set ``values`` for some periods and ``scales`` for other periods for the same shock. There can be
only one ``scales`` and ``values`` directive each for a given shock, so all affected periods must be set in one
statement.
*Example*
::
heteroskedastic_shocks;
var e1;
periods 86:87, 89:97;
scales 0.5, 0;
var e1;
periods 88;
values 0.1;
var e2;
periods 86:87 88:97;
values 0.04 0.01;
end;
.. matcomm:: get_shock_stderr_by_name ('EXOGENOUS_NAME');
|br| Given the name of an exogenous variable, returns its standard
deviation, as set by a previous ``shocks`` block.
.. matcomm:: set_shock_stderr_value ('EXOGENOUS_NAME', MATLAB_EXPRESSION);
|br| Sets the standard deviation of an exgonous variable. This does
essentially the same as setting the standard error via a ``shocks`` block,
except that it accepts arbitrary MATLAB/Octave expressions, and that it
works from MATLAB/Octave scripts.
Other general declarations
==========================
.. command:: dsample INTEGER [INTEGER];
|br| Reduces the number of periods considered in subsequent output commands.
.. _st-st:
Steady state
============
There are two ways of computing the steady state (i.e. the static
equilibrium) of a model. The first way is to let Dynare compute the
steady state using a nonlinear Newton-type solver; this should work
for most models, and is relatively simple to use. The second way is to
give more guidance to Dynare, using your knowledge of the model, by
providing it with a method to compute the steady state, either using a
`steady_state_model` block or writing matlab routine.
Finding the steady state with Dynare nonlinear solver
-----------------------------------------------------
.. command:: steady ;
steady (OPTIONS...);
|br| This command computes the steady state of a model using a
nonlinear Newton-type solver and displays it. When a steady state
file is used ``steady`` displays the steady state and checks that
it is a solution of the static model.
More precisely, it computes the equilibrium value of the
endogenous variables for the value of the exogenous variables
specified in the previous ``initval`` or ``endval`` block.
``steady`` uses an iterative procedure and takes as initial guess
the value of the endogenous variables set in the previous
``initval`` or ``endval`` block.
For complicated models, finding good numerical initial values for
the endogenous variables is the trickiest part of finding the
equilibrium of that model. Often, it is better to start with a
smaller model and add new variables one by one.
*Options*
.. _steady_maxit:
.. option:: maxit = INTEGER
Determines the maximum number of iterations used in the
non-linear solver. The default value of ``maxit`` is 50.
.. _steady_tolf:
.. option:: tolf = DOUBLE
Convergence criterion for termination based on the function
value. Iteration will cease when the residuals are smaller than
``tolf``. Default: ``eps^(1/3)``
.. _steady_tolx:
.. option:: tolx = DOUBLE
Convergence criterion for termination based on the step tolerance
along. Iteration will cease when the attempted step size is smaller than
``tolx``. Default: ``eps^(2/3)``
.. _solvalg:
.. option:: solve_algo = INTEGER
Determines the non-linear solver to use. Possible values for the
option are:
``0``
Use ``fsolve`` (under MATLAB, only available if you
have the Optimization Toolbox; always available under
Octave).
``1``
Use a Newton-like algorithm with line-search.
``2``
Splits the model into recursive blocks and solves each
block in turn using the same solver as value ``1``.
``3``
Use Chris Sims’ solver.
``4``
Splits the model into recursive blocks and solves each
block in turn using a trust-region solver with
autoscaling.
``5``
Newton algorithm with a sparse Gaussian elimination
(SPE) (requires ``bytecode`` option, see
:ref:`model-decl`).
``6``
Newton algorithm with a sparse LU solver at each
iteration (requires ``bytecode`` and/or ``block``
option, see :ref:`model-decl`).
``7``
Newton algorithm with a Generalized Minimal Residual
(GMRES) solver at each iteration (requires ``bytecode``
and/or ``block`` option, see :ref:`model-decl`).
``8``
Newton algorithm with a Stabilized Bi-Conjugate
Gradient (BICGSTAB) solver at each iteration (requires
bytecode and/or block option, see :ref:`model-decl`).
``9``
Trust-region algorithm with autoscaling (same as value ``4``,
but applied to the entire model, without splitting).
``10``
Levenberg-Marquardt mixed complementarity problem
(LMMCP) solver (*Kanzow and Petra (2004)*). The complementarity
conditions are specified with an ``mcp`` equation tag, see
:opt:`lmmcp`.
``11``
PATH mixed complementarity problem solver of *Ferris
and Munson (1999)*. The complementarity conditions are
specified with an ``mcp`` equation tag, see
:opt:`lmmcp`. Dynare only provides the interface for
using the solver. Due to licence restrictions, you have
to download the solver’s most current version yourself
from `http://pages.cs.wisc.edu/~ferris/path.html
<http://pages.cs.wisc.edu/~ferris/path.html>`__ and
place it in MATLAB’s search path.
``12``
Computes a block decomposition and then applies a Newton-type
solver on those smaller blocks rather than on the full
nonlinear system. This is similar to ``2``, but is typically
more efficient. The block decomposition is done at the
preprocessor level, which brings two benefits: it identifies
blocks that can be evaluated rather than solved; and evaluations
of the residual and Jacobian of the model are more efficient
because only the relevant elements are recomputed at every
iteration.
``14``
Computes a block decomposition and then applies a trust region
solver with autoscaling on those smaller blocks rather than on
the full nonlinear system. This is similar to ``4``, but is
typically more efficient. The block decomposition is done at
the preprocessor level, which brings two benefits: it
identifies blocks that can be evaluated rather than solved; and
evaluations of the residual and Jacobian of the model are more
efficient because only the relevant elements are recomputed at
every iteration.
|br| Default value is ``4``.
.. option:: homotopy_mode = INTEGER
Use a homotopy (or divide-and-conquer) technique to solve for
the steady state. If you use this option, you must specify a
``homotopy_setup`` block. This option can take three possible
values:
``1``
In this mode, all the parameters are changed
simultaneously, and the distance between the boundaries
for each parameter is divided in as many intervals as
there are steps (as defined by the ``homotopy_steps``
option); the problem is solved as many times as there
are steps.
``2``
Same as mode ``1``, except that only one parameter is
changed at a time; the problem is solved as many times
as steps times number of parameters.
``3``
Dynare tries first the most extreme values. If it
fails to compute the steady state, the interval
between initial and desired values is divided by two
for all parameters. Every time that it is impossible
to find a steady state, the previous interval is
divided by two. When it succeeds to find a steady
state, the previous interval is multiplied by two. In
that last case ``homotopy_steps`` contains the maximum
number of computations attempted before giving up.
.. option:: homotopy_steps = INTEGER
Defines the number of steps when performing a homotopy. See
``homotopy_mode`` option for more details.
.. option:: homotopy_force_continue = INTEGER
This option controls what happens when homotopy fails.
``0``
``steady`` fails with an error message
``1``
``steady`` keeps the values of the last homotopy step
that was successful and continues. **BE CAREFUL**:
parameters and/or exogenous variables are NOT at the
value expected by the user
|br| Default is ``0``.
.. option:: nocheck
Don’t check the steady state values when they are provided
explicitly either by a steady state file or a
``steady_state_model`` block. This is useful for models with
unit roots as, in this case, the steady state is not unique or
doesn’t exist.
.. _steady_markowitz:
.. option:: markowitz = DOUBLE
Value of the Markowitz criterion (in the interval :math:`(0,\infty)`) used to select the
pivot with sparse Gaussian elimination (``solve_algo = 5``). This criterion
governs the tradeoff between selecting the pivot resulting in the most
accurate solution (low ``markowitz`` values) and the one that preserves
maximum sparsity (high ``markowitz`` values). Default: 0.5.
.. option:: fsolve_options = (NAME, VALUE, ...)
A list of NAME and VALUE pairs. Can be used to set options for the
``fsolve`` routine, which is selected when ``solve_algo = 0`` (this
option has no effect for other values of ``solve_algo``). For the list
of available name/value pairs, see the documentation of ``fsolve`` in
the MATLAB or Octave manual. Note that Dynare already uses the values of
the ``maxit``, ``tolf`` and ``tolx`` options of the ``steady`` command
for initializing the corresponding options passed to ``fsolve``, so you
should not need to override those. Also note that you should not try to
override the value of the ``Jacobian`` or ``SpecifyObjectiveGradient``
option.
*Example*
See :ref:`init-term-cond`.
After computation, the steady state is available in the following variable:
.. matvar:: oo_.steady_state
Contains the computed steady state. Endogenous variables are
ordered in the order of declaration used in the ``var`` command (which
is also the order used in ``M_.endo_names``).
.. matcomm:: get_mean ('ENDOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );
Returns the steady of state of the given endogenous variable(s), as it is
stored in ``oo_.steady_state``. Note that, if the steady state has not yet
been computed with ``steady``, it will first try to compute it.
.. block:: homotopy_setup ;
This block is used to declare initial and final values when using
a homotopy method. It is used in conjunction with the option
``homotopy_mode`` of the steady command.
The idea of homotopy (also called divide-and-conquer by some
authors) is to subdivide the problem of finding the steady state
into smaller problems. It assumes that you know how to compute the
steady state for a given set of parameters, and it helps you
finding the steady state for another set of parameters, by
incrementally moving from one to another set of parameters.
The purpose of the ``homotopy_setup`` block is to declare the
final (and possibly also the initial) values for the parameters or
exogenous that will be changed during the homotopy. It should
contain lines of the form::
VARIABLE_NAME, EXPRESSION, EXPRESSION;
This syntax specifies the initial and final values of a given
parameter/exogenous.
There is an alternative syntax::
VARIABLE_NAME, EXPRESSION;
Here only the final value is specified for a given
parameter/exogenous; the initial value is taken from the
preceeding ``initval`` block.
A necessary condition for a successful homotopy is that Dynare
must be able to solve the steady state for the initial
parameters/exogenous without additional help (using the guess
values given in the ``initval`` block).
If the homotopy fails, a possible solution is to increase the
number of steps (given in ``homotopy_steps`` option of
``steady``).
*Example*
In the following example, Dynare will first compute the steady
state for the initial values (``gam=0.5`` and ``x=1``), and then
subdivide the problem into 50 smaller problems to find the steady
state for the final values (``gam=2`` and ``x=2``)::
var c k;
varexo x;
parameters alph gam delt bet aa;
alph=0.5;
delt=0.02;
aa=0.5;
bet=0.05;
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
end;
initval;
x = 1;
k = ((delt+bet)/(aa*x*alph))^(1/(alph-1));
c = aa*x*k^alph-delt*k;
end;
homotopy_setup;
gam, 0.5, 2;
x, 2;
end;
steady(homotopy_mode = 1, homotopy_steps = 50);
Providing the steady state to Dynare
------------------------------------
If you know how to compute the steady state for your model, you can
provide a MATLAB/Octave function doing the computation instead of
using ``steady``. Again, there are two options for doing that:
* The easiest way is to write a ``steady_state_model`` block, which
is described below in more details. See also ``fs2000.mod`` in the
``examples`` directory for an example. The steady state file
generated by Dynare will be called ``+FILENAME/steadystate.m.``
* You can write the corresponding MATLAB function by hand. If your
``.mod`` file is called ``FILENAME.mod``, the steady state file must be
called ``FILENAME_steadystate.m``. See
``NK_baseline_steadystate.m`` in the examples directory for an
example. This option gives a bit more flexibility (loops and
conditional structures can be used), at the expense of a heavier
programming burden and a lesser efficiency.
Note that both files allow to update parameters in each call of the
function. This allows for example to calibrate a model to a labor
supply of 0.2 in steady state by setting the labor disutility
parameter to a corresponding value (see ``NK_baseline_steadystate.m``
in the ``examples`` directory). They can also be used in estimation
where some parameter may be a function of an estimated parameter and
needs to be updated for every parameter draw. For example, one might
want to set the capital utilization cost parameter as a function of
the discount rate to ensure that capacity utilization is 1 in steady
state. Treating both parameters as independent or not updating one as
a function of the other would lead to wrong results. But this also
means that care is required. Do not accidentally overwrite your
parameters with new values as it will lead to wrong results.
.. block:: steady_state_model ;
|br| When the analytical solution of the model is known, this command
can be used to help Dynare find the steady state in a more
efficient and reliable way, especially during estimation where the
steady state has to be recomputed for every point in the parameter
space.
Each line of this block consists of a variable (either an
endogenous, a temporary variable or a parameter) which is assigned
an expression (which can contain parameters, exogenous at the
steady state, or any endogenous or temporary variable already
declared above). Each line therefore looks like::
VARIABLE_NAME = EXPRESSION;
Note that it is also possible to assign several variables at the
same time, if the main function in the right hand side is a
MATLAB/Octave function returning several arguments::
[ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION;
Dynare will automatically generate a steady state file (of the
form ``+FILENAME/steadystate.m``) using the information provided
in this block.
*Steady state file for deterministic models*
The ``steady_state_model`` block also works with deterministic
models. An ``initval`` block and, when necessary, an ``endval``
block, is used to set the value of the exogenous variables. Each
``initval`` or ``endval`` block must be followed by ``steady`` to
execute the function created by ``steady_state_model`` and set the
initial, respectively terminal, steady state.
*Example*
::
var m P c e W R k d n l gy_obs gp_obs y dA;
varexo e_a e_m;
parameters alp bet gam mst rho psi del;
...
// parameter calibration, (dynamic) model declaration, shock calibration...
...
steady_state_model;
dA = exp(gam);
gst = 1/dA; // A temporary variable
m = mst;
// Three other temporary variables
khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1));
xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1);
nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp );
n = xist/(nust+xist);
P = xist + nust;
k = khst*n;
l = psi*mst*n/( (1-psi)*(1-n) );
c = mst/P;
d = l - mst + 1;
y = k^alp*n^(1-alp)*gst^alp;
R = mst/bet;
// You can use MATLAB functions which return several arguments
[W, e] = my_function(l, n);
gp_obs = m/dA;
gy_obs = dA;
end;
steady;
.. _eq-tag-ss:
Replace some equations during steady state computations
-------------------------------------------------------
When there is no steady state file, Dynare computes the steady state
by solving the static model, i.e. the model from the ``.mod`` file
from which leads and lags have been removed.
In some specific cases, one may want to have more control over the way
this static model is created. Dynare therefore offers the possibility
to explicitly give the form of equations that should be in the static
model.
More precisely, if an equation is prepended by a ``[static]`` tag,
then it will appear in the static model used for steady state
computation, but that equation will not be used for other
computations. For every equation tagged in this way, you must tag
another equation with ``[dynamic]``: that equation will not be used
for steady state computation, but will be used for other computations.
This functionality can be useful on models with a unit root, where
there is an infinity of steady states. An equation (tagged
``[dynamic]``) would give the law of motion of the nonstationary
variable (like a random walk). To pin down one specific steady state,
an equation tagged ``[static]`` would affect a constant value to the
nonstationary variable. Another situation where the ``[static]`` tag
can be useful is when one has only a partial closed form solution for
the steady state.
*Example*
This is a trivial example with two endogenous variables. The second
equation takes a different form in the static model::
var c k;
varexo x;
...
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
[dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
[static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1));
end;
Getting information about the model
===================================
.. command:: check ;
check (OPTIONS...);
|br| Computes the eigenvalues of the model linearized around the
values specified by the last ``initval``, ``endval`` or ``steady``
statement. Generally, the eigenvalues are only meaningful if the
linearization is done around a steady state of the model. It is a
device for local analysis in the neighborhood of this steady
state.
A necessary condition for the uniqueness of a stable equilibrium
in the neighborhood of the steady state is that there are as many
eigenvalues larger than one in modulus as there are forward
looking variables in the system. An additional rank condition
requires that the square submatrix of the right Schur vectors
corresponding to the forward looking variables (jumpers) and to
the explosive eigenvalues must have full rank.
Note that the outcome may be different from what would be
suggested by ``sum(abs(oo_.dr.eigval))`` when eigenvalues are very
close to :opt:`qz_criterium <qz_criterium = DOUBLE>`.
*Options*
.. option:: solve_algo = INTEGER
See :ref:`solve_algo <solvalg>`, for the possible values and
their meaning.
.. option:: qz_zero_threshold = DOUBLE
Value used to test if a generalized eigenvalue is :math:`0/0`
in the generalized Schur decomposition (in which case the
model does not admit a unique solution). Default: ``1e-6``.
*Output*
``check`` returns the eigenvalues in the global variable ``oo_.dr.eigval``.
.. matvar:: oo_.dr.eigval
Contains the eigenvalues of the model, as computed by the ``check`` command.
.. command:: model_diagnostics ;
|br| This command performs various sanity checks on the model, and
prints a message if a problem is detected (missing variables at
current period, invalid steady state, singular Jacobian of static
model).
.. command:: model_info ;
model_info (OPTIONS...);
|br| This command provides information about the model. By default,
it will provide a list of predetermined state variables, forward-looking variables,
and purely static variables.
The command also allows to display information on the dynamic and static
versions of the block decomposition of the model:
* The normalization of the model: an endogenous variable is
attributed to each equation of the model (the dependent variable);
* The block structure of the model: for each block ``model_info``
indicates its type, size as well as the equation number(s) or name tags and
endogenous variables belonging to this block.
There are five different types of blocks depending on the
simulation method used:
* ``EVALUATE FORWARD``
In this case the block contains only equations where the
dependent variable :math:`j` attributed to the equation appears
contemporaneously on the left hand side and where no forward looking
endogenous variables appear. The block has the form:
:math:`y_{j,t} = f_j(y_t, y_{t-1}, \ldots, y_{t-k})`.
* ``EVALUATE BACKWARD``
The block contains only equations where the dependent variable :math:`j`
attributed to the equation appears contemporaneously on the left hand side
and where no backward looking endogenous variables
appear. The block has the form: :math:`y_{j,t} = f_j(y_t,
y_{t+1}, \ldots, y_{t+k})`.
* ``SOLVE BACKWARD x``
The block contains only equations where the dependent variable :math:`j`
attributed to the equation does not appear contemporaneously on the
left hand side and where no forward looking endogenous
variables appear. The block has the form: :math:`g_j(y_{j,t},
y_t, y_{t-1}, \ldots, y_{t-k})=0`. Here, ``x`` denotes the subtype of the block.
``x`` is equal to ``SIMPLE`` if the block has only one equation. If several equations
appear in the block, ``x`` is equal to ``COMPLETE``.
* ``SOLVE FORWARD x``
The block contains only equations where the dependent variable :math:`j`
attributed to the equation does not appear contemporaneously on the
left hand side and where no backward looking endogenous
variables appear. The block has the form: :math:`g_j(y_{j,t},
y_t, y_{t+1}, \ldots, y_{t+k})=0`. Here, ``x`` denotes the subtype of the block.
``x`` is equal to ``SIMPLE`` if the block has only one equation. If several equations
appear in the block, ``x`` is equal to ``COMPLETE``.
* ``SOLVE TWO BOUNDARIES x``
The block contains equations depending on both forward and
backward variables. The block looks like: :math:`g_j(y_{j,t},
y_t, y_{t-1}, \ldots, y_{t-k} ,y_t, y_{t+1}, \ldots,
y_{t+k})=0`. Here, ``x`` denotes the subtype of the block.
``x`` is equal to ``SIMPLE`` if the block has only one equation. If several equations
appear in the block, ``x`` is equal to ``COMPLETE``.
*Options*
.. option:: block_static
Prints out the block decomposition of the static model.
.. option:: block_dynamic
Prints out the block decomposition of the dynamic model.
.. option:: incidence
Displays the gross incidence matrix and the reordered incidence
matrix of the block decomposed model for the ``block_dynamic`` or ``block_static`` options.
.. command:: print_bytecode_dynamic_model ;
|br| Prints the equations and the Jacobian matrix of the dynamic
model stored in the bytecode binary format file. Can only be used
in conjunction with the ``bytecode`` option of the ``model``
block.
.. command:: print_bytecode_static_model ;
|br| Prints the equations and the Jacobian matrix of the static model
stored in the bytecode binary format file. Can only be used in
conjunction with the ``bytecode`` option of the ``model`` block.
.. _det-simul:
Deterministic simulation
========================
Perfect foresight
-----------------
When the framework is deterministic, Dynare can be used for models
with the assumption of perfect foresight. Typically, the system is
supposed to be in a state of equilibrium before a period ``1`` when the
news of a contemporaneous or of a future shock is learned by the
agents in the model. The purpose of the simulation is to describe the
reaction in anticipation of, then in reaction to the shock, until the
system returns to the old or to a new state of equilibrium. In most
models, this return to equilibrium is only an asymptotic phenomenon,
which one must approximate by an horizon of simulation far enough in
the future. Another exercise for which Dynare is well suited is to
study the transition path to a new equilibrium following a permanent
shock. For deterministic simulations, the numerical problem consists
of solving a nonlinear system of simultaneous equations in ``n`` endogenous
variables in ``T`` periods. Dynare offers several algorithms for solving
this problem, which can be chosen via the ``stack_solve_algo``
option. By default (``stack_solve_algo=0``), Dynare uses a Newton-type
method to solve the simultaneous equation system. Because the
resulting Jacobian is in the order of ``n`` by ``T`` and hence will be
very large for long simulations with many variables, Dynare makes use
of the sparse matrix capacities of MATLAB/Octave. A slower but
potentially less memory consuming alternative (``stack_solve_algo=1``)
is based on a Newton-type algorithm first proposed by *Laffargue
(1990)* and *Boucekkine (1995)*, which avoids ever storing the full
Jacobian. The details of the algorithm can be found in *Juillard
(1996)*. The third type of algorithms makes use of block decomposition
techniques (divide-and-conquer methods) that exploit the structure of
the model. The principle is to identify recursive and simultaneous
blocks in the model structure and use this information to aid the
solution process. These solution algorithms can provide a significant
speed-up on large models.
.. warning:: Be careful when employing auxiliary variables in the context
of perfect foresight computations. The same model may work for stochastic
simulations, but fail for perfect foresight simulations. The issue arises
when an equation suddenly only contains variables dated ``t+1`` (or ``t-1``
for that matter). In this case, the derivative in the last (first) period
with respect to all variables will be 0, rendering the stacked Jacobian singular.
*Example*
Consider the following specification of an Euler equation with log utility:
::
Lambda = beta*C(-1)/C;
Lambda(+1)*R(+1)= 1;
Clearly, the derivative of the second equation with respect to all endogenous
variables at time ``t`` is zero, causing ``perfect_foresight_solver`` to generally
fail. This is due to the use of the Lagrange multiplier ``Lambda`` as an auxiliary
variable. Instead, employing the identical
::
beta*C/C(+1)*R(+1)= 1;
will work.
.. command:: perfect_foresight_setup ;
perfect_foresight_setup (OPTIONS...);
|br| Prepares a perfect foresight simulation, by extracting the
information in the ``initval``, ``endval`` and ``shocks`` blocks
and converting them into simulation paths for exogenous and
endogenous variables.
This command must always be called before running the simulation
with ``perfect_foresight_solver``.
*Options*
.. option:: periods = INTEGER
Number of periods of the simulation.
.. option:: datafile = FILENAME
Used to specify path for all endogenous and exogenous variables.
Strictly equivalent to :comm:`initval_file`.
*Output*
The paths for the exogenous variables are stored into
``oo_.exo_simul``.
The initial and terminal conditions for the endogenous variables
and the initial guess for the path of endogenous variables are
stored into ``oo_.endo_simul``.
.. command:: perfect_foresight_solver ;
perfect_foresight_solver (OPTIONS...);
|br| Computes the perfect foresight (or deterministic) simulation
of the model.
Note that ``perfect_foresight_setup`` must be called before this
command, in order to setup the environment for the simulation.
If the perfect foresight solver cannot directly find the solution of the
problem, it subsequently tries a homotopy technique (unless the
``no_homotopy`` option is given). Concretely, this technique consists in
dividing the problem into smaller steps by diminishing the size of shocks
and increasing them progressively until the problem converges. Note that
the homotopy technique is not implemented for purely forward or backward
models.
*Options*
.. option:: maxit = INTEGER
Determines the maximum number of iterations used in the
non-linear solver. The default value of ``maxit`` is ``50``.
.. option:: tolf = DOUBLE
Convergence criterion for termination based on the function
value. Iteration will cease when it proves impossible to
improve the function value by more than ``tolf``. Default:
``1e-5``
.. option:: tolx = DOUBLE
Convergence criterion for termination based on the change in
the function argument. Iteration will cease when the solver
attempts to take a step that is smaller than ``tolx``. Default:
``1e-5``
.. option:: noprint
Don’t print anything. Useful for loops.
.. option:: print
Print results (opposite of ``noprint``).
.. option:: stack_solve_algo = INTEGER
Algorithm used for computing the solution. Possible values are:
``0``
Newton method to solve simultaneously all the equations for
every period, using sparse matrices (Default).
``1``
Use the Laffargue-Boucekkine-Juillard (LBJ) algorithm proposed
in *Juillard (1996)*. It is slower than ``stack_solve_algo=0``,
but may be less memory consuming on big models. Note that if the
``block`` option is used (see :ref:`model-decl`), a simple
Newton algorithm with sparse matrices is used for blocks which
are purely backward or forward (of type ``SOLVE BACKWARD`` or
``SOLVE FORWARD``, see :comm:`model_info`), since LBJ only makes
sense on blocks with both leads and lags (of type ``SOLVE TWO
BOUNDARIES``).
``2``
Use a Newton algorithm with a Generalized Minimal
Residual (GMRES) solver at each iteration (requires
``bytecode`` and/or ``block`` option, see
:ref:`model-decl`)
``3``
Use a Newton algorithm with a Stabilized Bi-Conjugate
Gradient (BICGSTAB) solver at each iteration (requires
``bytecode`` and/or ``block`` option, see
:ref:`model-decl`).
``4``
Use a Newton algorithm with an optimal path length at
each iteration (requires ``bytecode`` and/or ``block``
option, see :ref:`model-decl`).
``5``
Use a Newton algorithm with a sparse Gaussian
elimination (SPE) solver at each iteration (requires
``bytecode`` option, see :ref:`model-decl`).
``6``
Synonymous for ``stack_solve_algo=1``. Kept for historical
reasons.
``7``
Allows the user to solve the perfect foresight model
with the solvers available through option
``solve_algo`` (See :ref:`solve_algo <solvalg>` for a
list of possible values, note that values 5, 6, 7 and
8, which require ``bytecode`` and/or ``block`` options,
are not allowed). For instance, the following
commands::
perfect_foresight_setup(periods=400);
perfect_foresight_solver(stack_solve_algo=7, solve_algo=9)
trigger the computation of the solution with a trust
region algorithm.
.. option:: robust_lin_solve
Triggers the use of a robust linear solver for the default
``stack_solve_algo=0``.
.. option:: solve_algo
See :ref:`solve_algo <solvalg>`. Allows selecting the solver
used with ``stack_solve_algo=7``.
.. option:: no_homotopy
This option tells Dynare to not try a homotopy technique (as described
above) if the problem cannot be solved directly.
.. option:: homotopy_alt_starting_point
When the homotopy technique is tried (as described above), Dynare first
tries to reduce the size of the shock in order to get a successful
simulation on which to build upon for simulating a shock of the true
size. However, if an ``endval`` block is present (*i.e.* if the terminal
state differs from the initial state), there are two ways of reducing
the size of the shock. By default, Dynare will perform this reduction by
computing a simulation whose initial state is closer to the target
terminal state; in other words, it will implicitly modify the contents
of the ``initval`` and ``shocks`` blocks to make them closer to the the
contents of the ``endval`` block. If this option is set, Dynare will do
the opposite: it will implicitly modify the contents of the ``endval``
and ``shocks`` blocks to make them closer to the contents of the
``initval`` block.
.. option:: markowitz = DOUBLE
Value of the Markowitz criterion, used to select the
pivot. Only used when ``stack_solve_algo = 5``. Default:
``0.5``.
.. option:: minimal_solving_periods = INTEGER
Specify the minimal number of periods where the model has to be
solved, before using a constant set of operations for the
remaining periods. Only used when ``stack_solve_algo =
5``. Default: ``1``.
.. option:: lmmcp
Solves the perfect foresight model with a Levenberg-Marquardt
mixed complementarity problem (LMMCP) solver (*Kanzow and Petra
(2004)*), which allows to consider inequality constraints on
the endogenous variables (such as a ZLB on the nominal interest
rate or a model with irreversible investment). This option is
equivalent to ``stack_solve_algo=7`` **and**
``solve_algo=10``. Using the LMMCP solver avoids the need for min/max
operators and explicit complementary slackness conditions in the model
as they will typically introduce a singularity into the Jacobian. This is
done by setting the problem up as a mixed complementarity problem (MCP) of the form:
.. math::
LB = X &\Rightarrow F(X)>0\\
LB\leq X \leq UB &\Rightarrow F(X)=0\\
X =UB &\Rightarrow F(X)<0.
where :math:`X` denotes the vector of endogenous variables, :math:`F(X)` the equations
of the model, :math:`LB` denotes a lower bound, and :math:`UB` an upper bound. Such a setup
is implemented by attaching an equation tag (see :ref:`model-decl`)
with the ``mcp`` keyword to the affected equations. This tag states that
the equation to which the tag is attached has to hold unless the inequality
constraint within the tag is binding.
For instance, a ZLB on the nominal interest rate would be specified
as follows in the model block::
model;
...
[mcp = 'r > -1.94478']
r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e;
...
end;
where ``1.94478`` is the steady state level of the nominal
interest rate and ``r`` is the nominal interest rate in
deviation from the steady state. This construct implies that
the Taylor rule is operative, unless the implied interest rate
``r<=-1.94478``, in which case the ``r`` is fixed at
``-1.94478`` (thereby being equivalent to a complementary
slackness condition). By restricting the value of ``r`` coming
out of this equation, the ``mcp`` tag also avoids using
``max(r,-1.94478)`` for other occurrences of ``r`` in the rest
of the model. Two things are important to keep in mind. First, because the
``mcp`` tag effectively replaces a complementary slackness
condition, it cannot be simply attached to any
equation. Rather, it must be attached to the correct affected
equation as otherwise the solver will solve a different problem
than originally intended. Second, the sign of the residual of the dynamic
equation must conform to the MCP setup outlined above. In case of the ZLB,
we are dealing with a lower bound. Consequently, the dynamic equation
needs to return a positive residual. Dynare by default computes the residual
of an equation ``LHS=RHS`` as ``residual=LHS-RHS``, while an implicit equation
``LHS`` is interpreted as ``LHS=0``. For the above equation this implies
``residual= r - (rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e);``
which is correct, since it will be positive if the implied interest rate
``rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e`` is
below ``r=-1.94478``. In contrast, specifying the equation as
``rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e = r;```
would be wrong.
Note that in the current implementation, the content of the
``mcp`` equation tag is not parsed by the preprocessor. The
inequalities must therefore be as simple as possible: an
endogenous variable, followed by a relational operator,
followed by a number (not a variable, parameter or expression).
.. option:: endogenous_terminal_period
The number of periods is not constant across Newton iterations
when solving the perfect foresight model. The size of the
nonlinear system of equations is reduced by removing the
portion of the paths (and associated equations) for which the
solution has already been identified (up to the tolerance
parameter). This strategy can be interpreted as a mix of the
shooting and relaxation approaches. Note that round off errors
are more important with this mixed strategy (user should check
the reported value of the maximum absolute error). Only
available with option ``stack_solve_algo==0``.
.. option:: linear_approximation
Solves the linearized version of the perfect foresight
model. The model must be stationary. Only available with option
``stack_solve_algo==0`` or ``stack_solve_algo==7``.
*Output*
The simulated endogenous variables are available in global matrix
``oo_.endo_simul``.
.. command:: simul ;
simul (OPTIONS...);
|br| This command is deprecated. It is strictly equivalent
to a call to ``perfect_foresight_setup`` followed by a call to
``perfect_foresight_solver``.
*Options*
Accepts all the options of ``perfect_foresight_setup`` and
``perfect_foresight_solver``.
.. matvar:: oo_.endo_simul
|br| This variable stores the result of a deterministic simulation
(computed by ``perfect_foresight_solver`` or ``simul``) or of a
stochastic simulation (computed by ``stoch_simul`` with the
periods option or by ``extended_path``). The variables are
arranged row by row, in order of declaration (as in
``M_.endo_names``). Note that this variable also contains initial
and terminal conditions, so it has more columns than the value of the
``periods`` option: the first simulation period is in
column ``1+M_.maximum_lag``, and the total number of columns is
``M_.maximum_lag+periods+M_.maximum_lead``.
.. matvar:: oo_.exo_simul
|br| This variable stores the path of exogenous variables during a
simulation (computed by ``perfect_foresight_solver``, ``simul``,
``stoch_simul`` or ``extended_path``). The variables are arranged
in columns, in order of declaration (as in
``M_.exo_names``). Periods are in rows. Note that this convention
regarding columns and rows is the opposite of the convention for
``oo_.endo_simul``! Also note that this variable also contains initial
and terminal conditions, so it has more rows than the value of the
``periods`` option: the first simulation period is in row
``1+M_.maximum_lag``, and the total number of rows is
``M_.maximum_lag+periods+M_.maximum_lead``.
.. matvar:: M_.maximum_lag
|br| The maximum number of lags in the model. Note that this value is
computed on the model *after* the transformations related to auxiliary
variables, so in practice it is either 1 or 0 (the latter value corresponds
to a purely forward or static model).
.. matvar:: M_.maximum_lead
|br| The maximum number of leads in the model. Note that this value is
computed on the model *after* the transformations related to auxiliary
variables, so in practice it is either 1 or 0 (the latter value corresponds
to a purely backward or static model).
Perfect foresight with expectation errors
-----------------------------------------
The solution under perfect foresight that was presented in the previous section
makes the assumption that agents learn the complete path of future shocks in
period 1, without making any expectation errors.
One may however want to study a scenario where it turns out that agents make
expectation errors, in the sense that the path they had anticipated in period 1
does not realize exactly. More precisely, in some simulation periods, they may
receive new information that makes them revise their anticipation for the path
of future shocks. Also, under this scenario, it is assumed that agents
behave as under perfect foresight, *i.e.* they take their decisions as if there
was no uncertainty and they knew exactly the path of future shocks; the new
information that they may receive comes as a total surprise to them.
Such a scenario can be solved by Dynare using the
``perfect_foresight_with_expectation_errors_setup`` and
``perfect_foresight_with_expectation_errors_solver`` commands, alongside ``shocks``
and ``endval`` blocks which are given a special ``learnt_in`` option.
.. block:: shocks(learnt_in=INTEGER) ;
shocks(learnt_in=INTEGER,overwrite) ;
|br| The ``shocks(learnt_in=INTEGER)`` can be used to specify temporary
shocks that are learnt in a specific period. It should contain one or more
occurences of the following group of three lines, with the same semantics
as a regular :bck:`shocks` block::
var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
If the period in which information is learnt is greater or equal than 2,
then it is possible to specify the shock values in deviation with respect
to the values that were expected from the perspective of the previous
period. If the new information consists of an addition to the
previously-anticipated value, the ``values`` keyword can be replaced by the
``add`` keyword; similarly, if the new information consists of an addition to the
previously-anticipated value, the ``values`` keyword can be replaced by the
``multiply`` keyword.
The ``overwrite`` option says that this block cancels and replaces previous
``shocks`` blocks that have the same ``learnt_in`` option.
Note that a ``shocks(learnt_in=1)`` block is equivalent to a regular
:bck:`shocks` block.
*Example*
::
shocks(learnt_in=1);
var x;
periods 1:2 3:4 5;
values 1 1.2 1.4;
end;
shocks(learnt_in=2);
var x;
periods 3:4;
add 0.1;
end;
shocks(learnt_in=4);
var x;
periods 5;
multiply 2;
end;
This syntax means that:
- from the perspective of period 1, ``x`` is expected to be equal to 1 in
periods 1 and 2, to 1.2 in periods 3 and 4, and to 1.4 in period 5;
- from the perspective of periods 2 (and 3), ``x`` is expected to be
equal to 1 in period 2, to 1.3 in periods 3 and 4, and to 1.4 in period
5;
- from the perspective of periods 4 (and following), ``x`` is expected to
be equal to 1.3 in period 4, and to 2.8 in period 5.
.. block:: endval(learnt_in=INTEGER) ;
|br| The ``endval(learnt_in=INTEGER)`` can be used to specify terminal
conditions that are learnt in a specific period.
Note that an ``endval(learnt_in=1)`` block is equivalent to a regular
:bck:`endval` block.
It is possible to express the terminal condition by specifying the level of
the exogenous variable (using an equal symbol, as in a regular
:bck:`endval` blocks without the ``learnt_in`` option). But it is also
possible to express the terminal condition as an addition to the value
expected from the perspective of the previous previous period (using the
``+=`` operator), or as a multiplicative factor over that previously
expected value (using the ``*=`` operator).
*Example*
::
endval(learnt_in = 3);
x = 1.1;
y += 0.1;
z *= 2;
end;
This syntax means that, in period 3, the agents learn that:
- the terminal condition for ``x`` will be 1.1;
- the terminal condition for ``y`` will be 0.1 above the terminal
condition for ``y`` that was expected from the perspective of period 2;
- the terminal condition for ``z`` will be 2 times the terminal condition
for ``z`` that was expected from the perspective of period 2.
Those values will be the realized ones, unless there is another
``endval(learnt_in=p)`` block with ``p>3``.
.. command:: perfect_foresight_with_expectation_errors_setup ;
perfect_foresight_with_expectation_errors_setup (OPTIONS...);
|br| Prepares a perfect foresight simulation with expectation errors, by
extracting the contents of the ``initval``, ``endval`` and ``shocks``
blocks (the latter two types of blocks typically used with the
``learnt_in`` option); alternatively, the information about future shocks
can be given in a CSV file using the ``datafile`` option.
This command must always be called before running the simulation
with ``perfect_foresight_with_expectation_errors_solver``.
Note that this command makes the assumption that the terminal condition is
always a steady state. Hence, it will recompute the terminal steady state
as many times as the anticipation about the terminal condition changes. In
particular, the information about endogenous variables that may be given in
the ``endval`` block is ignored.
*Options*
.. option:: periods = INTEGER
Number of periods of the simulation.
.. option:: datafile = FILENAME
Used to specify the information about future shocks and their
anticipation through a CSV file, as an alternative to ``shocks`` and
``endval`` blocks.
The file has the following format:
- the first column is ignored (can be used to add descriptive labels)
- the first line contains names of exogenous variables
- the second line contains, in columns, indices of periods *at which*
expectations are formed; the information set used in a given period is
described by all the columns for which that line is equal to the
period index
- the subsequent lines correspond to the periods *for which*
expectations are formed, one period per line; each line gives the
values of present and future exogenous variables, as seen from the
period given in the second line
- the last line corresponds to the terminal condition for exogenous
variables, as anticipated in the various informational periods
If ``p`` is the value of the ``periods`` option and ``k`` is the number
of exogenous variables, then the CSV file has ``p+3`` lines and
``k×p+1`` columns.
Concretely, the value of a given exogenous in period ``t``, as anticipated
from period ``s``, is given in line ``t+2``, and in the column which has
the name of the variable on the first line and ``s`` on the second
line. Of course, values in cells corresponding to ``t<s`` are ignored.
.. option:: solve_algo = INTEGER
See :ref:`solve_algo <solvalg>`. Used when computing the terminal steady state.
.. option:: tolf = DOUBLE
See :ref:`tolf <steady_tolf>`. Used when computing the terminal steady state.
.. option:: tolx = DOUBLE
See :ref:`tolx <steady_tolx>`. Used when computing the terminal steady state.
.. option:: maxit = INTEGER
See :ref:`maxit <steady_maxit>`. Used when computing the terminal steady state.
.. option:: markowitz = DOUBLE
See :ref:`markowitz <steady_markowitz>`. Used when computing the terminal steady state.
*Output*
``oo_.exo_simul`` and ``oo_.endo_simul`` are initialized before the
simulation. Temporary shocks are stored in ``oo_.pfwee.shocks_info``,
terminal conditions for exogenous variables are stored in
``oo_.pfwee.terminal_info``, and terminal steady states are stored in
``oo_.pfwee.terminal_steady_state``.
*Example*
Here is a CSV file example that could be given to the ``datafile`` option
(adding some extra padding space for clarity):
::
Exogenous , x, x, x, x, x, x, x
Period (info), 1, 2, 3, 4, 5, 6, 7
Period 1 (real), 1.2, , , , , ,
Period 2 (real), 1, 1.3, , , , ,
Period 3 (real), 1, 1, 1.4, , , ,
Period 4 (real), 1, 1, 1, 1, , ,
Period 5 (real), 1, 1, 1, 1, 1, ,
Period 6 (real), 1, 1, 1, 1, 1, 1.1,
Period 7 (real), 1, 1, 1, 1, 1, 1.1, 1.1
Terminal (real), 1, 1.1, 1.2, 1.2, 1.2, 1.1, 1.1
In this example, there is only one exogenous variable (``x``), and 7
simulation periods. In the first period, agents learn a contemporary
shock (1.2), but anticipate no further shock. In period 2, they learn an
unexpected contemporary shock (1.3), and also a change in the terminal
condition (1.1). In period 3 again there is an unexpected contemporary
shock and a change in the terminal condition. No new information comes
in period 4 and 5. In period 6, an unexpected permanent shock is learnt.
No new information comes in period 7.
Alternatively, instead of using a CSV file, the same sequence of
information sets could be described using the following blocks:
::
initval;
x = 1;
end;
steady;
shocks(learnt_in = 1);
var x;
periods 1;
values 1.2;
end;
shocks(learnt_in = 2);
var x;
periods 2;
values 1.3;
end;
endval(learnt_in = 2);
x = 1.1;
end;
shocks(learnt_in = 3);
var x;
periods 3;
values 1.4;
end;
endval(learnt_in = 3);
x = 1.2;
end;
shocks(learnt_in = 6);
var x;
periods 6:7;
values 1.1;
end;
endval(learnt_in = 6);
x = 1.1;
end;
.. command:: perfect_foresight_with_expectation_solver ;
perfect_foresight_with_expectation_solver (OPTIONS...);
|br| Computes the perfect foresight simulation with expectation errors
of the model.
Note that ``perfect_foresight_with_expectation_errors_setup`` must be
called before this command, in order to setup the environment for the
simulation.
*Options*
This command accepts all the options of :comm:`perfect_foresight_solver`,
with the same semantics, plus the following ones:
.. option:: terminal_steady_state_as_guess_value
By default, the initial guess for the computation of the path of
endogenous is the initial steady state (when using the information set
from period 1) or the previously simulated path (when using an
information set that is different from that of period 1). When this
option is given, the initial guess is instead the terminal steady
state.
.. option:: constant_simulation_length
By default, every time the information set changes, the simulation with
the new information set is shorter than the previous one (because the
terminal date is getting closer). When this option is set, every new
simulation has the same length (as specified by the `periods`` option
of :comm:`perfect_foresight_with_expectation_errors_setup`; as a
consequence, the simulated paths as stored in ``oo_.endo_simul`` will
be longer when this option is set (if `s` is the last period in which
the information set is modified, then they will contain `s+periods-1`
periods, excluding initial and terminal conditions).
*Output*
The simulated paths of endogenous variables are available in
``oo_.endo_simul``.
.. matvar:: oo_.pfwee.shocks_info
|br| This variable stores the temporary shocks used during perfect
foresight simulations with expectation errors, after
:comm:`perfect_foresight_with_expectation_errors_setup` has been run. It is
a three-dimensional matrix: first dimension correspond to exogenous
variables (in declaration order); second dimension corresponds to real
time; third dimension corresponds to informational time. In other words,
the value of exogenous indexed ``k`` in period ``t``, as anticipated from
period ``s``, is stored in ``oo_.pfwee.shocks_info(k,t,s)``.
.. matvar:: oo_.pfwee.terminal_info
|br| This variable stores the terminal conditions for exogenous variables
used during perfect foresight simulations with expectation errors, after
:comm:`perfect_foresight_with_expectation_errors_setup` has been run. It is
a matrix, whose lines correspond to exogenous variables (in declaration
order), and whose columns correspond to informational time. In other words,
the terminal condition for exogenous indexed ``k``, as anticipated from
period ``s``, is stored in ``oo_.pfwee.terminal_info(k,s)``.
.. matvar:: oo_.pfwee.terminal_steady_state
|br| This variable stores the terminal steady states for endogenous
variables used during perfect foresight simulations with expectation
errors, after :comm:`perfect_foresight_with_expectation_errors_setup` has
been run. It is a matrix, whose lines correspond to endogenous variables
(in declaration order), and whose columns correspond to informational time.
In other words, the terminal steady state for endogenous indexed ``k``, as
anticipated from period ``s``, is stored in
``oo_.pfwee.terminal_steady_state(k,s)``.
.. _stoch-sol:
Stochastic solution and simulation
==================================
In a stochastic context, Dynare computes one or several simulations
corresponding to a random draw of the shocks.
The main algorithm for solving stochastic models relies on a Taylor
approximation, up to third order, of the expectation functions (see
*Judd (1996)*, *Collard and Juillard (2001a, 2001b)*, and
*Schmitt-Grohé and Uríbe (2004)*). The details of the
Dynare implementation of the first order solution are given in
*Villemot (2011)*. Such a solution is computed using the
``stoch_simul`` command.
As an alternative, it is possible to compute a simulation to a
stochastic model using the *extended path* method presented by *Fair
and Taylor (1983)*. This method is especially useful when there are
strong nonlinearities or binding constraints. Such a solution is
computed using the ``extended_path`` command.
.. _stoch-sol-simul:
Computing the stochastic solution
---------------------------------
.. command:: stoch_simul [VARIABLE_NAME...];
stoch_simul (OPTIONS...) [VARIABLE_NAME...];
|br| Solves a stochastic (i.e. rational expectations) model, using
perturbation techniques.
More precisely, ``stoch_simul`` computes a Taylor approximation of
the model around the deterministic steady state and solves of the
the decision and transition functions for the approximated
model. Using this, it computes impulse response functions and
various descriptive statistics (moments, variance decomposition,
correlation and autocorrelation coefficients). For correlated
shocks, the variance decomposition is computed as in the VAR
literature through a Cholesky decomposition of the covariance
matrix of the exogenous variables. When the shocks are correlated,
the variance decomposition depends upon the order of the variables
in the ``varexo`` command.
The Taylor approximation is computed around the steady state (see
:ref:`st-st`).
The IRFs are computed as the difference between the trajectory of
a variable following a shock at the beginning of period ``1`` and
its steady state value. More details on the computation of IRFs
can be found at `<https://archives.dynare.org/DynareWiki/IrFs>`__.
Variance decomposition, correlation, autocorrelation are only
displayed for variables with strictly positive variance. Impulse
response functions are only plotted for variables with response
larger than :math:`10^{-10}`.
Variance decomposition is computed relative to the sum of the
contribution of each shock. Normally, this is of course equal to
aggregate variance, but if a model generates very large variances,
it may happen that, due to numerical error, the two differ by a
significant amount. Dynare issues a warning if the maximum
relative difference between the sum of the contribution of each
shock and aggregate variance is larger than ``0.01%``.
The covariance matrix of the shocks is specified with the
``shocks`` command (see :ref:`shocks-exo`).
When a list of ``VARIABLE_NAME`` is specified, results are
displayed only for these variables.
*Options*
.. option:: ar = INTEGER
Order of autocorrelation coefficients to compute and to
print. Default: ``5``.
.. option:: drop = INTEGER
Number of points (burnin) dropped at the beginning of
simulation before computing the summary statistics. Note that
this option does not affect the simulated series stored in
``oo_.endo_simul`` and the workspace. Here, no periods are
dropped. Default: ``100``.
.. option:: hp_filter = DOUBLE
Uses HP filter with :math:`\lambda =` ``DOUBLE`` before
computing moments. If theoretical moments are requested, the
spectrum of the model solution is filtered following the
approach outlined in Uhlig (2001). Default: no filter.
.. option:: one_sided_hp_filter = DOUBLE
Uses the one-sided HP filter with :math:`\lambda =` ``DOUBLE``
described in *Stock and Watson (1999)* before computing
moments. This option is only available with simulated
moments. Default: no filter.
.. option:: bandpass_filter
Uses a bandpass filter with the default passband before
computing moments. If theoretical moments are requested, the
spectrum of the model solution is filtered using an ideal
bandpass filter. If empirical moments are requested, the
*Baxter and King (1999)* filter is used. Default: no filter.
.. option:: bandpass_filter = [HIGHEST_PERIODICITY LOWEST_PERIODICITY]
Uses a bandpass filter before computing moments. The passband
is set to a periodicity of to LOWEST_PERIODICITY,
e.g. :math:`6` to :math:`32` quarters if the model frequency is
quarterly. Default: ``[6,32]``.
.. option:: filtered_theoretical_moments_grid = INTEGER
When computing filtered theoretical moments (with either option
``hp_filter`` or option ``bandpass_filter``), this option governs the
number of points in the grid for the discrete Inverse Fast Fourier
Transform. It may be necessary to increase it for highly autocorrelated
processes. Default: ``512``.
.. option:: irf = INTEGER
Number of periods on which to compute the IRFs. Setting
``irf=0`` suppresses the plotting of IRFs. Default: ``40``.
.. option:: irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )
The exogenous variables for which to compute IRFs. Default:
all.
.. option:: relative_irf
Requests the computation of normalized IRFs. At first order,
the normal shock vector of size one standard deviation is
divided by the standard deviation of the current shock and
multiplied by 100. The impulse responses are hence the
responses to a unit shock of size 1 (as opposed to the regular
shock size of one standard deviation), multiplied by 100. Thus,
for a loglinearized model where the variables are measured in
percent, the IRFs have the interpretation of the percent
responses to a 100 percent shock. For example, a response of
400 of output to a TFP shock shows that output increases by 400
percent after a 100 percent TFP shock (you will see that TFP
increases by 100 on impact). Given linearity at ``order=1``, it
is straightforward to rescale the IRFs stored in ``oo_.irfs``
to any desired size. At higher order, the interpretation is
different. The ``relative_irf`` option then triggers the
generation of IRFs as the response to a 0.01 unit shock
(corresponding to 1 percent for shocks measured in percent) and
no multiplication with 100 is performed. That is, the normal
shock vector of size one standard deviation is divided by the
standard deviation of the current shock and divided by 100. For
example, a response of 0.04 of log output (thus measured in
percent of the steady state output level) to a TFP shock also
measured in percent then shows that output increases by 4
percent after a 1 percent TFP shock (you will see that TFP
increases by 0.01 on impact).
.. option:: irf_plot_threshold = DOUBLE
Threshold size for plotting IRFs. All IRFs for a particular
variable with a maximum absolute deviation from the steady
state smaller than this value are not displayed. Default:
``1e-10``.
.. option:: nocorr
Don’t print the correlation matrix (printing them is the
default).
.. option:: nodecomposition
Don’t compute (and don’t print) unconditional variance
decomposition.
.. option:: nofunctions
Don’t print the coefficients of the approximated solution
(printing them is the default).
.. option:: nomoments
Don’t print moments of the endogenous variables (printing them
is the default).
.. option:: nograph
Do not create graphs (which implies that they are not saved to
the disk nor displayed). If this option is not used, graphs
will be saved to disk (to the format specified by
``graph_format`` option, except if ``graph_format=none``) and
displayed to screen (unless ``nodisplay`` option is used).
.. option:: graph
Re-enables the generation of graphs previously shut off with
``nograph``.
.. option:: nodisplay
Do not display the graphs, but still save them to disk (unless
``nograph`` is used).
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
Specify the file format(s) for graphs saved to disk. Possible
values are ``eps`` (the default), ``pdf``, ``fig`` and ``none``
(under Octave, ``fig`` is unavailable). If the
file format is set equal to ``none``, the graphs are displayed
but not saved to the disk.
.. option:: noprint
See :opt:`noprint`.
.. option:: print
See :opt:`print`.
.. option:: order = INTEGER
Order of Taylor approximation. Note that for third order and above, the
``k_order_solver`` option is implied and only empirical moments are
available (you must provide a value for ``periods`` option). Default:
``2`` (except after an ``estimation`` command, in which case the default
is the value used for the estimation).
.. option:: k_order_solver
Use a k-order solver (implemented in C++) instead of the
default Dynare solver. This option is not yet compatible with
the ``bytecode`` option (see :ref:`model-decl`). Default:
disabled for order 1 and 2, enabled for order 3 and above.
.. option:: periods = INTEGER
If different from zero, empirical moments will be computed
instead of theoretical moments. The value of the option
specifies the number of periods to use in the
simulations. Values of the initval block, possibly recomputed
by ``steady``, will be used as starting point for the
simulation. The simulated endogenous variables are made
available to the user in a vector for each variable and in the
global matrix ``oo_.endo_simul`` (see
:mvar:`oo_.endo_simul`). The simulated exogenous variables are
made available in ``oo_.exo_simul`` (see
:mvar:`oo_.exo_simul`). Default: ``0``.
.. option:: qz_criterium = DOUBLE
Value used to split stable from unstable eigenvalues in
reordering the Generalized Schur decomposition used for solving
first order problems. Default: ``1.000001`` (except when
estimating with ``lik_init`` option equal to ``1``: the default
is ``0.999999`` in that case; see :ref:`estim`).
.. option:: qz_zero_threshold = DOUBLE
See :opt:`qz_zero_threshold <qz_zero_threshold = DOUBLE>`.
.. option:: replic = INTEGER
Number of simulated series used to compute the IRFs. Default:
``1`` if ``order=1``, and ``50`` otherwise.
.. option:: simul_replic = INTEGER
Number of series to simulate when empirical moments are
requested (i.e. ``periods`` :math:`>` 0). Note that if this
option is greater than 1, the additional series will not be
used for computing the empirical moments but will simply be
saved in binary form to the file ``FILENAME_simul`` in the
``FILENAME/Output`` folder. Default:
``1``.
.. option:: solve_algo = INTEGER
See :ref:`solve_algo <solvalg>`, for the possible values and
their meaning.
.. option:: aim_solver
Use the Anderson-Moore Algorithm (AIM) to compute the decision
rules, instead of using Dynare’s default method based on a
generalized Schur decomposition. This option is only valid for
first order approximation. See `AIM website`_ for more details
on the algorithm.
.. option:: conditional_variance_decomposition = INTEGER
conditional_variance_decomposition = [INTEGER1:INTEGER2]
conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]
Computes a conditional variance decomposition for the specified
period(s). The periods must be strictly positive. Conditional
variances are given by :math:`var(y_{t+k}\vert t)`. For period
1, the conditional variance decomposition provides the
decomposition of the effects of shocks upon impact.
The results are stored in
``oo_.conditional_variance_decomposition`` (see
:mvar:`oo_.conditional_variance_decomposition`). In the
presence of measurement error, the
``oo_.conditional_variance_decomposition`` field will contain
the variance contribution after measurement error has been
taken out, i.e. the decomposition will be conducted of the
actual as opposed to the measured variables. The variance
decomposition of the measured variables will be stored in
``oo_.conditional_variance_decomposition_ME`` (see
:mvar:`oo_.conditional_variance_decomposition_ME`). The
variance decomposition is only conducted, if theoretical
moments are requested, *i.e.* using the ``periods=0`` option.
Only available at ``order<3`` and without ``pruning``. In case of ``order=2``,
Dynare provides a second-order accurate
approximation to the true second moments based on the linear
terms of the second-order solution (see *Kim, Kim,
Schaumburg and Sims (2008)*). Note that the unconditional
variance decomposition *i.e.* at horizon infinity) is
automatically conducted if theoretical moments are requested
and if ``nodecomposition`` is not set (see
:mvar:`oo_.variance_decomposition`).
.. option:: pruning
Discard higher order terms when iteratively computing
simulations of the solution. At second order, Dynare uses the
algorithm of *Kim, Kim, Schaumburg and Sims (2008)*, while at
third order its generalization by *Andreasen,
Fernández-Villaverde and Rubio-Ramírez (2018)* is used.
Not available above third order. When specified, theoretical moments
are based on the pruned state space, i.e. the computation of second moments
uses all terms as in *Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018), page 10*
as opposed to simply providing a second-order accurate result based on the
linear solution as in *Kim, Kim, Schaumburg and Sims (2008)*.
.. option:: partial_information
Computes the solution of the model under partial information,
along the lines of *Pearlman, Currie and Levine (1986)*. Agents
are supposed to observe only some variables of the economy. The
set of observed variables is declared using the ``varobs``
command. Note that if ``varobs`` is not present or contains all
endogenous variables, then this is the full information case
and this option has no effect. More references can be found
`here <https://archives.dynare.org/DynareWiki/PartialInformation>`__ .
.. option:: sylvester = OPTION
Determines the algorithm used to solve the Sylvester equation
for block decomposed model. Possible values for OPTION are:
``default``
Uses the default solver for Sylvester equations
(``gensylv``) based on Ondra Kamenik’s algorithm (see
`here
<https://www.dynare.org/assets/dynare++/sylvester.pdf>`__
for more information).
``fixed_point``
Uses a fixed point algorithm to solve the Sylvester
equation (``gensylv_fp``). This method is faster than
the default one for large scale models.
|br| Default value is ``default``.
.. option:: sylvester_fixed_point_tol = DOUBLE
The convergence criterion used in the fixed point
Sylvester solver. Its default value is ``1e-12``.
.. option:: dr = OPTION
Determines the method used to compute the decision
rule. Possible values for OPTION are:
``default``
Uses the default method to compute the decision rule
based on the generalized Schur decomposition (see
*Villemot (2011)* for more information).
``cycle_reduction``
Uses the cycle reduction algorithm to solve the
polynomial equation for retrieving the coefficients
associated to the endogenous variables in the decision
rule. This method is faster than the default one for
large scale models.
``logarithmic_reduction``
Uses the logarithmic reduction algorithm to solve the
polynomial equation for retrieving the coefficients
associated to the endogenous variables in the decision
rule. This method is in general slower than the
``cycle_reduction``.
|br| Default value is ``default``.
.. option:: dr_cycle_reduction_tol = DOUBLE
The convergence criterion used in the cycle reduction
algorithm. Its default value is ``1e-7``.
.. option:: dr_logarithmic_reduction_tol = DOUBLE
The convergence criterion used in the logarithmic reduction
algorithm. Its default value is ``1e-12``.
.. option:: dr_logarithmic_reduction_maxiter = INTEGER
The maximum number of iterations used in the logarithmic
reduction algorithm. Its default value is ``100``.
.. option:: loglinear
See :ref:`loglinear <logl>`. Note that ALL variables are
log-transformed by using the Jacobian transformation, not only
selected ones. Thus, you have to make sure that your variables
have strictly positive steady states. ``stoch_simul`` will
display the moments, decision rules, and impulse responses for
the log-linearized variables. The decision rules saved in
``oo_.dr`` and the simulated variables will also be the ones
for the log-linear variables.
.. option:: tex
Requests the printing of results and graphs in TeX tables and
graphics that can be later directly included in LaTeX
files.
.. option:: dr_display_tol = DOUBLE
Tolerance for the suppression of small terms in the display of
decision rules. Rows where all terms are smaller than
``dr_display_tol`` are not displayed. Default value: ``1e-6``.
.. option:: contemporaneous_correlation
Saves the contemporaneous correlation between the endogenous
variables in ``oo_.contemporaneous_correlation``. Requires the
``nocorr`` option not to be set.
.. option:: spectral_density
Triggers the computation and display of the theoretical
spectral density of the (filtered) model variables. Results are
stored in ``oo_.SpectralDensity``, defined below. Default: do
not request spectral density estimates.
.. option:: hp_ngrid = INTEGER
Deprecated option. It has the same effect as
:opt:`filtered_theoretical_moments_grid <filtered_theoretical_moments_grid = INTEGER>`.
*Output*
This command sets ``oo_.dr``, ``oo_.mean``, ``oo_.var``, ``oo_.var_list``,
and ``oo_.autocorr``, which are described below.
If the ``periods`` option is present, sets ``oo_.skewness``,
``oo_.kurtosis``, and ``oo_.endo_simul`` (see
:mvar:`oo_.endo_simul`), and also saves the simulated variables in
MATLAB/Octave vectors of the global workspace with the same name
as the endogenous variables.
If option ``irf`` is different from zero, sets ``oo_.irfs`` (see
below) and also saves the IRFs in MATLAB/Octave vectors of the
global workspace (this latter way of accessing the IRFs is
deprecated and will disappear in a future version).
If the option ``contemporaneous_correlation`` is different from
``0``, sets ``oo_.contemporaneous_correlation``, which is
described below.
*Example*
::
shocks;
var e;
stderr 0.0348;
end;
stoch_simul;
Performs the simulation of the 2nd-order approximation of a
model with a single stochastic shock ``e``, with a standard
error of ``0.0348``.
*Example*
::
stoch_simul(irf=60) y k;
Performs the simulation of a model and displays impulse
response functions on 60 periods for variables ``y`` and
``k``.
.. matvar:: oo_.mean
|br| After a run of ``stoch_simul``, contains the mean of the
endogenous variables. Contains theoretical mean if the ``periods``
option is not present, and simulated mean otherwise. The variables
are arranged in declaration order.
.. matvar:: oo_.var
|br| After a run of ``stoch_simul``, contains the
variance-covariance of the endogenous variables. Contains
theoretical variance if the ``periods`` option is not present and simulated variance
otherwise. Only available for ``order<4``. At ``order=2`` it will be be
a second-order accurate approximation (i.e. ignoring terms of order 3 and 4 that would
arise when using the full second-order policy function). At ``order=3``, theoretical moments
are only available with ``pruning``. The variables are arranged in declaration order.
.. matvar:: oo_.var_list
The list of variables for which results are displayed.
.. matvar:: oo_.skewness
|br| After a run of ``stoch_simul`` contains the skewness
(standardized third moment) of the simulated variables if the
``periods`` option is present. The variables are arranged in
declaration order.
.. matvar:: oo_.kurtosis
|br| After a run of ``stoch_simul`` contains the excess kurtosis
(standardized fourth moment) of the simulated variables if the
``periods`` option is present. The variables are arranged in
declaration order.
.. matvar:: oo_.autocorr
|br| After a run of ``stoch_simul``, contains a cell array of the
autocorrelation matrices of the endogenous variables. The element
number of the matrix in the cell array corresponds to the order of
autocorrelation. The option ar specifies the number of
autocorrelation matrices available. Contains theoretical
autocorrelations if the ``periods`` option is not present and simulated
autocorrelations otherwise. Only available for ``order<4``. At ``order=2`` it will be be
a second-order accurate approximation. At ``order=3``, theoretical moments
are only available with ``pruning``. The field is only created if
stationary variables are present.
The element ``oo_.autocorr{i}(k,l)`` is equal to the correlation
between :math:`y^k_t` and :math:`y^l_{t-i}`, where :math:`y^k`
(resp. :math:`y^l`) is the :math:`k`-th (resp. :math:`l`-th)
endogenous variable in the declaration order.
Note that if theoretical moments have been requested,
``oo_.autocorr{i}`` is the same than ``oo_.gamma_y{i+1}``.
.. matvar:: oo_.gamma_y
|br| After a run of ``stoch_simul``, if theoretical moments have been
requested (i.e. if the ``periods`` option is not present), this
variable contains a cell array with the following values (where
``ar`` is the value of the option of the same name):
``oo_.gamma{1}``
Variance/covariance matrix.
``oo_.gamma{i+1}`` (for i=1:ar)
Autocorrelation function. See :mvar:`oo_.autocorr` for
more details. **Beware**, this is the autocorrelation
function, not the autocovariance function.
``oo_.gamma{ar+2}``
Unconditional variance decomposition, see
:mvar:`oo_.variance_decomposition`.
``oo_.gamma{ar+3}``
If a second order approximation has been requested,
contains the vector of the mean correction terms.
Only available at ``order<4``. In case ``order=2``, the
theoretical second moments are a second order accurate
approximation of the true second moments. See conditional_variance_decomposition.
At ``order=3``, theoretical moments are only available with ``pruning``.
.. matvar:: oo_.variance_decomposition
|br| After a run of ``stoch_simul`` when requesting theoretical
moments (``periods=0``), contains a matrix with the result of the
unconditional variance decomposition (i.e. at horizon
infinity). The first dimension corresponds to the endogenous
variables (in the order of declaration after the command or in
``M_.endo_names``) and the second dimension corresponds to
exogenous variables (in the order of declaration). Numbers are in
percent and sum up to 100 across columns. In the presence of
measurement error, the field will contain the variance
contribution after measurement error has been taken out, *i.e.*
the decomposition will be conducted of the actual as opposed to
the measured variables.
.. matvar:: oo_.variance_decomposition_ME
|br| Field set after a run of ``stoch_simul`` when requesting
theoretical moments (``periods=0``) if measurement error is
present. It is similar to :mvar:`oo_.variance_decomposition`, but
the decomposition will be conducted of the measured variables. The
field contains a matrix with the result of the unconditional
variance decomposition (*i.e.* at horizon infinity). The first
dimension corresponds to the observed endoogenous variables (in
the order of declaration after the command) and the second
dimension corresponds to exogenous variables (in the order of
declaration), with the last column corresponding to the
contribution of measurement error. Numbers are in percent and sum
up to 100 across columns.
.. matvar:: oo_.conditional_variance_decomposition
|br| After a run of ``stoch_simul`` with the
``conditional_variance_decomposition`` option, contains a
three-dimensional array with the result of the decomposition. The
first dimension corresponds to the endogenous variables (in the
order of declaration after the command or in ``M_.endo_names`` if
not specified), the second dimension corresponds to the forecast
horizons (as declared with the option), and the third dimension
corresponds to the exogenous variables (in the order of
declaration). In the presence of measurement error, the field will
contain the variance contribution after measurement error has been
taken out, *i.e.* the decomposition will be conductedof the actual
as opposed to the measured variables.
.. matvar:: oo_.conditional_variance_decomposition_ME
|br| Field set after a run of ``stoch_simul`` with the
``conditional_variance_decomposition`` option if measurement error
is present. It is similar to
:mvar:`oo_.conditional_variance_decomposition`, but the
decomposition will be conducted of the measured variables. It
contains a three-dimensional array with the result of the
decomposition. The first dimension corresponds to the endogenous
variables (in the order of declaration after the command or in
``M_.endo_names`` if not specified), the second dimension
corresponds to the forecast horizons (as declared with the
option), and the third dimension corresponds to the exogenous
variables (in the order of declaration), with the last column
corresponding to the contribution of the measurement error.
.. matvar:: oo_.contemporaneous_correlation
|br| After a run of ``stoch_simul`` with the
``contemporaneous_correlation option``, contains theoretical
contemporaneous correlations if the ``periods`` option is not
present, and simulated contemporaneous correlations otherwise.
Only available for ``order<4``. At ``order=2`` it will be be
a second-order accurate approximation. At ``order=3``, theoretical moments
are only available with ``pruning``. The variables
are arranged in declaration order.
.. matvar:: oo_.SpectralDensity
|br| After a run of ``stoch_simul`` with option
``spectral_density``, contains the spectral density of the model
variables. There will be a ``nvars`` by ``nfrequencies`` subfield
``freqs`` storing the respective frequency grid points ranging
from :math:`0` to :math:`2\pi` and a same sized subfield
``density`` storing the corresponding density.
.. matvar:: oo_.irfs
|br| After a run of ``stoch_simul`` with option ``irf`` different
from zero, contains the impulse responses, with the following
naming convention: `VARIABLE_NAME_SHOCK_NAME`.
For example, ``oo_.irfs.gnp_ea`` contains the effect on ``gnp`` of
a one-standard deviation shock on ``ea``.
.. matcomm:: get_irf ('EXOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );
|br| Given the name of an exogenous variables, returns the IRFs for the
requested endogenous variable(s), as they are stored in ``oo_.irfs``.
The approximated solution of a model takes the form of a set of
decision rules or transition equations expressing the current value of
the endogenous variables of the model as function of the previous
state of the model and shocks observed at the beginning of the
period. The decision rules are stored in the structure ``oo_.dr``
which is described below.
.. matvar:: oo_.dr
Structure storing the decision rules. The subfields for different
orders of approximation are explained below.
.. command:: extended_path ;
extended_path (OPTIONS...);
|br| Simulates a stochastic (i.e. rational expectations) model,
using the extended path method presented by *Fair and Taylor
(1983)*. Time series for the endogenous variables are generated by
assuming that the agents believe that there will no more shocks in
the following periods.
This function first computes a random path for the exogenous
variables (stored in ``oo_.exo_simul``, see :mvar:`oo_.exo_simul`)
and then computes the corresponding path for endogenous variables,
taking the steady state as starting point. The result of the
simulation is stored in ``oo_.endo_simul`` (see
:mvar:`oo_.endo_simul`). Note that this simulation approach does
not solve for the policy and transition equations but for paths
for the endogenous variables.
*Options*
.. option:: periods = INTEGER
The number of periods for which the simulation is to be
computed. No default value, mandatory option.
.. option:: solver_periods = INTEGER
The number of periods used to compute the solution of the
perfect foresight at every iteration of the algorithm. Default:
``200``.
.. option:: order = INTEGER
If order is greater than ``0`` Dynare uses a gaussian
quadrature to take into account the effects of future
uncertainty. If ``order`` :math:`=S` then the time series for
the endogenous variables are generated by assuming that the
agents believe that there will no more shocks after period
:math:`t+S`. This is an experimental feature and can be quite
slow. A non-zero value is not compatible with the ``bytecode``
option of the ``model`` block.
Default: ``0``.
.. option:: hybrid
Use the constant of the second order perturbation reduced form
to correct the paths generated by the (stochastic) extended
path algorithm.
.. option:: lmmcp
Solves the perfect foresight model with a Levenberg-Marquardt
mixed complementarity problem (LMMCP) solver (*Kanzow and Petra
(2004)*), which allows to consider inequality constraints on
the endogenous variables (such as a ZLB on the nominal interest
rate or a model with irreversible investment). For specifying the
necessary ``mcp`` tag, see :opt:`lmmcp`.
Typology and ordering of variables
----------------------------------
Dynare distinguishes four types of endogenous variables:
*Purely backward (or purely predetermined) variables*
Those that appear only at current and past period in the model,
but not at future period (i.e. at :math:`t` and :math:`t-1` but
not :math:`t+1`). The number of such variables is equal to
``M_.npred``.
*Purely forward variables*
Those that appear only at current and future period in the model,
but not at past period (i.e. at :math:`t` and :math:`t+1` but not
:math:`t-1`). The number of such variables is stored in
``M_.nfwrd``.
*Mixed variables*
Those that appear at current, past and future period in the model
(i.e. at :math:`t`, :math:`t+1` and :math:`t-1`). The number of
such variables is stored in ``M_.nboth``.
*Static variables*
Those that appear only at current, not past and future period in
the model (i.e. only at :math:`t`, not at :math:`t+1` or
:math:`t-1`). The number of such variables is stored in
``M_.nstatic``.
Note that all endogenous variables fall into one of these four
categories, since after the creation of auxiliary variables (see
:ref:`aux-variables`), all endogenous have at most one lead and one
lag. We therefore have the following identity:
.. code-block:: matlab
M_.npred + M_.both + M_.nfwrd + M_.nstatic = M_.endo_nbr
.. matvar:: M_.state_var
Vector of numerical indices identifying the state variables in the
vector of declared variables. ``M_.endo_names(M_.state_var)``
therefore yields the name of all variables that are states in
the model declaration, i.e. that show up with a lag.
Internally, Dynare uses two orderings of the endogenous variables: the
order of declaration (which is reflected in ``M_.endo_names``), and an
order based on the four types described above, which we will call the
DR-order (“DR” stands for decision rules). Most of the time, the
declaration order is used, but for elements of the decision rules, the
DR-order is used.
The DR-order is the following: static variables appear first, then
purely backward variables, then mixed variables, and finally purely
forward variables. Inside each category, variables are arranged
according to the declaration order.
.. matvar:: oo_.dr.order_var
This variables maps DR-order to declaration order.
.. matvar:: oo_.dr.inv_order_var
This variable contains the inverse map.
In other words, the k-th variable in the DR-order corresponds to the endogenous
variable numbered ``oo_.dr.order_var(k)`` in declaration
order. Conversely, k-th declared variable is numbered
``oo_.dr.inv_order_var(k)`` in DR-order.
Finally, the state variables of the model are the purely backward
variables and the mixed variables. They are ordered in DR-order when
they appear in decision rules elements. There are ``M_.nspred =
M_.npred + M_.nboth`` such variables. Similarly, one has ``M_.nsfwrd =
M_.nfwrd + M_.nboth``, and ``M_.ndynamic = M_.nfwrd + M_.nboth +
M_.npred``.
First-order approximation
-------------------------
The approximation has the stylized form:
.. math::
y_t = y^s + A y^h_{t-1} + B u_t
where :math:`y^s` is the steady state value of :math:`y` and
:math:`y^h_t=y_t-y^s`.
.. matvar:: oo.dr.state_var
Vector of numerical indices identifying the state variables in the
vector of declared variables, *given the current parameter values*
for which the decision rules have been computed. It may differ from
``M_.state_var`` in case a state variable drops from the model given
the current parameterization, because it only gets 0 coefficients in
the decision rules. See :mvar:`M_.state_var`.
The coefficients of the decision rules are stored as follows:
* :math:`y^s` is stored in ``oo_.dr.ys``. The vector rows correspond
to all endogenous in the declaration order.
* :math:`A` is stored in ``oo_.dr.ghx``. The matrix rows correspond to
all endogenous in DR-order. The matrix columns correspond to state
variables in DR-order, as given by ``oo_.dr.state_var``.
* :math:`B` is stored ``oo_.dr.ghu``. The matrix rows correspond to
all endogenous in DR-order. The matrix columns correspond to
exogenous variables in declaration order.
Of course, the shown form of the approximation is only stylized,
because it neglects the required different ordering in :math:`y^s` and
:math:`y^h_t`. The precise form of the approximation that shows the
way Dynare deals with differences between declaration and DR-order, is
.. math::
y_t(\mathrm{oo\_.dr.order\_var}) =
y^s(\mathrm{oo\_.dr.order\_var}) + A \cdot
y_{t-1}(\mathrm{oo\_.dr.order\_var(k2)}) -
y^s(\mathrm{oo\_.dr.order\_var(k2)}) + B\cdot u_t
where :math:`\mathrm{k2}` selects the state variables, :math:`y_t` and
:math:`y^s` are in declaration order and the coefficient matrices are
in DR-order. Effectively, all variables on the right hand side are
brought into DR order for computations and then assigned to
:math:`y_t` in declaration order.
Second-order approximation
--------------------------
The approximation has the form:
.. math::
y_t = y^s + 0.5 \Delta^2 + A y^h_{t-1} + B u_t + 0.5 C (y^h_{t-1}\otimes y^h_{t-1}) + 0.5 D (u_t \otimes u_t) + E (y^h_{t-1} \otimes u_t)
where :math:`y^s` is the steady state value of :math:`y`,
:math:`y^h_t=y_t-y^s`, and :math:`\Delta^2` is the shift effect of the
variance of future shocks. For the reordering required due to
differences in declaration and DR order, see the first order
approximation.
The coefficients of the decision rules are stored in the variables
described for first order approximation, plus the following variables:
* :math:`\Delta^2` is stored in ``oo_.dr.ghs2``. The vector rows
correspond to all endogenous in DR-order.
* :math:`C` is stored in ``oo_.dr.ghxx``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to the
Kronecker product of the vector of state variables in DR-order.
* :math:`D` is stored in ``oo_.dr.ghuu``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to the
Kronecker product of exogenous variables in declaration order.
* :math:`E` is stored in ``oo_.dr.ghxu``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to the
Kronecker product of the vector of state variables (in DR-order) by
the vector of exogenous variables (in declaration order).
Third-order approximation
-------------------------
The approximation has the form:
.. math::
y_t = y^s + G_0 + G_1 z_t + G_2 (z_t \otimes z_t) + G_3 (z_t \otimes z_t \otimes z_t)
where :math:`y^s` is the steady state value of :math:`y`, and
:math:`z_t` is a vector consisting of the deviation from the steady
state of the state variables (in DR-order) at date :math:`t-1`
followed by the exogenous variables at date :math:`t` (in declaration
order). The vector :math:`z_t` is therefore of size :math:`n_z` =
``M_.nspred`` + ``M_.exo_nbr``.
The coefficients of the decision rules are stored as follows:
* :math:`y^s` is stored in ``oo_.dr.ys``. The vector rows correspond
to all endogenous in the declaration order.
* :math:`G_0` is stored in ``oo_.dr.g_0``. The vector rows correspond
to all endogenous in DR-order.
* :math:`G_1` is stored in ``oo_.dr.g_1``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to
state variables in DR-order, followed by exogenous in declaration
order.
* :math:`G_2` is stored in ``oo_.dr.g_2``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to the
Kronecker product of state variables (in DR-order), followed by
exogenous (in declaration order). Note that the Kronecker product is
stored in a folded way, i.e. symmetric elements are stored only
once, which implies that the matrix has :math:`n_z(n_z+1)/2`
columns. More precisely, each column of this matrix corresponds to a
pair :math:`(i_1, i_2)` where each index represents an element of
:math:`z_t` and is therefore between :math:`1` and :math:`n_z`. Only
non-decreasing pairs are stored, i.e. those for which :math:`i_1
\leq i_2`. The columns are arranged in the lexicographical order of
non-decreasing pairs. Also note that for those pairs where
:math:`i_1 \neq i_2`, since the element is stored only once but
appears two times in the unfolded :math:`G_2` matrix, it must be
multiplied by 2 when computing the decision rules.
* :math:`G_3` is stored in ``oo_.dr.g_3``. The matrix rows correspond
to all endogenous in DR-order. The matrix columns correspond to the
third Kronecker power of state variables (in DR-order), followed by
exogenous (in declaration order). Note that the third Kronecker
power is stored in a folded way, i.e. symmetric elements are stored
only once, which implies that the matrix has
:math:`n_z(n_z+1)(n_z+2)/6` columns. More precisely, each column of
this matrix corresponds to a tuple :math:`(i_1, i_2, i_3)` where
each index represents an element of :math:`z_t` and is therefore
between :math:`1` and :math:`n_z`. Only non-decreasing tuples are
stored, i.e. those for which :math:`i_1 \leq i_2 \leq i_3`. The
columns are arranged in the lexicographical order of non-decreasing
tuples. Also note that for tuples that have three distinct indices
(i.e. :math:`i_1 \neq i_2` and :math:`i_1 \neq i_3` and :math:`i_2
\neq i_3`), since these elements are stored only once but appears
six times in the unfolded :math:`G_3` matrix, they must be
multiplied by 6 when computing the decision rules. Similarly, for
those tuples that have two equal indices (i.e. of the form
:math:`(a,a,b)` or :math:`(a,b,a)` or :math:`(b,a,a)`), since these
elements are stored only once but appears three times in the
unfolded :math:`G_3` matrix, they must be multiplied by 3 when
computing the decision rules.
Higher-order approximation
--------------------------
Higher-order approximations are simply a generalization of what is done at
order 3.
The steady state is stored in ``oo_.dr.ys`` and the constant correction is
stored in ``oo_.dr.g_0``. The coefficient for orders 1, 2, 3, 4… are
respectively stored in ``oo_.dr.g_0``, ``oo_.dr.g_1``, ``oo_.dr.g_2``,
``oo_.dr.g_3``, ``oo_.dr.g_4``… The columns of those matrices correspond to
multidimensional indices of state variables, in such a way that symmetric
elements are never repeated (for more details, see the description of
``oo_.dr.g_3`` in the third-order case).
Occasionally binding constraints (OCCBIN)
=========================================
Dynare allows simulating models with up to two occasionally-binding constraints by
relying on a piecewise linear solution as in *Guerrieri and Iacoviello (2015)*.
It also allows estimating such models employing either the inversion filter of
*Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019)* or the piecewise Kalman filter of
*Giovannini, Pfeiffer, and Ratto (2021)*. To trigger computations involving
occasionally-binding constraints requires
#. defining and naming the occasionally-binding constraints using an ``occbin_constraints`` block
#. specifying the model equations for the respective regimes in the ``model`` block using appropriate equation tags.
#. potentially specifying a sequence of surprise shocks using a ``shocks(surprise)`` block
#. setting up Occbin simulations or estimation with ``occbin_setup``
#. triggering a simulation with ``occbin_solver`` or running ``estimation`` or ``calib_smoother``.
All of these elements are discussed in the following.
.. block:: occbin_constraints ;
|br| The ``occbin_constraints`` block specifies the occasionally-binding constraints. It contains
one or two of the following lines:
name 'STRING'; bind EXPRESSION; [relax EXPRESSION;] [error_bind EXPRESSION;] [error_relax EXPRESSION;]
``STRING`` is the name of constraint that is used to reference the constraint in ``relax`` / ``bind``
equation tags to identify the respective regime (see below). The ``bind`` expression is mandatory and defines
a logical condition that is evaluated in the baseline/steady state regime to check whether the specified
constraint becomes binding. In contrast, the ``relax`` expression is optional and specifies a
logical condition that is evaluated in the binding regime to check whether the regime returns
to the baseline/steady state regime. If not specified, Dynare will simply check in the binding
regime whether the ``bind`` expression evaluates to false. However, there are cases
where the ``bind`` expression cannot be evaluated in the binding regime(s), because
the variables involved are constant by definition so that e.g. the value of the Lagrange
multiplier on the complementary slackness condition needs to be checked. In these cases,
it is necessary to provide an explicit condition that can be evaluated in the binding
regime that allows to check whether it should be left.
Note that the baseline regime denotes the steady state of the model where the economy will
settle in the long-run without shocks. For that matter, it may be one where e.g. a borrowing
constraint is binding. In that type of setup, the ``bind`` condition is used to specify the
condition when this borrowing constraint becomes non-binding so that the alternative regime
is entered.
Three things are important to keep in mind when specifying the expressions.
First, feasible expressions may only contain contemporaneous endogenous variables.
If you want to include leads/lags or exogenous variables, you need to define
an auxiliary variable. Second, Dynare will at the current stage not linearly
approximate the entered expressions. Because Occbin will work with a linearized
model, consistency will often require the user to enter a linearized constraint.
Otherwise, the condition employed for checking constraint violations may differ
from the one employed within model simulations based on the piecewise-linear
model solution. Third, in contrast to the original Occbin replication codes, the
variables used in expressions are not automatically demeaned, i.e. they refer to
the levels, not deviations from the steady state. To access the steady state
level of a variable, the ``STEADY_STATE()`` operator can be used.
Finally, it's worth keeping in mind that for each simulation period, Occbin will check
the respective conditions for whether the current regime should be left. Small numerical
differences from the cutoff point for a regime can sometimes lead to oscillations between
regimes and cause a spurious periodic solution. Such cases may be prevented by introducing
a small buffer between the two regimes, e.g.
::
occbin_constraints;
name 'ELB'; bind inom <= iss-1e8; relax inom > iss+1e-8;
end;
The ``error_bind`` and ``error_relax`` options are optional and allow specifying
numerical criteria for the size of the respective constraint violations employed
in numerical routines. By default, Dynare will simply use the absolute value of
the ``bind`` and ``relax`` inequalities. But occasionnally, user-specified
expressions perform better.
*Example*
::
occbin_constraints;
name 'IRR'; bind log_Invest-log(steady_state(Invest))<log(phi); relax Lambda<0;
name 'INEG'; bind log_Invest-log(steady_state(Invest))<0;
end;
IRR is a constraint for irreversible investment that becomes binding if investment drops below its
steady state by more than 0.025 percent in the non-binding regime. The constraint will be relaxed whenever
the associated Lagrange multiplier ``Lambda`` in the binding regime becomes negative. Note that the
constraint here takes on a linear form to be consistent with a piecewise linear model solution
The specification of the model equations belonging to the respective regimes is done in the ``model`` block,
with equation tags indicating to which regime a particular equation belongs. All equations that differ across
regimes must have a ``name`` tag attached to them that allows uniquely identifying different versions of the
same equation. The name of the constraints specified is then used in conjunction with a ``bind`` or ``relax``
tag to indicate to which regime a particular equation belongs. In case of more than one occasionally-binding
constraint, if an equation belongs to several regimes (e.g. both constraints binding), the
constraint name tags must be separated by a comma. If only one name tag is present,
the respective equation is assumed to hold for both states of the other constraint.
*Example*
::
[name='investment',bind='IRR,INEG']
(log_Invest - log(phi*steady_state(Invest))) = 0;
[name='investment',relax='IRR']
Lambda=0;
[name='investment',bind='IRR',relax='INEG']
(log_Invest - log(phi*steady_state(Invest))) = 0;
The three entered equations for the investment condition define the model
equation for all four possible combinations of the two constraints. The
first equation defines the model equation in the regime where both the
IRR and INEG constraint are binding. The second equation defines the
model equation for the regimes where the IRR constraint is non-binding,
regardless of whether the INEG constraint is binding or not. Finally,
the last equation defines the model equation for the final regime where the
IRR constraint is binding, but the INEG one is not.
.. block:: shocks(surprise) ;
shocks(surprise,overwrite);
|br| The ``shocks(surprise)`` block allows specifying a sequence of temporary changes in
the value of exogenous variables that in each period come as a surprise to agents, i.e.
are not anticipated. Note that to actually use the specified shocks in subsequent commands
like ``occbin_solver``, the block needs to be followed by a call to ``occbin_setup``.
The block mirrors the perfect foresight syntax in that it should contain one or more
occurrences of the following group of three lines::
var VARIABLE_NAME;
periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...;
values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
*Example* (with vector values and overwrite option)
::
shockssequence = randn(100,1)*0.02;
shocks(surprise,overwrite);
var epsilon;
periods 1:100;
values (shockssequence);
end;
.. command:: occbin_setup ;
occbin_setup (OPTIONS...);
|br| Prepares a simulation with occasionally binding constraints. This command
will also translate the contents of a ``shocks(surprise)`` block for use
in subsequent commands.
In order to conduct ``estimation`` with occasionally binding constraints, it needs to be
prefaced by a call to ``occbin_setup`` to trigger the use of either the inversion filter
or the piecewise Kalman filter (default). An issue that can arise in the context of
estimation is a structural shock dropping out of the model in a particular regime.
For example, at the zero lower bound on interest rates, the monetary policy shock
in the Taylor rule will not appear anymore. This may create a problem
of stochastic singularity if there are then more observables than shocks. To
avoid this issue, the data points for the zero interest rate should be set
to NaN and the standard deviation of the associated shock set to 0 for the
corresponding periods using the ``heteroskedastic_shocks`` block.
Note that models with unit roots will require the user to specify the ``diffuse_filter`` option as
otherwise Blanchard-Kahn errors will be triggered. For the piecewise Kalman filter, the
initialization steps in the diffuse filter will always rely on the model solved for the baseline
regime, without checking whether this is the actual regime in the first period(s).
*Example*
::
occbin_setup(likelihood_inversion_filter,smoother_inversion_filter);
estimation(smoother,heteroskedastic_filter,...);
The above piece of code sets up an estimation employing the inversion filter for both the likelihood
evaluation and the smoother, while also accounting for ``heteroskedastic_shocks`` using the
``heteroskedastic_filter`` option.
Be aware that Occbin has largely command-specific options, i.e. there are separate
options to control the behavior of Occbin when called by the smoother or when
computing the likelihood. These latter commands will not inherit the options
potentially previously set for simulations.
*Options*
.. option:: simul_periods = INTEGER
Number of periods of the simulation. Default: 100.
.. option:: simul_maxit = INTEGER
Maximum number of iterations when trying to find the regimes of the piecewise solution.
Default: 30.
.. option:: simul_check_ahead_periods = INTEGER
Number of periods for which to check ahead for return to the baseline regime.
This number should be chosen large enough, because Occbin requires the simulation
to return to the baseline regime at the end of time. Default: 200.
.. option:: simul_reset_check_ahead_periods
Allows to reset ``simul_check_ahead_periods`` to its specified value at the beginning
of each simulation period. Otherwise, the original value may permanently increase endogenously
at some point due to regimes that last very long in expectations. This may considerably slow
down convergence in subsequent periods. Default: not enabled.
.. option:: simul_max_check_ahead_periods = INTEGER
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods
(i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
``simul_check_ahead_periods`` ever further. Default: Inf.
.. option:: simul_curb_retrench
Instead of basing the initial regime guess for the current iteration on the last iteration, update
the guess only one period at a time. This will slow down the iterations, but may lead to
more robust convergence behavior. Default: not enabled.
.. option:: simul_periodic_solution
Accept a periodic solution where the solution alternates between two sets of results
across iterations, i.e. is not found to be unique. This is sometimes caused by spurious numerical errors
that lead to oscillations between regiems and may be prevented by allowing for a small buffer in regime
transitions. Default: not enabled.
.. option:: simul_debug
Provide additional debugging information during solving. Default: not enabled.
.. option:: smoother_periods = INTEGER
Number of periods employed during the simulation when called by the smoother
(equivalent of ``simul_periods``). Default: 100.
.. option:: smoother_maxit = INTEGER
Maximum number of iterations employed during the simulation when called by the smoother
(equivalent of ``simul_maxit``). Default: 30.
.. option:: smoother_check_ahead_periods = INTEGER
Number of periods for which to check ahead for return to the baseline regime during the
simulation when called by the smoother (equivalent of ``simul_check_ahead_periods``). Default: 200.
.. option:: smoother_max_check_ahead_periods = INTEGER
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods
(i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
``smoother_check_ahead_periods`` ever further. Equivalent of ``simul_max_check_ahead_periods``. Default: Inf.
.. option:: smoother_curb_retrench
Have the smoother invoke the ``simul_curb_retrench`` option during simulations.
Default: not enabled.
.. option:: smoother_periodic_solution
Accept periodic solution where solution alternates between two sets of results (equivalent of ``simul_periodic_solution``).
Default: not enabled.
.. option:: likelihood_periods = INTEGER
Number of periods employed during the simulation when computing the likelihood
(equivalent of ``simul_periods``). Default: 100.
.. option:: likelihood_maxit = INTEGER
Maximum number of iterations employed during the simulation when computing the likelihood
(equivalent of ``simul_maxit``). Default: 30.
.. option:: likelihood_check_ahead_periods = INTEGER
Number of periods for which to check ahead for return to the baseline regime during the
simulation when computing the likelihood (equivalent of ``simul_check_ahead_periods``). Default: 200.
.. option:: smoother_max_check_ahead_periods = INTEGER
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods
(i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
``likelihood_check_ahead_periods`` ever further. Equivalent of ``simul_max_check_ahead_periods``. Default: Inf.
.. option:: likelihood_curb_retrench
Have the likelihood computation invoke the ``simul_curb_retrench`` option during simulations.
Default: not enabled.
.. option:: likelihood_periodic_solution
Accept periodic solution where solution alternates between two sets of results (equivalent of ``simul_periodic_solution``).
Default: not enabled.
.. option:: likelihood_inversion_filter
Employ the inversion filter of *Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019)* when estimating
the model. Default: not enabled.
.. option:: likelihood_piecewise_kalman_filter
Employ the piecewise Kalman filter of *Giovannini, Pfeiffer, and Ratto (2021)* when estimating
the model. Note that this filter is incompatible with univariate Kalman filters, i.e. ``kalman_algo=2,4``.
Default: enabled.
.. option:: likelihood_max_kalman_iterations
Maximum number of iterations of the outer loop for the piecewise Kalman filter. Default: 10.
.. option:: smoother_inversion_filter
Employ the inversion filter of *Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019)* when running the
smoother. The underlying assumption is that the system starts at the steady state. In this case, the
inversion filter will provide the required smoother output. Default: not enabled.
.. option:: smoother_piecewise_kalman_filter
Employ the piecewise Kalman filter of *Giovannini, Pfeiffer, and Ratto (2021)* when running the
smoother. Default: enabled.
.. option:: filter_use_relaxation
Triggers relaxation within the guess and verify algorithm used in the update step of the piecewise
Kalman filter. When old and new guess regime differ to much, use a new guess closer to the previous guess.
In case of multiple solutions, tends to provide an occasionally binding regime with a shorter duration (typically preferable).
Specifying this option may slow down convergence. Default: not enabled.
*Output*
The paths for the exogenous variables are stored into
``options_.occbin.simul.SHOCKS``.
.. command:: occbin_solver ;
occbin_solver (OPTIONS...);
|br| Computes a simulation with occasionally-binding constraints based on
a piecewise-linear solution.
Note that ``occbin_setup`` must be called before this command in order for
the simulation to take into account previous ``shocks(surprise)`` blocks.
*Options*
.. option:: simul_periods = INTEGER
See :opt:`simul_periods <simul_periods = INTEGER>`.
.. option:: simul_maxit = INTEGER
See :opt:`simul_maxit <simul_maxit = INTEGER>`.
.. option:: simul_check_ahead_periods = INTEGER
See :opt:`simul_check_ahead_periods <simul_check_ahead_periods = INTEGER>`.
.. option:: simul_curb_retrench
See :opt:`simul_curb_retrench`.
.. option:: simul_debug
See :opt:`simul_debug`.
*Output*
The command outputs various objects into ``oo_.occbin``.
.. matvar:: oo_.occbin.simul.piecewise
|br| Matrix storing the simulations based on the piecewise-linear solution.
The variables are arranged by column, in order of declaration (as in
``M_.endo_names``), while the the rows correspond to the ``simul_periods``.
.. matvar:: oo_.occbin.simul.linear
|br| Matrix storing the simulations based on the linear solution, i.e. ignoring
the occasionally binding constraint(s). The variables are arranged column by column,
in order of declaration (as in ``M_.endo_names``), while the the rows correspond to
the ``simul_periods``.
.. matvar:: oo_.occbin.simul.shocks_sequence
|br| Matrix storing the shock sequence employed during the simulation. The shocks are arranged
column by column, with their order in ``M_.exo_names`` stored in ``oo_.occbin.exo_pos``. The
the rows correspond to the number of shock periods specified in a ``shocks(surprise)`` block, which
may be smaller than ``simul_periods``.
.. matvar:: oo_.occbin.simul.regime_history
|br| Structure storing information on the regime history, conditional on the shock that
happened in the respective period (stored along the rows). ``type`` is equal to either ``smoother``
or ``simul``, depending on whether the output comes from a run of simulations or the smoother.
The subfield ``regime`` contains
a vector storing the regime state, while the the subfield ``regimestart`` indicates the
expected start of the respective regime state. For example, if row 40 contains ``[1,0]`` for
``regime2`` and ``[1,6]`` for ``regimestart2``, it indicates that - after the shock in period 40
has occurred - the second constraint became binding (1) and is expected to revert to non-binding (0) after
six periods including the current one, i.e. period 45.
.. matvar:: oo_.occbin.simul.ys
|br| Vector of steady state values
.. command:: occbin_graph [VARIABLE_NAME...];
occbin_graph (OPTIONS...) [VARIABLE_NAME...];
|br| Plots a graph comparing the simulation results of the piecewise-linear solution
with the occasionally binding contraints to the linear solution ignoring the constraint.
*Options*
.. option:: noconstant
Omit the steady state in the graphs.
.. command:: occbin_write_regimes ;
occbin_write_regimes (OPTIONS...);
|br| Write the information on the regime history stored in ``oo_.occbin.simul.regime_history``
or ``oo_.occbin.smoother.regime_history`` into an Excel file stored in the ``FILENAME/Output`` folder.
*Options*
.. option:: periods = INTEGER
Number of periods for which to write the expected regime durations. Default: write all
available periods.
.. option:: filename = FILENAME
Name of the Excel file to write. Default: ``FILENAME_occbin_regimes``.
.. option:: simul
Selects the regime history from the last run of simulations. Default: enabled.
.. option:: smoother
Selects the regime history from the last run of the smoother. Default: use ``simul``.
.. _estim:
Estimation based on likelihood
==============================
Provided that you have observations on some endogenous variables, it
is possible to use Dynare to estimate some or all parameters. Both
maximum likelihood (as in *Ireland (2004)*) and Bayesian techniques
(as in *Fernández-Villaverde and Rubio-Ramírez (2004)*,
*Rabanal and Rubio-Ramirez (2003)*, *Schorfheide (2000)* or
*Smets and Wouters (2003)*) are available. Using Bayesian methods, it
is possible to estimate DSGE models, VAR models, or a combination of
the two techniques called DSGE-VAR.
Note that in order to avoid stochastic singularity, you must have at
least as many shocks or measurement errors in your model as you have
observed variables.
.. _varobs:
.. command:: varobs VARIABLE_NAME...;
|br| This command lists the name of observed endogenous variables
for the estimation procedure. These variables must be available in
the data file (see :ref:`estimation_cmd <estim-comm>`).
Alternatively, this command is also used in conjunction with the
``partial_information`` option of ``stoch_simul``, for declaring
the set of observed variables when solving the model under partial
information.
Only one instance of ``varobs`` is allowed in a model file. If one
needs to declare observed variables in a loop, the macro processor
can be used as shown in the second example below.
*Example*
::
varobs C y rr;
Declares endogenous variables ``C``, ``y`` and ``rr`` as
observed variables.
*Example* (with a macro processor loop)
::
varobs
@#for co in countries
GDP_@{co}
@#endfor
;
.. block:: observation_trends ;
|br| This block specifies linear trends for observed variables as
functions of model parameters. In case the ``loglinear`` option is
used, this corresponds to a linear trend in the logged
observables, i.e. an exponential trend in the level of the
observables.
Each line inside of the block should be of the form::
VARIABLE_NAME(EXPRESSION);
In most cases, variables shouldn’t be centered when
``observation_trends`` is used.
*Example*
::
observation_trends;
Y (eta);
P (mu/eta);
end;
.. block:: estimated_params ;
estimated_params (overwrite) ;
|br| This block lists all parameters to be estimated and specifies
bounds and priors as necessary.
Each line corresponds to an estimated parameter.
In a maximum likelihood or a method of moments estimation, each line follows this syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME
, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ];
In a Bayesian MCMC or a penalized method of moments estimation, each line follows this syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT
[, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE,
PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [,
PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ];
The first part of the line consists of one of the four following
alternatives:
* ``stderr VARIABLE_NAME``
Indicates that the standard error of the exogenous
variable VARIABLE_NAME, or of the observation
error/measurement errors associated with endogenous
observed variable VARIABLE_NAME, is to be estimated.
* ``corr VARIABLE_NAME1, VARIABLE_NAME2``
Indicates that the correlation between the exogenous
variables VARIABLE_NAME1 and VARIABLE_NAME2, or the
correlation of the observation errors/measurement errors
associated with endogenous observed variables
VARIABLE_NAME1 and VARIABLE_NAME2, is to be
estimated. Note that correlations set by previous
``shocks`` blocks or estimation commands are kept at their
value set prior to estimation if they are not estimated
again subsequently. Thus, the treatment is the same as in
the case of deep parameters set during model calibration
and not estimated.
* ``PARAMETER_NAME``
The name of a model parameter to be estimated
* ``DSGE_PRIOR_WEIGHT``
Special name for the weigh of the DSGE model in DSGE-VAR model.
The rest of the line consists of the following fields, some of
them being optional:
.. option:: INITIAL_VALUE
Specifies a starting value for the posterior mode optimizer or
the maximum likelihood estimation. If unset, defaults to the
prior mean.
.. option:: LOWER_BOUND
Specifies a lower bound for the parameter value in maximum
likelihood estimation. In a Bayesian estimation context, sets a
lower bound only effective while maximizing the posterior
kernel. This lower bound does not modify the shape of the prior
density, and is only aimed at helping the optimizer in
identifying the posterior mode (no consequences for the
MCMC). For some prior densities (namely inverse gamma, gamma,
uniform, beta or Weibull) it is possible to shift the support
of the prior distributions to the left or the right using
:opt:`prior_3rd_parameter <PRIOR_3RD_PARAMETER>`. In this case
the prior density is effectively modified (note that the
truncated Gaussian density is not implemented in Dynare). If
unset, defaults to minus infinity (ML) or the natural lower
bound of the prior (Bayesian estimation).
.. option:: UPPER_BOUND
Same as ``lower_bound``, but specifying an upper bound instead.
.. option:: PRIOR_SHAPE
A keyword specifying the shape of the prior density. The
possible values are: ``beta_pdf``, ``gamma_pdf``,
``normal_pdf``, ``uniform_pdf``, ``inv_gamma_pdf``,
``inv_gamma1_pdf``, ``inv_gamma2_pdf`` and
``weibull_pdf``. Note that ``inv_gamma_pdf`` is equivalent to
``inv_gamma1_pdf``.
.. option:: PRIOR_MEAN
The mean of the prior distribution.
.. option:: PRIOR_STANDARD_ERROR
The standard error of the prior distribution.
.. option:: PRIOR_3RD_PARAMETER
A third parameter of the prior used for generalized beta
distribution, generalized gamma, generalized Weibull and for
the uniform distribution. Default: ``0``.
.. option:: PRIOR_4TH_PARAMETER
A fourth parameter of the prior used for generalized beta
distribution and for the uniform distribution. Default: ``1``.
.. option:: SCALE_PARAMETER
A parameter specific scale parameter for the jumping
distribution’s covariance matrix of the Metropolis-Hasting
algorithm.
Note that INITIAL_VALUE, LOWER_BOUND, UPPER_BOUND, PRIOR_MEAN,
PRIOR_STANDARD_ERROR, PRIOR_3RD_PARAMETER, PRIOR_4TH_PARAMETER and
SCALE_PARAMETER can be any valid EXPRESSION. Some of them can be
empty, in which Dynare will select a default value depending on
the context and the prior shape.
In case of the uniform distribution, it can be specified either by
providing an upper and a lower bound using
:opt:`PRIOR_3RD_PARAMETER` and :opt:`PRIOR_4TH_PARAMETER` or via
mean and standard deviation using :opt:`PRIOR_MEAN`,
:opt:`PRIOR_STANDARD_ERROR`. The other two will automatically be
filled out. Note that providing both sets of hyperparameters will
yield an error message.
As one uses options more towards the end of the list, all previous
options must be filled: for example, if you want to specify
SCALE_PARAMETER, you must specify ``PRIOR_3RD_PARAMETER`` and
``PRIOR_4TH_PARAMETER``. Use empty values, if these parameters
don’t apply.
*Example*
::
corr eps_1, eps_2, 0.5, , , beta_pdf, 0, 0.3, -1, 1;
Sets a generalized beta prior for the correlation between
``eps_1`` and ``eps_2`` with mean ``0`` and variance
``0.3``. By setting ``PRIOR_3RD_PARAMETER`` to ``-1`` and
``PRIOR_4TH_PARAMETER`` to ``1`` the standard beta distribution
with support ``[0,1]`` is changed to a generalized beta with
support ``[-1,1]``. Note that LOWER_BOUND and UPPER_BOUND are
left empty and thus default to ``-1`` and ``1``,
respectively. The initial value is set to ``0.5``.
*Example*
::
corr eps_1, eps_2, 0.5, -0.5, 1, beta_pdf, 0, 0.3, -1, 1;
Sets the same generalized beta distribution as before, but now
truncates this distribution to ``[-0.5,1]`` through the use of
LOWER_BOUND and UPPER_BOUND.
*Parameter transformation*
Sometimes, it is desirable to estimate a transformation of a
parameter appearing in the model, rather than the parameter
itself. It is of course possible to replace the original parameter
by a function of the estimated parameter everywhere is the model,
but it is often unpractical.
In such a case, it is possible to declare the parameter to be
estimated in the parameters statement and to define the
transformation, using a pound sign (#) expression (see
:ref:`model-decl`).
*Example*
::
parameters bet;
model;
# sig = 1/bet;
c = sig*c(+1)*mpk;
end;
estimated_params;
bet, normal_pdf, 1, 0.05;
end;
It is possible to have several ``estimated_params`` blocks. By default,
subsequent blocks are concatenated with the previous ones; this can be
useful when building models in a modular fashion (see also
:bck:`estimated_params_remove` for that use case). However, if an
``estimated_params`` block has the ``overwrite`` option, its contents
becomes the new list of estimated parameters, cancelling previous blocks;
this can be useful when doing several estimations in a single ``.mod``
file.
.. block:: estimated_params_init ;
estimated_params_init (OPTIONS...);
|br| This block declares numerical initial values for the
optimizer when these ones are different from the prior mean. It
should be specified after the ``estimated_params`` block as
otherwise the specified starting values are overwritten by the
latter.
Each line has the following syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, INITIAL_VALUE;
*Options*
.. option:: use_calibration
For not specifically initialized parameters, use the deep
parameters and the elements of the covariance matrix specified
in the ``shocks`` block from calibration as starting values
for estimation. For components of the ``shocks`` block that
were not explicitly specified during calibration or which
violate the prior, the prior mean is used.
See :bck:`estimated_params`, for the meaning and syntax of the
various components.
.. block:: estimated_params_bounds ;
|br| This block declares lower and upper bounds for parameters in maximum likelihood estimation.
Each line has the following syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;
See :bck:`estimated_params`, for the meaning and syntax of the
various components.
.. block:: estimated_params_remove ;
|br| This block partially undoes the effect of a previous
:bck:`estimated_params` block, by removing some parameters from the estimation.
Each line has the following syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME;
.. _estim-comm:
.. command:: estimation [VARIABLE_NAME...];
estimation (OPTIONS...) [VARIABLE_NAME...];
|br| This command runs Bayesian or maximum likelihood estimation.
The following information will be displayed by the command:
* Results from posterior optimization (also for maximum likelihood)
* Marginal log data density
* Posterior mean and highest posterior density interval (shortest
credible set) from posterior simulation
* Convergence diagnostic table when only one MCM chain is used or
Metropolis-Hastings convergence graphs documented in *Pfeifer
(2014)* in case of multiple MCM chains
* Table with numerical inefficiency factors of the MCMC
* Graphs with prior, posterior, and mode
* Graphs of smoothed shocks, smoothed observation errors, smoothed
and historical variables
Note that the posterior moments, smoothed variables, k-step ahead
filtered variables and forecasts (when requested) will only be
computed on the variables listed after the ``estimation``
command. Alternatively, one can choose to compute these quantities
on all endogenous or on all observed variables (see
``consider_all_endogenous``, ``consider_all_endogenous_and_auxiliary``,
and ``consider_only_observed`` options
below). If no variable is listed after the estimation command,
then Dynare will interactively ask which variable set to use.
Also, during the MCMC (Bayesian estimation with ``mh_replic``
:math:`>0`) a (graphical or text) waiting bar is displayed showing
the progress of the Monte-Carlo and the current value of the
acceptance ratio. Note that if the ``load_mh_file`` option is used
(see below) the reported acceptance ratio does not take into
account the draws from the previous MCMC. In the literature there
is a general agreement for saying that the acceptance ratio should
be close to one third or one quarter. If this not the case, you
can stop the MCMC (``Ctrl-C``) and change the value of option
``mh_jscale`` (see below).
Note that by default Dynare generates random numbers using the
algorithm ``mt199937ar`` (i.e. Mersenne Twister method) with a
seed set equal to ``0``. Consequently the MCMCs in Dynare are
deterministic: one will get exactly the same results across
different Dynare runs (*ceteris paribus*). For instance, the
posterior moments or posterior densities will be exactly the
same. This behaviour allows to easily identify the consequences of
a change on the model, the priors or the estimation options. But
one may also want to check that across multiple runs, with
different sequences of proposals, the returned results are almost
identical. This should be true if the number of iterations
(i.e. the value of ``mh_replic``) is important enough to ensure
the convergence of the MCMC to its ergodic distribution. In this
case the default behaviour of the random number generators in not
wanted, and the user should set the seed according to the system
clock before the estimation command using the following command::
set_dynare_seed('clock');
so that the sequence of proposals will be different across different runs.
Finally, Dynare does not always properly distinguish between maximum
likelihood and Bayesian estimation in its field names. While there is
an important conceptual distinction between frequentist confidence intervals
and Bayesian highest posterior density intervals (HPDI) as well as between
posterior density and likelilhood, Dynare sometimes uses the Bayesian
terms as a stand-in in its display of maximum likelihood results. An
example is the storage of the output of the ``forecast`` option of
``estimation`` with ML, which will use ``HPDinf/HPDsup`` to denote
the confidence interval.
*Algorithms*
The Monte Carlo Markov Chain (MCMC) diagnostics are generated by
the estimation command if :opt:`mh_replic <mh_replic = INTEGER>`
is larger than 2000 and if option :opt:`nodiagnostic` is not
used. If :opt:`mh_nblocks <mh_nblocks = INTEGER>` is equal to one,
the convergence diagnostics of *Geweke (1992,1999)* is
computed. It uses a chi-square test to compare the means of the
first and last draws specified by :opt:`geweke_interval
<geweke_interval = [DOUBLE DOUBLE]>` after discarding the burn-in
of :opt:`mh_drop <mh_drop = DOUBLE>`. The test is computed using
variance estimates under the assumption of no serial correlation
as well as using tapering windows specified in :opt:`taper_steps
<taper_steps = [INTEGER1 INTEGER2 ...]>`. If :opt:`mh_nblocks
<mh_nblocks = INTEGER>` is larger than 1, the convergence
diagnostics of *Brooks and Gelman (1998)* are used instead. As
described in section 3 of *Brooks and Gelman (1998)* the
univariate convergence diagnostics are based on comparing pooled
and within MCMC moments (Dynare displays the second and third
order moments, and the length of the Highest Probability Density
interval covering 80% of the posterior distribution). Due to
computational reasons, the multivariate convergence diagnostic
does not follow *Brooks and Gelman (1998)* strictly, but rather
applies their idea for univariate convergence diagnostics to the
range of the posterior likelihood function instead of the
individual parameters. The posterior kernel is used to aggregate
the parameters into a scalar statistic whose convergence is then
checked using the *Brooks and Gelman (1998)* univariate
convergence diagnostic.
The inefficiency factors are computed as in *Giordano et
al.(2011)* based on Parzen windows as in e.g. *Andrews (1991)*.
*Options*
.. _dataf:
.. option:: datafile = FILENAME
The datafile: a ``.m`` file, a ``.mat`` file, a ``.csv`` file,
or a ``.xls/.xlsx`` file (under Octave, the `io
<https://octave.sourceforge.io/io/>`__ package from Octave-Forge
is required for the ``.csv`` and ``.xlsx`` formats and the
``.xls`` file extension is not supported). Note that the base
name (i.e. without extension) of the datafile has to be
different from the base name of the model file. If there are
several files named FILENAME, but with different file endings,
the file name must be included in quoted strings and provide
the file ending like::
estimation(datafile='../fsdat_simul.mat',...);
.. option:: dirname = FILENAME
Directory in which to store ``estimation`` output. To pass a
subdirectory of a directory, you must quote the
argument. Default: ``<mod_file>``.
.. option:: xls_sheet = QUOTED_STRING
The name of the sheet with the data in an Excel file.
.. option:: xls_range = RANGE
The range with the data in an Excel file. For example,
``xls_range=B2:D200``.
.. option:: nobs = INTEGER
The number of observations following :opt:`first_obs <first_obs
= [INTEGER1:INTEGER2]>` to be used. Default: all observations
in the file after ``first_obs``.
.. option:: nobs = [INTEGER1:INTEGER2]
Runs a recursive estimation and forecast for samples of size
ranging of ``INTEGER1`` to ``INTEGER2``. Option ``forecast``
must also be specified. The forecasts are stored in the
``RecursiveForecast`` field of the results structure (see
:mvar:`RecursiveForecast <oo_.RecursiveForecast>`). The
respective results structures ``oo_`` are saved in
``oo_recursive_`` (see :mvar:`oo_recursive_`) and are indexed
with the respective sample length.
.. option:: first_obs = INTEGER
The number of the first observation to be used. In case of
estimating a DSGE-VAR, ``first_obs`` needs to be larger than
the number of lags. Default: ``1``.
.. option:: first_obs = [INTEGER1:INTEGER2]
Runs a rolling window estimation and forecast for samples of
fixed size ``nobs`` starting with the first observation ranging
from ``INTEGER1`` to ``INTEGER2``. Option ``forecast`` must
also be specified. This option is incompatible with requesting
recursive forecasts using an expanding window (see :opt:`nobs
<nobs = [INTEGER1:INTEGER2]>`). The respective results
structures ``oo_`` are saved in ``oo_recursive_`` (see
:mvar:`oo_recursive_`) and are indexed with the respective
first observation of the rolling window.
.. option:: prefilter = INTEGER
A value of 1 means that the estimation procedure will demean
each data series by its empirical mean. If the :ref:`loglinear
<logl>` option without the :opt:`logdata` option is requested,
the data will first be logged and then demeaned. Default:
``0``, i.e. no prefiltering.
.. option:: presample = INTEGER
The number of observations after :opt:`first_obs <first_obs =
[INTEGER1:INTEGER2]>` to be skipped before evaluating the
likelihood. These presample observations do not enter the
likelihood, but are used as a training sample for starting the
Kalman filter iterations. This option is incompatible with
estimating a DSGE-VAR. Default: ``0``.
.. _logl:
.. option:: loglinear
Computes a log-linear approximation of the model instead of a
linear approximation. As always in the context of estimation,
the data must correspond to the definition of the variables
used in the model (see *Pfeifer (2013)* for more details on how
to correctly specify observation equations linking model
variables and the data). If you specify the loglinear option,
Dynare will take the logarithm of both your model variables and
of your data as it assumes the data to correspond to the
original non-logged model variables. The displayed posterior
results like impulse responses, smoothed variables, and moments
will be for the logged variables, not the original un-logged
ones. Default: computes a linear approximation.
.. option:: logdata
Dynare applies the :math:`log` transformation to the provided
data if a log-linearization of the model is requested
(:opt:`loglinear`) unless ``logdata`` option is used. This
option is necessary if the user provides data already in logs,
otherwise the :math:`log` transformation will be applied twice
(this may result in complex data).
.. option:: plot_priors = INTEGER
Control the plotting of priors.
``0``
No prior plot.
``1``
Prior density for each estimated parameter is
plotted. It is important to check that the actual shape
of prior densities matches what you have in
mind. Ill-chosen values for the prior standard density
can result in absurd prior densities.
|br| Default value is ``1``.
.. option:: nograph
See :opt:`nograph`.
.. option:: posterior_nograph
Suppresses the generation of graphs associated with Bayesian
IRFs (:opt:`bayesian_irf`), posterior smoothed objects
(:opt:`smoother`), and posterior forecasts (:opt:`forecast`).
.. option:: posterior_graph
Re-enables the generation of graphs previously shut off with
:opt:`posterior_nograph`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format <graph_format = ( FORMAT, FORMAT... )>`.
.. option:: no_init_estimation_check_first_obs
Do not check for stochastic singularity in first period. If used, `ESTIMATION CHECKS`
does not return an error if the check fails only in first observation.
This should only be used when observing stock variables (e.g. capital) in first period, on top of their associated flow (e.g. investment).
Using this option may lead to a crash or provide undesired/wrong results for badly specified problems
(e.g. the additional variable observed in first period is not predetermined).
For advanced use only.
.. option:: lik_init = INTEGER
Type of initialization of Kalman filter:
``1``
For stationary models, the initial matrix of variance
of the error of forecast is set equal to the
unconditional variance of the state variables.
``2``
For nonstationary models: a wide prior is used with an
initial matrix of variance of the error of forecast
diagonal with 10 on the diagonal (follows the
suggestion of *Harvey and Phillips(1979)*).
``3``
For nonstationary models: use a diffuse filter (use
rather the ``diffuse_filter`` option).
``4``
The filter is initialized with the fixed point of the
Riccati equation.
``5``
Use i) option 2 for the non-stationary elements by
setting their initial variance in the forecast error
matrix to 10 on the diagonal and all covariances to 0
and ii) option 1 for the stationary elements.
|br| Default value is 1. For advanced use only.
.. option:: conditional_likelihood
Do not use the kalman filter to evaluate the likelihood, but instead
evaluate the conditional likelihood, based on the first order reduced
form of the model, by assuming that the initial state vector is 0 for all
the endogenous variables. This approach requires that:
1. The number of structural innovations be equal to the number of observed variables.
2. The absence of measurement errors (as introduced by the Dynare
interface, see documentation about the :bck:`estimated_params` block).
3. The absence of missing observations.
The evaluation of the conditional likelihood is faster and more stable
than the evaluation of the likelihood with the Kalman filter. Also this
approach does not require special treatment for models with unit roots.
Note however that the conditional likelihood is sensitive to the choice
for the initial condition, which can be an issue if the data are
initially far from the steady state. This option is not compatible with
``analytical_derivation``.
.. option:: lik_algo = INTEGER
For internal use and testing only.
.. option:: conf_sig = DOUBLE
Level of significance of the confidence interval used for classical forecasting after
estimation. Default: 0.9.
.. option:: mh_conf_sig = DOUBLE
Confidence/HPD interval used for the computation of prior and
posterior statistics like: parameter distributions,
prior/posterior moments, conditional variance decomposition,
impulse response functions, Bayesian forecasting. Default:
``0.9``.
.. option:: mh_replic = INTEGER
Number of replications for each chain of the Metropolis-Hastings algorithm.
The number of draws should be sufficient to achieve convergence of the MCMC and
to meaningfully compute posterior objects. Default: ``20000``.
.. option:: sub_draws = INTEGER
Number of draws from the MCMC that are used to compute
posterior distribution of various objects (smoothed variable,
smoothed shocks, forecast, moments, IRF). The draws used to
compute these posterior moments are sampled uniformly in the
estimated empirical posterior distribution (i.e. draws of the
MCMC). ``sub_draws`` should be smaller than the total number of
MCMC draws available. Default:
``min(posterior_max_subsample_draws, (Total number of
draws)*(number of chains) )``.
.. option:: posterior_max_subsample_draws = INTEGER
Maximum number of draws from the MCMC used to compute posterior
distribution of various objects (smoothed variable, smoothed
shocks, forecast, moments, IRF), if not overriden by option
``sub_draws``. Default: ``1200``.
.. option:: mh_nblocks = INTEGER
Number of parallel chains for Metropolis-Hastings
algorithm. Default: ``2``.
.. option:: mh_drop = DOUBLE
The fraction of initially generated parameter vectors to be
dropped as a burn-in before using posterior
simulations. Default: ``0.5``.
.. option:: mh_jscale = DOUBLE
The scale parameter of the jumping distribution's covariance
matrix (Metropolis-Hastings or TaRB-algorithm). This option must be tuned to
obtain, ideally, an acceptance ratio of 25%-33%. Basically, the
idea is to increase the variance of the jumping distribution if
the acceptance ratio is too high, and decrease the same
variance if the acceptance ratio is too low. In some situations
it may help to consider parameter-specific values for this
scale parameter. This can be done in the
:bck:`estimated_params` block.
Note that ``mode_compute=6`` will tune the scale parameter to
achieve an acceptance rate of
:ref:`AcceptanceRateTarget<art>`. The resulting scale parameter
will be saved into a file named
``MODEL_FILENAME_mh_scale.mat`` in the ``FILENAME/Output`` folder.
This file can be loaded in
subsequent runs via the ``posterior_sampler_options`` option
:ref:`scale_file <scale-file>`. Both ``mode_compute=6`` and
``scale_file`` will overwrite any value specified in
``estimated_params`` with the tuned value. Default: ``2.38/sqrt(n)``.
Note also that for the Random Walk Metropolis Hastings
algorithm, it is possible to use option :opt:`mh_tune_jscale
<mh_tune_jscale [= DOUBLE]>`, to automatically tune the value
of ``mh_jscale``. In this case, the ``mh_jscale`` option must
not be used.
.. option:: mh_init_scale = DOUBLE (deprecated)
The scale to be used for drawing the initial value of the
Metropolis-Hastings chain. Generally, the starting points
should be overdispersed for the *Brooks and Gelman (1998)*
convergence diagnostics to be meaningful. Default:
``2*mh_jscale.``
It is important to keep in mind that ``mh_init_scale`` is set
at the beginning of Dynare execution, i.e. the default will not
take into account potential changes in ``mh_jscale`` introduced
by either ``mode_compute=6`` or the
``posterior_sampler_options`` option :ref:`scale_file<scale-file>`.
If ``mh_init_scale`` is too wide during
initalization of the posterior sampler so that 100 tested draws
are inadmissible (e.g. Blanchard-Kahn conditions are always
violated), Dynare will request user input of a new
``mh_init_scale`` value with which the next 100 draws will be
drawn and tested. If the :opt:`nointeractive` option has been
invoked, the program will instead automatically decrease
``mh_init_scale`` by 10 percent after 100 futile draws and try
another 100 draws. This iterative procedure will take place at
most 10 times, at which point Dynare will abort with an error
message.
.. option:: mh_init_scale_factor = DOUBLE
The multiple of ``mh_jscale`` used for drawing the initial value of the
Metropolis-Hastings chain. Generally, the starting points
should be overdispersed for the *Brooks and Gelman (1998)*
convergence diagnostics to be meaningful. Default:
``2``
If ``mh_init_scale_factor`` is too wide during
initalization of the posterior sampler so that 100 tested draws
are inadmissible (e.g. Blanchard-Kahn conditions are always
violated), Dynare will request user input of a new
``mh_init_scale_factor`` value with which the next 100 draws will be
drawn and tested. If the :opt:`nointeractive` option has been
invoked, the program will instead automatically decrease
``mh_init_scale_factor`` by 10 percent after 100 futile draws and try
another 100 draws. This iterative procedure will take place at
most 10 times, at which point Dynare will abort with an error
message.
.. option:: mh_tune_jscale [= DOUBLE]
Automatically tunes the scale parameter of the jumping
distribution's covariance matrix (Metropolis-Hastings), so that
the overall acceptance ratio is close to the desired
level. Default value is ``0.33``. It is not possible to
match exactly the desired acceptance ratio because of the
stochastic nature of the algorithm (the proposals and the
initial conditions of the markov chains if
``mh_nblocks>1``). This option is only available for the
Random Walk Metropolis Hastings algorithm. Must not be used in conjunction with
:opt:`mh_jscale = DOUBLE`.
.. option:: mh_tune_guess = DOUBLE
Specifies the initial value for the :opt:`mh_tune_jscale
<mh_tune_jscale [= DOUBLE]>` option. Default: ``2.39/sqrt(n)``. Must not
be set if :opt:`mh_tune_jscale <mh_tune_jscale [= DOUBLE]>` is
not used.
.. option:: mh_recover
Attempts to recover a Metropolis-Hastings simulation that
crashed prematurely, starting with the last available saved
``mh``-file. Shouldn’t be used together with ``load_mh_file``
or a different ``mh_replic`` than in the crashed run. Since
Dynare 4.5 the proposal density from the previous run will
automatically be loaded. In older versions, to assure a neat
continuation of the chain with the same proposal density, you
should provide the ``mode_file`` used in the previous run or
the same user-defined ``mcmc_jumping_covariance`` when using
this option. Note that under Octave, a neat continuation of the
crashed chain with the respective last random number generator
state is currently not supported.
.. option:: mh_posterior_mode_estimation
Skip optimizer-based mode-finding and instead compute the mode based
on a run of a MCMC. The MCMC will start at the prior mode and use the prior
variances to compute the inverse Hessian.
.. option:: mode_file = FILENAME
Name of the file containing previous value for the mode. When
computing the mode, Dynare stores the mode (``xparam1``) and
the hessian (``hh``, only if ``cova_compute=1``) in a file
called ``MODEL_FILENAME_mode.mat`` in the ``FILENAME/Output`` folder.
After a successful run of
the estimation command, the ``mode_file`` will be disabled to
prevent other function calls from implicitly using an updated
mode file. Thus, if the ``.mod`` file contains subsequent
``estimation`` commands, the ``mode_file`` option, if desired,
needs to be specified again.
.. option:: mode_compute = INTEGER | FUNCTION_NAME
Specifies the optimizer for the mode computation:
``0``
The mode isn’t computed. When the ``mode_file`` option
is specified, the mode is simply read from that file.
When ``mode_file`` option is not specified, Dynare
reports the value of the log posterior (log
likelihood) evaluated at the initial value of the
parameters.
When ``mode_file`` is not specified and there is no
``estimated_params`` block, but the ``smoother``
option is used, it is a roundabout way to compute the
smoothed value of the variables of a model with
calibrated parameters.
``1``
Uses ``fmincon`` optimization routine (available under
MATLAB if the Optimization Toolbox is installed; available
under Octave if the `optim
<https://octave.sourceforge.io/optim/>`__ package from
Octave-Forge, version 1.6 or above, is installed).
``2``
Uses the continuous simulated annealing global
optimization algorithm described in *Corana et
al.(1987)* and *Goffe et al.(1994)*.
``3``
Uses ``fminunc`` optimization routine (available under
MATLAB if the Optimization Toolbox is installed;
available under Octave if the `optim
<https://octave.sourceforge.io/optim/>`__ package from
Octave-Forge is installed).
``4``
Uses Chris Sims’s ``csminwel``.
``5``
Uses Marco Ratto’s ``newrat``. This value is not
compatible with non linear filters or DSGE-VAR
models. This is a slice optimizer: most iterations are
a sequence of univariate optimization step, one for
each estimated parameter or shock. Uses ``csminwel``
for line search in each step.
``6``
Uses a Monte-Carlo based optimization routine (see
`<https://archives.dynare.org/DynareWiki/MonteCarloOptimization>`__ for more details).
``7``
Uses ``fminsearch``, a simplex-based optimization
routine (available under MATLAB if the Optimization
Toolbox is installed; available under Octave if the
optim package from Octave-Forge is installed).
``8``
Uses Dynare implementation of the Nelder-Mead
simplex-based optimization routine (generally more
efficient than the MATLAB or Octave implementation
available with ``mode_compute=7``).
``9``
Uses the CMA-ES (Covariance Matrix Adaptation
Evolution Strategy) algorithm of *Hansen and Kern
(2004)*, an evolutionary algorithm for difficult
non-linear non-convex optimization.
``10``
Uses the ``simpsa`` algorithm, based on the
combination of the non-linear simplex and simulated
annealing algorithms as proposed by *Cardoso, Salcedo
and Feyo de Azevedo (1996)*.
``11``
This is not strictly speaking an optimization
algorithm. The (estimated) parameters are treated as
state variables and estimated jointly with the
original state variables of the model using a
nonlinear filter. The algorithm implemented in Dynare
is described in *Liu and West (2001)*, and works with
``k`` order local approximations of the model.
``12``
Uses the ``particleswarm`` optimization routine
(available under MATLAB if the Global Optimization
Toolbox is installed; not available under Octave).
``13``
Uses the ``lsqnonlin`` non-linear least squares optimization
routine (available under MATLAB if the Optimization Toolbox is
installed; available under Octave if the `optim
<https://octave.sourceforge.io/optim/>`__ package from
Octave-Forge is installed). Only supported for ``method_of_moments``.
``101``
Uses the SolveOpt algorithm for local nonlinear
optimization problems proposed by *Kuntsevich and
Kappel (1997)*.
``102``
Uses ``simulannealbnd`` optimization routine
(available under MATLAB if the Global Optimization
Toolbox is installed; not available under Octave)
``FUNCTION_NAME``
It is also possible to give a FUNCTION_NAME to this
option, instead of an INTEGER. In that case, Dynare
takes the return value of that function as the
posterior mode.
|br| Default value is ``4``.
.. option:: silent_optimizer
Instructs Dynare to run mode computing/optimization silently
without displaying results or saving files in between. Useful
when running loops.
.. option:: mcmc_jumping_covariance = OPTION
Tells Dynare which covariance to use for the proposal density
of the MCMC sampler. OPTION can be one of the following:
``hessian``
Uses the Hessian matrix computed at the mode.
``prior_variance``
Uses the prior variances. No infinite prior variances
are allowed in this case.
``identity_matrix``
Uses an identity matrix.
``FILENAME``
Loads an arbitrary user-specified covariance matrix
from ``FILENAME.mat``. The covariance matrix must be
saved in a variable named ``jumping_covariance``, must
be square, positive definite, and have the same
dimension as the number of estimated parameters.
Note that the covariance matrices are still scaled with
:opt:`mh_jscale <mh_jscale = DOUBLE>`. Default value is
``hessian``.
.. option:: mode_check
Tells Dynare to plot the posterior density for values around
the computed mode for each estimated parameter in turn. This is
helpful to diagnose problems with the optimizer. Note that for
``order>1`` the likelihood function resulting from the particle
filter is not differentiable anymore due to the resampling
step. For this reason, the ``mode_check`` plot may look wiggly.
.. option:: mode_check_neighbourhood_size = DOUBLE
Used in conjunction with option ``mode_check``, gives the width
of the window around the posterior mode to be displayed on the
diagnostic plots. This width is expressed in percentage
deviation. The ``Inf`` value is allowed, and will trigger a
plot over the entire domain (see also
``mode_check_symmetric_plots``). Default:``0.5``.
.. option:: mode_check_symmetric_plots = INTEGER
Used in conjunction with option ``mode_check``, if set to
``1``, tells Dynare to ensure that the check plots are
symmetric around the posterior mode. A value of ``0`` allows to
have asymmetric plots, which can be useful if the posterior
mode is close to a domain boundary, or in conjunction with
``mode_check_neighbourhood_size = Inf`` when the domain in not
the entire real line. Default: ``1``.
.. option:: mode_check_number_of_points = INTEGER
Number of points around the posterior mode where the posterior
kernel is evaluated (for each parameter). Default is ``20``.
.. option:: prior_trunc = DOUBLE
Probability of extreme values of the prior density that is
ignored when computing bounds for the parameters. Default:
``1e-32``.
.. option:: huge_number = DOUBLE
Value for replacing infinite values in the definition of
(prior) bounds when finite values are required for
computational reasons. Default: ``1e7``.
.. option:: load_mh_file
Tells Dynare to add to previous Metropolis-Hastings simulations
instead of starting from scratch. Since Dynare 4.5 the proposal
density from the previous run will automatically be loaded. In
older versions, to assure a neat continuation of the chain with
the same proposal density, you should provide the ``mode_file``
used in the previous run or the same user-defined
``mcmc_jumping_covariance`` when using this option. Shouldn’t
be used together with ``mh_recover``. Note that under Octave, a
neat continuation of the chain with the last random number
generator state of the already present draws is currently not
supported.
.. option:: load_results_after_load_mh
This option is available when loading a previous MCMC run
without adding additional draws, i.e. when ``load_mh_file`` is
specified with ``mh_replic=0``. It tells Dynare to load the
previously computed convergence diagnostics, marginal data
density, and posterior statistics from an existing ``_results``
file instead of recomputing them.
.. option:: mh_initialize_from_previous_mcmc
This option allows to pick initial values for new MCMC from a previous one,
where the model specification, the number of estimated parameters,
(some) prior might have changed (so a situation where ``load_mh_file`` would not work).
If an additional parameter is estimated, it is automatically initialized from prior_draw.
Note that, if this option is used to skip the optimization step, you should use a sampling method which does not require
a proposal density, like slice. Otherwise, optimization should always be done beforehand or a mode file with
an appropriate posterior covariance matrix should be used.
.. option:: mh_initialize_from_previous_mcmc_directory = FILENAME
If ``mh_initialize_from_previous_mcmc`` is set, users must provide here
the path to the standard FNAME folder from where to load prior definitions and
last MCMC values to be used to initialize the new MCMC.
Example: if previous project directory is ``/my_previous_dir`` and FNAME is ``mymodel``,
users should set the option as
``mh_initialize_from_previous_mcmc_directory = '/my_previous_dir/mymodel'``
Dynare will then look for the last record file into
``/my_previous_dir/mymodel/metropolis/mymodel_mh_history_<LAST>.mat``
and for the prior definition file into
``/my_previous_dir/mymodel/prior/definition.mat``
.. option:: mh_initialize_from_previous_mcmc_record = FILENAME
If ``mh_initialize_from_previous_mcmc`` is set, and whenever the standard file or directory tree
is not applicable to load initial values, users may directly provide here
the path to the record file from which to load
values to be used to initialize the new MCMC.
.. option:: mh_initialize_from_previous_mcmc_prior = FILENAME
If ``mh_initialize_from_previous_mcmc`` is set, and whenever the standard file or directory tree
is not applicable to load initial values, users may directly provide here
the path to the prior definition file, to get info in the priors used in previous MCMC.
.. option:: optim = (NAME, VALUE, ...)
A list of NAME and VALUE pairs. Can be used to set options for
the optimization routines. The set of available options depends
on the selected optimization routine (i.e. on the value of
option :opt:`mode_compute <mode_compute = INTEGER |
FUNCTION_NAME>`):
``1, 3, 7, 12, 13``
Available options are given in the documentation of the
MATLAB Optimization Toolbox or in Octave’s
documentation.
``2``
Available options are:
``'initial_step_length'``
Initial step length. Default: ``1``.
``'initial_temperature'``
Initial temperature. Default: ``15``.
``'MaxIter'``
Maximum number of function
evaluations. Default: ``100000``.
``'neps'``
Number of final function values used to decide
upon termination. Default: ``10``.
``'ns'``
Number of cycles. Default: ``10``.
``'nt'``
Number of iterations before temperature
reduction. Default: ``10``.
``'step_length_c'``
Step length adjustment. Default: ``0.1``.
``'TolFun'``
Stopping criteria. Default: ``1e-8``.
``'rt'``
Temperature reduction factor. Default: ``0.1``.
``'verbosity'``
Controls verbosity of display during
optimization, ranging from ``0`` (silent) to
``3`` (each function evaluation). Default:
``1``
``4``
Available options are:
``'InitialInverseHessian'``
Initial approximation for the inverse of the
Hessian matrix of the posterior kernel (or
likelihood). Obviously this approximation has
to be a square, positive definite and symmetric
matrix. Default: ``'1e-4*eye(nx)'``, where nx
is the number of parameters to be estimated.
``'MaxIter'``
Maximum number of iterations. Default: ``1000``.
``'NumgradAlgorithm'``
Possible values are ``2``, ``3`` and ``5``,
respectively, corresponding to the two, three
and five points formula used to compute the
gradient of the objective function (see
*Abramowitz and Stegun (1964)*). Values ``13``
and ``15`` are more experimental. If
perturbations on the right and the left
increase the value of the objective function
(we minimize this function) then we force the
corresponding element of the gradient to be
zero. The idea is to temporarily reduce the
size of the optimization problem. Default:
``2``.
``'NumgradEpsilon'``
Size of the perturbation used to compute
numerically the gradient of the objective
function. Default: ``1e-6``.
``'TolFun'``
Stopping criteria. Default: ``1e-7``.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``'SaveFiles'``
Controls saving of intermediate results during
optimization. Set to ``0`` to shut off
saving. Default: ``1``.
``5``
Available options are:
``'Hessian'``
Triggers three types of Hessian
computations. ``0``: outer product gradient; ``1``:
default Dynare Hessian routine; ``2``: ’mixed’
outer product gradient, where diagonal elements are
obtained using second order derivation formula and
outer product is used for correlation
structure. Both {0} and {2} options require
univariate filters, to ensure using maximum number
of individual densities and a positive definite
Hessian. Both {0} and {2} are quicker than default
Dynare numeric Hessian, but provide decent starting
values for Metropolis for large models (option {2}
being more accurate than {0}). Default: ``1``.
``'MaxIter'``
Maximum number of iterations. Default: ``1000``.
``'TolFun'``
Stopping criteria. Default: ``1e-5`` for numerical
derivatives, ``1e-7`` for analytic derivatives.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``'SaveFiles'``
Controls saving of intermediate results during
optimization. Set to ``0`` to shut off
saving. Default: ``1``.
``6``
Available options are:
.. _art:
``'AcceptanceRateTarget'``
A real number between zero and one. The scale
parameter of the jumping distribution is
adjusted so that the effective acceptance rate
matches the value of option
``'AcceptanceRateTarget'``. Default:
``1.0/3.0``.
``'InitialCovarianceMatrix'``
Initial covariance matrix of the jumping
distribution. It is also used to initialize the covariance
matrix during recursive updating. Default is ``'previous'`` if
option ``mode_file`` is used, ``'prior'``
otherwise. The user can also specify ``'identity'``, which will
use an identity matrix with a diagonal of 0.1.
``'nclimb-mh'``
Number of iterations in the last MCMC (climbing
mode). Default: ``200000``.
``'ncov-mh'``
Number of iterations used for updating the
covariance matrix of the jumping
distribution. Default: ``20000``.
``'nscale-mh'``
Maximum number of iterations used for adjusting
the scale parameter of the jumping
distribution. Default: ``200000``.
``'NumberOfMh'``
Number of MCMC run sequentially. Default: ``3``.
``8``
Available options are:
``'InitialSimplexSize'``
Initial size of the simplex, expressed as
percentage deviation from the provided initial
guess in each direction. Default: ``.05``.
``'MaxIter'``
Maximum number of iterations. Default: ``5000``.
``'MaxFunEvals'``
Maximum number of objective function
evaluations. No default.
``'MaxFunvEvalFactor'``
Set ``MaxFunvEvals`` equal to
``MaxFunvEvalFactor`` times the number of
estimated parameters. Default: ``500``.
``'TolFun'``
Tolerance parameter (w.r.t the objective
function). Default: ``1e-4``.
``'TolX'``
Tolerance parameter (w.r.t the
instruments). Default: ``1e-4``.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``9``
Available options are:
``'CMAESResume'``
Resume previous run. Requires the
``variablescmaes.mat`` from the last run. Set
to ``1`` to enable. Default: ``0``.
``'MaxIter'``
Maximum number of iterations.
``'MaxFunEvals'``
Maximum number of objective function
evaluations. Default: ``Inf``.
``'TolFun'``
Tolerance parameter (w.r.t the objective
function). Default: ``1e-7``.
``'TolX'``
Tolerance parameter (w.r.t the
instruments). Default: ``1e-7``.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``'SaveFiles'``
Controls saving of intermediate results during
optimization. Set to ``0`` to shut off
saving. Default: ``1``.
``10``
Available options are:
``'EndTemperature'``
Terminal condition w.r.t the temperature. When
the temperature reaches ``EndTemperature``, the
temperature is set to zero and the algorithm
falls back into a standard simplex
algorithm. Default: ``0.1``.
``'MaxIter'``
Maximum number of iterations. Default:
``5000``.
``'MaxFunvEvals'``
Maximum number of objective function
evaluations. No default.
``'TolFun'``
Tolerance parameter (w.r.t the objective
function). Default: ``1e-4``.
``'TolX'``
Tolerance parameter (w.r.t the
instruments). Default: ``1e-4``.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``101``
Available options are:
``'LBGradientStep'``
Lower bound for the stepsize used for the
difference approximation of gradients. Default:
``1e-11``.
``'MaxIter'``
Maximum number of iterations. Default: ``15000``
``'SpaceDilation'``
Coefficient of space dilation. Default: ``2.5``.
``'TolFun'``
Tolerance parameter (w.r.t the objective
function). Default: ``1e-6``.
``'TolX'``
Tolerance parameter (w.r.t the
instruments). Default: ``1e-6``.
``'verbosity'``
Controls verbosity of display during
optimization. Set to ``0`` to set to
silent. Default: ``1``.
``102``
Available options are given in the documentation of the
MATLAB Global Optimization Toolbox.
*Example*
To change the defaults of ``csminwel`` (``mode_compute=4``)::
estimation(..., mode_compute=4,optim=('NumgradAlgorithm',3,'TolFun',1e-5),...);
.. option:: nodiagnostic
Does not compute the convergence diagnostics for
Metropolis-Hastings. Default: diagnostics are computed and
displayed.
.. option:: bayesian_irf
Triggers the computation of the posterior distribution of
IRFs. The length of the IRFs are controlled by the ``irf``
option. Results are stored in ``oo_.PosteriorIRF.dsge`` (see
below for a description of this variable).
.. option:: relative_irf
See :opt:`relative_irf`.
.. option:: dsge_var = DOUBLE
Triggers the estimation of a DSGE-VAR model, where the weight
of the DSGE prior of the VAR model is calibrated to the value
passed (see *Del Negro and Schorfheide (2004)*). It represents
the ratio of dummy over actual observations. To assure that the
prior is proper, the value must be bigger than :math:`(k+n)/T`,
where :math:`k` is the number of estimated parameters,
:math:`n` is the number of observables, and :math:`T` is the
number of observations.
NB: The previous method of declaring ``dsge_prior_weight`` as
a parameter and then calibrating it is now deprecated and will
be removed in a future release of Dynare. Some of objects
arising during estimation are stored with their values at the
mode in ``oo_.dsge_var.posterior_mode``.
.. option:: dsge_var
Triggers the estimation of a DSGE-VAR model, where the weight
of the DSGE prior of the VAR model will be estimated (as in
*Adjemian et al.(2008)*). The prior on the weight of the DSGE
prior, ``dsge_prior_weight``, must be defined in the
``estimated_params`` section.
NB: The previous method of declaring ``dsge_prior_weight`` as
a parameter and then placing it in ``estimated_params`` is now
deprecated and will be removed in a future release of Dynare.
.. option:: dsge_varlag = INTEGER
The number of lags used to estimate a DSGE-VAR model. Default:
``4``.
.. option:: posterior_sampling_method = NAME
Selects the sampler used to sample from the posterior
distribution during Bayesian
estimation. Default:``’random_walk_metropolis_hastings’``.
``'random_walk_metropolis_hastings'``
Instructs Dynare to use the Random-Walk
Metropolis-Hastings. In this algorithm, the proposal
density is recentered to the previous draw in every
step.
``'tailored_random_block_metropolis_hastings'``
Instructs Dynare to use the Tailored randomized block
(TaRB) Metropolis-Hastings algorithm proposed by *Chib
and Ramamurthy (2010)* instead of the standard
Random-Walk Metropolis-Hastings. In this algorithm, at
each iteration the estimated parameters are randomly
assigned to different blocks. For each of these blocks
a mode-finding step is conducted. The inverse Hessian
at this mode is then used as the covariance of the
proposal density for a Random-Walk Metropolis-Hastings
step. If the numerical Hessian is not positive
definite, the generalized Cholesky decomposition of
*Schnabel and Eskow (1990)* is used, but without
pivoting. The TaRB-MH algorithm massively reduces the
autocorrelation in the MH draws and thus reduces the
number of draws required to representatively sample
from the posterior. However, this comes at a
computational cost as the algorithm takes more time to
run.
``'independent_metropolis_hastings'``
Use the Independent Metropolis-Hastings algorithm where
the proposal distribution - in contrast to the Random
Walk Metropolis-Hastings algorithm - does not depend on
the state of the chain.
``'slice'``
Instructs Dynare to use the Slice sampler of *Planas,
Ratto, and Rossi (2015)*. Note that ``'slice'`` is
incompatible with ``prior_trunc=0``.
Whereas one Metropolis-Hastings iteration requires one
evaluation of the posterior, one slice iteration requires :math:`neval`
evaluations, where as a rule of thumb :math:`neval=7\times npar` with
:math:`npar` denoting the number of estimated parameters. Spending
the same computational budget of :math:`N` posterior evaluations in the
slice sampler then implies setting ``mh_replic=N/neval``.
Note that the slice sampler will typically return less autocorrelated Monte Carlo Markov
Chain draws than the MH-algorithm. Its relative (in)efficiency can be investigated via
the reported inefficiency factors.
.. option:: posterior_sampler_options = (NAME, VALUE, ...)
A list of NAME and VALUE pairs. Can be used to set options for
the posterior sampling methods. The set of available options
depends on the selected posterior sampling routine (i.e. on the
value of option :opt:`posterior_sampling_method
<posterior_sampling_method = NAME>`):
``'random_walk_metropolis_hastings'``
Available options are:
.. _prop_distrib:
``'proposal_distribution'``
Specifies the statistical distribution used for the
proposal density.
``'rand_multivariate_normal'``
Use a multivariate normal distribution. This is the default.
``'rand_multivariate_student'``
Use a multivariate student distribution.
``'student_degrees_of_freedom'``
Specifies the degrees of freedom to be used with the
multivariate student distribution. Default: ``3``.
.. _usemhcov:
``'use_mh_covariance_matrix'``
Indicates to use the covariance matrix of the draws
from a previous MCMC run to define the covariance of
the proposal distribution. Requires the
:opt:`load_mh_file` option to be specified. Default:
``0``.
.. _scale-file:
``'scale_file'``
Provides the name of a ``_mh_scale.mat`` file storing
the tuned scale factor from a previous run of
``mode_compute=6``.
.. _savetmp:
``'save_tmp_file'``
Save the MCMC draws into a ``_mh_tmp_blck`` file at the
refresh rate of the status bar instead of just saving
the draws when the current ``_mh*_blck`` file is
full. Default: ``0``
``'independent_metropolis_hastings'``
Takes the same options as in the case of
``random_walk_metropolis_hastings``.
``'slice'``
``'rotated'``
Triggers rotated slice iterations using a covariance
matrix from initial burn-in iterations. Requires either
``use_mh_covariance_matrix`` or
``slice_initialize_with_mode``. Default: ``0``.
``'mode_files'``
For multimodal posteriors, provide the name of a file
containing a ``nparam`` by ``nmodes`` variable called
``xparams`` storing the different modes. This array
must have one column vector per mode and the estimated
parameters along the row dimension. With this info, the
code will automatically trigger the ``rotated`` and
``mode`` options. Default: ``[]``.
``'slice_initialize_with_mode'``
The default for slice is to set ``mode_compute=0`` and
start the chain(s) from a random location in the prior
space. This option first runs the mode-finder and then
starts the chain from the mode. Together with
``rotated``, it will use the inverse Hessian from the
mode to perform rotated slice iterations. Default:
``0``.
``'initial_step_size'``
Sets the initial size of the interval in the
stepping-out procedure as fraction of the prior
support, i.e. the size will be ``initial_step_size *
(UB-LB)``. ``initial_step_size`` must be a real number
in the interval ``[0,1]``. Default: ``0.8``.
``'use_mh_covariance_matrix'``
See :ref:`use_mh_covariance_matrix <usemhcov>`. Must be
used with ``'rotated'``. Default: ``0``.
``'save_tmp_file'``
See :ref:`save_tmp_file <savetmp>`. Default: ``1``.
``'tailored_random_block_metropolis_hastings'``
``'proposal_distribution'``
Specifies the statistical distribution used for the
proposal density. See :ref:`proposal_distribution <prop_distrib>`.
``new_block_probability = DOUBLE``
Specifies the probability of the next parameter
belonging to a new block when the random blocking in
the TaRB Metropolis-Hastings algorithm is
conducted. The higher this number, the smaller is the
average block size and the more random blocks are
formed during each parameter sweep. Default: ``0.25``.
``mode_compute = INTEGER``
Specifies the mode-finder run in every iteration for
every block of the TaRB Metropolis-Hastings
algorithm. See :opt:`mode_compute <mode_compute =
INTEGER | FUNCTION_NAME>`. Default: ``4``.
``optim = (NAME, VALUE,...)``
Specifies the options for the mode-finder used in the
TaRB Metropolis-Hastings algorithm. See :opt:`optim
<optim = (NAME, VALUE, ...)>`.
``'scale_file'``
See :ref:`scale_file <scale-file>`..
``'save_tmp_file'``
See :ref:`save_tmp_file <savetmp>`. Default: ``1``.
.. option:: moments_varendo
Triggers the computation of the posterior distribution of the
theoretical moments of the endogenous variables. Results are
stored in ``oo_.PosteriorTheoreticalMoments`` (see
:mvar:`oo_.PosteriorTheoreticalMoments`). The number of lags in
the autocorrelation function is controlled by the ``ar``
option.
.. option:: contemporaneous_correlation
See :opt:`contemporaneous_correlation`. Results are stored in
``oo_.PosteriorTheoreticalMoments``. Note that the ``nocorr``
option has no effect.
.. option:: no_posterior_kernel_density
Shuts off the computation of the kernel density estimator for
the posterior objects (see :ref:`density <dens>` field).
.. option:: conditional_variance_decomposition = INTEGER
conditional_variance_decomposition = [INTEGER1:INTEGER2]
conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]
Computes the posterior distribution of the conditional variance
decomposition for the specified period(s). The periods must be
strictly positive. Conditional variances are given by
:math:`var(y_{t+k}\vert t)`. For period 1, the conditional
variance decomposition provides the decomposition of the
effects of shocks upon impact. The results are stored in
``oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition``.. Note
that this option requires the option ``moments_varendo`` to be
specified. In the presence of measurement error, the field will
contain the variance contribution after measurement error has
been taken out, *i.e.* the decomposition will be conducted of the
actual as opposed to the measured variables. The variance
decomposition of the measured variables will be stored in
``oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecompositionME``.
.. option:: filtered_vars
Triggers the computation of the posterior distribution of
filtered endogenous variables/one-step ahead forecasts,
i.e. :math:`E_{t}{y_{t+1}}`. Results are stored in
``oo_.FilteredVariables`` (see below for a description of this
variable)
.. option:: smoother
Triggers the computation of the posterior distribution of
smoothed endogenous variables and shocks, i.e. the expected
value of variables and shocks given the information available
in all observations up to the final date
(:math:`E_{T}{y_t}`). Results are stored in
``oo_.SmoothedVariables``, ``oo_.SmoothedShocks`` and
``oo_.SmoothedMeasurementErrors``. Also triggers the
computation of ``oo_.UpdatedVariables``, which contains the
estimation of the expected value of variables given the
information available at the current date
(:math:`E_{t}{y_t}`). See below for a description of all these
variables.
.. option:: smoother_redux
Triggers a faster computation of the smoothed endogenous variables and shocks for large models.
It runs the smoother only for the state variables (i.e. with the same representation used for
likelihood computations) and computes the remaining variables ex-post.
Static unobserved objects (filtered, smoothed, updated, k-step ahead) are recovered, but there are
exceptions to a full recovery, depending on how static unobserved variables depend on the restricted
state space adopted. For example, lagged shocks which are ONLY used to recover NON-observed static
variables will not be recovered).
For such exceptions, only the following output is provided:
``FilteredVariablesKStepAhead``: will be fully recovered
``SmoothedVariables``, ``FilteredVariables``, ``UpdatedVariables``: recovered for all periods beyond period ``d+1``,
where ``d`` denotes the number of diffuse filtering steps.
``FilteredVariablesKStepAheadVariances``, ``Variance``, and ``State_uncertainty`` cannot be recovered, and ZERO is provided as output.
If you need variances for those variables, either do not set the option, or declare the variable as observed, using NaNs as data points.
.. option:: forecast = INTEGER
Computes the posterior distribution of a forecast on INTEGER
periods after the end of the sample used in estimation. If no
Metropolis-Hastings is computed, the result is stored in
variable ``oo_.forecast`` and corresponds to the forecast at
the posterior mode. If a Metropolis-Hastings is computed, the
distribution of forecasts is stored in variables
``oo_.PointForecast`` and ``oo_.MeanForecast``. See
:ref:`fore`, for a description of these variables.
.. option:: tex
See :opt:`tex`.
.. option:: kalman_algo = INTEGER
``0``
Automatically use the Multivariate Kalman Filter for
stationary models and the Multivariate Diffuse Kalman
Filter for non-stationary models.
``1``
Use the Multivariate Kalman Filter.
``2``
Use the Univariate Kalman Filter.
``3``
Use the Multivariate Diffuse Kalman Filter.
``4``
Use the Univariate Diffuse Kalman Filter.
Default value is ``0``. In case of missing observations of
single or all series, Dynare treats those missing values as
unobserved states and uses the Kalman filter to infer their
value (see e.g. *Durbin and Koopman (2012)*, Ch. 4.10) This
procedure has the advantage of being capable of dealing with
observations where the forecast error variance matrix becomes
singular for some variable(s). If this happens, the respective
observation enters with a weight of zero in the log-likelihood,
i.e. this observation for the respective variable(s) is dropped
from the likelihood computations (for details see *Durbin and
Koopman (2012)*, Ch. 6.4 and 7.2.5 and *Koopman and Durbin
(2000)*). If the use of a multivariate Kalman filter is
specified and a singularity is encountered, Dynare by default
automatically switches to the univariate Kalman filter for this
parameter draw. This behavior can be changed via the
:opt:`use_univariate_filters_if_singularity_is_detected
<use_univariate_filters_if_singularity_is_detected = INTEGER>`
option.
.. option:: fast_kalman_filter
Select the fast Kalman filter using Chandrasekhar recursions as
described by ``Herbst (2015)``. This setting is only used with
``kalman_algo=1`` or ``kalman_algo=3``. In case of using the
diffuse Kalman filter (``kalman_algo=3/lik_init=3``), the
observables must be stationary. This option is not yet
compatible with :opt:`analytic_derivation`.
.. option:: kalman_tol = DOUBLE
Numerical tolerance for determining the singularity of the
covariance matrix of the prediction errors during the Kalman
filter (minimum allowed reciprocal of the matrix condition
number). Default value is ``1e-10``.
.. option:: diffuse_kalman_tol = DOUBLE
Numerical tolerance for determining the singularity of the
covariance matrix of the prediction errors (:math:`F_{\infty}`)
and the rank of the covariance matrix of the non-stationary
state variables (:math:`P_{\infty}`) during the Diffuse Kalman
filter. Default value is ``1e-6``.
.. option:: filter_covariance
Saves the series of one step ahead error of forecast covariance
matrices. With Metropolis, they are saved in
:mvar:`oo_.FilterCovariance`, otherwise in
:mvar:`oo_.Smoother.Variance`. Saves also k-step ahead error of
forecast covariance matrices if ``filter_step_ahead`` is set.
.. option:: filter_step_ahead = [INTEGER1:INTEGER2]
filter_step_ahead = [INTEGER1 INTEGER2 ...]
Triggers the computation k-step ahead filtered values,
i.e. :math:`E_{t}{y_{t+k}}`. Stores results in
``oo_.FilteredVariablesKStepAhead``. Also stores 1-step ahead
values in
``oo_.FilteredVariables``. ``oo_.FilteredVariablesKStepAheadVariances``
is stored if ``filter_covariance``.
.. option:: filter_decomposition
Triggers the computation of the shock decomposition of the
above k-step ahead filtered values. Stores results in
``oo_.FilteredVariablesShockDecomposition``.
.. option:: smoothed_state_uncertainty
Triggers the computation of the variance of smoothed estimates,
i.e. :math:`var_T(y_t)`. Stores results in
``oo_.Smoother.State_uncertainty``.
.. option:: diffuse_filter
Uses the diffuse Kalman filter (as described in *Durbin and
Koopman (2012)* and *Koopman and Durbin (2003)* for the
multivariate and *Koopman and Durbin (2000)* for the univariate
filter) to estimate models with non-stationary observed
variables. This option will also reset the ``qz_criterium`` to
count unit root variables towards the stable variables. Trying to estimate
a model with unit roots will otherwise result in a Blanchard-Kahn error.
When ``diffuse_filter`` is used the ``lik_init`` option of
``estimation`` has no effect.
When there are nonstationary exogenous variables in a model,
there is no unique deterministic steady state. For instance, if
productivity is a pure random walk:
.. math::
a_t = a_{t-1} + e_t
any value of :math:`\bar a` of :math:`a` is a deterministic
steady state for productivity. Consequently, the model admits
an infinity of steady states. In this situation, the user must
help Dynare in selecting one steady state, except if zero is a
trivial model’s steady state, which happens when the ``linear``
option is used in the model declaration. The user can either
provide the steady state to Dynare using a
``steady_state_model`` block (or writing a steady state file)
if a closed form solution is available, see
:bck:`steady_state_model`, or specify some constraints on the
steady state, see
:ref:`equation_tag_for_conditional_steady_state <eq-tag-ss>`,
so that Dynare computes the steady state conditionally on some
predefined levels for the non stationary variables. In both
cases, the idea is to use dummy values for the steady state
level of the exogenous non stationary variables.
Note that the nonstationary variables in the model must be
integrated processes (their first difference or k-difference
must be stationary).
.. option:: heteroskedastic_filter
Runs filter, likelihood, and smoother using heteroskedastic definitions provided in
a ``heteroskedastic_shocks`` block.
.. option:: selected_variables_only
Only run the classical smoother on the variables listed just
after the ``estimation`` command. This option is incompatible
with requesting classical frequentist forecasts and will be
overridden in this case. When using Bayesian estimation, the
smoother is by default only run on the declared endogenous
variables. Default: run the smoother on all the declared
endogenous variables.
.. option:: cova_compute = INTEGER
When ``0``, the covariance matrix of estimated parameters is
not computed after the computation of posterior mode (or
maximum likelihood). This increases speed of computation in
large models during development, when this information is not
always necessary. Of course, it will break all successive
computations that would require this covariance
matrix. Otherwise, if this option is equal to ``1``, the
covariance matrix is computed and stored in variable ``hh`` of
``MODEL_FILENAME_mode.mat``. Default is ``1``.
.. option:: solve_algo = INTEGER
See :ref:`solve_algo <solvalg>`.
.. option:: order = INTEGER
Order of approximation around the deterministic steady
state. When greater than 1, the likelihood is evaluated with a
particle or nonlinear filter (see *Fernández-Villaverde and
Rubio-Ramírez (2005)*). Default is ``1``, i.e. the likelihood
of the linearized model is evaluated using a standard Kalman
filter.
.. option:: irf = INTEGER
See :opt:`irf <irf = INTEGER>`. Only used if
:opt:`bayesian_irf` is passed.
.. option:: irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )
See :opt:`irf_shocks <irf_shocks = ( VARIABLE_NAME [[,]
VARIABLE_NAME ...] )>`. Only used if :opt:`bayesian_irf` is
passed.
.. option:: irf_plot_threshold = DOUBLE
See :opt:`irf_plot_threshold <irf_plot_threshold =
DOUBLE>`. Only used if :opt:`bayesian_irf` is passed.
.. option:: aim_solver
See :opt:`aim_solver`.
.. option:: sylvester = OPTION
See :opt:`sylvester <sylvester = OPTION>`.
.. option:: sylvester_fixed_point_tol = DOUBLE
See :opt:`sylvester_fixed_point_tol <sylvester_fixed_point_tol
= DOUBLE>` .
.. option:: lyapunov = OPTION
Determines the algorithm used to solve the Lyapunov equation to
initialized the variance-covariance matrix of the Kalman filter
using the steady-state value of state variables. Possible
values for OPTION are:
``default``
Uses the default solver for Lyapunov equations based on
Bartels-Stewart algorithm.
``fixed_point``
Uses a fixed point algorithm to solve the Lyapunov
equation. This method is faster than the ``default``
one for large scale models, but it could require a
large amount of iterations.
``doubling``
Uses a doubling algorithm to solve the Lyapunov
equation (``disclyap_fast``). This method is faster
than the two previous one for large scale models.
``square_root_solver``
Uses a square-root solver for Lyapunov equations
(``dlyapchol``). This method is fast for large scale
models (available under MATLAB if the Control System
Toolbox is installed; available under Octave if the
`control <https://octave.sourceforge.io/control/>`__
package from Octave-Forge is installed)
Default value is ``default``.
.. option:: lyapunov_fixed_point_tol = DOUBLE
This is the convergence criterion used in the fixed point
Lyapunov solver. Its default value is ``1e-10``.
.. option:: lyapunov_doubling_tol = DOUBLE
This is the convergence criterion used in the doubling
algorithm to solve the Lyapunov equation. Its default value is
``1e-16``.
.. option:: use_penalized_objective_for_hessian
Use the penalized objective instead of the objective function
to compute numerically the hessian matrix at the mode. The
penalties decrease the value of the posterior density (or
likelihood) when, for some perturbations, Dynare is not able to
solve the model (issues with steady state existence, Blanchard
and Kahn conditions, ...). In pratice, the penalized and
original objectives will only differ if the posterior mode is
found to be near a region where the model is ill-behaved. By
default the original objective function is used.
.. option:: analytic_derivation
Triggers estimation with analytic gradient at ``order=1``.
The final hessian at the mode is also computed analytically.
Only works for stationary models without missing observations,
i.e. for ``kalman_algo<3``. Optimizers that rely on analytic
gradients are ``mode_compute=1,3,4,5,101``.
.. option:: ar = INTEGER
See :opt:`ar <ar = INTEGER>`. Only useful in conjunction with
option ``moments_varendo``.
.. option:: endogenous_prior
Use endogenous priors as in *Christiano, Trabandt and Walentin
(2011)*. The procedure is motivated by sequential Bayesian
learning. Starting from independent initial priors on the
parameters, specified in the ``estimated_params`` block, the
standard deviations observed in a "pre-sample", taken to be the
actual sample, are used to update the initial priors. Thus, the
product of the initial priors and the pre-sample likelihood of
the standard deviations of the observables is used as the new
prior (for more information, see the technical appendix of
*Christiano, Trabandt and Walentin (2011)*). This procedure
helps in cases where the regular posterior estimates, which
minimize in-sample forecast errors, result in a large
overprediction of model variable variances (a statistic that is
not explicitly targeted, but often of particular interest to
researchers).
.. option:: use_univariate_filters_if_singularity_is_detected = INTEGER
Decide whether Dynare should automatically switch to univariate
filter if a singularity is encountered in the likelihood
computation (this is the behaviour if the option is equal to
``1``). Alternatively, if the option is equal to ``0``, Dynare
will not automatically change the filter, but rather use a
penalty value for the likelihood when such a singularity is
encountered. Default: ``1``.
.. option:: keep_kalman_algo_if_singularity_is_detected
With the default
:opt:`use_univariate_filters_if_singularity_is_detected=1
<use_univariate_filters_if_singularity_is_detected = INTEGER>`,
Dynare will switch to the univariate Kalman filter when it
encounters a singular forecast error variance matrix during
Kalman filtering. Upon encountering such a singularity for the
first time, all subsequent parameter draws and computations
will automatically rely on univariate filter, i.e. Dynare will
never try the multivariate filter again. Use the
``keep_kalman_algo_if_singularity_is_detected`` option to have
the ``use_univariate_filters_if_singularity_is_detected`` only
affect the behavior for the current draw/computation.
.. option:: rescale_prediction_error_covariance
Rescales the prediction error covariance in the Kalman filter
to avoid badly scaled matrix and reduce the probability of a
switch to univariate Kalman filters (which are slower). By
default no rescaling is done.
.. option:: qz_zero_threshold = DOUBLE
See :opt:`qz_zero_threshold <qz_zero_threshold = DOUBLE>`.
.. option:: taper_steps = [INTEGER1 INTEGER2 ...]
Percent tapering used for the spectral window in the *Geweke
(1992,1999)* convergence diagnostics (requires
:opt:`mh_nblocks=1 <mh_nblocks = INTEGER>`). The tapering is
used to take the serial correlation of the posterior draws into
account. Default: ``[4 8 15]``.
.. option:: geweke_interval = [DOUBLE DOUBLE]
Percentage of MCMC draws at the beginning and end of the MCMC
chain taken to compute the *Geweke (1992,1999)* convergence
diagnostics (requires :opt:`mh_nblocks=1 <mh_nblocks =
INTEGER>`) after discarding the first :opt:`mh_drop = DOUBLE
<mh_drop>` percent of draws as a burnin. Default: [0.2 0.5].
.. option:: raftery_lewis_diagnostics
Triggers the computation of the *Raftery and Lewis (1992)*
convergence diagnostics. The goal is deliver the number of
draws required to estimate a particular quantile of the CDF
``q`` with precision ``r`` with a probability ``s``. Typically,
one wants to estimate the ``q=0.025`` percentile (corresponding
to a 95 percent HPDI) with a precision of 0.5 percent
(``r=0.005``) with 95 percent certainty (``s=0.95``). The
defaults can be changed via :opt:`raftery_lewis_qrs
<raftery_lewis_qrs = [DOUBLE DOUBLE DOUBLE]>`. Based on the
theory of first order Markov Chains, the diagnostics will
provide a required burn-in (``M``), the number of draws after
the burnin (``N``) as well as a thinning factor that would
deliver a first order chain (``k``). The last line of the table
will also deliver the maximum over all parameters for the
respective values.
.. option:: raftery_lewis_qrs = [DOUBLE DOUBLE DOUBLE]
Sets the quantile of the CDF ``q`` that is estimated with
precision ``r`` with a probability ``s`` in the *Raftery and
Lewis (1992)* convergence diagnostics. Default: ``[0.025 0.005
0.95]``.
.. option:: consider_all_endogenous
Compute the posterior moments, smoothed variables, k-step ahead
filtered variables and forecasts (when requested) on all the
endogenous variables. This is equivalent to manually listing
all the endogenous variables after the ``estimation`` command.
.. option:: consider_all_endogenous_and_auxiliary
Compute the posterior moments, smoothed variables, k-step ahead
filtered variables and forecasts (when requested) on all the
endogenous variables and the auxiliary variables introduced by the
preprocessor. This option is useful when e.g. running ``smoother2histval``
on the results of the Kalman smoother.
.. option:: consider_only_observed
Compute the posterior moments, smoothed variables, k-step ahead
filtered variables and forecasts (when requested) on all the
observed variables. This is equivalent to manually listing all
the observed variables after the ``estimation`` command.
.. option:: number_of_particles = INTEGER
Number of particles used when evaluating the likelihood of a
non linear state space model. Default: ``1000``.
.. option:: resampling = OPTION
Determines if resampling of the particles is done. Possible
values for OPTION are:
``none``
No resampling.
``systematic``
Resampling at each iteration, this is the default value.
``generic``
Resampling if and only if the effective sample size is
below a certain level defined by
:opt:`resampling_threshold <resampling_threshold =
DOUBLE>` * :opt:`number_of_particles
<number_of_particles = INTEGER>`.
.. option:: resampling_threshold = DOUBLE
A real number between zero and one. The resampling step is
triggered as soon as the effective number of particles is less
than this number times the total number of particles (as set by
:opt:`number_of_particles <number_of_particles =
INTEGER>`). This option is effective if and only if option
:opt:`resampling <resampling = OPTION>` has value ``generic``.
.. option:: resampling_method = OPTION
Sets the resampling method. Possible values for OPTION are:
``kitagawa``, ``stratified`` and ``smooth``.
.. option:: filter_algorithm = OPTION
Sets the particle filter algorithm. Possible values for OPTION
are:
``sis``
Sequential importance sampling algorithm, this is the
default value.
``apf``
Auxiliary particle filter.
``gf``
Gaussian filter.
``gmf``
Gaussian mixture filter.
``cpf``
Conditional particle filter.
``nlkf``
Use a standard (linear) Kalman filter algorithm with
the nonlinear measurement and state equations.
.. option:: proposal_approximation = OPTION
Sets the method for approximating the proposal
distribution. Possible values for OPTION are: ``cubature``,
``montecarlo`` and ``unscented``. Default value is
``unscented``.
.. option:: distribution_approximation = OPTION
Sets the method for approximating the particle
distribution. Possible values for OPTION are: ``cubature``,
``montecarlo`` and ``unscented``. Default value is
``unscented``.
.. option:: cpf_weights = OPTION
Controls the method used to update the weights in conditional
particle filter, possible values are ``amisanotristani``
(*Amisano et al. (2010)*) or ``murrayjonesparslow`` (*Murray et
al. (2013)*). Default value is ``amisanotristani``.
.. option:: nonlinear_filter_initialization = INTEGER
Sets the initial condition of the nonlinear filters. By default
the nonlinear filters are initialized with the unconditional
covariance matrix of the state variables, computed with the
reduced form solution of the first order approximation of the
model. If ``nonlinear_filter_initialization=2``, the nonlinear
filter is instead initialized with a covariance matrix
estimated with a stochastic simulation of the reduced form
solution of the second order approximation of the model. Both
these initializations assume that the model is stationary, and
cannot be used if the model has unit roots (which can be seen
with the :comm:`check` command prior to estimation). If the
model has stochastic trends, user must use
``nonlinear_filter_initialization=3``, the filters are then
initialized with an identity matrix for the covariance matrix
of the state variables. Default value is
``nonlinear_filter_initialization=1`` (initialization based on
the first order approximation of the model).
.. option:: particle_filter_options = (NAME, VALUE, ...)
A list of NAME and VALUE pairs. Can be used to set some fine-grained
options for the particle filter routines. The set of available options
depends on the selected filter routine.
More information on particle filter options is available at
`<https://git.dynare.org/Dynare/dynare/-/wikis/Particle-filters>`__.
Available options are:
``'pruning'``
Enable pruning for particle filter-related simulations. Default: ``false``.
``'liu_west_delta'``
Set the value for delta for the Liu/West online filter. Default: ``0.99``.
``'unscented_alpha'``
Set the value for alpha for unscented transforms. Default: ``1``.
``'unscented_beta'``
Set the value for beta for unscented transforms. Default: ``2``.
``'unscented_kappa'``
Set the value for kappa for unscented transforms. Default: ``1``.
``'initial_state_prior_std'``
Value of the diagonal elements for the initial covariance of the state
variables when employing ``nonlinear_filter_initialization=3``. Default: ``1``.
``'mixture_state_variables'``
Number of mixture components in the Gaussian-mixture filter (gmf)
for the state variables. Default: ``5``.
``'mixture_structural_shocks'``
Number of mixture components in the Gaussian-mixture filter (gmf)
for the structural shocks. Default: ``1``.
``'mixture_measurement_shocks'``
Number of mixture components in the Gaussian-mixture filter (gmf)
for the measurement errors. Default: ``1``.
*Note*
If no ``mh_jscale`` parameter is used for a parameter in
``estimated_params``, the procedure uses ``mh_jscale`` for all
parameters. If ``mh_jscale`` option isn’t set, the procedure uses
``0.2`` for all parameters. Note that if ``mode_compute=6`` is
used or the ``posterior_sampler_option`` called ``scale_file`` is
specified, the values set in ``estimated_params`` will be
overwritten.
*“Endogenous” prior restrictions*
It is also possible to impose implicit “endogenous” priors about
IRFs and moments on the model during estimation. For example, one
can specify that all valid parameter draws for the model must
generate fiscal multipliers that are bigger than 1 by specifying
how the IRF to a government spending shock must look like. The
prior restrictions can be imposed via ``irf_calibration`` and
``moment_calibration`` blocks (see :ref:`irf-momcal`). The way it
works internally is that any parameter draw that is inconsistent
with the “calibration” provided in these blocks is discarded,
i.e. assigned a prior density of 0. When specifying these blocks,
it is important to keep in mind that one won’t be able to easily
do ``model_comparison`` in this case, because the prior density
will not integrate to 1.
*Output*
After running estimation, the parameters ``M_.params`` and the
variance matrix ``M_.Sigma_e`` of the shocks are set to the mode
for maximum likelihood estimation or posterior mode computation
without Metropolis iterations. After estimation with Metropolis
iterations (option ``mh_replic > 0`` or option ``load_mh_file``
set) the parameters ``M_.params`` and the variance matrix
``M_.Sigma_e`` of the shocks are set to the posterior mean.
Depending on the options, ``estimation`` stores results in various
fields of the ``oo_`` structure, described below. In the following
variables, we will adopt the following shortcuts for specific
field names:
``MOMENT_NAME``
This field can take the following values:
``HPDinf``
Lower bound of a 90% HPD interval. [#f3]_
``HPDsup``
Upper bound of a 90% HPD interval.
``HPDinf_ME``
Lower bound of a 90% HPD interval [#f4]_ for
observables when taking measurement error into account
(see e.g. *Christoffel et al. (2010*), p.17).
``HPDsup_ME``
Upper bound of a 90% HPD interval for observables when
taking measurement error into account.
``Mean``
Mean of the posterior distribution.
``Median``
Median of the posterior distribution.
``Std``
Standard deviation of the posterior distribution.
``Variance``
Variance of the posterior distribution.
``deciles``
Deciles of the distribution.
.. _dens:
``density``
Non parametric estimate of the posterior density
following the approach outlined in *Skoeld and Roberts
(2003)*. First and second columns are respectively
abscissa and ordinate coordinates.
``ESTIMATED_OBJECT``
This field can take the following values:
``measurement_errors_corr``
Correlation between two measurement errors.
``measurement_errors_std``
Standard deviation of measurement errors.
``parameters``
Parameters.
``shocks_corr``
Correlation between two structural shocks.
``shocks_std``
Standard deviation of structural shocks.
.. matvar:: oo_.MarginalDensity.LaplaceApproximation
Variable set by the ``estimation`` command. Stores the marginal
data density based on the Laplace Approximation.
.. matvar:: oo_.MarginalDensity.ModifiedHarmonicMean
Variable set by the ``estimation command``, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Stores the
marginal data density based on *Geweke (1999)* Modified
Harmonic Mean estimator.
.. matvar:: oo_.posterior.optimization
Variable set by the ``estimation`` command if mode-finding is
used. Stores the results at the mode. Fields are of the form::
oo_.posterior.optimization.OBJECT
where OBJECT is one of the following:
``mode``
Parameter vector at the mode.
``Variance``
Inverse Hessian matrix at the mode or MCMC jumping
covariance matrix when used with the
:opt:`MCMC_jumping_covariance <mcmc_jumping_covariance
= OPTION>` option.
``log_density``
Log likelihood (ML)/log posterior density (Bayesian) at the
mode when used with ``mode_compute>0``.
.. matvar:: oo_.posterior.metropolis
Variable set by the ``estimation`` command if ``mh_replic>0`` is
used. Fields are of the form::
oo_.posterior.metropolis.OBJECT
where OBJECT is one of the following:
``mean``
Mean parameter vector from the MCMC.
``Variance``
Covariance matrix of the parameter draws in the MCMC.
.. matvar:: oo_.FilteredVariables
Variable set by the ``estimation`` command, if it is used with the
``filtered_vars`` option.
After an estimation without Metropolis, fields are of the form::
oo_.FilteredVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form::
oo_.FilteredVariables.MOMENT_NAME.VARIABLE_NAME
.. matvar:: oo_.FilteredVariablesKStepAhead
Variable set by the ``estimation`` command, if it is used with
the ``filter_step_ahead`` option. The k-steps are stored along
the rows while the columns indicate the respective
variables. The third dimension of the array provides the
observation for which the forecast has been made. For example,
if ``filter_step_ahead=[1 2 4]`` and ``nobs=200``, the element
(3,5,204) stores the four period ahead filtered value of
variable 5 computed at time t=200 for time t=204. The periods
at the beginning and end of the sample for which no forecasts
can be made, e.g. entries (1,5,1) and (1,5,204) in the
example, are set to zero. Note that in case of Bayesian
estimation the variables will be ordered in the order of
declaration after the estimation command (or in general
declaration order if no variables are specified here). In case
of running the classical smoother, the variables will always
be ordered in general declaration order. If the
:opt:`selected_variables_only` option is specified with the
classical smoother, non-requested variables will be simply
left out in this order.
.. matvar:: oo_.FilteredVariablesKStepAheadVariances
Variable set by the ``estimation`` command, if it is used with
the ``filter_step_ahead option``. It is a 4 dimensional array
where the k-steps are stored along the first dimension, while
the fourth dimension of the array provides the observation for
which the forecast has been made. The second and third
dimension provide the respective variables. For example, if
``filter_step_ahead=[1 2 4]`` and ``nobs=200``, the element
(3,4,5,204) stores the four period ahead forecast error
covariance between variable 4 and variable 5, computed at time
t=200 for time t=204. Padding with zeros and variable ordering
is analogous to ``oo_.FilteredVariablesKStepAhead``.
.. matvar:: oo_.Filtered_Variables_X_step_ahead
Variable set by the ``estimation`` command, if it is used with the
``filter_step_ahead option`` in the context of Bayesian
estimation. Fields are of the form::
oo_.Filtered_Variables_X_step_ahead.VARIABLE_NAME
The n-th entry stores the k-step ahead filtered variable computed
at time n for time n+k.
.. matvar:: oo_.FilteredVariablesShockDecomposition
Variable set by the ``estimation`` command, if it is used with
the ``filter_step_ahead`` option. The k-steps are stored along
the rows while the columns indicate the respective
variables. The third dimension corresponds to the shocks in
declaration order. The fourth dimension of the array provides
the observation for which the forecast has been made. For
example, if ``filter_step_ahead=[1 2 4]`` and ``nobs=200``,
the element (3,5,2,204) stores the contribution of the second
shock to the four period ahead filtered value of variable 5
(in deviations from the mean) computed at time t=200 for time
t=204. The periods at the beginning and end of the sample for
which no forecasts can be made, e.g. entries (1,5,1) and
(1,5,204) in the example, are set to zero. Padding with zeros
and variable ordering is analogous to
``oo_.FilteredVariablesKStepAhead``.
.. matvar:: oo_.PosteriorIRF.dsge
Variable set by the ``estimation`` command, if it is used with the
``bayesian_irf`` option. Fields are of the form::
oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME
.. matvar:: oo_.SmoothedMeasurementErrors
Variable set by the ``estimation`` command, if it is used with the
``smoother`` option. Fields are of the form::
oo_.SmoothedMeasurementErrors.VARIABLE_NAME
.. matvar:: oo_.SmoothedShocks
Variable set by the ``estimation`` command (if used with the
``smoother`` option), or by the ``calib_smoother`` command.
After an estimation without Metropolis, or if computed by
``calib_smoother``, fields are of the form::
oo_.SmoothedShocks.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form::
oo_.SmoothedShocks.MOMENT_NAME.VARIABLE_NAME
.. matvar:: oo_.SmoothedVariables
Variable set by the ``estimation`` command (if used with the
``smoother`` option), or by the ``calib_smoother`` command.
After an estimation without Metropolis, or if computed by
``calib_smoother``, fields are of the form::
oo_.SmoothedVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form::
oo_.SmoothedVariables.MOMENT_NAME.VARIABLE_NAME
.. matcomm:: get_smooth ('VARIABLE_NAME' [, 'VARIABLE_NAME']...);
Returns the smoothed values of the given endogenous or exogenous
variable(s), as they are stored in the ``oo_.SmoothedVariables``
and ``oo_.SmoothedShocks`` variables.
.. matvar:: oo_.UpdatedVariables
Variable set by the ``estimation`` command (if used with the
``smoother`` option), or by the ``calib_smoother``
command. Contains the estimation of the expected value of
variables given the information available at the current date.
After an estimation without Metropolis, or if computed by
``calib_smoother``, fields are of the form::
oo_.UpdatedVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form::
oo_.UpdatedVariables.MOMENT_NAME.VARIABLE_NAME
.. matcomm:: get_update ('VARIABLE_NAME' [, 'VARIABLE_NAME']...);
Returns the updated values of the given variable(s), as they are stored
in the ``oo_.UpdatedVariables`` variable.
.. matvar:: oo_.FilterCovariance
Three-dimensional array set by the ``estimation`` command if
used with the ``smoother`` and Metropolis, if the
``filter_covariance`` option has been requested. Contains the
series of one-step ahead forecast error covariance matrices
from the Kalman smoother. The ``M_.endo_nbr`` times
``M_.endo_nbr`` times ``T+1`` array contains the variables in
declaration order along the first two dimensions. The third
dimension of the array provides the observation for which the
forecast has been made. Fields are of the form::
oo_.FilterCovariance.MOMENT_NAME
Note that density estimation is not supported.
.. matvar:: oo_.Smoother.Variance
Three-dimensional array set by the ``estimation`` command (if
used with the ``smoother``) without Metropolis, or by the
``calib_smoother`` command, if the ``filter_covariance``
option has been requested. Contains the series of one-step
ahead forecast error covariance matrices from the Kalman
smoother. The ``M_.endo_nbr`` times ``M_.endo_nbr`` times
``T+1`` array contains the variables in declaration order
along the first two dimensions. The third dimension of the
array provides the observation for which the forecast has been
made.
.. matvar:: oo_.Smoother.State_uncertainty
Three-dimensional array set by the ``estimation`` command (if
used with the ``smoother`` option) without Metropolis, or by
the ``calib_smoother`` command, if the
``smoothed_state_uncertainty`` option has been
requested. Contains the series of covariance matrices for the
state estimate given the full data from the Kalman
smoother. The ``M_.endo_nbr`` times ``M_.endo_nbr`` times
``T`` array contains the variables in declaration order along
the first two dimensions. The third dimension of the array
provides the observation for which the smoothed estimate has
been made.
.. matvar:: oo_.Smoother.SteadyState
Variable set by the ``estimation`` command (if used with the
``smoother``) without Metropolis, or by the
``calib_smoother`` command. Contains the steady state
component of the endogenous variables used in the smoother in
order of variable declaration.
.. matvar:: oo_.Smoother.TrendCoeffs
Variable set by the ``estimation`` command (if used with the
``smoother``) without Metropolis, or by the ``calib_smoother``
command. Contains the trend coefficients of the observed
variables used in the smoother in order of declaration of the
observed variables.
.. matvar:: oo_.Smoother.Trend
Variable set by the ``estimation command`` (if used with the
``smoother`` option), or by the ``calib_smoother``
command. Contains the trend component of the variables used in
the smoother.
Fields are of the form::
oo_.Smoother.Trend.VARIABLE_NAME
.. matvar:: oo_.Smoother.Constant
Variable set by the ``estimation`` command (if used with the
``smoother`` option), or by the ``calib_smoother``
command. Contains the constant part of the endogenous
variables used in the smoother, accounting e.g. for the data
mean when using the prefilter option.
Fields are of the form::
oo_.Smoother.Constant.VARIABLE_NAME
.. matvar:: oo_.Smoother.loglinear
Indicator keeping track of whether the smoother was run with
the :ref:`loglinear <logl>` option and thus whether stored
smoothed objects are in logs.
.. matvar:: oo_.PosteriorTheoreticalMoments
Variable set by the ``estimation`` command, if it is used with the
``moments_varendo`` option. Fields are of the form::
oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME
where *THEORETICAL_MOMENT* is one of the following:
``covariance``
Variance-covariance of endogenous variables.
``contemporaneous_correlation``
Contemporaneous correlation of endogenous variables when the
:opt:`contemporaneous_correlation` option is specified.
``correlation``
Auto- and cross-correlation of endogenous variables. Fields
are vectors with correlations from 1 up to order
``options_.ar``.
.. _VarianceDecomposition:
``VarianceDecomposition``
Decomposition of variance (unconditional variance, i.e. at
horizon infinity). [#f5]_
``VarianceDecompositionME``
Same as `VarianceDecomposition`_, but contains
the decomposition of the measured as opposed to the
actual variable. The joint contribution of the
measurement error will be saved in a field named
``ME``.
.. _ConditionalVarianceDecomposition:
``ConditionalVarianceDecomposition``
Only if the ``conditional_variance_decomposition``
option has been specified. In the presence of
measurement error, the field will contain the variance
contribution after measurement error has been taken
out, i.e. the decomposition will be conducted of the
actual as opposed to the measured variables.
``ConditionalVarianceDecompositionME``
Only if the ``conditional_variance_decomposition``
option has been specified. Same as
`ConditionalVarianceDecomposition`_, but contains the
decomposition of the measured as opposed to the actual
variable. The joint contribution of the measurement
error will be saved in a field names ``ME``.
.. matvar:: oo_.posterior_density
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of
the form::
oo_.posterior_density.PARAMETER_NAME
.. matvar:: oo_.posterior_hpdinf
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of
the form::
oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_hpdsup
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of the
form::
oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_mean
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of the
form::
oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_mode
Variable set by the ``estimation`` command during
mode-finding. Fields are of the form::
oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_std_at_mode
Variable set by the ``estimation`` command during mode-finding. It
is based on the inverse Hessian at ``oo_.posterior_mode``. Fields
are of the form::
oo_.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_std
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of the
form::
oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_var
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of the
form::
oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME
.. matvar:: oo_.posterior_median
Variable set by the ``estimation`` command, if it is used with
``mh_replic > 0`` or ``load_mh_file`` option. Fields are of the
form::
oo_.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME
*Example*
Here are some examples of generated variables::
oo_.posterior_mode.parameters.alp
oo_.posterior_mean.shocks_std.ex
oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso
.. matvar:: oo_.dsge_var.posterior_mode
Structure set by the ``dsge_var`` option of the ``estimation``
command after mode_compute.
The following fields are saved:
``PHI_tilde``
Stacked posterior DSGE-BVAR autoregressive matrices at the
mode (equation (28) of *Del Negro and Schorfheide (2004)*).
``SIGMA_u_tilde``
Posterior covariance matrix of the DSGE-BVAR at the mode
(equation (29) of *Del Negro and Schorfheide (2004)*).
``iXX``
Posterior population moments in the DSGE-BVAR at the mode (
:math:`inv(\lambda T \Gamma_{XX}^*+ X'X)`).
``prior``
Structure storing the DSGE-BVAR prior.
``PHI_star``
Stacked prior DSGE-BVAR autoregressive matrices at the
mode (equation (22) of *Del Negro and Schorfheide
(2004)*).
``SIGMA_star``
Prior covariance matrix of the DSGE-BVAR at the mode
(equation (23) of *Del Negro and Schorfheide (2004)*).
``ArtificialSampleSize``
Size of the artifical prior sample ( :math:`inv(\lambda T)`).
``DF``
Prior degrees of freedom ( :math:`inv(\lambda T-k-n)`).
``iGXX_star``
Inverse of the theoretical prior “covariance” between
X and X (:math:`\Gamma_{xx}^*` in *Del Negro and
Schorfheide (2004)*).
.. matvar:: oo_.RecursiveForecast
Variable set by the ``forecast`` option of the ``estimation``
command when used with the nobs = [INTEGER1:INTEGER2] option (see
:opt:`nobs <nobs = [INTEGER1:INTEGER2]>`).
Fields are of the form::
oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME
where ``FORECAST_OBJECT`` is one of the following [#f6]_ :
``Mean``
Mean of the posterior forecast distribution.
``HPDinf/HPDsup``
Upper/lower bound of the 90% HPD interval taking into account
only parameter uncertainty (corresponding to
:mvar:`oo_.MeanForecast`).
``HPDTotalinf/HPDTotalsup``.
Upper/lower bound of the 90% HPD interval taking into account
both parameter and future shock uncertainty (corresponding to
:mvar:`oo_.PointForecast`)
``VARIABLE_NAME`` contains a matrix of the following size:
number of time periods for which forecasts are requested using
the ``nobs = [INTEGER1:INTEGER2]`` option times the number of
forecast horizons requested by the forecast option. i.e., the
row indicates the period at which the forecast is performed
and the column the respective k-step ahead forecast. The
starting periods are sorted in ascending order, not in
declaration order.
.. matvar:: oo_.convergence.geweke
Variable set by the convergence diagnostics of the ``estimation``
command when used with ``mh_nblocks=1`` option (see
:opt:`mh_nblocks <mh_nblocks = INTEGER>`).
Fields are of the form::
oo_.convergence.geweke.VARIABLE_NAME.DIAGNOSTIC_OBJECT
where *DIAGNOSTIC_OBJECT* is one of the following:
``posteriormean``
Mean of the posterior parameter distribution.
``posteriorstd``
Standard deviation of the posterior parameter distribution.
``nse_iid``
Numerical standard error (NSE) under the assumption of iid draws.
``rne_iid``
Relative numerical efficiency (RNE) under the assumption
of iid draws.
``nse_x``
Numerical standard error (NSE) when using an x% taper.
``rne_x``
Relative numerical efficiency (RNE) when using an x% taper.
``pooled_mean``
Mean of the parameter when pooling the beginning and end parts
of the chain specified in :opt:`geweke_interval
<geweke_interval = [DOUBLE DOUBLE]>` and weighting them with
their relative precision. It is a vector containing the
results under the iid assumption followed by the ones using
the ``taper_steps`` option (see :opt:`taper_steps <taper_steps
= [INTEGER1 INTEGER2 ...]>`).
``pooled_nse``
NSE of the parameter when pooling the beginning and end parts
of the chain and weighting them with their relative
precision. See ``pooled_mean``.
``prob_chi2_test``
p-value of a chi-squared test for equality of means in the
beginning and the end of the MCMC chain. See
``pooled_mean``. A value above 0.05 indicates that the null
hypothesis of equal means and thus convergence cannot be
rejected at the 5 percent level. Differing values along the
``taper_steps`` signal the presence of significant
autocorrelation in draws. In this case, the estimates using a
higher tapering are usually more reliable.
.. command:: unit_root_vars VARIABLE_NAME...;
|br| This command is deprecated. Use ``estimation`` option
``diffuse_filter`` instead for estimating a model with
non-stationary observed variables or ``steady`` option ``nocheck``
to prevent ``steady`` to check the steady state returned by your
steady state file.
Dynare also has the ability to estimate Bayesian VARs:
.. command:: bvar_density ;
|br| Computes the marginal density of an estimated BVAR model, using
Minnesota priors.
See ``bvar-a-la-sims.pdf``, which comes with Dynare distribution,
for more information on this command.
Estimation based on moments
===========================
Provided that you have observations on some endogenous variables, it
is possible to use Dynare to estimate some or all parameters using a
method of moments approach. Both the Simulated Method of Moments (SMM)
and the Generalized Method of Moments (GMM) are available. The general
idea is to minimize the distance between unconditional model moments
and corresponding data moments (so called orthogonality or moment
conditions). For SMM, Dynare computes model moments via stochastic
simulations based on the perturbation approximation up to any order,
whereas for GMM model moments are computed in closed-form based on the
pruned state-space representation of the perturbation solution up to third
order. The implementation of SMM is inspired by *Born and Pfeifer (2014)*
and *Ruge-Murcia (2012)*, whereas the one for GMM is adapted from
*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* and *Mutschler
(2018)*. Successful estimation heavily relies on the accuracy and efficiency of
the perturbation approximation, so it is advised to tune this as much as
possible (see :ref:`stoch-sol-simul`). The method of moments estimator is consistent
and asymptotically normally distributed given certain regularity conditions
(see *Duffie and Singleton (1993)* for SMM and *Hansen (1982)* for GMM).
For instance, it is required to have at least as many moment conditions as
estimated parameters (over-identified or just identified). Moreover, the
Jacobian of the moments with respect to the estimated parameters needs to
have full rank. :ref:`identification-analysis` helps to check this regularity condition.
In the over-identified case of declaring more moment conditions than estimated parameters, the
choice of :opt:`weighting_matrix <weighting_matrix = ['WM1','WM2',...,'WMn']>`
matters for the efficiency of the estimation, because the estimated
orthogonality conditions are random variables with unequal variances and
usually non-zero cross-moment covariances. A weighting matrix allows to
re-weight moments to put more emphasis on moment conditions that are
more informative or better measured (in the sense of having a smaller
variance). To achieve asymptotic efficiency, the weighting matrix needs to
be chosen such that, after appropriate scaling, it has a probability limit
proportional to the inverse of the covariance matrix of the limiting
distribution of the vector of orthogonality conditions. Dynare uses a
Newey-West-type estimator with a Bartlett kernel to compute an estimate of this
so-called optimal weighting matrix. Note that in this over-identified case,
it is advised to perform the estimation in at least two stages by setting
e.g. :opt:`weighting_matrix=['DIAGONAL','DIAGONAL'] <weighting_matrix = ['WM1','WM2',...,'WMn']>`
so that the computation of the optimal weighting matrix benefits from the
consistent estimation of the previous stages. The optimal weighting matrix
is used to compute standard errors and the J-test of overidentifying
restrictions, which tests whether the model and selection of moment
conditions fits the data sufficiently well. If the null hypothesis of a
"valid" model is rejected, then something is (most likely) wrong with either your model
or selection of orthogonality conditions.
In case the (presumed) global minimum of the moment distance function is
located in a region of the parameter space that
is typically considered unlikely (`dilemma of absurd parameters`), you may
opt to choose the :opt:`penalized_estimator <penalized_estimator>` option.
Similar to adding priors to the likelihood, this option incorporates prior
knowledge (i.e. the prior mean) as additional moment restrictions and
weights them by their prior precision to guide the minimization algorithm
to more plausible regions of the parameter space. Ideally, these regions are
characterized by only slightly worse values of the objective function. Note that
adding prior information comes at the cost of a loss in efficiency of the estimator.
.. command:: varobs VARIABLE_NAME...;
|br| Required. All variables used in the :bck:`matched_moments` block
need to be observable. See :ref:`varobs <varobs>` for more details.
.. block:: matched_moments ;
|br| This block specifies the product moments which are used in estimation.
Currently, only linear product moments (e.g.
:math:`E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]`)
are supported. For other functions like :math:`E[\log(y_t)e^{x_t}]` you
need to declare auxiliary endogenous variables.
Each line inside of the block should be of the form::
VARIABLE_NAME(LEAD/LAG)^POWER*VARIABLE_NAME(LEAD/LAG)^POWER*...*VARIABLE_NAME(LEAD/LAG)^POWER;
where `VARIABLE_NAME` is the name of a declared observable variable,
`LEAD/LAG` is either a negative integer for lags or a positive one
for leads, and `POWER` is a positive integer indicating the exponent on
the variable. You can omit `LEAD/LAG` equal to `0` or `POWER` equal to `1`.
*Example*
For :math:`E[c_t], E[y_t], E[c_t^2], E[c_t y_t], E[y_t^2], E[c_t c_{t+3}], E[y_{t+1}^2 c^3_{t-4}], E[c^3_{t-5} y_{t}^2]`
use the following block:
::
matched_moments;
c;
y;
c*c;
c*y;
y^2;
c*c(3);
y(1)^2*c(-4)^3;
c(-5)^3*y(0)^2;
end;
*Limitations*
1. For GMM, Dynare can only compute the theoretical mean, covariance, and
autocovariances (i.e. first and second moments). Higher-order moments are only supported for SMM.
2. By default, the product moments are not demeaned, unless the
:opt:`prefilter <prefilter = INTEGER>` option is set to 1. That is, by default,
`c*c` corresponds to :math:`E[c_t^2]` and not to :math:`Var[c_t]=E[c_t^2]-E[c_t]^2`.
*Output*
Dynare translates the :bck:`matched_moments` block into a cell array
``M_.matched_moments`` where:
* the first column contains a vector of indices for the chosen variables in declaration order
* the second column contains the corresponding vector of leads and lags
* the third column contains the corresponding vector of powers
During the estimation phase, Dynare will eliminate all redundant or duplicate
orthogonality conditions in ``M_.matched_moments`` and display which
conditions were removed. In the example above, this would be the case for the
last row, which is the same as the second-to-last one. The original block is
saved in ``M_.matched_moments_orig``.
.. block:: estimated_params ;
|br| Required. See :bck:`estimated_params` for the meaning and syntax.
.. block:: estimated_params_init ;
|br| See :bck:`estimated_params_init` for the meaning and syntax.
.. block:: estimated_params_bounds ;
|br| See :bck:`estimated_params_bounds` for the meaning and syntax.
.. command:: method_of_moments (OPTIONS...);
|br| This command runs the method of moments estimation. The following
information will be displayed in the command window:
* Overview of options chosen by the user
* Estimation results for each stage and iteration
* Value of minimized moment distance objective function
* Result of the J-test
* Table of data moments and estimated model moments
*Necessary options*
.. option:: mom_method = SMM|GMM
"Simulated Method of Moments" is triggered by `SMM` and
"Generalized Method of Moments" by `GMM`.
.. option:: datafile = FILENAME
The name of the file containing the data. See
:opt:`datafile <datafile = FILENAME>` for the meaning and syntax.
*Options common for SMM and GMM*
.. option:: order = INTEGER
Order of perturbation approximation. For GMM only orders 1|2|3 are
supported. For SMM, you can choose an arbitrary order. Note that the
order set in other functions will not overwrite the default.
Default: ``1``.
.. option:: pruning
Discard higher order terms when iteratively computing simulations
of the solution. See :opt:`pruning <pruning>` for more details.
Default: not set for SMM, always set for GMM.
.. option:: penalized_estimator
This option includes deviations of the estimated parameters from the
prior mean as additional moment restrictions and weights them by
their prior precision.
Default: not set.
.. option:: weighting_matrix = ['WM1','WM2',...,'WMn']
Determines the weighting matrix used at each estimation stage. The number of elements
will define the number of stages, i.e. ``weighting_matrix = ['DIAGONAL','DIAGONAL','OPTIMAL']``
performs a three-stage estimation. Possible values for ``WM`` are:
``IDENTITY_MATRIX``
Sets the weighting matrix equal to the identity matrix.
``OPTIMAL``
Uses the optimal weighting matrix computed by a Newey-West-type
estimate with a Bartlett kernel. At the first
stage, the data-moments are used as initial estimate of the
model moments, whereas at subsequent stages the previous estimate
of model moments will be used when computing
the optimal weighting matrix.
``DIAGONAL``
Uses the diagonal of the ``OPTIMAL`` weighting matrix. This choice
puts weights on the specified moments instead of on their linear combinations.
``FILENAME``
The name of the MAT-file (extension ``.mat``) containing a
user-specified weighting matrix. The file must include a positive definite
square matrix called ``weighting_matrix`` with both dimensions
equal to the number of orthogonality conditions.
Default value is ``['DIAGONAL','OPTIMAL']``.
.. option:: weighting_matrix_scaling_factor = DOUBLE
Scaling of weighting matrix in objective function. This value should be chosen to
obtain values of the objective function in a reasonable numerical range to prevent
over- and underflows.
Default: ``1``.
.. option:: bartlett_kernel_lag = INTEGER
Bandwidth of kernel for computing the optimal weighting matrix.
Default: ``20``.
.. option:: se_tolx = DOUBLE
Step size for numerical differentiation when computing standard
errors with a two-sided finite difference method.
Default: ``1e-5``.
.. option:: verbose
Display and store intermediate estimation results in ``oo_.mom``.
Default: not set.
*SMM-specific options*
.. option:: burnin = INTEGER
Number of periods dropped at the beginning of simulation.
Default: ``500``.
.. option:: bounded_shock_support
Trim shocks in simulations to :math:`\pm 2` standard deviations.
Default: not set.
.. option:: seed = INTEGER
Common seed used in simulations.
Default: ``24051986``.
.. option:: simulation_multiple = INTEGER
Multiple of data length used for simulation.
Default: ``7``.
*GMM-specific options*
.. option:: analytic_standard_errors
Compute standard errors using analytical derivatives of moments
with respect to estimated parameters.
Default: not set, i.e. standard errors are computed using a two-sided
finite difference method, see :opt:`se_tolx <se_tolx = DOUBLE>`.
*General options*
.. option:: dirname = FILENAME
Directory in which to store ``estimation`` output.
See :opt:`dirname <dirname = FILENAME>` for more details.
Default: ``<mod_file>``.
.. option:: graph_format = FORMAT
Specify the file format(s) for graphs saved to disk.
See :opt:`graph_format <graph_format = FORMAT>` for more details.
Default: ``eps``.
.. option:: nodisplay
See :opt:`nodisplay`. Default: not set.
.. option:: nograph
See :opt:`nograph`. Default: not set.
.. option:: noprint
See :opt:`noprint`. Default: not set.
.. option:: plot_priors = INTEGER
Control the plotting of priors.
See :opt:`plot_priors <plot_priors = INTEGER>` for more details.
Default: ``1``, i.e. plot priors.
.. option:: prior_trunc = DOUBLE
See :opt:`prior_trunc <prior_trunc = DOUBLE>` for more details.
Default: ``1e-10``.
.. option:: tex
See :opt:`tex`. Default: not set.
*Data options*
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
Default: ``1``.
.. option:: nobs = INTEGER
See :opt:`nobs <nobs = INTEGER>`.
Default: all observations are considered.
.. option:: prefilter = INTEGER
A value of 1 means that the estimation procedure will demean each data
series by its empirical mean and each model moment by its theoretical
mean. See :opt:`prefilter <prefilter = INTEGER>` for more details.
Default: ``0``, i.e. no prefiltering.
.. option:: logdata
See :opt:`logdata <logdata>`. Default: not set.
.. option:: xls_sheet = QUOTED_STRING
See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
.. option:: xls_range = RANGE
See :opt:`xls_range <xls_range = RANGE>`.
*Optimization options*
.. option:: huge_number = DOUBLE
See :opt:`huge_number <huge_number = DOUBLE>`.
Default: ``1e7``.
.. option:: mode_compute = INTEGER | FUNCTION_NAME
See :opt:`mode_compute <mode_compute = INTEGER | FUNCTION_NAME>`.
Default: ``13``, i.e. ``lsqnonlin`` if the MATLAB Optimization Toolbox or
the Octave optim-package are present, ``4``, i.e. ``csminwel`` otherwise.
.. option:: additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]
Vector of additional minimization algorithms run after
``mode_compute``. If :opt:`verbose` option is set, then the additional estimation
results are saved into the ``oo_.mom`` structure prefixed with `verbose_`.
Default: no additional optimization iterations.
.. option:: optim = (NAME, VALUE, ...)
See :opt:`optim <optim = (NAME, VALUE, ...)>`.
.. option:: silent_optimizer
See :opt:`silent_optimizer`.
Default: not set.
*Numerical algorithms options*
.. option:: aim_solver
See :opt:`aim_solver <aim_solver>`. Default: not set.
.. option:: k_order_solver
See :opt:`k_order_solver <k_order_solver>`.
Default: disabled for order 1 and 2, enabled for order 3 and above.
.. option:: dr = OPTION
See :opt:`dr <dr = OPTION>`. Default: ``default``, i.e. generalized
Schur decomposition.
.. option:: dr_cycle_reduction_tol = DOUBLE
See :opt:`dr_cycle_reduction_tol <dr_cycle_reduction_tol = DOUBLE>`.
Default: ``1e-7``.
.. option:: dr_logarithmic_reduction_tol = DOUBLE
See :opt:`dr_logarithmic_reduction_tol <dr_logarithmic_reduction_tol = DOUBLE>`.
Default: ``1e-12``.
.. option:: dr_logarithmic_reduction_maxiter = INTEGER
See :opt:`dr_logarithmic_reduction_maxiter <dr_logarithmic_reduction_maxiter = INTEGER>`.
Default: ``100``.
.. option:: lyapunov = OPTION
See :opt:`lyapunov <lyapunov = OPTION>`. Default: ``default``, i.e.
based on Bartlets-Stewart algorithm.
.. option:: lyapunov_complex_threshold = DOUBLE
See :opt:`lyapunov_complex_threshold <lyapunov_complex_threshold = DOUBLE>`.
Default: ``1e-15``.
.. option:: lyapunov_fixed_point_tol = DOUBLE
See :opt:`lyapunov_fixed_point_tol <lyapunov_fixed_point_tol = DOUBLE>`.
Default: ``1e-10``.
.. option:: lyapunov_doubling_tol = DOUBLE
See :opt:`lyapunov_doubling_tol <lyapunov_doubling_tol = DOUBLE>`.
Default: ``1e-16``.
.. option:: sylvester = OPTION
See :opt:`sylvester <sylvester = OPTION>`.
Default: ``default``, i.e. uses ``gensylv``.
.. option:: sylvester_fixed_point_tol = DOUBLE
See :opt:`sylvester_fixed_point_tol <sylvester_fixed_point_tol = DOUBLE>`.
Default: ``1e-12``.
.. option:: qz_criterium = DOUBLE
See :opt:`qz_criterium <qz_criterium = DOUBLE>`.
Default: ``0.999999`` as it is assumed that the observables are weakly
stationary.
.. option:: qz_zero_threshold = DOUBLE
See :opt:`qz_zero_threshold <qz_zero_threshold = DOUBLE>`.
Default: ``1e-6``.
.. option:: schur_vec_tol = DOUBLE
Tolerance level used to find nonstationary variables in Schur decomposition
of the transition matrix. Default: ``1e-11``.
.. option:: mode_check
Plots univariate slices through the moments distance objective function around the
computed minimum for each estimated parameter. This is
helpful to diagnose problems with the optimizer.
Default: not set.
.. option:: mode_check_neighbourhood_size = DOUBLE
See :opt:`mode_check_neighbourhood_size <mode_check_neighbourhood_size = DOUBLE>`.
Default: ``0.5``.
.. option:: mode_check_symmetric_plots = INTEGER
See :opt:`mode_check_symmetric_plots <mode_check_symmetric_plots = INTEGER>`.
Default: ``1``.
.. option:: mode_check_number_of_points = INTEGER
See :opt:`mode_check_number_of_points <mode_check_number_of_points = INTEGER>`.
Default: ``20``.
*Output*
``method_of_moments`` stores user options in a structure called
`options_mom_` in the global workspace. After running the estimation,
the parameters ``M_.params`` and the covariance matrices of the shocks
``M_.Sigma_e`` and of the measurement errors ``M_.H`` are set to the
parameters that minimize the quadratic moments distance objective
function. The estimation results are stored in the ``oo_.mom`` structure
with the following fields:
.. matvar:: oo_.mom.data_moments
Variable set by the ``method_of_moments`` command. Stores the mean
of the selected empirical moments of data. NaN values due to leads/lags
or missing data are omitted when computing the mean. Vector of dimension
equal to the number of orthogonality conditions.
.. matvar:: oo_.mom.m_data
Variable set by the ``method_of_moments`` command. Stores the selected
empirical moments at each point in time. NaN values due to leads/lags
or missing data are replaced by the corresponding mean of the moment.
Matrix of dimension time periods times number of orthogonality conditions.
.. matvar:: oo_.mom.Sw
Variable set by the ``method_of_moments`` command. Stores the
Cholesky decomposition of the currently used weighting matrix.
Square matrix of dimensions equal to the number of orthogonality
conditions.
.. matvar:: oo_.mom.model_moments
Variable set by the ``method_of_moments`` command. Stores the implied
selected model moments given the current parameter guess. Model moments
are computed in closed-form from the pruned state-space system for GMM,
whereas for SMM these are based on averages of simulated data. Vector of dimension equal
to the number of orthogonality conditions.
.. matvar:: oo_.mom.Q
Variable set by the ``method_of_moments`` command. Stores the scalar
value of the quadratic moment's distance objective function.
.. matvar:: oo_.mom.model_moments_params_derivs
Variable set by the ``method_of_moments`` command. Stores the analytically
computed Jacobian matrix of the derivatives of the model moments with
respect to the estimated parameters. Only for GMM with :opt:`analytic_standard_errors`.
Matrix with dimension equal to the number of orthogonality conditions
times number of estimated parameters.
.. matvar:: oo_.mom.gmm_stage_*_mode
.. matvar:: oo_.mom.smm_stage_*_mode
.. matvar:: oo_.mom.verbose_gmm_stage_*_mode
.. matvar:: oo_.mom.verbose_smm_stage_*_mode
Variables set by the ``method_of_moments`` command when estimating
with GMM or SMM. Stores the estimated values at stages 1, 2,....
The structures contain the following fields:
- ``measurement_errors_corr``: estimated correlation between two measurement errors
- ``measurement_errors_std``: estimated standard deviation of measurement errors
- ``parameters``: estimated model parameters
- ``shocks_corr``: estimated correlation between two structural shocks.
- ``shocks_std``: estimated standard deviation of structural shocks.
If the :opt:`verbose` option is set, additional fields prefixed with
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.
.. matvar:: oo_.mom.gmm_stage_*_std_at_mode
.. matvar:: oo_.mom.smm_stage_*_std_at_mode
.. matvar:: oo_.mom.verbose_gmm_stage_*_std_at_mode
.. matvar:: oo_.mom.verbose_smm_stage_*_std_at_mode
Variables set by the ``method_of_moments`` command when estimating
with GMM or SMM. Stores the estimated standard errors at stages 1, 2,....
The structures contain the following fields:
- ``measurement_errors_corr``: standard error of estimated correlation between two measurement errors
- ``measurement_errors_std``: standard error of estimated standard deviation of measurement errors
- ``parameters``: standard error of estimated model parameters
- ``shocks_corr``: standard error of estimated correlation between two structural shocks.
- ``shocks_std``: standard error of estimated standard deviation of structural shocks.
If the :opt:`verbose` option is set, additional fields prefixed with
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.
.. matvar:: oo_.mom.J_test
Variable set by the ``method_of_moments`` command. Structure where the
value of the test statistic is saved into a field called ``j_stat``, the
degress of freedom into a field called ``degrees_freedom`` and the p-value
of the test statistic into a field called ``p_val``.
Model Comparison
================
.. command:: model_comparison FILENAME[(DOUBLE)]...;
model_comparison (marginal_density = ESTIMATOR) FILENAME[(DOUBLE)]...;
|br| This command computes odds ratios and estimate a posterior density
over a collection of models (see e.g. *Koop (2003)*, Ch. 1). The
priors over models can be specified as the *DOUBLE* values,
otherwise a uniform prior over all models is assumed. In contrast
to frequentist econometrics, the models to be compared do not need
to be nested. However, as the computation of posterior odds ratios
is a Bayesian technique, the comparison of models estimated with
maximum likelihood is not supported.
It is important to keep in mind that model comparison of this type
is only valid with proper priors. If the prior does not integrate
to one for all compared models, the comparison is not valid. This
may be the case if part of the prior mass is implicitly truncated
because Blanchard and Kahn conditions (instability or
indeterminacy of the model) are not fulfilled, or because for some
regions of the parameters space the deterministic steady state is
undefined (or Dynare is unable to find it). The compared marginal
densities should be renormalized by the effective prior mass, but
this not done by Dynare: it is the user’s responsibility to make
sure that model comparison is based on proper priors. Note that,
for obvious reasons, this is not an issue if the compared marginal
densities are based on Laplace approximations.
*Options*
.. option:: marginal_density = ESTIMATOR
Specifies the estimator for computing the marginal data
density. *ESTIMATOR* can take one of the following two values:
``laplace`` for the Laplace estimator or
``modifiedharmonicmean`` for the *Geweke (1999)* Modified
Harmonic Mean estimator. Default value: ``laplace``
*Output*
The results are stored in ``oo_.Model_Comparison``, which is
described below.
*Example*
::
model_comparison my_model(0.7) alt_model(0.3);
This example attributes a 70% prior over ``my_model`` and 30%
prior over ``alt_model``.
.. matvar:: oo_.Model_Comparison
Variable set by the ``model_comparison`` command. Fields are of
the form::
oo_.Model_Comparison.FILENAME.VARIABLE_NAME
where FILENAME is the file name of the model and VARIABLE_NAME is
one of the following:
``Prior``
(Normalized) prior density over the model.
``Log_Marginal_Density``
Logarithm of the marginal data density.
``Bayes_Ratio``
Ratio of the marginal data density of the model relative
to the one of the first declared model
``Posterior_Model_Probability``
Posterior probability of the respective model.
Shock Decomposition
===================
.. command:: shock_decomposition [VARIABLE_NAME]...;
shock_decomposition (OPTIONS...) [VARIABLE_NAME]...;
|br| This command computes the historical shock decomposition for a
given sample based on the Kalman smoother, i.e. it decomposes the
historical deviations of the endogenous variables from their
respective steady state values into the contribution coming from
the various shocks. The ``variable_names`` provided govern for
which variables the decomposition is plotted.
Note that this command must come after either ``estimation`` (in
case of an estimated model) or ``stoch_simul`` (in case of a
calibrated model).
*Options*
.. option:: parameter_set = OPTION
Specify the parameter set to use for running the
smoother. Possible values for OPTION are:
* ``calibration``
* ``prior_mode``
* ``prior_mean``
* ``posterior_mode``
* ``posterior_mean``
* ``posterior_median``
* ``mle_mode``
Note that the parameter set used in subsequent commands like
``stoch_simul`` will be set to the specified
``parameter_set``. Default value: ``posterior_mean`` if
Metropolis has been run, ``mle_mode`` if MLE has been run.
.. option:: datafile = FILENAME
See :ref:`datafile <dataf>`. Useful when computing the shock
decomposition on a calibrated model.
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
.. option:: nobs = INTEGER
See :opt:`nobs <nobs = INTEGER>`.
.. option:: prefilter = INTEGER
See :opt:`prefilter <prefilter = INTEGER>`.
.. option:: loglinear
See :opt:`loglinear <loglinear>`.
.. option:: diffuse_kalman_tol = DOUBLE
See :opt:`diffuse_kalman_tol <diffuse_kalman_tol = DOUBLE>`.
.. option:: diffuse_filter
See :opt:`diffuse_filter <diffuse_filter>`.
.. option:: xls_sheet = QUOTED_STRING
See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
.. option:: xls_range = RANGE
See :opt:`xls_range <xls_range = RANGE>`.
.. option:: use_shock_groups [= NAME]
Uses shock grouping defined by the string instead of
individual shocks in the decomposition. The groups of shocks
are defined in the :bck:`shock_groups` block. If no group name is
given, ``default`` is assumed.
.. option:: colormap = VARIABLE_NAME
Controls the ``colormap`` used for the shocks decomposition graphs.
VARIABLE_NAME must be the name of a MATLAB/Octave variable that has
been declared beforehand and whose value will be passed to the
MATLAB/Octave ``colormap`` function (see the MATLAB/Octave manual for
the list of acceptable values).
.. option:: nograph
See :opt:`nograph`. Suppresses the display and creation only
within the ``shock_decomposition`` command, but does not
affect other commands. See :comm:`plot_shock_decomposition`
for plotting graphs.
.. option:: init_state = BOOLEAN
If equal to 0, the shock decomposition is computed conditional
on the smoothed state variables in period ``0``, i.e. the
smoothed shocks starting in period 1 are used. If equal to
``1``, the shock decomposition is computed conditional on the
smoothed state variables in period 1. Default: ``0``.
.. option:: with_epilogue
If set, then also compute the decomposition for variables declared in
the ``epilogue`` block (see :ref:`epilogue`).
*Output*
.. matvar:: oo_.shock_decomposition
The results are stored in the field
``oo_.shock_decomposition``, which is a three dimensional
array. The first dimension contains the ``M_.endo_nbr``
endogenous variables. The second dimension stores in the first
``M_.exo_nbr`` columns the contribution of the respective
shocks. Column ``M_.exo_nbr+1`` stores the contribution of the
initial conditions, while column ``M_.exo_nbr+2`` stores the
smoothed value of the respective endogenous variable in
deviations from their steady state, i.e. the mean and trends
are subtracted. The third dimension stores the time
periods. Both the variables and shocks are stored in the order
of declaration, i.e. ``M_.endo_names`` and ``M_.exo_names``,
respectively.
.. block:: shock_groups ;
shock_groups(OPTIONS...);
|br| Shocks can be regrouped for the purpose of shock
decomposition. The composition of the shock groups is written in a
block delimited by ``shock_groups`` and ``end``.
Each line defines a group of shocks as a list of exogenous variables::
SHOCK_GROUP_NAME = VARIABLE_1 [[,] VARIABLE_2 [,]...];
'SHOCK GROUP NAME' = VARIABLE_1 [[,] VARIABLE_2 [,]...];
*Options*
.. option:: name = NAME
Specifies a name for the following definition of shock
groups. It is possible to use several ``shock_groups`` blocks
in a model file, each grouping being identified by a different
name. This name must in turn be used in the
``shock_decomposition`` command. If no name is given, ``default`` is
used.
*Example*
::
varexo e_a, e_b, e_c, e_d;
...
shock_groups(name=group1);
supply = e_a, e_b;
'aggregate demand' = e_c, e_d;
end;
shock_decomposition(use_shock_groups=group1);
This example defines a shock grouping with the name
``group1``, containing a set of supply and demand shocks and
conducts the shock decomposition for these two groups.
.. command:: realtime_shock_decomposition [VARIABLE_NAME]...;
realtime_shock_decomposition (OPTIONS...) [VARIABLE_NAME]...;
|br| This command computes the realtime historical shock
decomposition for a given sample based on the Kalman smoother. For
each period :math:`T=[\texttt{presample},\ldots,\texttt{nobs}]`,
it recursively computes three objects:
* Real-time historical shock decomposition :math:`Y(t\vert T)`
for :math:`t=[1,\ldots,T]`, i.e. without observing data in
:math:`[T+1,\ldots,\texttt{nobs}]`. This results in a
standard shock decomposition being computed for each
additional datapoint becoming available after ``presample``.
* Forecast shock decomposition :math:`Y(T+k\vert T)` for
:math:`k=[1,\ldots,forecast]`, i.e. the :math:`k`-step ahead
forecast made for every :math:`T` is decomposed in its shock
contributions.
* Real-time conditional shock decomposition of the difference
between the real-time historical shock decomposition and the
forecast shock decomposition. If :opt:`vintage <vintage =
INTEGER>` is equal to ``0``, it computes the effect of
shocks realizing in period :math:`T`, i.e. decomposes
:math:`Y(T\vert T)-Y(T\vert T-1)`. Put differently, it
conducts a :math:`1`-period ahead shock decomposition from
:math:`T-1` to :math:`T`, by decomposing the update step of
the Kalman filter. If ``vintage>0`` and smaller than
``nobs``, the decomposition is conducted of the forecast
revision :math:`Y(T+k\vert T+k)-Y(T+k\vert T)`.
Like :comm:`shock_decomposition` it decomposes the historical
deviations of the endogenous variables from their respective
steady state values into the contribution coming from the various
shocks. The ``variable_names`` provided govern for which variables
the decomposition is plotted.
Note that this command must come after either ``estimation`` (in
case of an estimated model) or ``stoch_simul`` (in case of a
calibrated model).
*Options*
.. option:: parameter_set = OPTION
See :opt:`parameter_set <parameter_set = OPTION>` for
possible values.
.. option:: datafile = FILENAME
See :ref:`datafile <dataf>`.
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
.. option:: nobs = INTEGER
See :opt:`nobs <nobs = INTEGER>`.
.. option:: use_shock_groups [= NAME]
See :opt:`use_shock_groups <use_shock_groups [= NAME]>`.
.. option:: colormap = VARIABLE_NAME
See :opt:`colormap <colormap = VARIABLE_NAME>`.
.. option:: nograph
See :opt:`nograph`. Only shock decompositions are computed and
stored in ``oo_.realtime_shock_decomposition``,
``oo_.conditional_shock_decomposition`` and
``oo_.realtime_forecast_shock_decomposition`` but no plot is
made (See :comm:`plot_shock_decomposition`).
.. option:: presample = INTEGER
Data point above which recursive realtime shock
decompositions are computed, *i.e.* for
:math:`T=[\texttt{presample+1} \ldots \texttt{nobs}]`.
.. option:: forecast = INTEGER
Compute shock decompositions up to :math:`T+k` periods,
i.e. get shock contributions to k-step ahead forecasts.
.. option:: save_realtime = INTEGER_VECTOR
Choose for which vintages to save the full realtime shock
decomposition. Default: ``0``.
.. option:: fast_realtime = INTEGER
fast_realtime = [INTEGER1:INTEGER2]
fast_realtime = [INTEGER1 INTEGER2 ...]
Runs the smoother only for the data vintages provided
by the specified integer (vector).
.. option:: with_epilogue
See :opt:`with_epilogue`.
*Output*
.. matvar:: oo_.realtime_shock_decomposition
Structure storing the results of realtime historical
decompositions. Fields are three-dimensional arrays with the
first two dimension equal to the ones of
:mvar:`oo_.shock_decomposition`. The third dimension stores
the time periods and is therefore of size
``T+forecast``. Fields are of the form::
oo_.realtime_shock_decomposition.OBJECT
where OBJECT is one of the following:
``pool``
Stores the pooled decomposition, i.e. for every
real-time shock decomposition terminal period
:math:`T=[\texttt{presample},\ldots,\texttt{nobs}]` it
collects the last period’s decomposition
:math:`Y(T\vert T)` (see also
:comm:`plot_shock_decomposition`). The third dimension
of the array will have size ``nobs+forecast``.
``time_*``
Stores the vintages of realtime historical shock
decompositions if ``save_realtime`` is used. For
example, if ``save_realtime=[5]`` and ``forecast=8``,
the third dimension will be of size ``13``.
.. matvar:: oo_.realtime_conditional_shock_decomposition
Structure storing the results of real-time conditional
decompositions. Fields are of the form::
oo_.realtime_conditional_shock_decomposition.OBJECT
where OBJECT is one of the following:
``pool``
Stores the pooled real-time conditional shock
decomposition, i.e. collects the decompositions of
:math:`Y(T\vert T)-Y(T\vert T-1)` for the terminal
periods
:math:`T=[\texttt{presample},\ldots,\texttt{nobs}]`. The
third dimension is of size ``nobs``.
``time_*``
Store the vintages of :math:`k`-step conditional
forecast shock decompositions :math:`Y(t\vert T+k)`,
for :math:`t=[T \ldots T+k]`. See :opt:`vintage
<vintage = INTEGER>`. The third dimension is of size
``1+forecast``.
.. matvar:: oo_.realtime_forecast_shock_decomposition
Structure storing the results of realtime forecast
decompositions. Fields are of the form::
oo_.realtime_forecast_shock_decomposition.OBJECT
where ``OBJECT`` is one of the following:
``pool``
Stores the pooled real-time forecast decomposition of
the :math:`1`-step ahead effect of shocks on the
:math:`1`-step ahead prediction, i.e. :math:`Y(T\vert
T-1)`.
``time_*``
Stores the vintages of :math:`k`-step out-of-sample
forecast shock decompositions, i.e. :math:`Y(t\vert
T)`, for :math:`t=[T \ldots T+k]`. See :opt:`vintage
<vintage = INTEGER>`.
.. command:: plot_shock_decomposition [VARIABLE_NAME]...;
plot_shock_decomposition (OPTIONS...) [VARIABLE_NAME]...;
|br| This command plots the historical shock decomposition already
computed by ``shock_decomposition`` or
``realtime_shock_decomposition``. For that reason, it must come
after one of these commands. The ``variable_names`` provided
govern which variables the decomposition is plotted for.
Further note that, unlike the majority of Dynare commands, the
options specified below are overwritten with their defaults before
every call to ``plot_shock_decomposition``. Hence, if you want to
reuse an option in a subsequent call to
``plot_shock_decomposition``, you must pass it to the command
again.
*Options*
.. option:: use_shock_groups [= NAME]
See :opt:`use_shock_groups <use_shock_groups [= NAME]>`.
.. option:: colormap = VARIABLE_NAME
See :opt:`colormap <colormap = VARIABLE_NAME>`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: nograph
See :opt:`nograph`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format <graph_format = FORMAT>`.
.. option:: detail_plot
Plots shock contributions using subplots, one per shock (or
group of shocks). Default: not activated
.. option:: interactive
Under MATLAB, add uimenus for detailed group plots. Default:
not activated
.. option:: screen_shocks
For large models (i.e. for models with more than 16 shocks),
plots only the shocks that have the largest historical
contribution for chosen selected
``variable_names``. Historical contribution is ranked by the
mean absolute value of all historical contributions.
.. option:: steadystate
If passed, the the :math:`y`-axis value of the zero line in
the shock decomposition plot is translated to the steady state
level. Default: not activated
.. option:: type = qoq | yoy | aoa
For quarterly data, valid arguments are: ``qoq`` for
quarter-on-quarter plots, ``yoy`` for year-on-year plots of
growth rates, ``aoa`` for annualized variables, i.e. the value
in the last quarter for each year is plotted. Default value:
empty, i.e. standard period-on-period plots (``qoq`` for
quarterly data).
.. option:: fig_name = STRING
Specifies a user-defined keyword to be appended to the default
figure name set by ``plot_shock_decomposition``. This can
avoid to overwrite plots in case of sequential calls to
``plot_shock_decomposition``.
.. option:: write_xls
Saves shock decompositions to Excel file in the main
directory, named
``FILENAME_shock_decomposition_TYPE_FIG_NAME.xls``. This
option requires your system to be configured to be able to
write Excel files. [#f7]_
.. option:: realtime = INTEGER
Which kind of shock decomposition to plot. INTEGER can take
the following values:
* ``0``: standard historical shock decomposition. See
:comm:`shock_decomposition`.
* ``1``: realtime historical shock decomposition. See
:comm:`realtime_shock_decomposition`.
* ``2``: conditional realtime shock decomposition. See
:comm:`realtime_shock_decomposition`.
* ``3``: realtime forecast shock decomposition. See
:comm:`realtime_shock_decomposition`.
If no vintage is requested, i.e. ``vintage=0`` then the pooled
objects from :comm:`realtime_shock_decomposition` will be
plotted and the respective vintage otherwise. Default: ``0``.
.. option:: vintage = INTEGER
Selects a particular data vintage in
:math:`[presample,\ldots,nobs]` for which to plot the results
from :comm:`realtime_shock_decomposition` selected via the
:opt:`realtime <realtime = INTEGER>` option. If the standard
historical shock decomposition is selected (``realtime=0``),
``vintage`` will have no effect. If ``vintage=0`` the pooled
objects from :comm:`realtime_shock_decomposition` will be
plotted. If ``vintage>0``, it plots the shock decompositions
for vintage :math:`T=\texttt{vintage}` under the following
scenarios:
* ``realtime=1``: the full vintage shock decomposition
:math:`Y(t\vert T)` for :math:`t=[1,\ldots,T]`
* ``realtime=2``: the conditional forecast shock
decomposition from :math:`T`, i.e. plots
:math:`Y(T+j\vert T+j)` and the shock contributions
needed to get to the data :math:`Y(T+j)` conditional on
:math:`T=` vintage, with
:math:`j=[0,\ldots,\texttt{forecast}]`.
* ``realtime=3``: plots unconditional forecast shock
decomposition from :math:`T`, i.e. :math:`Y(T+j\vert
T)`, where :math:`T=\texttt{vintage}` and
:math:`j=[0,\ldots,\texttt{forecast}]`.
Default: ``0``.
.. option:: plot_init_date = DATE
If passed, plots decomposition using ``plot_init_date`` as initial period.
Default: first observation in estimation
.. option:: plot_end_date = DATE
If passed, plots decomposition using ``plot_end_date`` as last period.
Default: last observation in estimation
.. option:: diff
If passed, plot the decomposition of the first difference of the list of variables.
If used in combination with :opt:`flip`, the ``diff`` operator is first applied.
Default: not activated
.. option:: flip
If passed, plot the decomposition of the opposite of the list of variables.
If used in combination with :opt:`diff`, the ``diff`` operator is first applied.
Default: not activated
.. option:: max_nrows
Maximum number of rows in the subplot layout of detailed shock
decomposition graphs. Note that columns are always 3. Default: 6
.. option:: with_epilogue
See :opt:`with_epilogue`.
.. option:: init2shocks
init2shocks = NAME
Use the information contained in an :bck:`init2shocks` block, in order
to attribute initial conditions to shocks. The name of the block can be
explicitly given, otherwise it defaults to the ``default`` block.
.. block:: init2shocks ;
init2shocks (OPTIONS...);
|br| This blocks gives the possibility of attributing the initial condition
of endogenous variables to the contribution of exogenous variables in the
shock decomposition.
For example, in an AR(1) process, the contribution of the initial condition
on the process variable can naturally be assigned to the innovation of the
process.
Each line of the block should have the syntax::
VARIABLE_1 [,] VARIABLE_2;
Where VARIABLE_1 is an endogenous variable whose initial condition
will be attributed to the exogenous VARIABLE_2.
The information contained in this block is used by the
:comm:`plot_shock_decomposition` command when given the ``init2shocks``
option.
*Options*
.. option:: name = NAME
Specifies a name for the block, that can be referenced from
``plot_shock_decomposition``, so that several such blocks can coexist
in a single model file. If the name is unspecified, it defaults to
``default``.
*Example*
::
var y y_s R pie dq pie_s de A y_obs pie_obs R_obs;
varexo e_R e_q e_ys e_pies e_A;
...
model;
dq = rho_q*dq(-1)+e_q;
A = rho_A*A(-1)+e_A;
...
end;
...
init2shocks;
dq e_q;
A e_A;
end;
shock_decomposition(nograph);
plot_shock_decomposition(init2shocks) y_obs R_obs pie_obs dq de;
In this example, the initial conditions of ``dq`` and ``A`` will
be respectively attributed to ``e_q`` and ``e_A``.
.. command:: initial_condition_decomposition [VARIABLE_NAME]...;
initial_condition_decomposition (OPTIONS...) [VARIABLE_NAME]...;
|br| This command computes and plots the decomposition of the effect of
smoothed initial conditions of state variables. The ``variable_names`` provided
govern which variables the decomposition is plotted for.
Further note that, unlike the majority of Dynare commands, the
options specified below are overwritten with their defaults before
every call to ``initial_condition_decomposition``. Hence, if you want to
reuse an option in a subsequent call to
``initial_condition_decomposition``, you must pass it to the command
again.
*Options*
.. option:: colormap = VARIABLE_NAME
See :opt:`colormap <colormap = VARIABLE_NAME>`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format <graph_format = FORMAT>`.
.. option:: detail_plot
Plots shock contributions using subplots, one per shock (or
group of shocks). Default: not activated
.. option:: steadystate
If passed, the the :math:`y`-axis value of the zero line in
the shock decomposition plot is translated to the steady state
level. Default: not activated
.. option:: type = qoq | yoy | aoa
For quarterly data, valid arguments are: ``qoq`` for
quarter-on-quarter plots, ``yoy`` for year-on-year plots of
growth rates, ``aoa`` for annualized variables, i.e. the value
in the last quarter for each year is plotted. Default value:
empty, i.e. standard period-on-period plots (``qoq`` for
quarterly data).
.. option:: fig_name = STRING
Specifies a user-defined keyword to be appended to the default
figure name set by ``plot_shock_decomposition``. This can
avoid to overwrite plots in case of sequential calls to
``plot_shock_decomposition``.
.. option:: write_xls
Saves shock decompositions to Excel file in the main
directory, named
``FILENAME_shock_decomposition_TYPE_FIG_NAME_initval.xls``. This
option requires your system to be configured to be able to
write Excel files. [#f7]_
.. option:: plot_init_date = DATE
If passed, plots decomposition using ``plot_init_date`` as initial period.
Default: first observation in estimation
.. option:: plot_end_date = DATE
If passed, plots decomposition using ``plot_end_date`` as last period.
Default: last observation in estimation
.. option:: diff
If passed, plot the decomposition of the first difference of the list of variables.
If used in combination with :opt:`flip`, the ``diff`` operator is first applied.
Default: not activated
.. option:: flip
If passed, plot the decomposition of the opposite of the list of variables.
If used in combination with :opt:`diff`, the ``diff`` operator is first applied.
Default: not activated
.. command:: squeeze_shock_decomposition [VARIABLE_NAME]...;
|br| For large models, the size of the information stored by shock
decompositions (especially various settings of realtime decompositions) may
become huge. This command allows to squeeze this information in two
possible ways:
* Automatic (default): only the variables for which plotting has been
explicitly required with ``plot_shock_decomposition`` will have their
decomposition left in ``oo_`` after this command is run;
* If a list of variables is passed to the command, then only those
variables will have their decomposition left in ``oo_`` after this
command is run.
Calibrated Smoother
===================
Dynare can also run the smoother on a calibrated model:
.. command:: calib_smoother [VARIABLE_NAME]...;
calib_smoother (OPTIONS...) [VARIABLE_NAME]...;
|br| This command computes the smoothed variables (and possible
the filtered variables) on a calibrated model.
A datafile must be provided, and the observable variables declared
with ``varobs``. The smoother is based on a first-order
approximation of the model.
By default, the command computes the smoothed variables and shocks
and stores the results in ``oo_.SmoothedVariables`` and
``oo_.SmoothedShocks``. It also fills ``oo_.UpdatedVariables``.
*Options*
.. option:: datafile = FILENAME
See :ref:`datafile <dataf>`.
.. option:: filtered_vars
Triggers the computation of filtered variables. See
:opt:`filtered_vars`, for more details.
.. option:: filter_step_ahead = [INTEGER1:INTEGER2]
See :opt:`filter_step_ahead <filter_step_ahead = [INTEGER1:INTEGER2]>`.
.. option:: prefilter = INTEGER
See :opt:`prefilter <prefilter = INTEGER>`.
.. option:: parameter_set = OPTION
See :opt:`parameter_set <parameter_set = OPTION>` for
possible values. Default: ``calibration``.
.. option:: loglinear
See :ref:`loglinear <logl>`.
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
.. option:: filter_decomposition
See :opt:`filter_decomposition`.
.. option:: filter_covariance
See :opt:`filter_covariance`.
.. option:: smoother_redux
See :opt:`smoother_redux`.
.. option:: kalman_algo = INTEGER
See :opt:`kalman_algo <kalman_algo = INTEGER>`.
.. option:: diffuse_filter = INTEGER
See :opt:`diffuse_filter`.
.. option:: diffuse_kalman_tol = DOUBLE
See :opt:`diffuse_kalman_tol <diffuse_kalman_tol = DOUBLE>`.
.. option:: xls_sheet = QUOTED_STRING
See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
.. option:: xls_range = RANGE
See :opt:`xls_range <xls_range = RANGE>`.
.. option:: heteroskedastic_filter
See :opt:`heteroskedastic_filter`.
.. option:: nobs = INTEGER
nobs = [INTEGER1:INTEGER2]
See :opt:`nobs <nobs = INTEGER>`.
.. _fore:
Forecasting
===========
On a calibrated model, forecasting is done using the ``forecast``
command. On an estimated model, use the ``forecast`` option of
``estimation`` command.
It is also possible to compute forecasts on a calibrated or estimated
model for a given constrained path of the future endogenous
variables. This is done, from the reduced form representation of the
DSGE model, by finding the structural shocks that are needed to match
the restricted paths. Use :comm:`conditional_forecast`,
:bck:`conditional_forecast_paths` and :comm:`plot_conditional_forecast` for
that purpose.
Finally, it is possible to do forecasting with a Bayesian VAR using
the :comm:`bvar_forecast` command.
.. command:: forecast [VARIABLE_NAME...];
forecast (OPTIONS...) [VARIABLE_NAME...];
|br| This command computes a simulation of a stochastic model from
an arbitrary initial point.
When the model also contains deterministic exogenous shocks, the
simulation is computed conditionally to the agents knowing the
future values of the deterministic exogenous variables.
``forecast`` must be called after ``stoch_simul``.
``forecast`` plots the trajectory of endogenous variables. When a
list of variable names follows the command, only those variables
are plotted. A 90% confidence interval is plotted around the mean
trajectory. Use option ``conf_sig`` to change the level of the
confidence interval.
*Options*
.. option:: periods = INTEGER
Number of periods of the forecast. Default: ``5``.
.. _confsig:
.. option:: conf_sig = DOUBLE
Level of significance for confidence interval. Default: ``0.90``.
.. option:: nograph
See :opt:`nograph`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format = FORMAT`.
*Initial Values*
``forecast`` computes the forecast taking as initial values the
values specified in ``histval`` (see :bck:`histval`). When no
``histval`` block is present, the initial values are the one
stated in ``initval``. When ``initval`` is followed by command
``steady``, the initial values are the steady state (see
:comm:`steady`).
*Output*
The results are stored in ``oo_.forecast``, which is described below.
*Example*
::
varexo_det tau;
varexo e;
...
shocks;
var e; stderr 0.01;
var tau;
periods 1:9;
values -0.15;
end;
stoch_simul(irf=0);
forecast;
.. matvar:: oo_.forecast
Variable set by the ``forecast`` command, or by the
``estimation`` command if used with the ``forecast`` option
and ML or if no Metropolis-Hastings has been computed (in that case,
the forecast is computed for the posterior mode). Fields are
of the form::
oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME
where ``FORECAST_MOMENT`` is one of the following:
``HPDinf``
Lower bound of a 90% HPD interval [#f8]_ of forecast
due to parameter uncertainty, but ignoring the effect
of measurement error on observed variables. In case of ML,
it stores the lower bound of the confidence interval.
``HPDsup``
Upper bound of a 90% HPD forecast interval due to
parameter uncertainty, but ignoring the effect of
measurement error on observed variables. In case of ML,
it stores the upper bound of the confidence interval.
``HPDinf_ME``
Lower bound of a 90% HPD interval [#f9]_ of forecast
for observed variables due to parameter uncertainty
and measurement error. In case of ML,
it stores the lower bound of the confidence interval.
``HPDsup_ME``
Upper bound of a 90% HPD interval of forecast for
observed variables due to parameter uncertainty and
measurement error. In case of ML,
it stores the upper bound of the confidence interval.
``Mean``
Mean of the posterior distribution of forecasts.
.. matvar:: oo_.PointForecast
Set by the ``estimation`` command, if it is used with the
``forecast`` option and if either ``mh_replic > 0`` or the
``load_mh_file`` option are used.
Contains the distribution of forecasts taking into account the
uncertainty about both parameters and shocks.
Fields are of the form::
oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME
.. matvar:: oo_.MeanForecast
Set by the ``estimation`` command, if it is used with the
``forecast`` option and if either ``mh_replic > 0`` or
``load_mh_file`` option are used.
Contains the distribution of forecasts where the uncertainty
about shocks is averaged out. The distribution of forecasts
therefore only represents the uncertainty about parameters.
Fields are of the form::
oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME
.. command:: conditional_forecast (OPTIONS...);
|br| This command computes forecasts on an estimated or calibrated
model for a given constrained path of some future endogenous
variables. This is done using the reduced form first order
state-space representation of the DSGE model by finding the
structural shocks that are needed to match the restricted
paths. Consider the augmented state space representation that
stacks both predetermined and non-predetermined variables into a
vector :math:`y_{t}`:
.. math::
y_t=Ty_{t-1}+R\varepsilon_t
Both :math:`y_t` and :math:`\varepsilon_t` are split up into controlled and
uncontrolled ones, and we assume without loss of generality that the
constrained endogenous variables and the controlled shocks come first :
.. math::
\begin{pmatrix}
y_{c,t}\\
y_{u,t}
\end{pmatrix}
=
\begin{pmatrix}
T_{c,c} & T_{c,u}\\
T_{u,c} & T_{u,u}
\end{pmatrix}
\begin{pmatrix}
y_{c,t-1}\\
y_{u,t-1}
\end{pmatrix}
+
\begin{pmatrix}
R_{c,c} & R_{c,u}\\
R_{u,c} & R_{u,u}
\end{pmatrix}
\begin{pmatrix}
\varepsilon_{c,t}\\
\varepsilon_{u,t}
\end{pmatrix}
where matrices :math:`T` and :math:`R` are partitioned consistently with the
vectors of endogenous variables and innovations. Provided that matrix
:math:`R_{c,c}` is square and full rank (a necessary condition is that the
number of free endogenous variables matches the number of free innovations),
given :math:`y_{c,t}`, :math:`\varepsilon_{u,t}` and :math:`y_{t-1}` the
first block of equations can be solved for :math:`\varepsilon_{c,t}`:
.. math::
\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t} - T_{c,u}y_{u,t} - R_{c,u}\varepsilon_{u,t}\bigr)
and :math:`y_{u,t}` can be updated by evaluating the second block of equations:
.. math::
y_{u,t} = T_{u,c}y_{c,t-1} + T_{u,u}y_{u,t-1} + R_{u,c}\varepsilon_{c,t} + R_{u,u}\varepsilon_{u,t}
By iterating over these two blocks of equations, we can build a forecast for
all the endogenous variables in the system conditional on paths for a subset of the
endogenous variables. If the distribution of the free innovations
:math:`\varepsilon_{u,t}` is provided (*i.e.* some of them have positive
variances) this exercise is replicated (the number of replication is
controlled by the option :opt:`replic` described below) by drawing different
sequences of free innovations. The result is a predictive distribution for
the uncontrolled endogenous variables, :math:`y_{u,t}`, that Dynare will use to report
confidence bands around the point conditional forecast.
A few things need to be noted. First, the controlled
exogenous variables are set to zero for the uncontrolled periods. This implies
that there is no forecast uncertainty arising from these exogenous variables
in uncontrolled periods. Second, by making use of the first order state
space solution, even if a higher-order approximation was performed, the
conditional forecasts will be based on a first order approximation. Since
the controlled exogenous variables are identified on the basis of the
reduced form model (*i.e.* after solving for the expectations), they are
unforeseen shocks from the perspective of the agents in the model. That is,
agents expect the endogenous variables to return to their respective steady
state levels but are surprised in each period by the realisation of shocks
keeping the endogenous variables along a predefined (unexpected) path.
Fourth, if the structural innovations are correlated, because the calibrated
or estimated covariance matrix has non zero off diagonal elements, the
results of the conditional forecasts will depend on the ordering of the
innovations (as declared after ``varexo``). As in VAR models, a Cholesky
decomposition is used to factorise the covariance matrix and identify
orthogonal impulses. It is preferable to declare the correlations in the
model block (explicitly imposing the identification restrictions), unless
you are satisfied with the implicit identification restrictions implied by
the Cholesky decomposition.
This command has to be called after ``estimation`` or ``stoch_simul``.
Use :bck:`conditional_forecast_paths` block to give the list of
constrained endogenous, and their constrained future path. Option
``controlled_varexo`` is used to specify the structural shocks
which will be matched to generate the constrained path.
Use :comm:`plot_conditional_forecast` to graph the results.
*Options*
.. option:: parameter_set = OPTION
See :opt:`parameter_set <parameter_set = OPTION>` for
possible values. No default value, mandatory option.
.. option:: controlled_varexo = (VARIABLE_NAME...)
Specify the exogenous variables to use as control
variables. No default value, mandatory option.
.. option:: periods = INTEGER
Number of periods of the forecast. Default:
``40``. ``periods`` cannot be smaller than the number of
constrained periods.
.. option:: replic = INTEGER
Number of simulations used to compute the conditional forecast uncertainty. Default: ``5000``.
.. option:: conf_sig = DOUBLE
Level of significance for confidence interval. Default: ``0.80``.
*Output*
The results are stored in ``oo_.conditional_forecast``, which is described below.
*Example*
::
var y a;
varexo e u;
...
estimation(...);
conditional_forecast_paths;
var y;
periods 1:3, 4:5;
values 2, 5;
var a;
periods 1:5;
values 3;
end;
conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = 3000);
plot_conditional_forecast(periods = 10) a y;
.. matvar:: oo_.conditional_forecast.cond
Variable set by the ``conditional_forecast`` command. It
stores the conditional forecasts. Fields are ``periods+1`` by
``1`` vectors storing the steady state (time 0) and the
subsequent ``periods`` forecasts periods. Fields are of the
form::
oo_.conditional_forecast.cond.FORECAST_MOMENT.VARIABLE_NAME
where FORECAST_MOMENT is one of the following:
``Mean``
Mean of the conditional forecast distribution.
``ci``
Confidence interval of the conditional forecast
distribution. The size corresponds to ``conf_sig``.
.. matvar:: oo_.conditional_forecast.uncond
Variable set by the ``conditional_forecast`` command. It stores
the unconditional forecasts. Fields are of the form::
oo_.conditional_forecast.uncond.FORECAST_MOMENT.VARIABLE_NAME
.. matvar:: forecasts.instruments
Variable set by the ``conditional_forecast command``. Stores
the names of the exogenous instruments.
.. matvar:: oo_.conditional_forecast.controlled_variables
Variable set by the ``conditional_forecast`` command. Stores
the position of the constrained endogenous variables in
declaration order.
.. matvar:: oo_.conditional_forecast.controlled_exo_variables
Variable set by the ``conditional_forecast`` command. Stores
the values of the controlled exogenous variables underlying
the conditional forecasts to achieve the constrained
endogenous variables. Fields are ``[number of constrained
periods]`` by ``1`` vectors and are of the form::
oo_.conditional_forecast.controlled_exo_variables.FORECAST_MOMENT.SHOCK_NAME
.. matvar:: oo_.conditional_forecast.graphs
Variable set by the ``conditional_forecast`` command. Stores
the information for generating the conditional forecast plots.
.. block:: conditional_forecast_paths ;
|br| Describes the path of constrained endogenous, before calling
``conditional_forecast``. The syntax is similar to deterministic
shocks in ``shocks``, see ``conditional_forecast`` for an example.
The syntax of the block is the same as for the deterministic
shocks in the ``shocks`` blocks (see :ref:`shocks-exo`). Note that
you need to specify the full path for all constrained endogenous
variables between the first and last specified period. If an
intermediate period is not specified, a value of 0 is
assumed. That is, if you specify only values for periods 1 and 3,
the values for period 2 will be 0. Currently, it is not possible
to have uncontrolled intermediate periods.
It is however possible to have different number of controlled
periods for different variables. In that case, the order of
declaration of endogenous controlled variables and of
``controlled_varexo`` matters: if the second endogenous variable is
controlled for less periods than the first one, the second
``controlled_varexo`` isn't set for the last periods.
In case of the presence of ``observation_trends``, the specified
controlled path for these variables needs to include the trend
component. When using the :ref:`loglinear <logl>` option, it is
necessary to specify the logarithm of the controlled variables.
.. block:: filter_initial_state ;
|br| This block specifies the initial values of the endogenous states at the beginning
of the Kalman filter recursions. That is, if the Kalman filter recursion starts with
time t=1 being the first observation, this block provides the state estimate at time 0
given information at time 0, :math:`E_0(x_0)`. If nothing is specified, the initial condition is assumed to be at
the steady state (which is the unconditional mean for a stationary model).
This block is terminated by ``end;``.
Each line inside of the block should be of the form::
VARIABLE_NAME(INTEGER)=EXPRESSION;
``EXPRESSION`` is any valid expression returning a numerical value and can contain parameter values. This
allows specifying relationships that will be honored during estimation. ``INTEGER`` refers to the lag
with which a variable appears. By convention in Dynare, period 1 is the first period. Going backwards in time,
the first period before the start of the simulation is period 0, then period -1, and so on.
Note that the ``filter_initial_state`` block does not take non-state variables.
*Example*
::
filter_initial_state;
k(0)= ((1/bet-(1-del))/alp)^(1/(alp-1))*l_ss;
P(0)=2.5258;
m(0)= mst;
end;
.. command:: plot_conditional_forecast [VARIABLE_NAME...];
plot_conditional_forecast (periods = INTEGER) [VARIABLE_NAME...];
|br| Plots the conditional (plain lines) and unconditional (dashed
lines) forecasts.
To be used after ``conditional_forecast``.
*Options*
.. option:: periods = INTEGER
Number of periods to be plotted. Default: equal to periods in
``conditional_forecast``. The number of periods declared in
``plot_conditional_forecast`` cannot be greater than the one
declared in ``conditional_forecast``.
.. command:: bvar_forecast ;
|br| This command computes (out-of-sample) forecasts for an
estimated BVAR model, using Minnesota priors.
See ``bvar-a-la-sims.pdf``, which comes with Dynare distribution,
for more information on this command.
If the model contains strong non-linearities or if some perfectly
expected shocks are considered, the forecasts and the conditional
forecasts can be computed using an extended path method. The forecast
scenario describing the shocks and/or the constrained paths on some
endogenous variables should be build. The first step is the forecast
scenario initialization using the function ``init_plan``:
.. matcomm:: HANDLE = init_plan (DATES);
Creates a new forecast scenario for a forecast period (indicated
as a dates class, see :ref:`dates class members
<dates-members>`). This function return a handle on the new
forecast scenario.
The forecast scenario can contain some simple shocks on the exogenous
variables. This shocks are described using the function
``basic_plan``:
.. matcomm:: HANDLE = basic_plan (HANDLE, `VAR_NAME', `SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE | [DOUBLE | EXPR [DOUBLE | EXPR] ] );
Adds to the forecast scenario a shock on the exogenous variable
indicated between quotes in the second argument. The shock type
has to be specified in the third argument between quotes:
’surprise’ in case of an unexpected shock or ’perfect_foresight’
for a perfectly anticipated shock. The fourth argument indicates
the period of the shock using a dates class (see :ref:`dates class
members <dates-members>`). The last argument is the shock path
indicated as a MATLAB vector of double. This function return the
handle of the updated forecast scenario.
The forecast scenario can also contain a constrained path on an
endogenous variable. The values of the related exogenous variable
compatible with the constrained path are in this case computed. In
other words, a conditional forecast is performed. This kind of shock
is described with the function ``flip_plan``:
.. matcomm:: HANDLE = flip_plan (HANDLE, `VAR_NAME', `VAR_NAME', `SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE | [DOUBLE | EXPR [DOUBLE | EXPR] ] );
Adds to the forecast scenario a constrained path on the endogenous
variable specified between quotes in the second argument. The
associated exogenous variable provided in the third argument
between quotes, is considered as an endogenous variable and its
values compatible with the constrained path on the endogenous
variable will be computed. The nature of the expectation on the
constrained path has to be specified in the fourth argument
between quotes: ’surprise’ in case of an unexpected path or
’perfect_foresight’ for a perfectly anticipated path. The fifth
argument indicates the period where the path of the endogenous
variable is constrained using a dates class (see :ref:`dates class
members <dates-members>`). The last argument contains the
constrained path as a MATLAB vector of double. This function
return the handle of the updated forecast scenario.
Once the forecast scenario if fully described, the forecast is
computed with the command ``det_cond_forecast``:
.. matcomm:: DSERIES = det_cond_forecast (HANDLE[, DSERIES [, DATES]]);
Computes the forecast or the conditional forecast using an
extended path method for the given forecast scenario (first
argument). The past values of the endogenous and exogenous
variables provided with a dseries class (see :ref:`dseries class
members <dseries-members>`) can be indicated in the second
argument. By default, the past values of the variables are equal
to their steady-state values. The initial date of the forecast can
be provided in the third argument. By default, the forecast will
start at the first date indicated in the ``init_plan
command``. This function returns a dset containing the historical
and forecast values for the endogenous and exogenous variables.
*Example*
::
% conditional forecast using extended path method
% with perfect foresight on r path
var y r;
varexo e u;
...
smoothed = dseries('smoothed_variables.csv');
fplan = init_plan(2013Q4:2029Q4);
fplan = flip_plan(fplan, 'y', 'u', 'surprise', 2013Q4:2014Q4, [1 1.1 1.2 1.1 ]);
fplan = flip_plan(fplan, 'r', 'e', 'perfect_foresight', 2013Q4:2014Q4, [2 1.9 1.9 1.9 ]);
dset_forecast = det_cond_forecast(fplan, smoothed);
plot(dset_forecast.{'y','u'});
plot(dset_forecast.{'r','e'});
.. command:: smoother2histval ;
smoother2histval(OPTIONS...);
The purpose of this command is to construct initial conditions
(for a subsequent simulation) that are the smoothed values of a
previous estimation.
More precisely, after an estimation run with the ``smoother``
option, ``smoother2histval`` will extract the smoothed values
(from ``oo_.SmoothedVariables``, and possibly from
``oo_.SmoothedShocks`` if there are lagged exogenous), and will
use these values to construct initial conditions (as if they had
been manually entered through ``histval``).
*Options*
.. option:: period = INTEGER
Period number to use as the starting point for the subsequent
simulation. It should be between 1 and the number of
observations that were used to produce the smoothed
values. Default: the last observation.
.. option:: infile = FILENAME
Load the smoothed values from a ``_results.mat`` file created
by a previous Dynare run. Default: use the smoothed values
currently in the global workspace.
.. option:: invars = ( VARIABLE_NAME [VARIABLE_NAME ...] )
A list of variables to read from the smoothed values. It can
contain state endogenous variables, and also exogenous
variables having a lag. Default: all the state endogenous
variables, and all the exogenous variables with a lag.
.. option:: outfile = FILENAME
Write the initial conditions to a file. Default: write the
initial conditions in the current workspace, so that a
simulation can be performed.
.. option:: outvars = ( VARIABLE_NAME [VARIABLE_NAME ...] )
A list of variables which will be given the initial
conditions. This list must have the same length than the list
given to ``invars``, and there will be a one-to-one mapping
between the two list. Default: same value as option
``invars``.
*Use cases*
There are three possible ways of using this command:
* Everything in a single file: run an estimation with a
smoother, then run ``smoother2histval`` (without the
``infile`` and ``outfile`` options), then run a stochastic
simulation.
* In two files: in the first file, run the smoother and then
run ``smoother2histval`` with the ``outfile`` option; in the
second file, run ``histval_file`` to load the initial
conditions, and run a (deterministic or stochastic)
simulation.
* In two files: in the first file, run the smoother; in the
second file, run ``smoother2histval`` with the ``infile``
option equal to the ``_results.mat`` file created by the
first file, and then run a (deterministic or stochastic)
simulation.
Optimal policy
==============
Dynare has tools to compute optimal policies for various types of
objectives. You can either solve for optimal policy under
commitment with ``ramsey_model``, for optimal policy under discretion
with ``discretionary_policy`` or for optimal simple rules with ``osr``
(also implying commitment).
.. command:: planner_objective MODEL_EXPRESSION ;
|br| This command declares the policy maker objective, for use
with ``ramsey_model`` or ``discretionary_policy``.
You need to give the one-period objective, not the discounted
lifetime objective. The discount factor is given by the
``planner_discount`` option of ``ramsey_model`` and
``discretionary_policy``. The objective function can only contain
current endogenous variables and no exogenous ones. This
limitation is easily circumvented by defining an appropriate
auxiliary variable in the model.
With ``ramsey_model``, you are not limited to quadratic
objectives: you can give any arbitrary nonlinear expression.
With ``discretionary_policy``, the objective function must be quadratic.
.. command:: evaluate_planner_objective;
evaluate_planner_objective (OPTIONS...);
This command computes, displays, and stores the value of the
planner objective function under Ramsey policy or discretion in
``oo_.planner_objective_value``. It will provide both unconditional welfare
and welfare conditional on the initial (i.e. period 0) values of the endogenous
and exogenous state variables inherited by the planner. In a deterministic context,
the respective initial values are set using ``initval`` or ``histval`` (depending
on the exact context).
In a stochastic context, if no initial state values have
been specified with ``histval``, their values are taken to be the steady state
values. Because conditional welfare is computed conditional on optimal
policy by the planner in the first endogenous period (period 1), it is conditional
on the information set in the period 1. This information set includes both the
predetermined states inherited from period 0 (specified via ``histval`` for both
endogenous and lagged exogenous states) as well as the period 1 values of
the exogenous shocks. The latter are specified using the perfect foresight syntax
of the ``shocks`` block.
At the current stage, the stochastic context does not support the ``pruning`` option.
At ``order>3``, only the computation of conditional welfare with steady state Lagrange
multipliers is supported. Note that at ``order=2``, the output is based on the second-order
accurate approximation of the variance stored in ``oo_.var``.
*Options*
.. option:: periods = INTEGER
The value of the option specifies the number of periods to use in the
simulations in the computation of unconditional welfare at higher order.
Default: ``10000``.
.. option:: drop = INTEGER
The number of burn-in draws out of ``periods`` discarded before computing
the unconditional welfare at higher order. Default: ``1000``.
*Example (stochastic context)*
::
var a ...;
varexo u;
model;
a = rho*a(-1)+u+u(-1);
...
end;
histval;
u(0)=1;
a(0)=-1;
end;
shocks;
var u; stderr 0.008;
var u;
periods 1;
values 1;
end;
evaluate_planner_objective;
.. matvar:: oo_.planner_objective_value.unconditional
Scalar storing the value of unconditional welfare. In a perfect foresight context,
it corresponds to welfare in the long-run, approximated as welfare in the terminal
simulation period.
.. matvar:: oo_.planner_objective_value.conditional
In a perfect foresight context, this field will be a scalar storing the value of
welfare conditional on the specified initial condition and zero initial Lagrange
multipliers.
In a stochastic context, it will have two subfields:
.. matvar:: oo_.planner_objective_value.conditional.steady_initial_multiplier
Stores the value of the planner objective when the initial
Lagrange multipliers associated with the planner’s problem are set
to their steady state values (see :comm:`ramsey_policy`).
.. matvar:: oo_.planner_objective_value.conditional.zero_initial_multiplier
Stores the value of the planner objective when the initial
Lagrange multipliers associated with the planner’s problem are set
to 0, i.e. it is assumed that the planner exploits its
ability to surprise private agents in the first period of
implementing Ramsey policy. This value corresponds to the planner
implementing optimal policy for the first time and committing not to
re-optimize in the future.
Optimal policy under commitment (Ramsey)
----------------------------------------
Dynare allows to automatically compute optimal policy choices of a Ramsey planner
who takes the specified private sector equilibrium conditions into account and commits
to future policy choices. Doing so requires specifying the private sector equilibrium
conditions in the ``model`` block and a ``planner_objective`` as well as potentially some
``instruments`` to facilitate computations.
.. warning:: Be careful when employing forward-looking auxiliary variables in the context
of timeless perspective Ramsey computations. They may alter the problem the Ramsey
planner will solve for the first period, although they seemingly leave the private
sector equilibrium unaffected. The reason is the planner optimizes with respect to variables
dated ``t`` and takes the value of time 0 variables as given, because they are predetermined.
This set of initially predetermined variables will change with forward-looking definitions.
Thus, users are strongly advised to use model-local variables instead.
*Example*
Consider a perfect foresight example where the Euler equation for the
return to capital is given by
::
1/C=beta*1/C(+1)*(R(+1)+(1-delta))
The job of the Ramsey planner in period ``1`` is to choose :math:`C_1` and :math:`R_1`, taking as given
:math:`C_0`. The above equation may seemingly equivalently be written as
::
1/C=beta*1/C(+1)*(R_cap);
R_cap=R(+1)+(1-delta);
due to perfect foresight. However, this changes the problem of the Ramsey planner in the first period
to choosing :math:`C_1` and :math:`R_1`, taking as given both :math:`C_0` and :math:`R^{cap}_0`. Thus,
the relevant return to capital in the Euler equation of the first period is not a
choice of the planner anymore due to the forward-looking nature of the definition in the second line!
A correct specification would be to instead define ``R_cap`` as a model-local variable:
::
1/C=beta*1/C(+1)*(R_cap);
#R_cap=R(+1)+(1-delta);
.. command:: ramsey_model (OPTIONS...);
|br| This command computes the First Order Conditions for maximizing
the policy maker objective function subject to the constraints
provided by the equilibrium path of the private economy.
The planner objective must be declared with the :comm:`planner_objective` command.
This command only creates the expanded model, it doesn’t perform
any computations. It needs to be followed by other instructions to
actually perform desired computations. Examples are calls to ``steady``
to compute the steady state of the Ramsey economy, to ``stoch_simul``
with various approximation orders to conduct stochastic simulations based on
perturbation solutions, to ``estimation`` in order to estimate models
under optimal policy with commitment, and to perfect foresight simulation
routines.
See :ref:`aux-variables`, for an explanation of how Lagrange
multipliers are automatically created.
*Options*
This command accepts the following options:
.. option:: planner_discount = EXPRESSION
Declares or reassigns the discount factor of the central
planner ``optimal_policy_discount_factor``. Default: ``1.0``.
.. option:: planner_discount_latex_name = LATEX_NAME
Sets the LaTeX name of the ``optimal_policy_discount_factor``
parameter.
.. option:: instruments = (VARIABLE_NAME,...)
Declares instrument variables for the computation of the
steady state under optimal policy. Requires a
``steady_state_model`` block or a ``_steadystate.m`` file. See
below.
*Steady state*
Dynare takes advantage of the fact that the Lagrange multipliers
appear linearly in the equations of the steady state of the model
under optimal policy. Nevertheless, it is in general very
difficult to compute the steady state with simply a numerical
guess in ``initval`` for the endogenous variables.
It greatly facilitates the computation, if the user provides an
analytical solution for the steady state (in
``steady_state_model`` block or in a ``_steadystate.m`` file). In
this case, it is necessary to provide a steady state solution
CONDITIONAL on the value of the instruments in the optimal policy
problem and declared with the option ``instruments``. The initial value
of the instrument for steady state finding in this case is set with
``initval``. Note that computing and displaying steady state values
using the ``steady`` command or calls to ``resid`` must come after
the ``ramsey_model`` statement and the ``initval`` block.
Note that choosing the instruments is partly a matter of interpretation and
you can choose instruments that are handy from a mathematical
point of view but different from the instruments you would refer
to in the analysis of the paper. A typical example is choosing
inflation or nominal interest rate as an instrument.
.. block:: ramsey_constraints ;
|br| This block lets you define constraints on the variables in
the Ramsey problem. The constraints take the form of a variable,
an inequality operator (> or <) and a constant.
*Example*
::
ramsey_constraints;
i > 0;
end;
.. command:: ramsey_policy [VARIABLE_NAME...];
ramsey_policy (OPTIONS...) [VARIABLE_NAME...];
|br| This command is deprecated and formally equivalent to the calling sequence
::
ramsey_model;
stoch_simul;
evaluate_planner_objective;
It computes an approximation of the
policy that maximizes the policy maker’s objective function
subject to the constraints provided by the equilibrium path of the
private economy and under commitment to this optimal policy. The
Ramsey policy is computed by approximating the equilibrium system
around the perturbation point where the Lagrange multipliers are
at their steady state, i.e. where the Ramsey planner acts as if
the initial multipliers had been set to 0 in the distant past,
giving them time to converge to their steady state
value. Consequently, the optimal decision rules are computed
around this steady state of the endogenous variables and the
Lagrange multipliers.
Note that the variables in the list after the ``ramsey_policy``
or ``stoch_simul`` command can also contain multiplier names, but
in a case-sensititve way (e.g. ``MULT_1``). In that case, Dynare
will for example display the IRFs of the respective multipliers
when ``irf>0``.
The planner objective must be declared with the :comm:`planner_objective` command.
*Options*
This command accepts all options of ``stoch_simul``, plus:
.. option:: planner_discount = EXPRESSION
See :opt:`planner_discount <planner_discount = EXPRESSION>`.
.. option:: instruments = (VARIABLE_NAME,...)
Declares instrument variables for the computation of the
steady state under optimal policy. Requires a
``steady_state_model`` block or a ``_steadystate.m`` file. See
below.
*Output*
This command generates all the output variables of
``stoch_simul``. For specifying the initial values for the
endogenous state variables (except for the Lagrange multipliers),
see above.
*Steady state*
See :comm:`Ramsey steady state <ramsey_model>`.
Optimal policy under discretion
-------------------------------
.. command:: discretionary_policy [VARIABLE_NAME...];
discretionary_policy (OPTIONS...) [VARIABLE_NAME...];
|br| This command computes an approximation of the optimal policy
under discretion. The algorithm implemented is essentially an LQ
solver, and is described by *Dennis (2007)*.
You must ensure that your objective is quadratic. Regarding the model, it must
either be linear or solved at first order with an analytical steady state provided.
In the first case, you should set the ``linear`` option of the
``model`` block.
It is possible to use the :comm:`estimation` command after the
``discretionary_policy`` command, in order to estimate the model with
optimal policy under discretion and :comm:`evaluate_planner_objective`
to compute welfare.
*Options*
This command accepts the same options as ``ramsey_policy``, plus:
.. option:: discretionary_tol = NON-NEGATIVE DOUBLE
Sets the tolerance level used to assess convergence of the
solution algorithm. Default: ``1e-7``.
.. option:: maxit = INTEGER
Maximum number of iterations. Default: ``3000``.
Optimal Simple Rules (OSR)
--------------------------
.. command:: osr [VARIABLE_NAME...];
osr (OPTIONS...) [VARIABLE_NAME...];
|br| This command computes optimal simple policy rules for
linear-quadratic problems of the form:
.. math::
\min_\gamma E(y'_tWy_t)
such that:
.. math::
A_1 E_ty_{t+1}+A_2 y_t+ A_3 y_{t-1}+C e_t=0
where:
* :math:`E` denotes the unconditional expectations operator;
* :math:`\gamma` are parameters to be optimized. They must be
elements of the matrices :math:`A_1`, :math:`A_2`,
:math:`A_3`, i.e. be specified as parameters in the
``params`` command and be entered in the ``model`` block;
* :math:`y` are the endogenous variables, specified in the
``var`` command, whose (co)-variance enters the loss
function;
* :math:`e` are the exogenous stochastic shocks, specified in
the ``varexo``- ommand;
* :math:`W` is the weighting matrix;
The linear quadratic problem consists of choosing a subset of
model parameters to minimize the weighted (co)-variance of a
specified subset of endogenous variables, subject to a linear law
of motion implied by the first order conditions of the model. A
few things are worth mentioning. First, :math:`y` denotes the
selected endogenous variables’ deviations from their steady state,
i.e. in case they are not already mean 0 the variables entering
the loss function are automatically demeaned so that the centered
second moments are minimized. Second, ``osr`` only solves linear
quadratic problems of the type resulting from combining the
specified quadratic loss function with a first order approximation
to the model’s equilibrium conditions. The reason is that the
first order state-space representation is used to compute the
unconditional (co)-variances. Hence, ``osr`` will automatically
select ``order=1``. Third, because the objective involves
minimizing a weighted sum of unconditional second moments, those
second moments must be finite. In particular, unit roots in
:math:`y` are not allowed.
The subset of the model parameters over which the optimal simple
rule is to be optimized, :math:`\gamma`, must be listed with
``osr_params``.
The weighting matrix :math:`W` used for the quadratic objective
function is specified in the ``optim_weights`` block. By attaching
weights to endogenous variables, the subset of endogenous
variables entering the objective function, :math:`y`, is
implicitly specified.
The linear quadratic problem is solved using the numerical
optimizer specified with :opt:`opt_algo <opt_algo = INTEGER>`.
*Options*
The ``osr`` command will subsequently run ``stoch_simul`` and
accepts the same options, including restricting the endogenous
variables by listing them after the command, as ``stoch_simul``
(see :ref:`stoch-sol`) plus
.. option:: opt_algo = INTEGER
Specifies the optimizer for minimizing the objective
function. The same solvers as for ``mode_compute`` (see
:opt:`mode_compute <mode_compute = INTEGER | FUNCTION_NAME>`)
are available, except for ``5``, ``6``, and ``10``.
.. option:: optim = (NAME, VALUE, ...)
A list of NAME`` and VALUE pairs. Can be used to set options
for the optimization routines. The set of available options
depends on the selected optimization routine (i.e. on the
value of option :opt:`opt_algo <opt_algo = INTEGER>`). See
:opt:`optim <optim = (NAME, VALUE, ...)>`.
.. option:: maxit = INTEGER
Determines the maximum number of iterations used in
``opt_algo=4``. This option is now deprecated and will be
removed in a future release of Dynare. Use ``optim`` instead
to set optimizer-specific values. Default: ``1000``.
.. option:: tolf = DOUBLE
Convergence criterion for termination based on the function
value used in ``opt_algo=4``. Iteration will cease when it
proves impossible to improve the function value by more than
tolf. This option is now deprecated and will be removed in a
future release of Dynare. Use ``optim`` instead to set
optimizer-specific values. Default: ``1e-7``.
.. option:: silent_optimizer
See :opt:`silent_optimizer`.
.. option:: huge_number = DOUBLE
Value for replacing the infinite bounds on parameters by
finite numbers. Used by some optimizers for numerical reasons
(see :opt:`huge_number <huge_number = DOUBLE>`). Users need to
make sure that the optimal parameters are not larger than this
value. Default: ``1e7``.
The value of the objective is stored in the variable
``oo_.osr.objective_function`` and the value of parameters at the
optimum is stored in ``oo_.osr.optim_params``. See below for more
details.
After running ``osr`` the parameters entering the simple rule will
be set to their optimal value so that subsequent runs of
``stoch_simul`` will be conducted at these values.
.. command:: osr_params PARAMETER_NAME...;
|br| This command declares parameters to be optimized by ``osr``.
.. block:: optim_weights ;
|br| This block specifies quadratic objectives for optimal policy problems.
More precisely, this block specifies the nonzero elements of the
weight matrix :math:`W` used in the quadratic form of the
objective function in ``osr``.
An element of the diagonal of the weight matrix is given by a line
of the form::
VARIABLE_NAME EXPRESSION;
An off-the-diagonal element of the weight matrix is given
by a line of the form::
VARIABLE_NAME, VARIABLE_NAME EXPRESSION;
*Example*
::
var y inflation r;
varexo y_ inf_;
parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_;
delta = 0.44;
kappa = 0.18;
alpha = 0.48;
sigma = -0.06;
gammarr = 0;
gammax0 = 0.2;
gammac0 = 1.5;
gamma_y_ = 8;
gamma_inf_ = 3;
model(linear);
y = delta * y(-1) + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_;
inflation = alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_;
r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_;
end;
shocks;
var y_; stderr 0.63;
var inf_; stderr 0.4;
end;
optim_weights;
inflation 1;
y 1;
y, inflation 0.5;
end;
osr_params gammax0 gammac0 gamma_y_ gamma_inf_;
osr y;
.. block:: osr_params_bounds ;
|br| This block declares lower and upper bounds for parameters in
the optimal simple rule. If not specified the optimization is
unconstrained.
Each line has the following syntax::
PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;
Note that the use of this block requires the use of a constrained
optimizer, i.e. setting :opt:`opt_algo <opt_algo = INTEGER>` to
``1``, ``2``, ``5`` or ``9``.
*Example*
::
osr_params_bounds;
gamma_inf_, 0, 2.5;
end;
osr(opt_algo=9) y;
.. matvar:: oo_.osr.objective_function
After an execution of the ``osr`` command, this variable contains
the value of the objective under optimal policy.
.. matvar:: oo_.osr.optim_params
After an execution of the ``osr`` command, this variable contains
the value of parameters at the optimum, stored in fields of the
form ``oo_.osr.optim_params.PARAMETER_NAME``.
.. matvar:: M_.osr.param_names
After an execution of the ``osr`` command, this cell contains the
names of the parameters.
.. matvar:: M_.osr.param_indices
After an execution of the ``osr`` command, this vector contains
the indices of the OSR parameters in ``M_.params``.
.. matvar:: M_.osr.param_bounds
After an execution of the ``osr`` command, this two by number of
OSR parameters matrix contains the lower and upper bounds of the
parameters in the first and second column, respectively.
.. matvar:: M_.osr.variable_weights
After an execution of the ``osr`` command, this sparse matrix
contains the weighting matrix associated with the variables in the
objective function.
.. matvar:: M_.osr.variable_indices
After an execution of the ``osr`` command, this vector contains
the indices of the variables entering the objective function in
``M_.endo_names``.
Sensitivity and identification analysis
=======================================
Dynare provides an interface to the global sensitivity analysis (GSA)
toolbox (developed by the Joint Research Center (JRC) of the European
Commission), which is now part of the official Dynare
distribution. The GSA toolbox can be used to answer the following
questions:
1. What is the domain of structural coefficients assuring the
stability and determinacy of a DSGE model?
2. Which parameters mostly drive the fit of, e.g., GDP and which
the fit of inflation? Is there any conflict between the optimal
fit of one observed series versus another?
3. How to represent in a direct, albeit approximated, form the
relationship between structural parameters and the reduced form
of a rational expectations model?
The discussion of the methodologies and their application is described
in *Ratto (2008)*.
With respect to the previous version of the toolbox, in order to work
properly, the GSA toolbox no longer requires that the Dynare
estimation environment is set up.
Performing sensitivity analysis
-------------------------------
.. command:: dynare_sensitivity ;
dynare_sensitivity(OPTIONS...);
|br| This command triggers sensitivity analysis on a DSGE model.
.. _sampl-opt:
*Sampling Options*
.. option:: Nsam = INTEGER
Size of the Monte-Carlo sample. Default: ``2048``.
.. option:: ilptau = INTEGER
If equal to ``1``, use :math:`LP_\tau` quasi-Monte-Carlo. If
equal to ``0``, use LHS Monte-Carlo. Default: ``1``.
.. option:: pprior = INTEGER
If equqal to ``1``, sample from the prior distributions. If
equal to ``0``, sample from the multivariate normal
:math:`N(\bar{\theta},\Sigma)`, where :math:`\bar{\theta}` is
the posterior mode and :math:`\Sigma=H^{-1}`, :math:`H` is the
Hessian at the mode. Default: ``1``.
.. option:: prior_range = INTEGER
If equal to ``1``, sample uniformly from prior ranges. If
equal to ``0``, sample from prior distributions. Default:
``1``.
.. option:: morris = INTEGER
If equal to ``0``, ANOVA mapping (Type I error) If equal to
``1``, Screening analysis (Type II error). If equal to ``2``,
Analytic derivatives (similar to Type II error, only valid
when identification=1). Default: ``1`` when
``identification=1``, ``0`` otherwise.
.. option:: morris_nliv = INTEGER
Number of levels in Morris design. Default: ``6``.
.. option:: morris_ntra = INTEGER
Number trajectories in Morris design. Default: ``20``.
.. option:: ppost = INTEGER
If equal to ``1``, use Metropolis posterior sample. If equal
to ``0``, do not use Metropolis posterior sample. Default:
``0``.
NB: This overrides any other sampling option.
.. option:: neighborhood_width = DOUBLE
When ``pprior=0`` and ``ppost=0``, allows for the sampling of
parameters around the value specified in the ``mode_file``, in
the range :math:`\texttt{xparam1} \pm \left \vert
\texttt{xparam1} \times \texttt{neighborhood\_width} \right
\vert`. Default: ``0``.
*Stability Mapping Options*
.. option:: stab = INTEGER
If equal to ``1``, perform stability mapping. If equal to
``0``, do not perform stability mapping. Default: ``1``.
.. option:: load_stab = INTEGER
If equal to ``1``, load a previously created sample. If equal
to ``0``, generate a new sample. Default: ``0``.
.. option:: alpha2_stab = DOUBLE
Critical value for correlations :math:`\rho` in filtered
samples: plot couples of parmaters with
:math:`\left\vert\rho\right\vert>` ``alpha2_stab``. Default:
``0``.
.. option:: pvalue_ks = DOUBLE
The threshold :math:`pvalue` for significant
Kolmogorov-Smirnov test (i.e. plot parameters with
:math:`pvalue<` ``pvalue_ks``). Default: ``0.001``.
.. option:: pvalue_corr = DOUBLE
The threshold :math:`pvalue` for significant correlation in
filtered samples (i.e. plot bivariate samples when
:math:`pvalue<` ``pvalue_corr``). Default: ``1e-5``.
*Reduced Form Mapping Options*
.. option:: redform = INTEGER
If equal to ``1``, prepare Monte-Carlo sample of reduced form
matrices. If equal to ``0``, do not prepare Monte-Carlo sample
of reduced form matrices. Default: ``0``.
.. option:: load_redform = INTEGER
If equal to ``1``, load previously estimated mapping. If equal
to ``0``, estimate the mapping of the reduced form
model. Default: ``0``.
.. option:: logtrans_redform = INTEGER
If equal to ``1``, use log-transformed entries. If equal to
``0``, use raw entries. Default: ``0``.
.. option:: threshold_redform = [DOUBLE DOUBLE]
The range over which the filtered Monte-Carlo entries of the
reduced form coefficients should be analyzed. The first number
is the lower bound and the second is the upper bound. An empty
vector indicates that these entries will not be
filtered. Default: empty.
.. option:: ksstat_redform = DOUBLE
Critical value for Smirnov statistics :math:`d` when reduced
form entries are filtered. Default: ``0.001``.
.. option:: alpha2_redform = DOUBLE
Critical value for correlations :math:`\rho` when reduced form
entries are filtered. Default: ``1e-5``.
.. option:: namendo = (VARIABLE_NAME...)
List of endogenous variables. ‘:’ indicates all endogenous
variables. Default: empty.
.. option:: namlagendo = (VARIABLE_NAME...)
List of lagged endogenous variables. ‘:’ indicates all lagged
endogenous variables. Analyze entries [namendo :math:`\times`
namlagendo] Default: empty.
.. option:: namexo = (VARIABLE_NAME...)
List of exogenous variables. ‘:’ indicates all exogenous
variables. Analyze entries [namendo :math:`\times`
namexo]. Default: empty.
*RMSE Options*
.. option:: rmse = INTEGER
If equal to ``1``, perform RMSE analysis. If equal to ``0``,
do not perform RMSE analysis. Default: ``0``.
.. option:: load_rmse = INTEGER
If equal to ``1``, load previous RMSE analysis. If equal to
``0``, make a new RMSE analysis. Default: ``0``.
.. option:: lik_only = INTEGER
If equal to ``1``, compute only likelihood and posterior. If
equal to ``0``, compute RMSE’s for all observed
series. Default: ``0``.
.. option:: var_rmse = (VARIABLE_NAME...)
List of observed series to be considered. ‘:’ indicates all
observed variables. Default: ``varobs``.
.. option:: pfilt_rmse = DOUBLE
Filtering threshold for RMSE’s. Default: ``0.1``.
.. option:: istart_rmse = INTEGER
Value at which to start computing RMSE’s (use ``2`` to avoid
big intitial error). Default: ``presample+1``.
.. option:: alpha_rmse = DOUBLE
Critical value for Smirnov statistics :math:`d`: plot
parameters with :math:`d>` ``alpha_rmse``. Default: ``0.001``.
.. option:: alpha2_rmse = DOUBLE
Critical value for correlation :math:`\rho`: plot couples of
parmaters with :math:`\left\vert\rho\right\vert=`
``alpha2_rmse``. Default: ``1e-5``.
.. option:: datafile = FILENAME
See :ref:`datafile <dataf>`.
.. option:: nobs = INTEGER
nobs = [INTEGER1:INTEGER2]
See :opt:`nobs <nobs = INTEGER>`.
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
.. option:: prefilter = INTEGER
See :opt:`prefilter <prefilter = INTEGER>`.
.. option:: presample = INTEGER
See :opt:`presample <presample = INTEGER>`.
.. option:: nograph
See :opt:`nograph`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format <graph_format = FORMAT>`.
.. option:: conf_sig = DOUBLE
See :ref:`conf_sig <confsig>`.
.. option:: loglinear
See :ref:`loglinear <logl>`.
.. option:: mode_file = FILENAME
See :opt:`mode_file <mode_file = FILENAME>`.
.. option:: kalman_algo = INTEGER
See :opt:`kalman_algo <kalman_algo = INTEGER>`.
*Identification Analysis Options*
.. option:: identification = INTEGER
If equal to ``1``, performs identification analysis (forcing
``redform=0`` and ``morris=1``) If equal to ``0``, no
identification analysis. Default: ``0``.
.. option:: morris = INTEGER
See :opt:`morris <morris = INTEGER>`.
.. option:: morris_nliv = INTEGER
See :opt:`morris_nliv <morris_nliv = INTEGER>`.
.. option:: morris_ntra = INTEGER
See :opt:`morris_ntra <morris_ntra = INTEGER>`.
.. option:: load_ident_files = INTEGER
Loads previously performed identification analysis. Default: ``0``.
.. option:: useautocorr = INTEGER
Use autocorrelation matrices in place of autocovariance
matrices in moments for identification analysis. Default:
``0``.
.. option:: ar = INTEGER
Maximum number of lags for moments in identification
analysis. Default: ``1``.
.. option:: diffuse_filter = INTEGER
See :opt:`diffuse_filter`.
.. _irf-momcal:
IRF/Moment calibration
----------------------
The ``irf_calibration`` and ``moment_calibration`` blocks allow
imposing implicit “endogenous” priors about IRFs and moments on the
model. The way it works internally is that any parameter draw that is
inconsistent with the “calibration” provided in these blocks is
discarded, i.e. assigned a prior density of ``0``. In the context of
``dynare_sensitivity``, these restrictions allow tracing out which
parameters are driving the model to satisfy or violate the given
restrictions.
IRF and moment calibration can be defined in ``irf_calibration`` and
``moment_calibration`` blocks:
.. block:: irf_calibration ;
irf_calibration (OPTIONS...);
|br| This block allows defining IRF calibration criteria and is
terminated by ``end;``. To set IRF sign restrictions, the
following syntax is used::
VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, -;
VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, +;
To set IRF restrictions with specific intervals, the following
syntax is used::
VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION];
VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION];
When ``(INTEGER:INTEGER)`` is used, the restriction is considered
to be fulfilled by a logical OR. A list of restrictions must
always be fulfilled with logical AND.
*Options*
.. option:: relative_irf
See :opt:`relative_irf`.
*Example*
::
irf_calibration;
y(1:4), e_ys, [-50, 50]; //[first year response with logical OR]
@#for ilag in 21:40
R_obs(@{ilag}), e_ys, [0, 6]; //[response from 5th to 10th years with logical AND]
@#endfor
end;
.. block:: moment_calibration ;
moment_calibration (OPTIONS...);
|br| This block allows defining moment calibration criteria. This
block is terminated by ``end;``, and contains lines of the form::
VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), [EXPRESSION, EXPRESSION];
VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), +/-;
VARIABLE_NAME1, VARIABLE_NAME2(+/-(INTEGER:INTEGER)), [EXPRESSION, EXPRESSION];
VARIABLE_NAME1, VARIABLE_NAME2((-INTEGER:+INTEGER)), [EXPRESSION, EXPRESSION];
When ``(INTEGER:INTEGER)`` is used, the restriction is considered
to be fulfilled by a logical OR. A list of restrictions must
always be fulfilled with logical AND. The moment restrictions generally apply to
auto- and cross-correlations between variables. The only exception is a restriction
on the unconditional variance of an endogenous variable, specified as shown in
the example below.
*Example*
::
moment_calibration;
y_obs,y_obs, [0.5, 1.5]; //[unconditional variance]
y_obs,y_obs(-(1:4)), +; //[sign restriction for first year autocorrelation with logical OR]
@#for ilag in -2:2
y_obs,R_obs(@{ilag}), -; //[-2:2 cross correlation with logical AND]
@#endfor
@#for ilag in -4:4
y_obs,pie_obs(@{ilag}), -; //[-4_4 cross correlation with logical AND]
@#endfor
end;
.. _identification-analysis:
Performing identification analysis
----------------------------------
.. command:: identification ;
identification (OPTIONS...);
|br| This command triggers:
1. Theoretical identification analysis based on
* moments as in *Iskrev (2010)*
* spectral density as in *Qu and Tkachenko (2012)*
* minimal system as in *Komunjer and Ng (2011)*
* reduced-form solution and linear rational expectation model
as in *Ratto and Iskrev (2011)*
Note that for orders 2 and 3, all identification checks are based on the pruned
state space system as in *Mutschler (2015)*. That is, theoretical moments and
spectrum are computed from the pruned ABCD-system, whereas the minimal system
criteria is based on the first-order system, but augmented by the theoretical
(pruned) mean at order 2 or 3.
2. Identification strength analysis based on (theoretical or simulated) curvature of
moment information matrix as in *Ratto and Iskrev (2011)*
3. Parameter checks based on nullspace and multicorrelation coefficients to
determine which (combinations of) parameters are involved
*General Options*
.. option:: order = 1|2|3
Order of approximation. At orders 2 and 3 identification is based on the
pruned state space system. Note that the order set in other functions does
not overwrite the default.
Default: ``1``.
.. option:: parameter_set = OPTION
See :opt:`parameter_set <parameter_set = OPTION>` for
possible values. Default: ``prior_mean``.
.. option:: prior_mc = INTEGER
Size of Monte-Carlo sample.
Default: ``1``.
.. option:: prior_range = INTEGER
Triggers uniform sample within the range implied by the prior
specifications (when ``prior_mc>1``).
Default: ``0``.
.. option:: advanced = INTEGER
If set to ``1``, shows a more detailed analysis, comprised of
an analysis for the linearized rational expectation model as
well as the associated reduced form solution. Further performs
a bruteforce search of the groups of parameters best reproducing
the behavior of each single parameter. The maximum dimension of
the group searched is triggered by ``max_dim_cova_group``.
Default: ``0``.
.. option:: max_dim_cova_group = INTEGER
In the brute force search (performed when ``advanced=1``) this
option sets the maximum dimension of groups of parameters that
best reproduce the behavior of each single model parameter.
Default: ``2``.
.. option:: gsa_sample_file = INTEGER|FILENAME
If equal to ``0``, do not use sample file. If equal to ``1``,
triggers gsa prior sample. If equal to ``2``, triggers gsa
Monte-Carlo sample (i.e. loads a sample corresponding to
``pprior=0`` and ``ppost=0`` in the ``dynare_sensitivity``
options). If equal to ``FILENAME`` uses the provided path to
a specific user defined sample file.
Default: ``0``.
.. option:: diffuse_filter
Deals with non-stationary cases. See :opt:`diffuse_filter`.
*Numerical Options*
.. option:: analytic_derivation_mode = INTEGER
Different ways to compute derivatives either analytically or numerically.
Possible values are:
* ``0``: efficient sylvester equation method to compute
analytical derivatives
* ``1``: kronecker products method to compute analytical
derivatives (only at order=1)
* ``-1``: numerical two-sided finite difference method
to compute all identification Jacobians (numerical tolerance
level is equal to ``options_.dynatol.x``)
* ``-2``: numerical two-sided finite difference method
to compute derivatives of steady state and dynamic
model numerically, the identification Jacobians are
then computed analytically (numerical tolerance
level is equal to ``options_.dynatol.x``)
Default: ``0``.
.. option:: normalize_jacobians = INTEGER
If set to ``1``: Normalize Jacobian matrices by rescaling
each row by its largest element in absolute value.
Normalize Gram (or Hessian-type) matrices by transforming
into correlation-type matrices.
Default: ``1``
.. option:: tol_rank = DOUBLE
Tolerance level used for rank computations.
Default: ``1.e-10``.
.. option:: tol_deriv = DOUBLE
Tolerance level for selecting non-zero columns in Jacobians.
Default: ``1.e-8``.
.. option:: tol_sv = DOUBLE
Tolerance level for selecting non-zero singular values.
Default: ``1.e-3``.
.. option:: schur_vec_tol = DOUBLE
See :opt:`schur_vec_tol <schur_vec_tol = DOUBLE>`.
*Identification Strength Options*
.. option:: no_identification_strength
Disables computations of identification strength analysis
based on sample information matrix.
.. option:: periods = INTEGER
When the analytic Hessian is not available (i.e. with missing
values or diffuse Kalman filter or univariate Kalman filter),
this triggers the length of stochastic simulation to compute
Simulated Moments Uncertainty. Default: ``300``.
.. option:: replic = INTEGER
When the analytic Hessian is not available, this triggers the
number of replicas to compute Simulated Moments
Uncertainty. Default: ``100``.
*Moments Options*
.. option:: no_identification_moments
Disables computations of identification check based on
Iskrev (2010)'s J, i.e. derivative of first two moments.
.. option:: ar = INTEGER
Number of lags of computed autocovariances/autocorrelations
(theoretical moments) in Iskrev (2010)'s J criteria.
Default: ``1``.
.. option:: useautocorr = INTEGER
If equal to ``1``, compute derivatives of autocorrelation. If
equal to ``0``, compute derivatives of
autocovariances. Default: ``0``.
*Spectrum Options*
.. option:: no_identification_spectrum
Disables computations of identification check based on
*Qu and Tkachenko (2012)*'s G, i.e. Gram matrix of derivatives of
first moment plus outer product of derivatives of spectral density.
.. option:: grid_nbr = INTEGER
Number of grid points in [-pi;pi] to approximate the integral
to compute Qu and Tkachenko (2012)'s G criteria.
Default: ``5000``.
*Minimal State Space System Options*
.. option:: no_identification_minimal
Disables computations of identification check based on
*Komunjer and Ng (2011)*'s D, i.e. minimal state space system
and observational equivalent spectral density transformations.
*Misc Options*
.. option:: nograph
See :opt:`nograph`.
.. option:: nodisplay
See :opt:`nodisplay`.
.. option:: graph_format = FORMAT
graph_format = ( FORMAT, FORMAT... )
See :opt:`graph_format <graph_format = FORMAT>`.
.. option:: tex
See :opt:`tex`.
*Debug Options*
.. option:: load_ident_files = INTEGER
If equal to ``1``, allow Dynare to load previously computed
analyzes. Default: ``0``.
.. option:: lik_init = INTEGER
See :opt:`lik_init <lik_init = INTEGER>`.
.. option:: kalman_algo = INTEGER
See :opt:`kalman_algo <kalman_algo = INTEGER>`.
.. option:: no_identification_reducedform
Disables computations of identification check based on
steady state and reduced-form solution.
.. option:: checks_via_subsets = INTEGER
If equal to ``1``: finds problematic parameters in a bruteforce
fashion: It computes the rank of the Jacobians for all possible
parameter combinations. If the rank condition is not fullfilled,
these parameter sets are flagged as non-identifiable.
The maximum dimension of the group searched is triggered by
``max_dim_subsets_groups``.
Default: ``0``.
.. option:: max_dim_subsets_groups = INTEGER
Sets the maximum dimension of groups of parameters for which
the above bruteforce search is performed.
Default: ``4``.
Types of analysis and output files
----------------------------------
The sensitivity analysis toolbox includes several types of
analyses. Sensitivity analysis results are saved locally in
``<mod_file>/gsa``, where ``<mod_file>.mod`` is the name of the Dynare
model file.
Sampling
^^^^^^^^
The following binary files are produced:
* ``<mod_file>_prior.mat``: this file stores information about the
analyses performed sampling from the prior, i.e. ``pprior=1``
and ``ppost=0``;
* ``<mod_file>_mc.mat``: this file stores information about the
analyses performed sampling from multivariate normal,
i.e. ``pprior=0`` and ``ppost=0``;
* ``<mod_file>_post.mat``: this file stores information about
analyses performed using the Metropolis posterior sample,
i.e. ``ppost=1``.
Stability Mapping
^^^^^^^^^^^^^^^^^
Figure files produced are of the form ``<mod_file>_prior_*.fig`` and
store results for stability mapping from prior Monte-Carlo samples:
* ``<mod_file>_prior_stable.fig``: plots of the Smirnov test and
the correlation analyses confronting the cdf of the sample
fulfilling Blanchard-Kahn conditions (blue color) with the cdf
of the rest of the sample (red color), i.e. either instability
or indeterminacy or the solution could not be found (e.g. the
steady state solution could not be found by the solver);
* ``<mod_file>_prior_indeterm.fig``: plots of the Smirnov test and
the correlation analyses confronting the cdf of the sample
producing indeterminacy (red color) with the cdf of the rest of
the sample (blue color);
* ``<mod_file>_prior_unstable.fig``: plots of the Smirnov test and
the correlation analyses confronting the cdf of the sample
producing explosive roots (red color) with the cdf of the rest
of the sample (blue color);
* ``<mod_file>_prior_wrong.fig``: plots of the Smirnov test and
the correlation analyses confronting the cdf of the sample where
the solution could not be found (e.g. the steady state solution
could not be found by the solver - red color) with the cdf of
the rest of the sample (blue color);
* ``<mod_file>_prior_calib.fig``: plots of the Smirnov test and
the correlation analyses splitting the sample fulfilling
Blanchard-Kahn conditions, by confronting the cdf of the sample
where IRF/moment restrictions are matched (blue color) with the
cdf where IRF/moment restrictions are NOT matched (red color);
Similar conventions apply for ``<mod_file>_mc_*.fig`` files, obtained
when samples from multivariate normal are used.
IRF/Moment restrictions
^^^^^^^^^^^^^^^^^^^^^^^
The following binary files are produced:
* ``<mod_file>_prior_restrictions.mat``: this file stores
information about the IRF/moment restriction analysis performed
sampling from the prior ranges, i.e. ``pprior=1`` and
``ppost=0``;
* ``<mod_file>_mc_restrictions.mat``: this file stores information
about the IRF/moment restriction analysis performed sampling
from multivariate normal, i.e. ``pprior=0`` and ``ppost=0``;
* ``<mod_file>_post_restrictions.mat``: this file stores
information about IRF/moment restriction analysis performed
using the Metropolis posterior sample, i.e. ``ppost=1``.
Figure files produced are of the form
``<mod_file>_prior_irf_calib_*.fig`` and
``<mod_file>_prior_moment_calib_*.fig`` and store results for mapping
restrictions from prior Monte-Carlo samples:
* ``<mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_<PERIOD>.fig``:
plots of the Smirnov test and the correlation analyses splitting
the sample fulfilling Blanchard-Kahn conditions, by confronting
the cdf of the sample where the individual IRF restriction
``<ENDO_NAME>`` vs. ``<EXO_NAME>`` at period(s) ``<PERIOD>`` is
matched (blue color) with the cdf where the IRF restriction is
NOT matched (red color)
* ``<mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig``:
plots of the Smirnov test and the correlation analyses splitting
the sample fulfilling Blanchard-Kahn conditions, by confronting
the cdf of the sample where ALL the individual IRF restrictions
for the same couple ``<ENDO_NAME>`` vs. ``<EXO_NAME>`` are
matched (blue color) with the cdf where the IRF restriction is
NOT matched (red color)
* ``<mod_file>_prior_irf_restrictions.fig``: plots visual
information on the IRF restrictions compared to the actual Monte
Carlo realization from prior sample.
* ``<mod_file>_prior_moment_calib_<ENDO_NAME1>_vs_<ENDO_NAME2>_<LAG>.fig``:
plots of the Smirnov test and the correlation analyses splitting
the sample fulfilling Blanchard-Kahn conditions, by confronting
the cdf of the sample where the individual acf/ccf moment
restriction ``<ENDO_NAME1>`` vs. ``<ENDO_NAME2>`` at lag(s)
``<LAG>`` is matched (blue color) with the cdf where the IRF
restriction is NOT matched (red color)
* ``<mod_file>_prior_moment_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig``:
plots of the Smirnov test and the correlation analyses splitting
the sample fulfilling Blanchard-Kahn conditions, by confronting
the cdf of the sample where ALL the individual acf/ccf moment
restrictions for the same couple ``<ENDO_NAME1>``
vs. ``<ENDO_NAME2>`` are matched (blue color) with the cdf where
the IRF restriction is NOT matched (red color)
* ``<mod_file>_prior_moment_restrictions.fig``: plots visual
information on the moment restrictions compared to the actual
Monte Carlo realization from prior sample.
Similar conventions apply for ``<mod_file>_mc_*.fig`` and
``<mod_file>_post_*.fig`` files, obtained when samples from
multivariate normal or from posterior are used.
Reduced Form Mapping
^^^^^^^^^^^^^^^^^^^^
When the option ``threshold_redform`` is not set, or it is empty (the
default), this analysis estimates a multivariate smoothing spline
ANOVA model (the ’mapping’) for the selected entries in the transition
matrix of the shock matrix of the reduce form first order solution of
the model. This mapping is done either with prior samples or with MC
samples with ``neighborhood_width``. Unless ``neighborhood_width`` is
set with MC samples, the mapping of the reduced form solution forces
the use of samples from prior ranges or prior distributions, i.e.:
``pprior=1`` and ``ppost=0``. It uses 250 samples to optimize
smoothing parameters and 1000 samples to compute the fit. The rest of
the sample is used for out-of-sample validation. One can also load a
previously estimated mapping with a new Monte-Carlo sample, to look at
the forecast for the new Monte-Carlo sample.
The following synthetic figures are produced:
* ``<mod_file>_redform_<endo name>_vs_lags_*.fig``: shows bar
charts of the sensitivity indices for the ten most important
parameters driving the reduced form coefficients of the selected
endogenous variables (``namendo``) versus lagged endogenous
variables (``namlagendo``); suffix ``log`` indicates the results
for log-transformed entries;
* ``<mod_file>_redform_<endo name>_vs_shocks_*.fig``: shows bar
charts of the sensitivity indices for the ten most important
parameters driving the reduced form coefficients of the selected
endogenous variables (``namendo``) versus exogenous variables
(``namexo``); suffix ``log`` indicates the results for
log-transformed entries;
* ``<mod_file>_redform_gsa(_log).fig``: shows bar chart of all
sensitivity indices for each parameter: this allows one to
notice parameters that have a minor effect for any of the
reduced form coefficients.
Detailed results of the analyses are shown in the subfolder
``<mod_file>/gsa/redform_prior`` for prior samples and in
``<mod_file>/gsa/redform_mc`` for MC samples with option
``neighborhood_width``, where the detailed results of the estimation
of the single functional relationships between parameters
:math:`\theta` and reduced form coefficient (denoted as :math:`y`
hereafter) are stored in separate directories named as:
* ``<namendo>_vs_<namlagendo>``, for the entries of the transition matrix;
* ``<namendo>_vs_<namexo>``, for entries of the matrix of the shocks.
The following files are stored in each directory (we stick with prior
sample but similar conventions are used for MC samples):
* ``<mod_file>_prior_<namendo>_vs_<namexo>.fig``: histogram and
CDF plot of the MC sample of the individual entry of the shock
matrix, in sample and out of sample fit of the ANOVA model;
* ``<mod_file>_prior_<namendo>_vs_<namexo>_map_SE.fig``: for
entries of the shock matrix it shows graphs of the estimated
first order ANOVA terms :math:`y = f(\theta_i)` for each deep
parameter :math:`\theta_i`;
* ``<mod_file>_prior_<namendo>_vs_<namlagendo>.fig``: histogram
and CDF plot of the MC sample of the individual entry of the
transition matrix, in sample and out of sample fit of the ANOVA
model;
* ``<mod_file>_prior_<namendo>_vs_<namlagendo>_map_SE.fig``: for
entries of the transition matrix it shows graphs of the
estimated first order ANOVA terms :math:`y = f(\theta_i)` for
each deep parameter :math:`\theta_i`;
* ``<mod_file>_prior_<namendo>_vs_<namexo>_map.mat``,
``<mod_file>_<namendo>_vs_<namlagendo>_map.mat``: these files
store info in the estimation;
When option ``logtrans_redform`` is set, the ANOVA estimation is
performed using a log-transformation of each y. The ANOVA mapping is
then transformed back onto the original scale, to allow comparability
with the baseline estimation. Graphs for this log-transformed case,
are stored in the same folder in files denoted with the ``_log``
suffix.
When the option ``threshold_redform`` is set, the analysis is
performed via Monte Carlo filtering, by displaying parameters that
drive the individual entry ``y`` inside the range specified in
``threshold_redform``. If no entry is found (or all entries are in the
range), the MCF algorithm ignores the range specified in
``threshold_redform`` and performs the analysis splitting the MC
sample of ``y`` into deciles. Setting ``threshold_redform=[-inf inf]``
triggers this approach for all ``y``’s.
Results are stored in subdirectories of ``<mod_file>/gsa/redform_prior`` named
* ``<mod_file>_prior_<namendo>_vs_<namlagendo>_threshold``, for
the entries of the transition matrix;
* ``<mod_file>_prior_<namendo>_vs_<namexo>_threshold``, for
entries of the matrix of the shocks.
The files saved are named:
* ``<mod_file>_prior_<namendo>_vs_<namexo>_threshold.fig``, ``<mod_file>_<namendo>_vs_<namlagendo>_threshold.fig``: graphical outputs;
* ``<mod_file>_prior_<namendo>_vs_<namexo>_threshold.mat``, ``<mod_file>_<namendo>_vs_<namlagendo>_threshold.mat``: info on the analysis;
RMSE
^^^^
The RMSE analysis can be performed with different types of sampling options:
1. When ``pprior=1`` and ``ppost=0``, the toolbox analyzes the
RMSEs for the Monte-Carlo sample obtained by sampling
parameters from their prior distributions (or prior ranges):
this analysis provides some hints about what parameter drives
the fit of which observed series, prior to the full estimation;
2. When ``pprior=0`` and ``ppost=0``, the toolbox analyzes the
RMSEs for a multivariate normal Monte-Carlo sample, with
covariance matrix based on the inverse Hessian at the optimum:
this analysis is useful when maximum likelihood estimation is
done (i.e. no Bayesian estimation);
3. When ``ppost=1`` the toolbox analyzes the RMSEs for the
posterior sample obtained by Dynare’s Metropolis procedure.
The use of cases 2 and 3 requires an estimation step beforehand. To
facilitate the sensitivity analysis after estimation, the
``dynare_sensitivity`` command also allows you to indicate some
options of the ``estimation command``. These are:
* ``datafile``
* ``nobs``
* ``first_obs``
* ``prefilter``
* ``presample``
* ``nograph``
* ``nodisplay``
* ``graph_format``
* ``conf_sig``
* ``loglinear``
* ``mode_file``
Binary files produced my RMSE analysis are:
* ``<mod_file>_prior_*.mat``: these files store the filtered and
smoothed variables for the prior Monte-Carlo sample, generated
when doing RMSE analysis (``pprior=1`` and ``ppost=0``);
* ``<mode_file>_mc_*.mat``: these files store the filtered and
smoothed variables for the multivariate normal Monte-Carlo
sample, generated when doing RMSE analysis (``pprior=0`` and
``ppost=0``).
Figure files <mod_file>_rmse_*.fig store results for the RMSE analysis.
* ``<mod_file>_rmse_prior*.fig``: save results for the analysis
using prior Monte-Carlo samples;
* ``<mod_file>_rmse_mc*.fig``: save results for the analysis using
multivariate normal Monte-Carlo samples;
* ``<mod_file>_rmse_post*.fig``: save results for the analysis
using Metropolis posterior samples.
The following types of figures are saved (we show prior sample to fix
ideas, but the same conventions are used for multivariate normal and
posterior):
* ``<mod_file>_rmse_prior_params_*.fig``: for each parameter,
plots the cdfs corresponding to the best 10% RMSEs of each
observed series (only those cdfs below the significance
threshold ``alpha_rmse``);
* ``<mod_file>_rmse_prior_<var_obs>_*.fig``: if a parameter
significantly affects the fit of ``var_obs``, all possible
trade-off’s with other observables for same parameter are
plotted;
* ``<mod_file>_rmse_prior_<var_obs>_map.fig``: plots the MCF
analysis of parameters significantly driving the fit the
observed series ``var_obs``;
* ``<mod_file>_rmse_prior_lnlik*.fig``: for each observed series,
plots in BLUE the cdf of the log-likelihood corresponding to the
best 10% RMSEs, in RED the cdf of the rest of the sample and in
BLACK the cdf of the full sample; this allows one to see the
presence of some idiosyncratic behavior;
* ``<mod_file>_rmse_prior_lnpost*.fig``: for each observed series,
plots in BLUE the cdf of the log-posterior corresponding to the
best 10% RMSEs, in RED the cdf of the rest of the sample and in
BLACK the cdf of the full sample; this allows one to see
idiosyncratic behavior;
* ``<mod_file>_rmse_prior_lnprior*.fig``: for each observed
series, plots in BLUE the cdf of the log-prior corresponding to
the best 10% RMSEs, in RED the cdf of the rest of the sample and
in BLACK the cdf of the full sample; this allows one to see
idiosyncratic behavior;
* ``<mod_file>_rmse_prior_lik.fig``: when ``lik_only=1``, this
shows the MCF tests for the filtering of the best 10%
log-likelihood values;
* ``<mod_file>_rmse_prior_post.fig``: when ``lik_only=1``, this
shows the MCF tests for the filtering of the best 10%
log-posterior values.
Screening Analysis
^^^^^^^^^^^^^^^^^^
Screening analysis does not require any additional options with
respect to those listed in :ref:`Sampling Options <sampl-opt>`. The
toolbox performs all the analyses required and displays results.
The results of the screening analysis with Morris sampling design are
stored in the subfolder ``<mod_file>/gsa/screen``. The data file
``<mod_file>_prior`` stores all the information of the analysis
(Morris sample, reduced form coefficients, etc.).
Screening analysis merely concerns reduced form coefficients. Similar
synthetic bar charts as for the reduced form analysis with Monte-Carlo
samples are saved:
* ``<mod_file>_redform_<endo name>_vs_lags_*.fig``: shows bar
charts of the elementary effect tests for the ten most important
parameters driving the reduced form coefficients of the selected
endogenous variables (``namendo``) versus lagged endogenous
variables (``namlagendo``);
* ``<mod_file>_redform_<endo name>_vs_shocks_*.fig``: shows bar
charts of the elementary effect tests for the ten most important
parameters driving the reduced form coefficients of the selected
endogenous variables (``namendo``) versus exogenous variables
(``namexo``);
* ``<mod_file>_redform_screen.fig``: shows bar chart of all
elementary effect tests for each parameter: this allows one to
identify parameters that have a minor effect for any of the
reduced form coefficients.
Identification Analysis
^^^^^^^^^^^^^^^^^^^^^^^
Setting the option ``identification=1``, an identification analysis
based on theoretical moments is performed. Sensitivity plots are
provided that allow to infer which parameters are most likely to be
less identifiable.
Prerequisite for properly running all the identification routines, is
the keyword ``identification``; in the Dynare model file. This keyword
triggers the computation of analytic derivatives of the model with
respect to estimated parameters and shocks. This is required for
option ``morris=2``, which implements *Iskrev (2010)* identification
analysis.
For example, the placing::
identification;
dynare_sensitivity(identification=1, morris=2);
in the Dynare model file triggers identification analysis using
analytic derivatives as in *Iskrev (2010)*, jointly with the mapping
of the acceptable region.
The identification analysis with derivatives can also be triggered by
the single command::
identification;
This does not do the mapping of acceptable regions for the model and
uses the standard random sampler of Dynare. Additionally, using only
``identification;`` adds two additional identification checks: namely,
of *Qu and Tkachenko (2012)* based on the spectral density and of
*Komunjer and Ng (2011)* based on the minimal state space system.
It completely offsets any use of the sensitivity analysis toolbox.
Markov-switching SBVAR
======================
Given a list of variables, observed variables and a data file, Dynare
can be used to solve a Markov-switching SBVAR model according to
*Sims, Waggoner and Zha (2008)*. [#f10]_ Having done this, you can
create forecasts and compute the marginal data density, regime
probabilities, IRFs, and variance decomposition of the model.
The commands have been modularized, allowing for multiple calls to the
same command within a ``<mod_file>.mod`` file. The default is to use
``<mod_file>`` to tag the input (output) files used (produced) by the
program. Thus, to call any command more than once within a
``<mod_file>.mod`` file, you must use the ``*_tag`` options described
below.
.. command:: markov_switching (OPTIONS...);
|br| Declares the Markov state variable information of a
Markov-switching SBVAR model.
*Options*
.. option:: chain = INTEGER
The Markov chain considered. Default: ``none``.
.. option:: number_of_regimes = INTEGER
Specifies the total number of regimes in the Markov
Chain. This is a required option.
.. option:: duration = DOUBLE | [ROW VECTOR OF DOUBLES]
The duration of the regimes or regimes. This is a required
option. When passed a scalar real number, it specifies the
average duration for all regimes in this chain. When passed a
vector of size equal ``number_of_regimes``, it specifies the
average duration of the associated regimes
(``1:number_of_regimes``) in this chain. An absorbing state
can be specified through the :opt:`restrictions <restrictions
= [[ROW VECTOR OF 3 DOUBLES],[ROW VECTOR OF 3 DOUBLES],...]>`
option.
.. option:: restrictions = [[ROW VECTOR OF 3 DOUBLES],[ROW VECTOR OF 3 DOUBLES],...]
Provides restrictions on this chain’s regime transition
matrix. Its vector argument takes three inputs of the form:
``[current_period_regime, next_period_regime,
transition_probability]``.
The first two entries are positive integers, and the third is
a non-negative real in the set [0,1]. If restrictions are
specified for every transition for a regime, the sum of the
probabilities must be 1. Otherwise, if restrictions are not
provided for every transition for a given regime the sum of
the provided transition probabilities msut be <1. Regardless
of the number of lags, the restrictions are specified for
parameters at time ``t`` since the transition probability for
a parameter at t is equal to that of the parameter at ``t-1``.
In case of estimating a MS-DSGE model, [#f11]_ in addition the
following options are allowed:
.. option:: parameters = [LIST OF PARAMETERS]
This option specifies which parameters are controlled by this
Markov Chain.
.. option:: number_of_lags = DOUBLE
Provides the number of lags that each parameter can take
within each regime in this chain.
*Example*
::
markov_switching(chain=1, duration=2.5, restrictions=[[1,3,0],[3,1,0]]);
Specifies a Markov-switching BVAR with a first chain with 3
regimes that all have a duration of 2.5 periods. The
probability of directly going from regime 1 to regime 3 and
vice versa is 0.
*Example*
::
markov_switching(chain=2, number_of_regimes=3, duration=[0.5, 2.5, 2.5],
parameter=[alpha, rho], number_of_lags=2, restrictions=[[1,3,0],[3,3,1]]);
Specifies a Markov-switching DSGE model with a second chain
with 3 regimes that have durations of 0.5, 2.5, and 2.5
periods, respectively. The switching parameters are ``alpha``
and ``rho``. The probability of directly going from regime 1
to regime 3 is 0, while regime 3 is an absorbing state.
.. command:: svar (OPTIONS...);
|br| Each Markov chain can control the switching of a set of
parameters. We allow the parameters to be divided equation by
equation and by variance or slope and intercept.
*Options*
.. option:: coefficients
Specifies that only the slope and intercept in the given
equations are controlled by the given chain. One, but not
both, of ``coefficients`` or ``variances`` must
appear. Default: ``none``.
.. option:: variances
Specifies that only variances in the given equations are
controlled by the given chain. One, but not both, of
``coefficients`` or ``variances`` must appear. Default:
``none``.
.. option:: equations
Defines the equation controlled by the given chain. If not
specified, then all equations are controlled by
``chain``. Default: ``none``.
.. option:: chain = INTEGER
Specifies a Markov chain defined by
:comm:`markov_switching`. Default: ``none``.
.. command:: sbvar (OPTIONS...);
|br| To be documented. For now, see the wiki:
`<https://archives.dynare.org/DynareWiki/SbvarOptions>`__
*Options*
``datafile``,
``freq``,
``initial_year``,
``initial_subperiod``,
``final_year``,
``final_subperiod``,
``data``,
``vlist``,
``vlistlog``,
``vlistper``,
``restriction_fname``,
``nlags``,
``cross_restrictions``,
``contemp_reduced_form``,
``real_pseudo_forecast``,
``no_bayesian_prior``,
``dummy_obs``,
``nstates``,
``indxscalesstates``,
``alpha``,
``beta``,
``gsig2_lmdm``,
``q_diag``,
``flat_prior``,
``ncsk``,
``nstd``,
``ninv``,
``indxparr``,
``indxovr``,
``aband``,
``indxap``,
``apband``,
``indximf``,
``indxfore``,
``foreband``,
``indxgforhat``,
``indxgimfhat``,
``indxestima``,
``indxgdls``,
``eq_ms``,
``cms``,
``ncms``,
``eq_cms``,
``tlindx``,
``tlnumber``,
``cnum``,
``forecast``,
``coefficients_prior_hyperparameters``
.. block:: svar_identification ;
|br| This block is terminated by ``end;`` and contains lines of the form::
UPPER_CHOLESKY;
LOWER_CHOLESKY;
EXCLUSION CONSTANTS;
EXCLUSION LAG INTEGER; EQUATION INTEGER, VARIABLE_NAME [[,] VARIABLE_NAME...];
RESTRICTION EQUATION INTEGER, EXPRESSION = EXPRESSION;
To be documented. For now, see the wiki:
`<https://archives.dynare.org/DynareWiki/MarkovSwitchingInterface>`__
.. command:: ms_estimation (OPTIONS...);
|br| Triggers the creation of an initialization file for, and the
estimation of, a Markov-switching SBVAR model. At the end of the
run, the :math:`A^0`, :math:`A^+`, :math:`Q` and :math:`\zeta`
matrices are contained in the ``oo_.ms`` structure.
*General Options*
.. option:: file_tag = FILENAME
The portion of the filename associated with this run. This
will create the model initialization file,
``init_<file_tag>.dat``. Default: ``<mod_file>``.
.. option:: output_file_tag = FILENAME
The portion of the output filename that will be assigned to
this run. This will create, among other files,
``est_final_<output_file_tag>.out``,
``est_intermediate_<output_file_tag>.out``. Default:
``<file_tag>``.
.. option:: no_create_init
Do not create an initialization file for the model. Passing
this option will cause the *Initialization Options* to be
ignored. Further, the model will be generated from the output
files associated with the previous estimation run
(i.e. ``est_final_<file_tag>.out``,
``est_intermediate_<file_tag>.out`` or
``init_<file_tag>.dat``, searched for in sequential
order). This functionality can be useful for continuing a
previous estimation run to ensure convergence was reached or
for reusing an initialization file. NB: If this option is not
passed, the files from the previous estimation run will be
overwritten. Default: off (i.e. create initialization file)
*Initialization Options*
.. option:: coefficients_prior_hyperparameters = [DOUBLE1 DOUBLE2 ... DOUBLE6]
Sets the hyper parameters for the model. The six elements of
the argument vector have the following interpretations:
``1``
Overall tightness for :math:`A^0` and :math:`A^+`.
``2``
Relative tightness for :math:`A^+`.
``3``
Relative tightness for the constant term.
``4``
Tightness on lag decay (range: 1.2 - 1.5); a faster decay
produces better inflation process.
``5``
Weight on nvar sums of coeffs dummy observations (unit roots).
``6``
Weight on single dummy initial observation including constant.
Default: ``[1.0 1.0 0.1 1.2 1.0 1.0]``
.. option:: freq = INTEGER | monthly | quarterly | yearly
Frequency of the data (e.g. ``monthly, 12``). Default: ``4``.
.. option:: initial_year = INTEGER
The first year of data. Default: ``none``.
.. option:: initial_subperiod = INTEGER
The first period of data (i.e. for quarterly data, an integer
in ``[1,4]``). Default: ``1``.
.. option:: final_year = INTEGER
The last year of data. Default: Set to encompass entire dataset.
.. option:: final_subperiod = INTEGER
The final period of data (i.e. for monthly data, an integer in
``[1,12]``. Default: When final_year is also missing, set to
encompass entire dataset; when ``final_year`` is indicated,
set to the maximum number of subperiods given the frequency
(i.e. 4 for quarterly data, 12 for monthly,...).
.. option:: datafile = FILENAME
See :ref:`datafile <dataf>`.
.. option:: xls_sheet = QUOTED_STRING
See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
.. option:: xls_range = RANGE
See :opt:`xls_range <xls_range = RANGE>`.
.. option:: nlags = INTEGER
The number of lags in the model. Default: ``1``.
.. option:: cross_restrictions
Use cross :math:`A^0` and :math:`A^+` restrictions. Default: ``off``.
.. option:: contemp_reduced_form
Use contemporaneous recursive reduced form. Default: ``off``.
.. option:: no_bayesian_prior
Do not use Bayesian prior. Default: ``off`` (i.e. use Bayesian prior).
.. option:: alpha = INTEGER
Alpha value for squared time-varying structural shock
lambda. Default: ``1``.
.. option:: beta = INTEGER
Beta value for squared time-varying structural shock
lambda. Default: ``1``.
.. option:: gsig2_lmdm = INTEGER
The variance for each independent :math:`\lambda` parameter
under ``SimsZha`` restrictions. Default: ``50^2``.
.. option:: specification = sims_zha | none
This controls how restrictions are imposed to reduce the
number of parameters. Default: ``Random Walk``.
*Estimation Options*
.. option:: convergence_starting_value = DOUBLE
This is the tolerance criterion for convergence and refers to
changes in the objective function value. It should be rather
loose since it will gradually be tightened during
estimation. Default: ``1e-3``.
.. option:: convergence_ending_value = DOUBLE
The convergence criterion ending value. Values much smaller
than square root machine epsilon are probably
overkill. Default: ``1e-6``.
.. option:: convergence_increment_value = DOUBLE
Determines how quickly the convergence criterion moves from
the starting value to the ending value. Default: ``0.1``.
.. option:: max_iterations_starting_value = INTEGER
This is the maximum number of iterations allowed in the
hill-climbing optimization routine and should be rather small
since it will gradually be increased during
estimation. Default: ``50``.
.. option:: max_iterations_increment_value = DOUBLE
Determines how quickly the maximum number of iterations is
increased. Default: ``2``.
.. option:: max_block_iterations = INTEGER
The parameters are divided into blocks and optimization
proceeds over each block. After a set of blockwise
optimizations are performed, the convergence criterion is
checked and the blockwise optimizations are repeated if the
criterion is violated. This controls the maximum number of
times the blockwise optimization can be performed. Note that
after the blockwise optimizations have converged, a single
optimization over all the parameters is performed before
updating the convergence value and maximum number of
iterations. Default: ``100``.
.. option:: max_repeated_optimization_runs = INTEGER
The entire process described by :opt:`max_block_iterations
<max_block_iterations = INTEGER>` is repeated until
improvement has stopped. This is the maximum number of times
the process is allowed to repeat. Set this to ``0`` to not
allow repetitions. Default: ``10``.
.. option:: function_convergence_criterion = DOUBLE
The convergence criterion for the objective function when
``max_repeated_optimizations_runs`` is positive. Default:
``0.1``.
.. option:: parameter_convergence_criterion = DOUBLE
The convergence criterion for parameter values when
``max_repeated_optimizations_runs`` is positive. Default:
``0.1``.
.. option:: number_of_large_perturbations = INTEGER
The entire process described by :opt:`max_block_iterations
<max_block_iterations = INTEGER>` is repeated with random
starting values drawn from the posterior. This specifies the
number of random starting values used. Set this to ``0`` to
not use random starting values. A larger number should be
specified to ensure that the entire parameter space has been
covered. Default: ``5``.
.. option:: number_of_small_perturbations = INTEGER
The number of small perturbations to make after the large
perturbations have stopped improving. Setting this number much
above ``10`` is probably overkill. Default: ``5``.
.. option:: number_of_posterior_draws_after_perturbation = INTEGER
The number of consecutive posterior draws to make when
producing a small perturbation. Because the posterior draws
are serially correlated, a small number will result in a small
perturbation. Default: ``1``.
.. option:: max_number_of_stages = INTEGER
The small and large perturbation are repeated until
improvement has stopped. This specifies the maximum number of
stages allowed. Default: ``20``.
.. option:: random_function_convergence_criterion = DOUBLE
The convergence criterion for the objective function when
``number_of_large_perturbations`` is positive. Default:
``0.1``.
.. option:: random_parameter_convergence_criterion = DOUBLE
The convergence criterion for parameter values when
``number_of_large_perturbations`` is positive. Default:
``0.1``.
*Example*
::
ms_estimation(datafile=data, initial_year=1959, final_year=2005,
nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0);
ms_estimation(file_tag=second_run, datafile=data, initial_year=1959,
final_year=2005, nlags=4, max_repeated_optimization_runs=1,
max_number_of_stages=0);
ms_estimation(file_tag=second_run, output_file_tag=third_run,
no_create_init, max_repeated_optimization_runs=5,
number_of_large_perturbations=10);
.. command:: ms_simulation ;
ms_simulation (OPTIONS...);
|br| Simulates a Markov-switching SBVAR model.
*Options*
.. option:: file_tag = FILENAME
The portion of the filename associated with the
``ms_estimation`` run. Default: ``<mod_file>``.
.. option:: output_file_tag = FILENAME
The portion of the output filename that will be assigned to
this run. Default: ``<file_tag>``.
.. option:: mh_replic = INTEGER
The number of draws to save. Default: ``10,000``.
.. option:: drop = INTEGER
The number of burn-in draws. Default: ``0.1*mh_replic*thinning_factor``.
.. option:: thinning_factor = INTEGER
The total number of draws is equal to
``thinning_factor*mh_replic+drop``. Default: ``1``.
.. option:: adaptive_mh_draws = INTEGER
Tuning period for Metropolis-Hastings draws. Default: ``30,000``.
.. option:: save_draws
Save all elements of :math:`A^0`, :math:`A^+`, :math:`Q`, and
:math:`\zeta`, to a file named ``draws_<<file_tag>>.out`` with
each draw on a separate line. A file that describes how these
matrices are laid out is contained in
``draws_header_<<file_tag>>.out``. A file called
``load_flat_file.m`` is provided to simplify loading the saved
files into the corresponding variables ``A0``, ``Aplus``,
``Q``, and ``Zeta`` in your MATLAB/Octave workspace. Default:
``off``.
*Example*
::
ms_simulation(file_tag=second_run);
ms_simulation(file_tag=third_run, mh_replic=5000, thinning_factor=3);
.. command:: ms_compute_mdd ;
ms_compute_mdd (OPTIONS...);
|br| Computes the marginal data density of a Markov-switching
SBVAR model from the posterior draws. At the end of the run, the
Muller and Bridged log marginal densities are contained in the
``oo_.ms`` structure.
*Options*
.. option:: file_tag = FILENAME
See :opt:`file_tag <file_tag = FILENAME>`.
.. option:: output_file_tag = FILENAME
See :opt:`output_file_tag <output_file_tag = FILENAME>`.
.. option:: simulation_file_tag = FILENAME
The portion of the filename associated with the simulation
run. Default: ``<file_tag>``.
.. option:: proposal_type = INTEGER
The proposal type:
``1``
Gaussian.
``2``
Power.
``3``
Truncated Power.
``4``
Step.
``5``
Truncated Gaussian.
Default: ``3``
.. option:: proposal_lower_bound = DOUBLE
The lower cutoff in terms of probability. Not used for
``proposal_type`` in ``[1,2]``. Required for all other
proposal types. Default: ``0.1``.
.. option:: proposal_upper_bound = DOUBLE
The upper cutoff in terms of probability. Not used for
``proposal_type`` equal to ``1``. Required for all other
proposal types. Default: ``0.9``.
.. option:: mdd_proposal_draws = INTEGER
The number of proposal draws. Default: ``100,000``.
.. option:: mdd_use_mean_center
Use the posterior mean as center. Default: ``off``.
.. command:: ms_compute_probabilities ;
ms_compute_probabilities (OPTIONS...);
|br| Computes smoothed regime probabilities of a Markov-switching SBVAR
model. Output ``.eps`` files are contained in
``<output_file_tag/Output/Probabilities>``.
*Options*
.. option:: file_tag = FILENAME
See :opt:`file_tag <file_tag = FILENAME>`.
.. option:: output_file_tag = FILENAME
See :opt:`output_file_tag <output_file_tag = FILENAME>`.
.. option:: filtered_probabilities
Filtered probabilities are computed instead of
smoothed. Default: ``off``.
.. option:: real_time_smoothed
Smoothed probabilities are computed based on time ``t``
information for :math:`0\le t\le nobs`. Default: ``off``
.. command:: ms_irf ;
ms_irf (OPTIONS...);
|br| Computes impulse response functions for a Markov-switching SBVAR
model. Output ``.eps`` files are contained in
``<output_file_tag/Output/IRF>``, while data files are contained
in ``<output_file_tag/IRF>``.
*Options*
.. option:: file_tag = FILENAME
See :opt:`file_tag <file_tag = FILENAME>`.
.. option:: output_file_tag = FILENAME
See :opt:`output_file_tag <output_file_tag = FILENAME>`.
.. option:: simulation_file_tag = FILENAME
See :opt:`simulation_file_tag <simulation_file_tag = FILENAME>`.
.. option:: horizon = INTEGER
The forecast horizon. Default: ``12``.
.. option:: filtered_probabilities
Uses filtered probabilities at the end of the sample as
initial conditions for regime probabilities. Only one of
``filtered_probabilities``, ``regime`` and ``regimes`` may be
passed. Default: ``off``.
.. option:: error_band_percentiles = [DOUBLE1 ...]
The percentiles to compute. Default: ``[0.16 0.50 0.84]``. If
``median`` is passed, the default is ``[0.5]``.
.. option:: shock_draws = INTEGER
The number of regime paths to draw. Default: ``10,000``.
.. option:: shocks_per_parameter = INTEGER
The number of regime paths to draw under parameter
uncertainty. Default: ``10``.
.. option:: thinning_factor = INTEGER
Only :math:`1/ \texttt{thinning\_factor}` of the draws in
posterior draws file are used. Default: ``1``.
.. option:: free_parameters = NUMERICAL_VECTOR
A vector of free parameters to initialize theta of the
model. Default: use estimated parameters
.. option:: parameter_uncertainty
Calculate IRFs under parameter uncertainty. Requires that
``ms_simulation`` has been run. Default: ``off``.
.. option:: regime = INTEGER
Given the data and model parameters, what is the ergodic
probability of being in the specified regime. Only one of
``filtered_probabilities``, ``regime`` and ``regimes`` may be
passed. Default: ``off``.
.. option:: regimes
Describes the evolution of regimes. Only one of
``filtered_probabilities``, ``regime`` and ``regimes`` may be
passed. Default: ``off``.
.. option:: median
A shortcut to setting
``error_band_percentiles=[0.5]``. Default: ``off``.
.. command:: ms_forecast ;
ms_forecast (OPTIONS...);
|br| Generates forecasts for a Markov-switching SBVAR
model. Output ``.eps`` files are contained in
``<output_file_tag/Output/Forecast>``, while data files are
contained in ``<output_file_tag/Forecast>``.
*Options*
.. option:: file_tag = FILENAME
See :opt:`file_tag <file_tag = FILENAME>`.
.. option:: output_file_tag = FILENAME
See :opt:`output_file_tag <output_file_tag = FILENAME>`.
.. option:: simulation_file_tag = FILENAME
See :opt:`simulation_file_tag <simulation_file_tag = FILENAME>`.
.. option:: data_obs_nbr = INTEGER
The number of data points included in the output. Default: ``0``.
.. option:: error_band_percentiles = [DOUBLE1 ...]
See :opt:`error_band_percentiles <error_band_percentiles =
[DOUBLE1 ...]>`.
.. option:: shock_draws = INTEGER
See :opt:`shock_draws <shock_draws = INTEGER>`.
.. option:: shocks_per_parameter = INTEGER
See :opt:`shocks_per_parameter <shocks_per_parameter = INTEGER>`.
.. option:: thinning_factor = INTEGER
See :opt:`thinning_factor <thinning_factor = INTEGER>`.
.. option:: free_parameters = NUMERICAL_VECTOR
See :opt:`free_parameters <free_parameters = NUMERICAL_VECTOR>`.
.. option:: parameter_uncertainty
See :opt:`parameter_uncertainty`.
.. option:: regime = INTEGER
See :opt:`regime <regime = INTEGER>`.
.. option:: regimes
See :opt:`regimes`.
.. option:: median
See :opt:`median`.
.. option:: horizon = INTEGER
See :opt:`horizon <horizon = INTEGER>`.
.. command:: ms_variance_decomposition ;
ms_variance_decomposition (OPTIONS...);
|br| Computes the variance decomposition for a Markov-switching
SBVAR model. Output ``.eps`` files are contained in
``<output_file_tag/Output/Variance_Decomposition>``, while data
files are contained in
``<output_file_tag/Variance_Decomposition>``.
*Options*
.. option:: file_tag = FILENAME
See :opt:`file_tag <file_tag = FILENAME>`.
.. option:: output_file_tag = FILENAME
See :opt:`output_file_tag <output_file_tag = FILENAME>`.
.. option:: simulation_file_tag = FILENAME
See :opt:`simulation_file_tag <simulation_file_tag = FILENAME>`.
.. option:: horizon = INTEGER
See :opt:`horizon <horizon = INTEGER>`.
.. option:: filtered_probabilities
See :opt:`filtered_probabilities`.
.. option:: no_error_bands
Do not output percentile error bands (i.e. compute
mean). Default: ``off`` (i.e. output error bands)
.. option:: error_band_percentiles = [DOUBLE1 ...]
See :opt:`error_band_percentiles <error_band_percentiles =
[DOUBLE1 ...]>`.
.. option:: shock_draws = INTEGER
See :opt:`shock_draws <shock_draws = INTEGER>`.
.. option:: shocks_per_parameter = INTEGER
See :opt:`shocks_per_parameter <shocks_per_parameter = INTEGER>`.
.. option:: thinning_factor = INTEGER
See :opt:`thinning_factor <thinning_factor = INTEGER>`.
.. option:: free_parameters = NUMERICAL_VECTOR
See :opt:`free_parameters <free_parameters = NUMERICAL_VECTOR>`.
.. option:: parameter_uncertainty
See :opt:`parameter_uncertainty`.
.. option:: regime = INTEGER
See :opt:`regime <regime = INTEGER>`.
.. option:: regimes
See :opt:`regimes`.
.. _epilogue:
Epilogue Variables
==================
.. block:: epilogue ;
The epilogue block is useful for computing output variables of interest that
may not be necessarily defined in the model (e.g. various kinds of real/nominal
shares or relative prices, or annualized variables out of a quarterly model).
It can also provide several advantages in terms of computational efficiency and
flexibility:
- You can calculate variables in the epilogue block after smoothers/simulations
have already been run without adding the new definitions and equations and
rerunning smoothers/simulations. Even posterior smoother subdraws can be
recycled for computing epilogue variables without rerunning subdraws with the
new definitions and equations.
- You can also reduce the state space dimension in data
filtering/smoothing. Assume, for example, you want annualized variables as
outputs. If you define an annual growth rate in a quarterly model, you need
lags up to order 7 of the associated quarterly variable; in a medium/large
scale model this would just blow up the state dimension and increase by a
huge amount the computing time of a smoother.
The ``epilogue`` block is terminated by ``end;`` and contains lines of the
form:
NAME = EXPRESSION;
*Example*
::
epilogue;
// annualized level of y
ya = exp(y)+exp(y(-1))+exp(y(-2))+exp(y(-3));
// annualized growth rate of y
gya = ya/ya(-4)-1;
end;
.. _semi-strutural:
Semi-structural models
======================
Dynare provides tools for semi-structural models, in the vain of the FRB/US
model (see *Brayton and Tinsley (1996)*), where expectations are not necessarily
model consistent but based on a VAR auxiliary model. In the following, it is
assumed that each equation is written as ``VARIABLE = EXPRESSION`` or
``T(VARIABLE) = EXPRESSION`` where ``T(VARIABLE)`` stands for a transformation
of an endogenous variable (``log`` or ``diff``). This representation, where each
equation determines the endogenous variable on the LHS, can be exploited when
simulating the model (see algorithms 12 and 14 in :ref:`solve_algo <solvalg>`)
and is mandatory to define auxiliary models used for computing expectations (see
below).
Auxiliary models
----------------
The two auxiliary models defined in this section are linear backward-looking models
used to form expectations. Both models can be recast as VAR(1)-processes and
therefore offer isomorphic ways of specifying the expectations process, but differ
in their convenience of specifying features like cointegration and error correction.
``var_model`` directly specifies a VAR, while ``trend_component_model`` allows to define
a trend target to which the endogenous variables may be attracted in the long-run
(i.e. an error correction model).
.. command:: var_model (OPTIONS...);
|br| Picks equations in the ``model`` block to form a VAR model. This model can
be used as an auxiliary model in ``var_expectation_model`` or
``pac_model``. It must be of the following form:
.. math ::
Y_t = \mathbf{c} + \sum_{i=1}^p A_i Y_{t-i} + \varepsilon_t
or
.. math ::
A_0 Y_t = \mathbf{c} + \sum_{i=1}^p A_i Y_{t-i} + \varepsilon_t
if the VAR is structural (see below), where :math:`Y_t` and
:math:`\varepsilon_t` are :math:`n\times 1` vectors, :math:`\mathbf{c}` is a
:math:`n\times 1` vector of parameters, :math:`A_i` (:math:`i=0,\ldots,p`)
are :math:`n\times n` matrices of parameters, and :math:`A_0` is non
singular square matrix. Vector :math:`\mathbf{c}` and matrices :math:`A_i`
(:math:`i=0,\ldots,p`) are set by Dynare by parsing the equations in the
``model`` block. Then, Dynare builds a VAR(1)-companion form model for
:math:`\mathcal{Y}_t = (1, Y_t, \ldots, Y_{t-p+1})'` as:
.. math ::
\begin{pmatrix}
1\\
Y_t\\
Y_{t-1}\\
\vdots\\
\vdots\\
Y_{t-p+1}
\end{pmatrix}
=
\underbrace{
\begin{pmatrix}
1 & 0_n' & \ldots & \ldots & \ldots & 0_n'\\
\mathbf{c} & A_1 & A_2 & \ldots & \ldots & A_p\\
0_n & I_n & O_n & \ldots & \ldots & O_n\\
0_n & O_n & I_n & O_n & \ldots & O_n\\
\vdots & O_n & \ddots & \ddots & \ddots & \vdots \\
0_n & O_n & \ldots & O_n & I_n & O_n
\end{pmatrix}}_{\mathcal{C}}
\begin{pmatrix}
1\\
Y_{t-1}\\
Y_{t-2}\\
\vdots\\
\vdots\\
Y_{t-p}
\end{pmatrix}
+
\underbrace{
\begin{pmatrix}
0\\
\varepsilon_t\\
0_n\\
\vdots\\
\vdots\\
0_n
\end{pmatrix}}_{\mathcal{\epsilon}_t}
assuming that we are dealing with a reduced form VAR (otherwise, the right-hand
side would additionally be premultiplied by :math:`A_0^{-1}.` to obtain the reduced
for representation). If the VAR does not have a constant, we remove the first
line of the system and the first column of the companion matrix
:math:`\mathcal{C}.` Dynare only saves the companion matrix, since that
is the only information required to compute the expectations.
.. matvar:: oo_.var.MODEL_NAME.CompanionMatrix
Reduced form companion matrix of the ``var_model``.
*Options*
.. option:: model_name = STRING
Name of the VAR model, which will be referenced in ``var_expectation_model`` or ``pac_model``
as an ``auxiliary_model``. Needs to be a valid MATLAB field name.
.. option:: eqtags = [QUOTED_STRING[, QUOTED_STRING[, ...]]]
List of equations in the ``model`` block (referenced using the equation tag ``name``) used to build the VAR model.
.. option:: structural
By default the VAR model is not structural, *i.e.* each equation must
contain exactly one contemporaneous variable (on the LHS). If the
``structural`` option is provided then any variable defined in the system
can appear at time :math:`t` in each equation. Internally Dynare will
rewrite this model as a reduced form VAR (by inverting the implied matrix :math:`A_0`).
*Example*
::
var_model(model_name = toto, eqtags = [ 'X', 'Y', 'Z' ]);
model;
[ name = 'X' ]
x = a*x(-1) + b*x(-2) + c*z(-2) + e_x;
[ name = 'Z' ]
z = f*z(-1) + e_z;
[ name = 'Y' ]
y = d*y(-2) + e*z(-1) + e_y;
end;
.. command:: trend_component_model (OPTIONS...);
|br| Picks equations in the model block to form a trend component model. This
model can be used as an auxiliary model in ``var_expectation_model`` or
``pac_model``. It must be of the following form:
.. math ::
\begin{cases}
\Delta X_t &= A_0 (X_{t-1}-C_0 Z_{t-1}) + \sum_{i=1}^p A_i \Delta X_{t-i} + \varepsilon_t\\
Z_t &= Z_{t-1} + \eta_t
\end{cases}
where :math:`X_t` and :math:`Z_t` are :math:`n\times 1` and :math:`m\times 1` vectors
of endogenous variables. :math:`Z_t` defines the trend target to whose linear combination
:math:`C_0 Z_t` the endogenous variables :math:`X_t` will be attracted, provided the implied
error correction matrix :math:`A_0` is negative definite. :math:`\varepsilon_t` and :math:`\eta_t` are :math:`n\times 1`
and :math:`m\times 1` vectors of exogenous variables, :math:`A_i` (:math:`i=0,\ldots,p`)
are :math:`n\times n` matrices of parameters, and :math:`C_0` is a :math:`n\times m` matrix.
This model can also be cast into a VAR(1) model by first rewriting it in levels. Let
:math:`Y_t = (X_t',Z_t')'` and :math:`\zeta_t = (\varepsilon_t',\eta_t')'`. Then we have:
.. math ::
Y_t = \sum_{i=1}^{p+1} B_i Y_{t-i} + \zeta_t
with
.. math ::
B_1 = \begin{pmatrix}
I_n+A_0+A_1 & -\Lambda\\
O_{m,n} & I_m
\end{pmatrix}
where :math:`\Lambda = A_0C_0`,
.. math ::
B_i = \begin{pmatrix}
A_i-A_{i-1} & O_{n,m}\\
O_{m,n} & O_m
\end{pmatrix}
for :math:`i=2,\ldots,p`, and
.. math ::
B_{p+1} = \begin{pmatrix}
-A_p & O_{n,m}\\
O_{m,n} & O_m
\end{pmatrix}
This VAR(p+1) in levels can again be rewritten as a VAR(1)-companion model form.
.. matvar:: oo_.trend_component.MODEL_NAME.CompanionMatrix
Reduced form companion matrix of the ``trend_component_model``.
*Options*
.. option:: model_name = STRING
Name of the trend component model, will be referenced in ``var_expectation_model``
or ``pac_model`` as an ``auxiliary_model``. Needs to be a valid MATLAB field name.
.. option:: eqtags = [QUOTED_STRING[, QUOTED_STRING[, ...]]]
List of equations in the ``model`` block (referenced using the equation tag ``name``) used to build the trend component model.
.. option:: targets = [QUOTED_STRING[, QUOTED_STRING[, ...]]]
List of targets, corresponding to the variables in vector :math:`Z_t`, referenced using the equation tag ``name``) of the associated equation in the ``model`` block. ``target`` must be a subset of ``eqtags``.
*Example*
::
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
model;
[name='eq:x1']
diff(x1) = a_x1_0*(x1(-1)-x1bar(-1))+a_x1_0_*(x2(-1)-x2bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
[name='eq:x2']
diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
[name='eq:x1bar']
x1bar = x1bar(-1) + ex1bar;
[name='eq:x2bar']
x2bar = x2bar(-1) + ex2bar;
end;
VAR expectations
----------------
Suppose we wish to forecast a variable :math:`y_t` and that
:math:`y_t` is an element of vector of variables :math:`\mathcal{Y}_t` whose law of
motion is described by a VAR(1) model :math:`\mathcal{Y}_t =
\mathcal{C}\mathcal{Y}_{t-1}+\epsilon_t`. More generally, :math:`y_t` may
be a linear combination of the scalar variables in
:math:`\mathcal{Y}_t`. Let the vector :math:`\alpha` be such that
:math:`y_t = \alpha'\mathcal{Y}_t` (:math:`\alpha` is a selection
vector if :math:`y_t` is a variable in :math:`\mathcal{Y}_t`, *i.e.* a
column of an identity matrix, or an arbitrary vector defining the
weights of a linear combination). Then the best prediction, in the sense of the minimisation of the RMSE, for
:math:`y_{t+h}` given the information set at :math:`t-\tau` (which we assume to include all observables
up to time :math:`t-\tau`, :math:`\mathcal{Y}_{\underline{t-\tau}}`) is:
.. math ::
y_{t+h|t-\tau} = \mathbb E[y_{t+h}|\mathcal{Y}_{\underline{t-\tau}}] = \alpha\mathcal{C}^{h+\tau} \mathcal{Y}_{t-\tau}
In a semi-structural model, variables appearing in :math:`t+h` (*e.g.*
the expected output gap in a dynamic IS curve or expected inflation in a
(New Keynesian) Phillips curve) will be replaced by the expectation implied by an auxiliary VAR
model. Another use case is for the computation of permanent
incomes. Typically, consumption will depend on something like:
.. math ::
\sum_{h=0}^{\infty} \beta^h y_{t+h|t-\tau}
Assuming that $0<\beta<1$ and knowing the limit of geometric series, the conditional expectation of this variable can be evaluated based on the same auxiliary model:
.. math ::
\mathbb E \left[\sum_{h=0}^{\infty} \beta^h y_{t+h}\Biggl| \mathcal{Y}_{\underline{t-\tau}}\right] = \alpha \mathcal{C}^\tau(I-\beta\mathcal{C})^{-1}\mathcal{Y}_{t-\tau}
More generally, it is possible to consider finite discounted sums.
.. command:: var_expectation_model (OPTIONS...);
|br| Declares a model used to forecast an endogenous variable or linear
combination of variables in :math:`t+h`. More generally, the same model can
be used to forecast the discounted flow of a variable or a linear expression of variables:
.. math ::
\sum_{h=a}^b \mathbb \beta^{h-\tau}\mathbb E[y_{t+h}|\mathcal{Y}_{\underline{t-\tau}}]
where :math:`(a,b)\in\mathbb N^2` with :math:`a<b`, :math:`\beta\in(0,1]` is a discount factor,
and :math:`\tau` is a finite positive integer.
*Options*
.. option:: model_name = STRING
Name of the VAR based expectation model, which will be referenced in the ``model`` block.
.. option:: auxiliary_model = STRING
Name of the associated auxiliary model, defined with ``var_model`` or ``trend_component_model``.
.. option:: expression = VARIABLE_NAME | EXPRESSION
Name of the variable or expression (linear combination of variables) to be expected.
.. option:: discount = PARAMETER_NAME | DOUBLE
Discount factor (:math:`\beta`).
.. option:: horizon = INTEGER | [INTEGER:INTEGER]
The upper limit :math:`b` of the horizon :math:`h` (in which case :math:`a=0`), or range of periods
:math:`a:b` over which the discounted sum is computed (the upper bound can be ``Inf``).
.. option:: time_shift = INTEGER
Shift of the information set (:math:`\tau`), default value is 0.
.. operator:: var_expectation (NAME_OF_VAR_EXPECTATION_MODEL);
|br| This operator is used instead of a leaded variable, e.g. ``X(1)``, in the
``model`` block to substitute a model-consistent forecast with a forecast
based on a VAR model.
*Example*
::
var_model(model_name=toto, eqtags=['X', 'Y', 'Z']);
var_expectation_model(model_name=varexp, expression=x, auxiliary_model_name=toto, horizon=1, discount=beta);
model;
[name='X']
x = a*x(-1) + b*x(-2) + c*z(-2) + e_x;
[name='Z']
z = f*z(-1) + e_z;
[name='Y']
y = d*y(-2) + e*z(-1) + e_y;
foo = .5*foo(-1) + var_expectation(varexp);
end;
In this example ``var_expectation(varexp)`` stands for the one step ahead expectation of ``x``, as a replacement for ``x(1)``.
.. matcomm:: var_expectation.initialize(NAME_OF_VAR_EXPECTATION_MODEL);
|br| Initialise the ``var_expectation_model`` by building the companion matrix
of the associated auxiliary ``var_model``. Needs to be executed before attempts to simulate or
estimate the model.
|br|
.. matcomm:: var_expectation.update(NAME_OF_VAR_EXPECTATION_MODEL);
|br| Update/compute the reduced form parameters of ``var_expectation_model``. Needs to be executed
before attempts to simulate or estimate the model and requires the auxiliary ``var_model`` to have
previously been initialized.
|br|
*Example (continued)*
::
var_expectation.initialize('varexp');
var_expectation.update('varexp');
.. warning:: Changes to the parameters of the underlying auxiliary ``var_model`` require calls to
``var_expectation.initialize`` and ``var_expectation.update`` to become effective. Changes to the
``var_expectation_model`` or its associated parameters require a call to ``var_expectation.update``.
PAC equation
------------
In its simplest form, a PAC equation breaks down changes in a variable of
interest :math:`y` into three contributions: (*i*) the lagged deviation from a
target :math:`y^{\star}`, (*ii*) the lagged changes in the variable :math:`y`,
and (*iii*) the expected changes in the target :math:`y^{\star}`:
.. math ::
\Delta y_t = a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i} +\varepsilon_t
*Brayton et alii (2000)* shows how such an equation can be derived from the
minimisation of a quadratic cost function penalising expected deviations from
the target and non-smoothness of :math:`y`, where future costs are discounted
(with discount factor :math:`\beta`). They also show that the parameters
:math:`(d_i)_{i\in\mathbb N}` are non-linear functions of the :math:`m`
parameters :math:`a_i` and the discount factor :math:`\beta`. To simulate or
estimate this equation we need to figure out how to determine the expected
changes of the target. This can be done as in the previous section using VAR
based expectations, or considering model consistent expectations (MCE).
To ensure that the endogenous variable :math:`y` is equal to its target
:math:`y^{\star}` in the (deterministic) long run, *i.e.* that the error
correction term is zero in the long run, we can optionally add a growth neutrality
correction to this equation. Suppose that :math:`g` is the long run growth rate, for
:math:`y` and :math:`y^{\star}`, then in the long run (assuming that the data
are in logs) we must have:
.. math ::
g = a_0(y^{\star}_{\infty}-y_{\infty}) + g\sum_{i=1}^{m-1} a_i + g\sum_{i=0}^{\infty} d_i
\Leftrightarrow a_0(y^{\star}_{\infty}-y_{\infty}) = \left(1-\sum_{i=1}^{m-1} a_i-\sum_{i=0}^{\infty} d_i\right) g
Unless additional restrictions are placed on the coefficients
:math:`(a_i)_{i=0}^{m-1}`, i.e. on the form of the minimised cost function, there is
no reason for the right-hand side to be zero. Instead, we can optionally add the
right hand side to the PAC equation, to ensure that the error correction term is
asymptotically zero.
The PAC equations can be generalised by adding exogenous variables. This can be
done in two, non exclusive, manners. We can replace the PAC equation by a convex
combination of the original PAC equation (derived from an optimisation program)
and a linear expression involving exogenous variables (referred as the rule of thumb part as
opposed to the part derived from the minimisation of a cost function; not to be confused with
exogenous shocks):
.. math ::
\Delta y_t = \lambda \left(a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}\right) + (1-\lambda)\gamma'X_t +\varepsilon_t
where :math:`\lambda\in[0,1]` is the weight of the pure PAC equation,
:math:`\gamma` is a :math:`k\times 1` vector of parameters, and :math:`X_t` a
:math:`k\times 1` vector of variables in the rule of thumb part. Or we can
simply add the exogenous variables to the PAC equation (without the weight
:math:`\lambda`):
.. math ::
\Delta y_t = a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i} + \gamma'X_t +\varepsilon_t
.. command:: pac_model (OPTIONS...);
|br| Declares a PAC model. A ``.mod`` file can have more than one PAC model or PAC equation, but each PAC equation must be associated to a different PAC model.
*Options*
.. option:: model_name = STRING
Name of the PAC model, will be referenced in the ``model`` block.
.. option:: auxiliary_model = STRING
Name of the associated auxiliary model, defined with ``var_model`` or
``trend_component_model``, to compute the VAR based expectations for the
expected changes in the target, *i.e.* to evaluate
:math:`\sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}`. The infinite sum
will then be replaced by a linear combination of the variables involved in
the companion representation of the auxiliary model. The weights defining
the linear combination are nonlinear functions of the
:math:`(a_i)_{i=0}^{m-1}` coefficients in the PAC equation. This option is
not mandatory, if absent Dynare understands that the expected changes of the
target have to be computed under the MCE assumption. This is done by
rewriting recursively the infinite sum as shown in equation 10 of *Brayton et alii (2000)*.
.. option:: discount = PARAMETER_NAME | DOUBLE
Discount factor (:math:`\beta`) for future expected costs appearing in the
definition of the cost function.
.. option:: growth = PARAMETER_NAME | VARIABLE_NAME | EXPRESSION | DOUBLE
If present a growth neutrality correction is added to the PAC equation. The
user must ensure that the provided value (or long term level if a variable
or expression is given) is consistent with the asymptotic growth rate of the
endogenous variable.
.. operator:: pac_expectation (NAME_OF_PAC_MODEL);
|br| This operator is used instead of the infinite sum,
:math:`\sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}`, in a PAC equation
defined in the ``model`` block. Depending on the assumption regarding the
formation of expectations, it will be replaced by a linear combination of
the variables involved in the companion representation of the auxiliary model
or by a recursive forward equation.
|br|
.. matcomm:: pac.initialize(NAME_OF_PAC_MODEL);
.. matcomm:: pac.update(NAME_OF_PAC_MODEL);
|br| Same as in the previous section for the VAR expectations, initialise the
PAC model, by building the companion matrix of the auxiliary model, and
computes the reduced form parameters of the PAC equation (the weights in the
linear combination of the variables involved in the companion representation
of the auxiliary model, or the parameters of the recursive representation of
the infinite sum in the MCE case).
*Example*
::
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
pac_model(auxiliary_model_name=toto, discount=beta, growth=diff(x1(-1)), model_name=pacman);
model;
[name='eq:y']
y = rho_1*y(-1) + rho_2*y(-2) + ey;
[name='eq:x1']
diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
[name='eq:x2']
diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
[name='eq:x1bar']
x1bar = x1bar(-1) + ex1bar;
[name='eq:x2bar']
x2bar = x2bar(-1) + ex2bar;
[name='zpac']
diff(z) = e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman) + ez;
end;
pac.initialize('pacman');
pac.update.expectation('pacman');
Estimation of a PAC equation
----------------------------
The PAC equation, introduced in the previous section, can be estimated. This
equation is nonlinear with respect to the estimated parameters
:math:`(a_i)_{i=0}^{m-1}`, since the reduced form parameters (in the computation
of the infinite sum) are nonlinear functions of the autoregressive parameters
and the error correction parameter. *Brayton et alii (2000)* shows how to
estimate the PAC equation by iterative OLS. Although this approach is
implemented in Dynare, mainly for comparison purposes, we also propose NLS
estimation, which is much preferable (asymptotic properties of NLS being more
solidly grounded).
Note that it is currently not feasible to estimate the PAC equation jointly with
the remaining parameters of the model using e.g. Bayesian techniques. Thus, estimation
of the PAC equation can only be conducted conditional on the values of the parameters
of the auxiliary model.
.. warning:: The estimation routines described below require the option
``json=compute`` be passed to the preprocessor (via the command line
or at the top of the ``.mod`` file, see :ref:`dyn-invoc`).
.. matcomm:: pac.estimate.nls(EQNAME, GUESS, DATA, RANGE[, ALGO]);
.. matcomm:: pac.estimate.iterative_ols(EQNAME, GUESS, DATA, RANGE);
|br| Trigger the NLS or iterative OLS estimation of a PAC
equation. ``EQNAME`` is a row char array designating the PAC
equation to be estimated (the PAC equation must have a name
specified with an equation tag). ``DATA`` is a ``dseries`` object
containing the data required for the estimation (*i.e.* data for
all the endogenous and exogenous variables in the equation). The
residual values of the PAC equation (which correspond to a defined
`varexo`) must also be a member of ``DATA``,
but filled with ``NaN`` values. ``RANGE`` is a ``dates`` object
defining the time span of the sample. ``ALGO`` is a row char array
used to select the method (or minimisation algorithm) for NLS.
Possible values are : ``'fmincon'``, ``'fminunc'``,
``'fminsearch'``, ``'lsqnonlin'``, ``'particleswarm'``,
``'csminwel'``, ``'simplex'``, ``'annealing'``, and
``'GaussNewton'``. The first four algorithms require the Mathworks
Optimisation toolbox. The fifth algorithm requires the Mathworks
Global Optimisation toolbox. When the optimisation algorithm
allows it, we impose constraints on the error correction
parameter, which must be positive and smaller than 1 (it the case
for ``'fmincon'``, ``'lsqnonlin'``, ``'particleswarm'``, and
``'annealing'``). The default optimisation algorithm is
``'csminwel'``. ``GUESS`` is a structure containing the initial
guess values for the estimated parameters. Each field is the name
of a parameter in the PAC equation and holds the initial guess for
this parameter. If some parameters are calibrated, then they
should not be members of the ``GUESS`` structure (and values have
to be provided in the ``.mod`` file before the call to the
estimation routine).
For the NLS routine the estimation results are displayed in a
table after the estimation. For both the NLS and iterative OLS
routines, the results are saved in ``oo_`` (under the fields
``nls`` or ``iterative_ols``). Also, the values of the parameters
are updated in ``M_.params``.
*Example (continued)*
::
// Set the initial guess for the estimated parameters
eparams.e_c_m = .9;
eparams.c_z_1 = .5;
eparams.c_z_2 = .2;
// Define the dataset used for estimation
edata = TrueData;
edata.ez = dseries(NaN); // Set to NaN the residual of the equation.
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+200, 'annealing');
.. warning:: The specification of `GUESS` and `DATA` involves the use of structures.
As such, their subfields will not be cleared across Dynare runs as the structures
stay in the workspace. Be careful to clear these structures from the memory
(e.g. within the ``.mod`` file) when e.g. changing which parameters are calibrated.
Displaying and saving results
=============================
Dynare has comments to plot the results of a simulation and to save the results.
.. command:: rplot VARIABLE_NAME...;
|br| Plots the simulated path of one or several variables, as
stored in ``oo_.endo_simul`` by either
``perfect_foresight_solver``, ``simul`` (see :ref:`det-simul`) or
``stoch_simul`` with option ``periods`` (see
:ref:`stoch-sol`). The variables are plotted in levels.
.. command:: dynatype (FILENAME) [VARIABLE_NAME...];
|br| This command prints the listed endogenous or exogenous variables in a text file named
FILENAME. If no VARIABLE_NAME is listed, all endogenous variables
are printed.
.. command:: dynasave (FILENAME) [VARIABLE_NAME...];
|br| This command saves the listed endogenous or exogenous variables in a binary file
named FILENAME. If no VARIABLE_NAME is listed, all endogenous
variables are saved.
In MATLAB or Octave, variables saved with the ``dynasave`` command
can be retrieved by the command::
load(FILENAME,'-mat')
.. _macro-proc-lang:
Macro processing language
=========================
It is possible to use “macro” commands in the ``.mod`` file for performing
tasks such as: including modular source files, replicating blocks of equations
through loops, conditionally executing some code, writing indexed sums or
products inside equations...
The Dynare macro-language provides a new set of *macro-commands* which can be
used in ``.mod`` files. It features:
* File inclusion
* Loops (``for`` structure)
* Conditional inclusion (``if/then/else`` structures)
* Expression substitution
This macro-language is totally independent of the basic
Dynare language, and is processed by a separate component of the
Dynare pre-processor. The macro processor transforms a ``.mod`` file
with macros into a ``.mod`` file without macros (doing
expansions/inclusions), and then feeds it to the Dynare parser. The
key point to understand is that the macro processor only does text
substitution (like the C preprocessor or the PHP language). Note that
it is possible to see the output of the macro processor by using the
``savemacro`` option of the ``dynare`` command (see :ref:`dyn-invoc`).
The macro processor is invoked by placing *macro directives* in the ``.mod``
file. Directives begin with an at-sign followed by a pound sign (``@#``). They
produce no output, but give instructions to the macro processor. In most cases,
directives occupy exactly one line of text. If needed, two backslashes (``\\``)
at the end of the line indicate that the directive is continued on the next
line. Macro directives following ``//`` are not interpreted by the macro
processor. For historical reasons, directives in commented blocks, *ie*
surrounded by ``/*`` and ``*/``, are interpreted by the macro processor. The
user should not rely on this behavior. The main directives are:
* ``@#includepath``, paths to search for files that are to be included,
* ``@#include``, for file inclusion,
* ``@#define``, for defining a macro processor variable,
* ``@#if, @#ifdef, @#ifndef, @#elseif, @#else, @#endif`` for conditional statements,
* ``@#for, @#endfor`` for constructing loops.
The macro processor maintains its own list of variables (distinct from model
variables and MATLAB/Octave variables). These macro-variables are assigned
using the ``@#define`` directive and can be of the following basic types:
boolean, real, string, tuple, function, and array (of any of the previous
types).
.. _macro-exp:
Macro expressions
-----------------
Macro-expressions can be used in two places:
* Inside macro directives, directly;
* In the body of the ``.mod`` file, between an at-sign and curly
braces (like ``@{expr}``): the macro processor will substitute
the expression with its value
It is possible to construct macro-expressions that can be assigned to
macro-variables or used within a macro-directive. The expressions are
constructed using literals of the basic types (boolean, real, string, tuple,
array), comprehensions, macro-variables, macro-functions, and standard
operators.
.. note::
Elsewhere in the manual, MACRO_EXPRESSION designates an expression
constructed as explained in this section.
.. rubric:: Boolean
The following operators can be used on booleans:
* Comparison operators: ``==, !=``
* Logical operators: ``&&, ||, !``
.. rubric:: Real
The following operators can be used on reals:
* Arithmetic operators: ``+, -, *, /, ^``
* Comparison operators: ``<, >, <=, >=, ==, !=``
* Logical operators: ``&&, ||, !``
* Ranges with an increment of ``1``: ``REAL1:REAL2`` (for example, ``1:4``
is equivalent to real array ``[1, 2, 3, 4]``).
.. versionchanged:: 4.6 Previously, putting brackets around the arguments
to the colon operator (e.g. ``[1:4]``) had no effect. Now, ``[1:4]``
will create an array containing an array (i.e. ``[ [1, 2, 3, 4] ]``).
* Ranges with user-defined increment: ``REAL1:REAL2:REAL3`` (for example,
``6:-2.1:-1`` is equivalent to real array ``[6, 3.9, 1.8, -0.3]``).
* Functions: ``max, min, mod, exp, log, log10, sin, cos, tan, asin, acos,
atan, sqrt, cbrt, sign, floor, ceil, trunc, erf, erfc, gamma, lgamma,
round, normpdf, normcdf``. NB ``ln`` can be used instead of ``log``
.. rubric:: String
String literals have to be enclosed by **double** quotes (like ``"name"``).
The following operators can be used on strings:
* Comparison operators: ``<, >, <=, >=, ==, !=``
* Concatenation of two strings: ``+``
* Extraction of substrings: if ``s`` is a string, then ``s[3]`` is
a string containing only the third character of ``s``, and
``s[4:6]`` contains the characters from 4th to 6th
* Function: ``length``
.. rubric:: Tuple
Tuples are enclosed by parenthesis and elements separated by commas (like
``(a,b,c)`` or ``(1,2,3)``).
The following operators can be used on tuples:
* Comparison operators: ``==, !=``
* Functions: ``empty, length``
.. rubric:: Array
Arrays are enclosed by brackets, and their elements are separated
by commas (like ``[1,[2,3],4]`` or ``["US", "FR"]``).
The following operators can be used on arrays:
* Comparison operators: ``==, !=``
* Dereferencing: if ``v`` is an array, then ``v[2]`` is its 2nd element
* Concatenation of two arrays: ``+``
* Set union of two arrays: ``|``
* Set intersection of two arrays: ``&``
* Difference ``-``: returns the first operand from which the
elements of the second operand have been removed.
* Cartesian product of two arrays: ``*``
* Cartesian product of one array N times: ``^N``
* Extraction of sub-arrays: e.g. ``v[4:6]``
* Testing membership of an array: ``in`` operator (for example:
``"b"`` in ``["a", "b", "c"]`` returns ``1``)
* Functions: ``empty, sum, length``
.. rubric:: Comprehension
Comprehension syntax is a shorthand way to make arrays from other arrays. There
are three different ways the comprehension syntax can be employed: `filtering`,
`mapping`, and `filtering and mapping`.
**Filtering**
Filtering allows one to choose those elements from an array for which a certain
condition hold.
*Example*
Create a new array, choosing the even numbers from the array ``1:5``::
[ i in 1:5 when mod(i,2) == 0 ]
would result in::
[2, 4]
**Mapping**
Mapping allows you to apply a transformation to every element of an array.
*Example*
Create a new array, squaring all elements of the array ``1:5``::
[ i^2 for i in 1:5 ]
would result in::
[1, 4, 9, 16, 25]
**Filtering and Mapping**
Combining the two preceding ideas would allow one to apply a transformation to
every selected element of an array.
*Example*
Create a new array, squaring all even elements of the array ``1:5``::
[ i^2 for i in 1:5 when mod(i,2) == 0]
would result in::
[4, 16]
*Further Examples*
::
[ (j, i+1) for (i,j) in (1:2)^2 ]
[ (j, i+1) for (i,j) in (1:2)*(1:2) when i < j ]
would result in::
[(1, 2), (2, 2), (1, 3), (2, 3)]
[(2, 2)]
.. rubric:: Function
Functions can be defined in the macro processor using the ``@#define``
directive (see below). A function is evaluated at the time it is invoked, not
at define time. Functions can be included in expressions and the operators that
can be combined with them depend on their return type.
.. rubric:: Checking variable type
Given a variable name or literal, you can check the type it evaluates to using
the following functions: ``isboolean``, ``isreal``, ``isstring``, ``istuple``,
and ``isarray``.
*Examples*
+---------------------+------------+
| **Code** | **Output** |
+=====================+============+
| ``isboolean(0)`` | ``false`` |
+---------------------+------------+
| ``isboolean(true)`` | ``true`` |
+---------------------+------------+
| ``isreal("str")`` | ``false`` |
+---------------------+------------+
.. rubric:: Casting between types
Variables and literals of one type can be cast into another type. Some type
changes are straightforward (e.g. changing a `real` to a `string`) whereas
others have certain requirements (e.g. to cast an `array` to a `real` it must
be a one element array containing a type that can be cast to `real`).
*Examples*
+----------------------------------+------------+
| **Code** | **Output** |
+==================================+============+
| ``(bool) -1.1`` | ``true`` |
+----------------------------------+------------+
| ``(bool) 0`` | ``false`` |
+----------------------------------+------------+
| ``(real) "2.2"`` | ``2.2`` |
+----------------------------------+------------+
| ``(tuple) [3.3]`` | ``(3.3)`` |
+----------------------------------+------------+
| ``(array) 4.4`` | ``[4.4]`` |
+----------------------------------+------------+
| ``(real) [5.5]`` | ``5.5`` |
+----------------------------------+------------+
| ``(real) [6.6, 7.7]`` | ``error`` |
+----------------------------------+------------+
| ``(real) "8.8 in a string"`` | ``error`` |
+----------------------------------+------------+
Casts can be used in expressions:
*Examples*
+---------------------------+------------+
| **Code** | **Output** |
+===========================+============+
| ``(bool) 0 && true`` | ``false`` |
+---------------------------+------------+
| ``(real) "1" + 2`` | ``3`` |
+---------------------------+------------+
| ``(string) (3 + 4)`` | ``"7"`` |
+---------------------------+------------+
| ``(array) 5 + (array) 6`` | ``[5, 6]`` |
+---------------------------+------------+
Macro directives
----------------
.. macrodir:: @#includepath "PATH"
@#includepath MACRO_EXPRESSION
|br| This directive adds the path contained in PATH to the list of those to
search when looking for a ``.mod`` file specified by ``@#include``. If
provided with a MACRO_EXPRESSION argument, the argument must evaluate to a
string. Note that these paths are added *after* any paths passed using
:opt:`-I <-I\<\<path\>\>>`.
*Example*
::
@#includepath "/path/to/folder/containing/modfiles"
@#includepath folders_containing_mod_files
.. macrodir:: @#include "FILENAME"
@#include MACRO_EXPRESSION
|br| This directive simply includes the content of another file in its
place; it is exactly equivalent to a copy/paste of the content of the
included file. If provided with a MACRO_EXPRESSION argument, the argument
must evaluate to a string. Note that it is possible to nest includes
(i.e. to include a file from an included file). The file will be searched
for in the current directory. If it is not found, the file will be searched
for in the folders provided by :opt:`-I <-I\<\<path\>\>>` and
``@#includepath``.
*Example*
::
@#include "modelcomponent.mod"
@#include location_of_modfile
.. macrodir:: @#define MACRO_VARIABLE
@#define MACRO_VARIABLE = MACRO_EXPRESSION
@#define MACRO_FUNCTION = MACRO_EXPRESSION
|br| Defines a macro-variable or macro function.
*Example*
::
@#define var // Equals true
@#define x = 5 // Real
@#define y = "US" // String
@#define v = [ 1, 2, 4 ] // Real array
@#define w = [ "US", "EA" ] // String array
@#define u = [ 1, ["EA"] ] // Mixed array
@#define z = 3 + v[2] // Equals 5
@#define t = ("US" in w) // Equals true
@#define f(x) = " " + x + y // Function `f` with argument `x`
// returns the string ' ' + x + 'US'
*Example*
::
@#define x = 1
@#define y = [ "B", "C" ]
@#define i = 2
@#define f(x) = x + " + " + y[i]
@#define i = 1
model;
A = @{y[i] + f("D")};
end;
The latter is strictly equivalent to::
model;
A = BD + B;
end;
.. macrodir:: @#if MACRO_EXPRESSION
@#ifdef MACRO_VARIABLE
@#ifndef MACRO_VARIABLE
@#elseif MACRO_EXPRESSION
@#else
@#endif
|br| Conditional inclusion of some part of the ``.mod`` file. The lines
between ``@#if``, ``@#ifdef``, or ``@#ifndef`` and the next ``@#elseif``,
``@#else`` or ``@#endif`` is executed only if the condition evaluates to
``true``. Following the ``@#if`` body, you can zero or more ``@#elseif``
branches. An ``@#elseif`` condition is only evaluated if the preceding
``@#if`` or ``@#elseif`` condition evaluated to ``false``. The ``@#else``
branch is optional and is only evaluated if all ``@#if`` and ``@#elseif``
statements evaluate to false.
Note that when using ``@#ifdef``, the condition will evaluate to ``true``
if the MACRO_VARIABLE has been previously defined, regardless of its
value. Conversely, ``@#ifndef`` will evaluate to true if the MACRO_VARIABLE
has not yet been defined.
Note that when using ``@#elseif`` you can check whether or not a variable
has been defined by using the ``defined`` operator. Hence, to enter the
body of an ``@#elseif`` branch if the variable ``X`` has not been defined,
you would write: ``@#elseif !defined(X)``.
Note that if a real appears as the result of the MACRO_EXPRESSION, it
will be interpreted as a boolean; a value of ``0`` is interpreted as ``false``,
otherwise it is interpreted as ``true``. Further note that because of the
imprecision of reals, extra care must be taken when testing them in the
MACRO_EXPRESSION. For example, ``exp(log(5)) == 5`` will evaluate to
``false``. Hence, when comparing real values, you should generally use a
zero tolerance around the value desired, e.g. ``exp(log(5)) > 5-1e-14 &&
exp(log(5)) < 5+1e-14``
*Example*
Choose between two alternative monetary policy rules using a
macro-variable::
@#define linear_mon_pol = false // 0 would be treated the same
...
model;
@#if linear_mon_pol
i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar);
@#else
i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2;
@#endif
...
end;
This would result in::
...
model;
i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2;
...
end;
*Example*
Choose between two alternative monetary policy rules using a
macro-variable. The only difference between this example and the
previous one is the use of ``@#ifdef`` instead of ``@#if``. Even though
``linear_mon_pol`` contains the value ``false`` because ``@#ifdef`` only
checks that the variable has been defined, the linear monetary policy
is output::
@#define linear_mon_pol = false // 0 would be treated the same
...
model;
@#ifdef linear_mon_pol
i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar);
@#else
i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2;
@#endif
...
end;
This would result in::
...
model;
i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar);
...
end;
.. macrodir:: @#for MACRO_VARIABLE in MACRO_EXPRESSION
@#for MACRO_VARIABLE in MACRO_EXPRESSION when MACRO_EXPRESSION
@#for MACRO_TUPLE in MACRO_EXPRESSION
@#for MACRO_TUPLE in MACRO_EXPRESSION when MACRO_EXPRESSION
@#endfor
|br| Loop construction for replicating portions of the ``.mod``
file. Note that this construct can enclose variable/parameters
declaration, computational tasks, but not a model declaration.
*Example*
::
model;
@#for country in [ "home", "foreign" ]
GDP_@{country} = A * K_@{country}^a * L_@{country}^(1-a);
@#endfor
end;
The latter is equivalent to::
model;
GDP_home = A * K_home^a * L_home^(1-a);
GDP_foreign = A * K_foreign^a * L_foreign^(1-a);
end;
*Example*
::
model;
@#for (i, j) in ["GDP"] * ["home", "foreign"]
@{i}_@{j} = A * K_@{j}^a * L_@{j}^(1-a);
@#endfor
end;
The latter is equivalent to::
model;
GDP_home = A * K_home^a * L_home^(1-a);
GDP_foreign = A * K_foreign^a * L_foreign^(1-a);
end;
*Example*
::
@#define countries = ["US", "FR", "JA"]
@#define nth_co = "US"
model;
@#for co in countries when co != nth_co
(1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co};
@#endfor
E_@{nth_co} = 1;
end;
The latter is equivalent to::
model;
(1+i_FR) = (1+i_US) * E_FR(+1) / E_FR;
(1+i_JA) = (1+i_US) * E_JA(+1) / E_JA;
E_US = 1;
end;
.. macrodir:: @#echo MACRO_EXPRESSION
|br| Asks the preprocessor to display some message on standard
output. The argument must evaluate to a string.
.. macrodir:: @#error MACRO_EXPRESSION
|br| Asks the preprocessor to display some error message on standard
output and to abort. The argument must evaluate to a string.
.. macrodir:: @#echomacrovars
@#echomacrovars MACRO_VARIABLE_LIST
@#echomacrovars(save) MACRO_VARIABLE_LIST
|br| Asks the preprocessor to display the value of all macro variables up
until this point. If the ``save`` option is passed, then values of the
macro variables are saved to
``options_.macrovars_line_<<line_numbers>>``. If ``NAME_LIST`` is passed,
only display/save variables and functions with that name.
*Example*
::
@#define A = 1
@#define B = 2
@#define C(x) = x*2
@#echomacrovars A C D
The output of the command above is::
Macro Variables:
A = 1
Macro Functions:
C(x) = (x * 2)
Typical usages
--------------
Modularization
^^^^^^^^^^^^^^
The ``@#include`` directive can be used to split ``.mod`` files into
several modular components.
Example setup:
``modeldesc.mod``
Contains variable declarations, model equations, and shocks declarations.
``simul.mod``
Includes ``modeldesc.mod``, calibrates parameter,s and runs
stochastic simulations.
``estim.mod``
Includes ``modeldesc.mod``, declares priors on parameters, and runs
Bayesian estimation.
Dynare can be called on ``simul.mod`` and ``estim.mod`` but it makes
no sense to run it on ``modeldesc.mod``.
The main advantage is that you don't have to copy/paste the whole model (at the
beginning) or changes to the model (during development).
Indexed sums of products
^^^^^^^^^^^^^^^^^^^^^^^^
The following example shows how to construct a moving average::
@#define window = 2
var x MA_x;
...
model;
...
MA_x = @{1/(2*window+1)}*(
@#for i in -window:window
+x(@{i})
@#endfor
);
...
end;
After macro processing, this is equivalent to::
var x MA_x;
...
model;
...
MA_x = 0.2*(
+x(-2)
+x(-1)
+x(0)
+x(1)
+x(2)
);
...
end;
Multi-country models
^^^^^^^^^^^^^^^^^^^^
Here is a skeleton example for a multi-country model::
@#define countries = [ "US", "EA", "AS", "JP", "RC" ]
@#define nth_co = "US"
@#for co in countries
var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ...;
parameters a_@{co} ...;
varexo ...;
@#endfor
model;
@#for co in countries
Y_@{co} = K_@{co}^a_@{co} * L_@{co}^(1-a_@{co});
...
@#if co != nth_co
(1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; // UIP relation
@#else
E_@{co} = 1;
@#endif
@#endfor
end;
Endogeneizing parameters
^^^^^^^^^^^^^^^^^^^^^^^^
When calibrating the model, it may be useful to consider a parameter as an
endogenous variable (and vice-versa).
For example, suppose production is defined by a CES function:
.. math::
y = \left(\alpha^{1/\xi} \ell^{1-1/\xi}+(1-\alpha)^{1/\xi}k^{1-1/\xi}\right)^{\xi/(\xi-1)}
and the labor share in GDP is defined as:
.. math::
\textrm{lab\_rat} = (w \ell)/(p y)
In the model, :math:`\alpha` is a (share) parameter and ``lab_rat`` is an
endogenous variable.
It is clear that calibrating :math:`\alpha` is not straightforward;
on the contrary, we have real world data for ``lab_rat`` and it
is clear that these two variables are economically linked.
The solution is to use a method called *variable flipping*, which
consists in changing the way of computing the steady state. During
this computation, :math:`\alpha` will be made an endogenous variable
and ``lab_rat`` will be made a parameter. An economically relevant
value will be calibrated for ``lab_rat``, and the solution algorithm
will deduce the implied value for :math:`\alpha`.
An implementation could consist of the following files:
``modeqs.mod``
This file contains variable declarations and model equations. The
code for the declaration of :math:`\alpha` and ``lab_rat`` would
look like::
@#if steady
var alpha;
parameter lab_rat;
@#else
parameter alpha;
var lab_rat;
@#endif
``steady.mod``
This file computes the steady state. It begins with::
@#define steady = 1
@#include "modeqs.mod"
Then it initializes parameters (including ``lab_rat``, excluding
:math:`\alpha`), computes the steady state (using guess values for
endogenous, including :math:`\alpha`), then saves values of
parameters and endogenous at steady state in a file, using the
``save_params_and_steady_state`` command.
``simul.mod``
This file computes the simulation. It begins with::
@#define steady = 0
@#include "modeqs.mod"
Then it loads values of parameters and endogenous at steady state
from file, using the ``load_params_and_steady_state`` command, and
computes the simulations.
MATLAB/Octave loops versus macro processor loops
------------------------------------------------
Suppose you have a model with a parameter :math:`\rho` and you want
to run simulations for three values: :math:`\rho = 0.8, 0.9,
1`. There are several ways of doing this:
*With a MATLAB/Octave loop*
::
rhos = [ 0.8, 0.9, 1];
for i = 1:length(rhos)
rho = rhos(i);
stoch_simul(order=1);
end
Here the loop is not unrolled, MATLAB/Octave manages the
iterations. This is interesting when there are a lot of iterations.
*With a macro processor loop (case 1)*
::
rhos = [ 0.8, 0.9, 1];
@#for i in 1:3
rho = rhos(@{i});
stoch_simul(order=1);
@#endfor
This is very similar to the previous example, except that the loop
is unrolled. The macro processor manages the loop index but not
the data array (``rhos``).
*With a macro processor loop (case 2)*
::
@#for rho_val in [ 0.8, 0.9, 1]
rho = @{rho_val};
stoch_simul(order=1);
@#endfor
The advantage of this method is that it uses a shorter syntax, since the list
of values is directly given in the loop construct. The inconvenience is that
you can not reuse the macro array in MATLAB/Octave.
.. _verbatim:
Verbatim inclusion
==================
Pass everything contained within the verbatim block to the
``<mod_file>.m`` file.
.. block:: verbatim ;
|br| By default, whenever Dynare encounters code that is not
understood by the parser, it is directly passed to the
preprocessor output.
In order to force this behavior you can use the ``verbatim``
block. This is useful when the code you want passed to the
driver file contains tokens recognized by the Dynare
preprocessor.
*Example*
::
verbatim;
% Anything contained in this block will be passed
% directly to the driver file, including comments
var = 1;
end;
Misc commands
=============
.. command:: set_dynare_seed (INTEGER)
set_dynare_seed (`default')
set_dynare_seed (`clock')
set_dynare_seed (`reset')
set_dynare_seed (`ALGORITHM', INTEGER)
|br| Sets the seed used for random number generation. It is
possible to set a given integer value, to use a default value, or
to use the clock (by using the latter, one will therefore get
different results across different Dynare runs). The ``reset``
option serves to reset the seed to the value set by the last
``set_dynare_seed`` command. On MATLAB 7.8 or above, it is also
possible to choose a specific algorithm for random number
generation; accepted values are ``mcg16807``, ``mlfg6331_64``,
``mrg32k3a``, ``mt19937ar`` (the default), ``shr3cong`` and
``swb2712``.
.. command:: save_params_and_steady_state (FILENAME);
|br| For all parameters, endogenous and exogenous variables,
stores their value in a text file, using a simple name/value
associative table.
* for parameters, the value is taken from the last parameter
initialization.
* for exogenous, the value is taken from the last ``initval``
block.
* for endogenous, the value is taken from the last steady
state computation (or, if no steady state has been computed,
from the last ``initval`` block).
Note that no variable type is stored in the file, so that the
values can be reloaded with ``load_params_and_steady_state`` in a
setup where the variable types are different.
The typical usage of this function is to compute the steady-state
of a model by calibrating the steady-state value of some
endogenous variables (which implies that some parameters must be
endogeneized during the steady-state computation).
You would then write a first ``.mod`` file which computes the
steady state and saves the result of the computation at the end of
the file, using ``save_params_and_steady_state``.
In a second file designed to perform the actual simulations, you
would use ``load_params_and_steady_state`` just after your
variable declarations, in order to load the steady state
previously computed (including the parameters which had been
endogeneized during the steady state computation).
The need for two separate ``.mod`` files arises from the fact that
the variable declarations differ between the files for steady
state calibration and for simulation (the set of endogenous and
parameters differ between the two); this leads to different
``var`` and ``parameters`` statements.
Also note that you can take advantage of the ``@#include``
directive to share the model equations between the two files (see
:ref:`macro-proc-lang`).
.. command:: load_params_and_steady_state (FILENAME);
|br| For all parameters, endogenous and exogenous variables, loads
their value from a file created with
``save_params_and_steady_state``.
* for parameters, their value will be initialized as if they
had been calibrated in the ``.mod`` file.
* for endogenous and exogenous variables, their value will be
initialized as they would have been from an ``initval``
block .
This function is used in conjunction with
``save_params_and_steady_state``; see the documentation of that
function for more information.
.. command:: compilation_setup (OPTIONS);
When the :opt:`use_dll` option is present, Dynare uses the GCC compiler that
was distributed with it to compile the static and dynamic C files produced
by the preprocessor. You can use this option to change the compiler, flags,
and libraries used.
*Options*
.. option:: compiler = FILENAME
The path to the compiler.
.. option:: substitute_flags = QUOTED_STRING
The flags to use instead of the default flags.
.. option:: add_flags = QUOTED_STRING
The flags to use in addition to the default flags. If
``substitute_flags`` is passed, these flags are added to the flags
specified there.
.. option:: substitute_libs = QUOTED_STRING
The libraries to link against instead of the default libraries.
.. option:: add_libs = QUOTED_STRING
The libraries to link against in addition to the default libraries. If
``substitute_libs`` is passed, these libraries are added to the
libraries specified there.
.. matcomm:: dynare_version ;
|br| Output the version of Dynare that is currently being used
(i.e. the one that is highest on the MATLAB/Octave path).
.. matcomm:: write_latex_definitions ;
|br| Writes the names, LaTeX names and long names of
model variables to tables in a file named
``<<M_.fname>>_latex_definitions.tex``. Requires the following
LaTeX packages: ``longtable``.
.. matcomm:: write_latex_parameter_table ;
|br| Writes the LaTeX names, parameter names, and long
names of model parameters to a table in a file named
``<<M_.fname>>_latex_parameters.tex.`` The command writes the
values of the parameters currently stored. Thus, if parameters are
set or changed in the steady state computation, the command should
be called after a ``steady`` command to make sure the parameters were
correctly updated. The long names can be used to add parameter
descriptions. Requires the following LaTeX packages:
``longtable, booktabs``.
.. matcomm:: write_latex_prior_table ;
|br| Writes descriptive statistics about the prior distribution to
a LaTeX table in a file named
``<<M_.fname>>_latex_priors_table.tex``. The command writes the
prior definitions currently stored. Thus, this command must be
invoked after the ``estimated_params`` block. If priors are
defined over the measurement errors, the command must also be
preceeded by the declaration of the observed variables (with
``varobs``). The command displays a warning if no prior densities
are defined (ML estimation) or if the declaration of the observed
variables is missing. Requires the following LaTeX
packages: ``longtable, booktabs``.
.. matcomm:: collect_latex_files ;
|br| Writes a LaTeX file named
``<<M_.fname>>_TeX_binder.tex`` that collects all TeX output
generated by Dynare into a file. This file can be compiled using
``pdflatex`` and automatically tries to load all required
packages. Requires the following LaTeX packages:
``breqn``, ``psfrag``, ``graphicx``, ``epstopdf``, ``longtable``,
``booktabs``, ``caption``, ``float,`` ``amsmath``, ``amsfonts``,
and ``morefloats``.
.. _io: https://octave.sourceforge.io/io/
.. _AIM website: https://www.federalreserve.gov/econres/ama-index.htm
.. rubric:: Footnotes
.. [#feol] A ``.mod`` file must have lines that end with a line feed character,
which is not commonly visible in text editors. Files created on
Windows and Unix-based systems have always conformed to this
requirement, as have files created on OS X and macOS. Files created
on old, pre-OS X Macs used carriage returns as end of line
characters. If you get a Dynare parsing error of the form ``ERROR:
<<mod file>>: line 1, cols 341-347: syntax error,...`` and there's
more than one line in your ``.mod`` file, know that it uses the
carriage return as an end of line character. To get more helpful
error messages, the carriage returns should be changed to line
feeds.
.. [#f1] Note that arbitrary MATLAB or Octave expressions can be put
in a ``.mod`` file, but those expressions have to be on
separate lines, generally at the end of the file for
post-processing purposes. They are not interpreted by Dynare,
and are simply passed on unmodified to MATLAB or
Octave. Those constructions are not addresses in this
section.
.. [#f2] In particular, for big models, the compilation step can be
very time-consuming, and use of this option may be
counter-productive in those cases.
.. [#f3] See options :ref:`conf_sig <confsig>` and :opt:`mh_conf_sig <mh_conf_sig = DOUBLE>`
to change the size of the HPD interval.
.. [#f4] See options :ref:`conf_sig <confsig>` () and :opt:`mh_conf_sig <mh_conf_sig = DOUBLE>`
to change the size of the HPD interval.
.. [#f5] When the shocks are correlated, it is the decomposition of
orthogonalized shocks via Cholesky decomposition according to
the order of declaration of shocks (see :ref:`var-decl`)
.. [#f6] See :opt:`forecast <forecast = INTEGER>` for more information.
.. [#f7] In case of Excel not being installed,
`<https://mathworks.com/matlabcentral/fileexchange/38591-xlwrite--generate-xls-x--files-without-excel-on-mac-linux-win>`__
may be helpful.
.. [#f8] See option :ref:`conf_sig <confsig>` to change the size of
the HPD interval.
.. [#f9] See option :ref:`conf_sig <confsig>` to change the size of
the HPD interval.
.. [#f10] If you want to align the paper with the description herein,
please note that :math:`A` is :math:`A^0` and :math:`F` is
:math:`A^+`.
.. [#f11] An example can be found at
`<https://git.dynare.org/Dynare/dynare/blob/master/tests/ms-dsge/test_ms_dsge.mod>`__.
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