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Document composite targets in PAC equation.

dprior
Stéphane Adjemian (Argos) 2024-01-29 15:20:13 +01:00
parent 9f9f4a99ba
commit 8eab48aa5e
Signed by: stepan
GPG Key ID: 295C1FE89E17EB3C
2 changed files with 156 additions and 18 deletions
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169
doc/manual/source/the-model-file.rst
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@ -14709,7 +14709,7 @@ simply add the exogenous variables to the PAC equation (without the weight
``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
will then be replaced by a linear combination, defined by a vector :math:`h`, 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
@ -14729,6 +14729,16 @@ simply add the exogenous variables to the PAC equation (without the weight
or expression is given) is consistent with the asymptotic growth rate of the
endogenous variable.
.. option:: kind = dd | dl
Instructs Dynare how to compute the vector :math:`h`, the weights
defining the linear combination of the companion VAR
variables. The default value ``dd`` must be used if the target
appears in first difference in the auxiliary model, see equation
(A.79) in *Brayton et alii (2000)*, while value ``dl`` must be
used if the target shows up in level in the auxiliary model,
equation (A.74) in *Brayton et alii (2000)*.
.. operator:: pac_expectation (NAME_OF_PAC_MODEL);
@ -14739,7 +14749,89 @@ simply add the exogenous variables to the PAC equation (without the weight
the variables involved in the companion representation of the auxiliary model
or by a recursive forward equation.
|br|
The PAC equation target can be composite and defined as a weighted sum
of stationary and non stationary components. Such a target requires an
additional equation in the model block, with the target variable on
the left hand-side and the components in the right hand-side. Each
component must be an endogenous variable in the auxiliary model. The
characteristics of each component must be described in the
``pac_target_info`` block, see below, and the
``pac_target_nonstationary`` operator must be used in the error
correction term of the PAC equation to link the target to the provided
description. Note that composite targets make only sense if the
auxiliary model is not a trend component model (where all the
variables are non stationary).
.. block:: pac_target_info (NAME_OF_PAC_MODEL);
|br| This block enables the user to provide the properties of each
component of a target in PAC models with a composite target. The
``NAME_OF_PAC_MODEL`` argument refers to a PAC model (must match
the value of option ``model_name`` in the declaration of a PAC
model).
On the first line of the block, the name of the composite target
variable must be provided using the following syntax::
target VARIABLE_NAME ;
where ``VARIABLE_NAME`` is a declared endogenous variable, its
associated equation is not part of the auxiliary model but all the
components (the variables on the right hand-side) must be defined
in the auxiliary model. Next, the following line declares the name
of the auxilary variable that will appear in the error correction
term, this variable contains only the non stationary components of
the target::
auxname_target_nonstationary NAME ;
The block should contain the following group of lines for each
stationary component::
component STATIONARY_VARIABLE_NAME ;
kind ll ;
auxname AUX_VAR_NAME ;
where ``STATIONARY_VARIABLE_NAME`` is the name of a stationary
variable appearing in the right hand-side of the equation defining
the target ``VARIABLE_NAME``. The second line instructs Dynare that
the component appears in levels in the auxiliary model and in the
PAC expectations. The third line specifies the name of the
auxiliary variable created by Dynare for the component of the PAC
expectation related to ``STATIONARY_VARIABLE_NAME``.
The block should contain the following group of lines for each
nonstationary component::
component NONSTATIONARY_VARIABLE_NAME ;
kind dd | dl ;
auxname AUX_VAR_NAME ;
growth PARAMETER_NAME | VARIABLE_NAME | EXPRESSION | DOUBLE ;
where ``NONSTATIONARY_VARIABLE_NAME`` is the name of a
nonstationary variable appearing in the right hand-side of the
equation defining the target ``VARIABLE_NAME``. The second line
instructs Dynare on how to calculate the weights that define the linear
combination of the companion VAR variables. Use value ``dd`` if the
target appears in first difference in the auxiliary model, or
``dl`` if the target shows up in level in the auxiliary model. The
third line sets the name of the auxiliary variable created by
Dynare for the component of the PAC expectation related to
``NONSTATIONARY_VARIABLE_NAME``. The fourth line is mandatory if a
growth neutrality correction is required. The provided value for
this option must be consistent with the asymptotic growth rate of
the PAC endogenous variable.
.. operator:: pac_target_nonstationary (NAME_OF_PAC_MODEL);
|br| This operator is only required in presence of a composite
target in the PAC equation. The operator, used in the error
correction term of the PAC equation, selects the non stationary
components of the target.
.. matcomm:: pac.initialize(NAME_OF_PAC_MODEL);
.. matcomm:: pac.update(NAME_OF_PAC_MODEL);
@ -14752,33 +14844,33 @@ simply add the exogenous variables to the PAC equation (without the weight
the infinite sum in the MCE case).
*Example*
*Example (trend component auxiliary model)*
::
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);
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
model;
[name='eq:y']
y = rho_1*y(-1) + rho_2*y(-2) + ey;
[name='eq:y']
y = (1-rho_1-rho_2)*diff(x2(-1)) + 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: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: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:x1bar']
x1bar = x1bar(-1) + ex1bar;
[name='eq:x2bar']
x2bar = x2bar(-1) + ex2bar;
[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;
[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;
@ -14787,6 +14879,51 @@ simply add the exogenous variables to the PAC equation (without the weight
pac.update.expectation('pacman');
*Example (VAR auxiliary model and composite target)*
::
var_model(model_name=toto, eqtags=['eq:x', 'eq:y']);
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
pac_target_info(pacman);
target v;
auxname_target_nonstationary vns;
component y;
auxname pv_y_;
kind ll;
component x;
growth diff(x(-1));
auxname pv_dx_;
kind dd;
end;
model;
[name='eq:y']
y = a_y_1*y(-1) + a_y_2*diff(x(-1)) + b_y_1*y(-2) + b_y_2*diff(x(-2)) + ey ;
[name='eq:x']
diff(x) = b_x_1*y(-2) + b_x_2*diff(x(-1)) + ex ;
[name='eq:v']
v = x + d_y*y ; // Composite PAC target, no residuals here only variables defined in the auxiliary model.
[name='zpac']
diff(z) = e_c_m*(pac_target_nonstationary(pacman)-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
----------------------------

5
doc/manual/utils/dynare_lex.py
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@ -60,7 +60,7 @@ class DynareLexer(RegexLexer):
"addSeries","addParagraph","addVspace","write","compile")
operators = (
"STEADY_STATE","EXPECTATION","var_expectation","pac_expectation")
"STEADY_STATE","EXPECTATION","var_expectation","pac_expectation","pac_target_nonstationary")
macro_dirs = (
"@#includepath", "@#include", "@#define", "@#if",
@ -83,7 +83,8 @@ class DynareLexer(RegexLexer):
'osr_params_bounds','ramsey_constraints','irf_calibration',
'moment_calibration','identification','svar_identification',
'matched_moments','occbin_constraints','surprise','overwrite','bind','relax',
'verbatim','end','node','cluster','paths','hooks'), prefix=r'\b', suffix=r'\s*\b'),Keyword.Reserved),
'verbatim','end','node','cluster','paths','hooks','target','pac_target_info','auxname_target_nonstationary',
'component', 'growth', 'auxname', 'kind'), prefix=r'\b', suffix=r'\s*\b'),Keyword.Reserved),
# FIXME: Commands following multiline comments are not highlighted properly.
(words(commands + report_commands,