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Fix description of the conditional_forecast command. 

Closes #1844
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Stéphane Adjemian (Charybdis) 2022-02-24 20:33:47 +01:00
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commit c9825c803a
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108
doc/manual/source/the-model-file.rst
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@ -9907,7 +9907,7 @@ the :comm:`bvar_forecast` command.
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 an augmented state space representation that
paths. Consider the augmented state space representation that
stacks both predetermined and non-predetermined variables into a
vector :math:`y_{t}`:
@ -9915,43 +9915,83 @@ the :comm:`bvar_forecast` command.
y_t=Ty_{t-1}+R\varepsilon_t
Both :math:`y_t` and :math:`\varepsilon_t` are split up into
controlled and uncontrolled ones to get:
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::
y_t(contr\_vars)=Ty_{t-1}(contr\_vars)+R(contr\_vars,uncontr\_shocks)\varepsilon_t(uncontr\_shocks) \\
+ R(contr\_vars,contr\_shocks)\varepsilon_t(contr\_shocks)
\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}
which can be solved algebraically for :math:`\varepsilon_t(contr\_shocks)`.
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}`:
Using these controlled shocks, the state-space representation can
be used for forecasting. A few things need to be noted. First, it
is assumed that controlled exogenous variables are fully under
control of the policy maker for all forecast periods and not just
for the periods where the endogenous variables are controlled. For
all uncontrolled periods, the controlled exogenous variables are
assumed to be 0. 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. Third, although controlled exogenous variables are
taken as instruments perfectly under the control of the
policy-maker, they are nevertheless random and unforeseen shocks
from the perspective of the households. That is, households are in
each period surprised by the realization of a shock that keeps the
controlled endogenous variables at their respective level. Fourth,
keep in mind that 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
factorize 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.
.. 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``.
@ -9982,7 +10022,7 @@ the :comm:`bvar_forecast` command.
.. option:: replic = INTEGER
Number of simulations. Default: ``5000``.
Number of simulations used to compute the conditional forecast uncertainty. Default: ``5000``.
.. option:: conf_sig = DOUBLE

97
matlab/mcforecast3.m
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@ -1,43 +1,57 @@
function [forcs, e]= mcforecast3(cL,H,mcValue,shocks,forcs,T,R,mv,mu)
% [forcs, e] = mcforecast3(cL,H,mcValue,shocks,forcs,T,R,mv,mu)
function [forcs, e] = mcforecast3(cL, H, mcValue, shocks, forcs, T, R, mv, mu)
% Computes the shock values for constrained forecasts necessary to keep
% endogenous variables at their constrained paths
%
% INPUTS
% o cL [scalar] number of controlled periods
% o H [scalar] number of forecast periods
% o mcValue [n_controlled_vars by cL double] paths for constrained variables
% o shocks [nexo by H double] shock values draws (with zeros for controlled_varexo)
% o forcs [n_endovars by H+1 double] matrix of endogenous variables storing the inital condition
% o T [n_endovars by n_endovars double] transition matrix of the state equation.
% o R [n_endovars by n_exo double] matrix relating the endogenous variables to the innovations in the state equation.
% o mv [n_controlled_exo by n_endovars boolean] indicator vector selecting constrained endogenous variables
% o mu [n_controlled_vars by nexo boolean] indicator vector selecting controlled exogenous variables
% OUTPUTS
% o forcs [n_endovars by H+1 double] matrix of forecasted endogenous variables
% o e [nexo by H double] matrix of exogenous variables
% INPUTS:
% - cL [integer] scalar, number of controlled periods
% - H [integer] scalar, number of forecast periods
% - mcValue [double] n_controlled_vars*cL array, paths for constrained variables
% - shocks [double] n_controlled_vars*cL array, shock values draws (with zeros for controlled_varexo)
% - forcs [double] n_endovars*(H+1) matrix of endogenous variables storing the inital condition
% - T [double] n_endovars*n_endovars array, transition matrix of the state equation.
% - R [double] n_endovars*n_exo array, matrix relating the endogenous variables to the innovations in the state equation.
% - mv [logical] n_controlled_exo*n_endovars array, indicator selecting constrained endogenous variables
% - mu [logical] n_controlled_vars*nexo array, indicator selecting controlled exogenous variables
%
% OUTPUTS:
% - forcs [double] n_endovars*(H+1) array, forecasted endogenous variables
% - e [double] nexo*H array, exogenous variables
%
% ALGORITHM:
%
% Algorithm:
% Relies on state-space form:
% y_t=T*y_{t-1}+R*shocks(:,t)
% Shocks are split up into shocks_uncontrolled and shockscontrolled while
% the endogenous variables are also split up into controlled and
% uncontrolled ones to get:
% y_t(controlled_vars_index)=T*y_{t-1}(controlled_vars_index)+R(controlled_vars_index,uncontrolled_shocks_index)*shocks_uncontrolled_t
% + R(controlled_vars_index,controlled_shocks_index)*shocks_controlled_t
%
% This is then solved to get:
% shocks_controlled_t=(y_t(controlled_vars_index)-(T*y_{t-1}(controlled_vars_index)+R(controlled_vars_index,uncontrolled_shocks_index)*shocks_uncontrolled_t)/R(controlled_vars_index,controlled_shocks_index)
% yₜ = T yₜ₋₁ + R εₜ
%
% Variable number of controlled vars are allowed in different
% periods. Missing control information are indicated by NaN in
% y_t(controlled_vars_index).
% Both yₜ, the vector of endogenous variables, and εₜ 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 :
%
% After obtaining the shocks, and for uncontrolled periods, the state-space representation
% y_t=T*y_{t-1}+R*shocks(:,t)
% ⎧ y₁ₜ ⎫ ⎧ T₁₁ T₁₂ ⎫ ⎧ y₁ₜ₋₁ ⎫ ⎧ R₁₁ R₁₂ ⎫ ⎧ ε₁ₜ ⎫
% ⎩ y₂ₜ ⎭ = ⎩ T₂₁ T₂₂ ⎭ ⎩ y₂ₜ₋₁ ⎭ + ⎩ R₂₁ R₂₂ ⎭ ⎩ ε₂ₜ ⎭
%
% where matrices T and R are partitioned consistently with the
% vectors of endogenous variables and innovations. Provided that matrix
% R₁₁ is square and full rank (a necessary condition is that the
% number of free endogenous variables matches the number of free innovations),
% given y₁ₜ, ε₂ₜ and yₜ₋₁ the first block of equations can be solved for ε₁ₜ:
%
% ε₁ₜ = R₁₁ \ ( y₁ₜ - T₁₁y₁ₜ₋₁ - T₁₂y₂ₜ₋₁ - R₁₂ε₂ₜ )
%
% and y₂ₜ can be updated by evaluating the second block of equations:
%
% y₂ₜ = T₂₁y₁ₜ₋₁ + T₂₂y₂ₜ₋₁ + R₂₁ε₁ₜ + R₂₂ε₂ₜ
%
% 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. This exercise is replicated by drawing different
% sequences of free innovations. The result is a predictive distribution for
% the uncontrolled endogenous variables, y₂ₜ, that Dynare will use to report
% confidence bands around the point conditional forecast.
% is used for forecasting
%
% Copyright (C) 2006-2017 Dynare Team
% Copyright © 2006-2022 Dynare Team
%
% This file is part of Dynare.
%
@ -54,15 +68,14 @@ function [forcs, e]= mcforecast3(cL,H,mcValue,shocks,forcs,T,R,mv,mu)
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
if cL
e = zeros(size(mcValue,1),cL);
for t = 1:cL
% missing conditional values are indicated by NaN
k = find(isfinite(mcValue(:,t)));
e(k,t) = inv(mv(k,:)*R*mu(:,k))*(mcValue(k,t)-mv(k,:)*T*forcs(:,t)-mv(k,:)*R*shocks(:,t));
forcs(:,t+1) = T*forcs(:,t)+R*(mu(:,k)*e(k,t)+shocks(:,t));
if cL
e = zeros(size(mcValue,1),cL);
for t = 1:cL % Loop over the two blocks of equations
k = find(isfinite(mcValue(:,t))); % missing conditional values are indicated by NaN
e(k,t) = inv(mv(k,:)*R*mu(:,k))*(mcValue(k,t)-mv(k,:)*T*forcs(:,t)-mv(k,:)*R*shocks(:,t));
forcs(:,t+1) = T*forcs(:,t)+R*(mu(:,k)*e(k,t)+shocks(:,t));
end
end
end
for t = cL+1:H
forcs(:,t+1) = T*forcs(:,t)+R*shocks(:,t);
end
for t = cL+1:H
forcs(:,t+1) = T*forcs(:,t)+R*shocks(:,t);
end