diff --git a/doc/manual/source/the-model-file.rst b/doc/manual/source/the-model-file.rst
index f553ad507457607bb5dc9372a4ec4634ff8cb3f8..ffe4da7fb3eed6b61e7263858ab45d4f44fa533a 100644
--- a/doc/manual/source/the-model-file.rst
+++ b/doc/manual/source/the-model-file.rst
@@ -9775,7 +9775,7 @@ 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}`:
@@ -9783,43 +9783,83 @@ 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)
-
- which can be solved algebraically for :math:`\varepsilon_t(contr\_shocks)`.
-
- 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.
+ \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``.
@@ -9850,7 +9890,7 @@ 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