From 44fe31c73f6e2070b2fc761614d039627afcf82c Mon Sep 17 00:00:00 2001
From: Johannes Pfeifer <jpfeifer@gmx,de>
Date: Wed, 19 Mar 2014 20:01:06 +0100
Subject: [PATCH] Clarify OSR in manual
---
doc/dynare.texi | 152 +++++++++++++++++++++++++++++++++---------------
1 file changed, 105 insertions(+), 47 deletions(-)
diff --git a/doc/dynare.texi b/doc/dynare.texi
index 1e207bed8a..175e727e09 100644
--- a/doc/dynare.texi
+++ b/doc/dynare.texi
@@ -6057,7 +6057,7 @@ This command computes optimal simple policy rules for
linear-quadratic problems of the form:
@quotation
-@math{\max_\gamma E(y'_tWy_t)}
+@math{\min_\gamma E(y'_tWy_t)}
@end quotation
such that:
@@ -6067,86 +6067,144 @@ such that:
where:
-@itemize
+@itemize
+@item @math{E} denotes the unconditional expectations operator;
-@item
-@math{\gamma} are parameters to be optimized. They must be elements of matrices
-@math{A_1}, @math{A_2}, @math{A_3};
+@item @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 @code{params}-command and be entered in the
+@code{model}-block;
-@item
-@math{y} are the endogenous variables;
+@item @math{y} are the endogenous variables, specified in the
+@code{var}-command, whose (co)-variance enters the loss function;
-@item
-@math{e} are the exogenous stochastic shocks;
-@end itemize
+@item @math{e} are the exogenous stochastic shocks, specified in the
+@code{var_exo}-command;
-The parameters to be optimized must be listed with @code{osr_params}.
+@item @math{W} is the weighting matrix;
-The quadratic objectives must be listed with @code{optim_weights}.
+@end itemize
-This problem is solved using the numerical optimizer @code{csminwel} of Chris Sims.
+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, @code{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, @code{osr} will automatically select
+@code{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 @code{osr_params}.
+
+The weighting matrix @math{W} used for the quadratic objective function
+is specified in the @code{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
+@code{csminwel} of Chris Sims.
@optionshead
-The @code{osr} command will subsequently run @code{stoch_simul}
-and accepts the same options, including restricting the endogenous variables by listing them after the command, as @code{stoch_simul}
+The @code{osr} command will subsequently run @code{stoch_simul} and
+accepts the same options, including restricting the endogenous variables
+by listing them after the command, as @code{stoch_simul}
(@pxref{Computing the stochastic solution}) plus
@table @code
-@item maxit = @var{INTEGER}
-Determines the maximum number of iterations used in the non-linear solver. Default: @code{1000}
+@item maxit = @var{INTEGER} Determines the maximum number of iterations
+used in the non-linear solver. Default: @code{1000}
-@item tolf = @var{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: @code{1e-7}
-@end table
+@item tolf = @var{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: @code{1e-7} @end
+table
The value of the objective is stored in the variable
@code{oo_.osr.objective_function} and the value of parameters at the
optimum is stored in @code{oo_.osr.optim_params}. See below for more
details.
-@end deffn
+After running @code{osr} the parameters entering the simple rule will be
+set to their optimal value so that subsequent runs of @code{stoch_simul}
+will be conducted at these values.
-@anchor{osr_params}
-@deffn Command osr_params @var{PARAMETER_NAME}@dots{};
-This command declares parameters to be optimized by @code{osr}.
@end deffn
-@anchor{optim_weights}
-@deffn Block optim_weights ;
+@anchor{osr_params} @deffn Command osr_params
+@var{PARAMETER_NAME}@dots{}; This command declares parameters to be
+optimized by @code{osr}. @end deffn
+
+@anchor{optim_weights} @deffn Block optim_weights ;
This block specifies quadratic objectives for optimal policy problems
-More precisely, this block specifies the nonzero elements of the
-quadratic weight matrices for the objectives in @code{osr}.
+More precisely, this block specifies the nonzero elements of the weight
+matrix @math{W} used in the quadratic form of the objective function in
+@code{osr}.
-An element of the diagonal of the weight matrix is given by a line of
-the form:
-@example
-@var{VARIABLE_NAME} @var{EXPRESSION};
-@end example
+An element of the diagonal of the weight matrix is given by a line of the
+form: @example @var{VARIABLE_NAME} @var{EXPRESSION}; @end example
An off-the-diagonal element of the weight matrix is given by a line of
-the form:
-@example
-@var{VARIABLE_NAME}, @var{VARIABLE_NAME} @var{EXPRESSION};
-@end example
+the form: @example @var{VARIABLE_NAME}, @var{VARIABLE_NAME}
+@var{EXPRESSION}; @end example
@end deffn
-@defvr {MATLAB/Octave variable} oo_.osr.objective_function
-After an execution of the @code{osr} command, this variable contains
-the value of the objective under optimal policy.
-@end defvr
+@examplehead
-@defvr {MATLAB/Octave variable} oo_.osr.optim_params
-After an execution of the @code{osr} command, this variable contains
-the value of parameters at the optimum, stored in fields of the form
-@code{oo_.osr.optim_params.@var{PARAMETER_NAME}}.
-@end defvr
+@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;
+@end example
+
+@defvr {MATLAB/Octave variable} oo_.osr.objective_function
+After an execution of the @code{osr} command, this variable contains the value of
+the objective under optimal policy. @end defvr
+
+@defvr {MATLAB/Octave variable} oo_.osr.optim_params
+After an execution of the @code{osr} command, this variable contains the value of parameters
+at the optimum, stored in fields of the form
+@code{oo_.osr.optim_params.@var{PARAMETER_NAME}}. @end defvr
@anchor{Ramsey}
--
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