Clarify OSR in manual
Johannes Pfeifer
parent
12026b98a5
commit
44fe31c73f
150
doc/dynare.texi
150
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, specified in the
|
||||
@code{var_exo}-command;
|
||||
|
||||
@item @math{W} is the weighting matrix;
|
||||
|
||||
@item
|
||||
@math{e} are the exogenous stochastic shocks;
|
||||
@end itemize
|
||||
|
||||
The parameters to be optimized must be listed with @code{osr_params}.
|
||||
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 quadratic objectives must be listed with @code{optim_weights}.
|
||||
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}.
|
||||
|
||||
This problem is solved using the numerical optimizer @code{csminwel} of Chris Sims.
|
||||
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.
|
||||
|
||||
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.
|
||||
|
||||
@end deffn
|
||||
|
||||
@anchor{osr_params}
|
||||
@deffn Command osr_params @var{PARAMETER_NAME}@dots{};
|
||||
This command declares parameters to be optimized by @code{osr}.
|
||||
@end deffn
|
||||
@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{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}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue