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Dynare has tools to compute optimal policies for various types of
objectives. You can either solve for optimal policy under commitment
with ramsey_policy
, for optimal policy under discretion with
discretionary_policy
or for optimal simple rule with
osr
.
Description
This command computes optimal simple policy rules for linear-quadratic problems of the form:
such that:
where:
params
-command and be entered in the
model
-block;
var
-command, whose (co)-variance enters the loss function;
var_exo
-command;
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, 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,
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, osr
will automatically select
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 are not allowed.
The subset of the model parameters over which the optimal simple rule is
to be optimized, , must be listed with
osr_params
.
The weighting matrix used for the quadratic objective function
is specified in the
optim_weights
-block. By attaching weights to
endogenous variables, the subset of endogenous variables entering the
objective function, , is implicitly specified.
The linear quadratic problem is solved using the numerical optimizer
csminwel
of Chris Sims.
Options
The osr
command will subsequently run stoch_simul
and
accepts the same options, including restricting the endogenous variables
by listing them after the command, as stoch_simul
(see section Computing the stochastic solution) plus
maxit = INTEGER Determines the maximum number of iterations
used in the non-linear solver. Default: 1000
tolf = 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: 1e-7
The value of the objective is stored in the variable
oo_.osr.objective_function
, which is described below.
After running osr
the parameters entering the simple rule will be
set to their optimal value so that subsequent runs of stoch_simul
will be conducted at these values.
This command declares parameters to be optimized by osr
.
This block specifies quadratic objectives for optimal policy problems
More precisely, this block specifies the nonzero elements of the weight
matrix used in the quadratic form of the objective function in
osr
.
An element of the diagonal of the weight matrix is given by a line of the form:
VARIABLE_NAME EXPRESSION; |
An off-the-diagonal element of the weight matrix is given by a line of the form:
VARIABLE_NAME, VARIABLE_NAME EXPRESSION; |
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; |
After an execution of the osr
command, this variable contains the value of
the objective under optimal policy.
Description
This command computes the First Order Conditions for maximizing the policy maker objective function subject to the constraints provided by the equilibrium path of the economy.
The planner objective must be declared with the planner_objective
command.
This command only creates the expanded model, it doesn’t perform any computations. It needs to be followed by other instructions to actually perfrom desired computations. Note that it is the only way to perform perfect foresight simulation of the Ramsey policy problem.
See section Auxiliary variables, for an explanation of how Lagrange multipliers are automatically created.
Options
This command accepts the following options:
planner_discount = EXPRESSION
Declares the discount factor of the central planner. Default: 1.0
instruments = (VARIABLE_NAME,…)
Declares instrument variables for the computation of the steady state
under optimal policy. Requires a steady_state_model
block or a
…_steadystate.m
file. See below.
Steady state
Dynare takes advantage of the fact that the Lagrange multipliers appear
linearly in the equations of the steady state of the model under optimal
policy. Nevertheless, it is in general very difficult to compute the
steady state with simply a numerical guess in initval
for the
endogenous variables.
It greatly facilitates the computation, if the user provides an
analytical solution for the steady state (in steady_state_model
block or in a …_steadystate.m
file). In this case, it is
necessary to provide a steady state solution CONDITIONAL on the value
of the instruments in the optimal policy problem and declared with
option instruments
. Note that choosing the instruments is
partly a matter of interpretation and you can choose instruments that
are handy from a mathematical point of view but different from the
instruments you would refer to in the analysis of the paper. A typical
example is choosing inflation or nominal interest rate as an
instrument.
Description
This command computes the first order approximation of the policy that maximizes the policy maker objective function submitted to the constraints provided by the equilibrium path of the economy.
The planner objective must be declared with the planner_objective
command.
See section Auxiliary variables, for an explanation of how this operator is handled internally and how this affects the output.
Options
This command accepts all options of stoch_simul
, plus:
planner_discount = EXPRESSION
Declares the discount factor of the central planner. Default: 1.0
instruments = (VARIABLE_NAME,…)
Declares instrument variables for the computation of the steady state
under optimal policy. Requires a steady_state_model
block or a
…_steadystate.m
file. See below.
Note that only first order approximation is available (i.e.
order=1
must be specified).
Output
This command generates all the output variables of stoch_simul
.
In addition, it stores the value of planner objective function under
Ramsey policy in oo_.planner_objective_value
.
Steady state
Dynare takes advantage of the fact that the Lagrange multipliers appear
linearly in the equations of the steady state of the model under optimal
policy. Nevertheless, it is in general very difficult to compute the
steady state with simply a numerical guess in initval
for the
endogenous variables.
It greatly facilitates the computation, if the user provides an
analytical solution for the steady state (in steady_state_model
block or in a …_steadystate.m
file). In this case, it is
necessary to provide a steady state solution CONDITIONAL on the value
of the instruments in the optimal policy problem and declared with
option instruments
. Note that choosing the instruments is
partly a matter of interpretation and you can choose instruments that
are handy from a mathematical point of view but different from the
instruments you would refer to in the analysis of the paper. A typical
example is choosing inflation or nominal interest rate as an
instrument.
Description
This command computes an approximation of the optimal policy under discretion. The algorithm implemented is essentially an LQ solver, and is described by Dennis (2007).
You should ensure that your model is linear and your objective is
quadratic. Also, you should set the linear
option of the
model
block.
Options
This command accepts the same options than ramsey_policy
, plus:
discretionary_tol = NON-NEGATIVE DOUBLE
Sets the tolerance level used to assess convergence of the solution
algorithm. Default: 1e-7
.
maxit = INTEGER
Maximum number of iterations. Default: 3000
.
This command declares the policy maker objective, for use with
ramsey_policy
or discretionary_policy
.
You need to give the one-period objective, not the discounted lifetime
objective. The discount factor is given by the planner_discount
option of ramsey_policy
and discretionary_policy
. The
objective function can only contain current endogenous variables and no
exogenous ones. This limitation is easily circumvented by defining an
appropriate auxiliary variable in the model.
With ramsey_policy
, you are not limited to quadratic
objectives: you can give any arbitrary nonlinear expression.
With discretionary_policy
, the objective function must be quadratic.
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