Verified Commit 4c0f8ec6 authored by Willi Mutschler's avatar Willi Mutschler Committed by Stéphane Adjemian
Browse files

:book:Added documentation for method of moments

parent 8d024117
......@@ -3634,6 +3634,8 @@ strong nonlinearities or binding constraints. Such a solution is
computed using the ``extended_path`` command.
.. _stoch-sol-simul:
Computing the stochastic solution
---------------------------------
......@@ -4622,8 +4624,8 @@ elements are never repeated (for more details, see the description of
.. _estim:
Estimation
==========
Estimation based on likelihood
==============================
Provided that you have observations on some endogenous variables, it
is possible to use Dynare to estimate some or all parameters. Both
......@@ -4640,6 +4642,8 @@ observed variables.
The estimation using a first order approximation can benefit from the
block decomposition of the model (see :opt:`block`).
.. _varobs:
.. command:: varobs VARIABLE_NAME...;
|br| This command lists the name of observed endogenous variables
......@@ -4699,7 +4703,6 @@ block decomposition of the model (see :opt:`block`).
P (mu/eta);
end;
.. block:: estimated_params ;
|br| This block lists all parameters to be estimated and specifies
......@@ -4707,12 +4710,12 @@ block decomposition of the model (see :opt:`block`).
Each line corresponds to an estimated parameter.
In a maximum likelihood estimation, each line follows this syntax::
In a maximum likelihood or a method of moments estimation, each line follows this syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME
, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ];
In a Bayesian estimation, each line follows this syntax::
In a Bayesian MCMC or a penalized method of moments estimation, each line follows this syntax::
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT
[, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE,
......@@ -7633,8 +7636,585 @@ Dynare also has the ability to estimate Bayesian VARs:
See ``bvar-a-la-sims.pdf``, which comes with Dynare distribution,
for more information on this command.
Estimation based on moments
===========================
Provided that you have observations on some endogenous variables, it
is possible to use Dynare to estimate some or all parameters using a
method of moments approach. Both the Simulated Method of Moments (SMM)
and the Generalized Method of Moments (GMM) are available. The general
idea is to minimize the distance between unconditional model moments
and corresponding data moments (so called orthogonality or moment
conditions). For SMM Dynare computes model moments via stochastic
simulations based on the perturbation approximation up to any order,
whereas for GMM model moments are computed in closed-form based on the
pruned state-space representation of the perturbation solution up to third
order. The implementation of SMM is inspired by *Born and Pfeifer (2014)*
and *Ruge-Murcia (2012)*, whereas the one for GMM is adapted from
*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* and *Mutschler
(2018)*. The estimation heavily relies on the accuracy and efficiency of
the perturbation approximation, so it is advised to tune this as much as
possible (see :ref:`stoch-sol-simul`). The estimator is consistent and
asymptotically normally distributed given certain regularity conditions
(see *Duffie and Singleton (1993)* for SMM and *Hansen (1982)* for GMM).
For instance, it is required to have at least as many moment conditions as
estimated parameters. Moreover, the Jacobian of the moments with respect to
the estimated parameters needs to be full rank. :ref:`identification-analysis`
helps evaluating this regularity condition.
In case you declare more moment conditions than estimated parameters, the
choice of :opt:`weighting_matrix <weighting_matrix = ['WM1','WM2',...,'WMn']>`
matters for the efficiency of the estimation, because the estimated
orthogonality conditions are random variables with unequal variances and
usually non-zero cross-moment covariances. Using a weighting matrix you can
re-weigh moments to pay more attention to orthogonality conditions that are
more informative or better measured (in the sense of having a smaller
variance). To achieve asymptotic efficiency, the weighting matrix needs to
be chosen such that, after appropriate scaling, it has probability limit
proportional to the inverse of the covariance matrix of the limiting
distribution of the vector of orthogonality conditions. Dynare uses a
Newey-West estimator with a Bartlett kernel to compute an estimate of this
so-called optimal weighting matrix. Moreover, in this over-identified case,
it is advised to perform the estimation in at least two stages by setting
e.g. :opt:`weighting_matrix=['DIAGONAL','DIAGONAL'] <weighting_matrix = ['WM1','WM2',...,'WMn']>`
so that the computation of the optimal weighting matrix benefits from the
consistent estimation of the previous stages. The optimal weighting matrix
is used to compute standard errors and the J-test of overidentifying
restrictions which tests whether the model and selection of moment
conditions fits the data sufficiently well. If the null hypothesis of a
"valid" model is rejected, then something is wrong with either your model
or selection of orthogonality conditions.
In case the global minimum is found in a region of the parameter space that
is typically considered unlikely (`dilemma of absurd parameters`), you may
opt to choose the :opt:`penalized_estimator <penalized_estimator>` option.
Similar to adding priors to the likelihood, this option includes prior
knowledge (i.e. the prior mean) as additional moment restrictions and
weighs them by their prior precision to guide the minimization algorithm
in more plausible regions of the parameter space. Ideally, these are
characterized by slightly worse values of the objective function. Note that
this comes at the cost of a loss in efficiency of the estimator.
|br|
.. command:: varobs VARIABLE_NAME...;
Required. All variables used in the :bck:`matched_moments` block
need to be observable. See :ref:`varobs <varobs>` for more details.
|br|
.. block:: matched_moments ;
This block specifies the product moments which are used in estimation.
Currently, only linear product moments (e.g.
:math:`E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]`)
are supported. For other functions like :math:`E[log(y_t)e^{x_t}]` you
need to declare auxiliary endogenous variables.
Each line inside of the block should be of the form::
VARIABLE_NAME(LEAD/LAG)^POWER*VARIABLE_NAME(LEAD/LAG)^POWER*...*VARIABLE_NAME(LEAD/LAG)^POWER;
where `VARIABLE_NAME` is the name of a declared observable variable,
`LEAD/LAG` is either a negative integer for lags or a positive one
for leads, and `POWER` is a positive integer indicating the exponent on
the variable. You can omit `LEAD/LAG` equal to `0` or `POWER` equal to `1`.
*Example*
For :math:`E[c_t], E[y_t], E[c_t^2], E[c_t y_t], E[y_t^2], E[c_t c_{t+3}], E[y_{t+1}^2 c^3_{t-4}], E[c^3_{t-5} y_{t}^2]`
use the following block:
::
matched_moments;
c;
y;
c*c;
c*y;
y^2;
c*c(3);
y(1)^2*c(-4)^3;
c(-5)^3*y(0)^2;
end;
*Limitations*
1. For GMM Dynare can only compute the theoretical mean, covariance and
autocovariances. Higher-order moments are only supported for SMM.
2. The product moments are not demeaned by default, unless the
:opt:`prefilter <prefilter = INTEGER>` option is set to 1. That is, by default,
`c*c` corresponds to :math:`E[c_t^2]` and not to :math:`Var[c_t]=E[c_t^2]-E[c_t]^2`.
*Output*
Dynare translates the :bck:`matched_moments` block into a cell array
``M_.matched_moments`` where:
* the first column contains a vector of indices for the chosen variables in declaration order
* the second column contains the corresponding vector of leads and lags
* the third column contains the corresponding vector of powers
During the estimation phase Dynare gets rid of redundant or duplicate
orthogonality conditions in ``M_.matched_moments`` and tells you which
conditions are removed. In the example above it would get grid of the
last row. The original block stays available in ``M_.matched_moments_orig``.
|br|
.. block:: estimated_params ;
Required. See :bck:`estimated_params` for the meaning and syntax.
|br|
.. block:: estimated_params_init ;
See :bck:`estimated_params_init` for the meaning and syntax.
|br|
.. block:: estimated_params_bounds ;
See :bck:`estimated_params_bounds` for the meaning and syntax.
|br|
.. command:: method_of_moments (OPTIONS...);
This command runs the method of moments estimation. The following
information will be displayed in the command window:
* Overview of options chosen by the user
* Estimation results for each stage and iteration
* Value of minimized moment distance objective function
* Result of J-test
* Table of data moments and estimated model moments
*Necessary options*
.. option:: mom_method = SMM|GMM
"Simulated Method of Moments" is triggered by `SMM` and
"Generalized Method of Moments" by `GMM`.
.. option:: datafile = FILENAME
The name of the file containing the data. See
:opt:`datafile <datafile = FILENAME>` for the meaning and syntax.
*Common options for SMM and GMM*
.. option:: order = INTEGER
Order of perturbation approximation. For GMM only orders 1|2|3 are
supported. For SMM you can choose an arbitrary order. Note that the
order set in other functions does not overwrite the default.
Default: ``1``.
.. option:: pruning
Discard higher order terms when iteratively computing simulations
of the solution. See :opt:`pruning <pruning>` for more details.
Default: not set for SMM, always set for GMM.
.. option:: penalized_estimator
This option includes deviations of the estimated parameters from the
prior mean as additional moment restrictions and weighs them by
their prior precision.
Default: not set.
.. option:: weighting_matrix = ['WM1','WM2',...,'WMn']
Determines the weighting matrix used at each estimation stage. Note
that this defines the number of stages, i.e. ``weighting_matrix = ['DIAGONAL','DIAGONAL','OPTIMAL']``
performs a three-stage estimation. Possible values for ``WM`` are:
``IDENTITY_MATRIX``
Sets the weighting matrix equal to the identity matrix.
``OPTIMAL``
Uses the optimal weighting matrix that is computed by a
Newey-West estimate with a Bartlett kernel. At the first
stage the data-moments are used as initial estimate of the
model moments, whereas at subsequent stages the previous
state estimate of model moments is used when computing
the optimal weighting matrix.
``DIAGONAL``
Uses the diagonal of the ``OPTIMAL`` weighting matrix.
``FILENAME``
The name of the M-file (extension ``.m``) containing a
user-specified weighting matrix. The file must include a
square matrix called `weighting_matrix` with both dimensions
equal to the number of orthogonality conditions.
Default value is ``['DIAGONAL','OPTIMAL']``.
.. option:: weighting_matrix_scaling_factor = DOUBLE
Scaling of weighting matrix in objective function.
Default: ``1``.
.. option:: bartlett_kernel_lag = INTEGER
Bandwidth of kernel for computing the optimal weighting matrix.
Default: ``20``.
.. option:: se_tolx = DOUBLE
Step size of numerical differentiation when computing standard
errors numerically.
Default: ``1e-5``.
.. option:: verbose
Display and store intermediate estimation results in ``oo_.mom``.
Default: not set.
*SMM-specific options*
.. option:: burnin = INTEGER
Number of periods dropped at the beginning of simulation.
Default: ``500``.
.. option:: bounded_shock_support
Trim shocks in simulations to :math:`\pm 2` standard deviations.
Default: not set.
.. option:: seed = INTEGER
Common seed used in simulations.
Default: ``24051986``.
.. option:: simulation_multiple = INTEGER
Multiple of data length used for simulation.
Default: ``7``.
*GMM-specific options*
.. option:: analytic_standard_errors
Compute standard errors using analytical derivatives of moments
with respect to estimated parameters.
Default: not set, i.e. standard errors are computed using a two-sided
finite difference method, see :opt:`se_tolx <se_tolx = DOUBLE>`.
*General options*
.. option:: dirname
Directory in which to store ``estimation`` output.
See :opt:`dirname <dirname = FILENAME>` for more details.
Default: ``<mod_file>``.
.. option:: graph_format = FORMAT
Specify the file format(s) for graphs saved to disk.
See :opt:`graph_format <graph_format = FORMAT>` for more details.
Default: ``eps``.
.. option:: nodisplay
See :opt:`nodisplay`. Default: not set.
.. option:: nograph
See :opt:`nograph`. Default: not set.
.. option:: noprint
See :opt:`noprint`. Default: not set.
.. option:: plot_priors = INTEGER
Control the plotting of priors.
See :opt:`plot_priors <plot_priors = INTEGER>` for more details.
Default: ``1``, i.e. plot priors.
.. option:: prior_trunc = DOUBLE
See :opt:`prior_trunc <prior_trunc = DOUBLE>` for more details.
Default: ``1e-10``.
.. option:: tex
See :opt:`tex`. Default: not set.
*Data options*
.. option:: first_obs = INTEGER
See :opt:`first_obs <first_obs = INTEGER>`.
Default: ``1``.
.. option:: nobs = INTEGER
See :opt:`nobs <nobs = INTEGER>`.
Default: all observations are considered.
.. option:: prefilter = INTEGER
A value of 1 means that the estimation procedure will demean each data
series by its empirical mean and each model moment by its theoretical
mean. See :opt:`prefilter <prefilter = INTEGER>` for more details.
Default: `0`, i.e. no prefiltering.
.. option:: logdata
See :opt:`logdata <logdata>`. Default: not set.
.. option:: xls_sheet = QUOTED_STRING
See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
.. option:: xls_range = RANGE
See :opt:`xls_range <xls_range = RANGE>`.
*Optimization options*
.. option:: huge_number = DOUBLE
See :opt:`huge_number <huge_number = DOUBLE>`.
Default: ``1e7``.
.. option:: mode_compute = INTEGER | FUNCTION_NAME
See :opt:`mode_compute <mode_compute = INTEGER | FUNCTION_NAME>`.
Default: ``13``, i.e. ``lsqnonlin``.
.. option:: additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]
Vector of additional minimization algorithms run after
``mode_compute``. If :opt:`verbose` option is set, then the additional estimation
results are saved into the ``oo_.mom`` structure prefixed with `verbose_`.
Default: no additional optimization iterations.
.. option:: optim = (NAME, VALUE, ...)
See :opt:`optim <optim = (NAME, VALUE, ...)>`.
.. option:: silent_optimizer
See :opt:`silent_optimizer`.
Default: not set.
*Numerical algorithms options*
.. option:: aim_solver
See :opt:`aim_solver <aim_solver>`. Default: not set.
.. option:: k_order_solver
See :opt:`k_order_solver <k_order_solver>`.
Default: disabled for order 1 and 2, enabled for order 3 and above.
.. option:: dr = OPTION
See :opt:`dr <dr = OPTION>`. Default: ``default``, i.e. generalized
Schur decomposition.
.. option:: dr_cycle_reduction_tol = DOUBLE
See :opt:`dr_cycle_reduction_tol <dr_cycle_reduction_tol = DOUBLE>`.
Default: ``1e-7``.
.. option:: dr_logarithmic_reduction_tol = DOUBLE
See :opt:`dr_logarithmic_reduction_tol <dr_logarithmic_reduction_tol = DOUBLE>`.
Default: ``1e-12``.
.. option:: dr_logarithmic_reduction_maxiter = INTEGER
See :opt:`dr_logarithmic_reduction_maxiter <dr_logarithmic_reduction_maxiter = INTEGER>`.
Default: ``100``.
.. option:: lyapunov = OPTION
See :opt:`lyapunov <lyapunov = OPTION>`. Default: ``default``, i.e.
based on Bartlets-Stewart algorithm.
.. option:: lyapunov_complex_threshold = DOUBLE
See :opt:`lyapunov_complex_threshold <lyapunov_complex_threshold = DOUBLE>`.
Default: ``1e-15``.
.. option:: lyapunov_fixed_point_tol = DOUBLE
See :opt:`lyapunov_fixed_point_tol <lyapunov_fixed_point_tol = DOUBLE>`.
Default: ``1e-10``.
.. option:: lyapunov_doubling_tol = DOUBLE
See :opt:`lyapunov_doubling_tol <lyapunov_doubling_tol = DOUBLE>`.
Default: ``1e-16``.
.. option:: sylvester = OPTION
See :opt:`sylvester <sylvester = OPTION>`.
Default: ``default``, i.e. uses ``gensylv``.
.. option:: sylvester_fixed_point_tol = DOUBLE
See :opt:`sylvester_fixed_point_tol <sylvester_fixed_point_tol = DOUBLE>`.
Default: ``1e-12``.
.. option:: qz_criterium = DOUBLE
See :opt:`qz_criterium <qz_criterium = DOUBLE>`.
Default: ``0.999999`` as it is assumed that the observables are weakly
stationary.
.. option:: qz_zero_threshold = DOUBLE
See :opt:`qz_zero_threshold <qz_zero_threshold = DOUBLE>`.
Default: ``1e-6``.
.. option:: schur_vec_tol = DOUBLE
Tolerance level used to find nonstationary variables in Schur decomposition
of the transition matrix. Default: ``1e-11``.
.. option:: mode_check
Plot the moments distance objective function for values around the
computed minimum for each estimated parameter in turn. This is
helpful to diagnose problems with the optimizer.
Default: not set.
.. option:: mode_check_neighbourhood_size = DOUBLE
See :opt:`mode_check_neighbourhood_size <mode_check_neighbourhood_size = DOUBLE>`.
Default: ``0.5``.
.. option:: mode_check_symmetric_plots = INTEGER
See :opt:`mode_check_symmetric_plots <mode_check_symmetric_plots = INTEGER>`.
Default: ``1``.
.. option:: mode_check_number_of_points = INTEGER
See :opt:`mode_check_number_of_points <mode_check_number_of_points = INTEGER>`.
Default: ``20``.
*Output*
``method_of_moments`` stores user options in a structure called
`options_mom_` in the global workspace. After running the estimation,
the parameters ``M_.params`` and the covariance matrices of the shocks
``M_.Sigma_e`` and of the measurement errors ``M_.H`` are set to the
parameters that minimize the quadratic moments distance objective
function. The estimation results are stored in the ``oo_.mom`` structure
with the following fields:
.. matvar:: oo_.mom.data_moments
Variable set by the ``method_of_moments`` command. Stores the mean
of the selected empirical moments of data. NaN values due to leads/lags
or missing data are omitted when computing the mean. Vector of dimension
equal to the number of orthogonality conditions.
.. matvar:: oo_.mom.m_data
Variable set by the ``method_of_moments`` command. Stores the selected
empirical moments at each point in time. NaN values due to leads/lags
or missing data are replaced by the corresponding mean of the moment.
Matrix of dimension time periods times number of orthogonality conditions.
.. matvar:: oo_.mom.Sw
Variable set by the ``method_of_moments`` command. Stores the
Cholesky decomposition of the currently used weighting matrix.
Square matrix of dimensions equal to the number of orthogonality
conditions.
.. matvar:: oo_.mom.model_moments
Variable set by the ``method_of_moments`` command. Stores the implied
selected model moments given the current parameter guess. Model moments
are computed in closed-form from the pruned state-space system for GMM,
whereas for SMM these are based on averages of simulated data. Vector of dimension equal
to the number of orthogonality conditions.
.. matvar:: oo_.mom.Q
Variable set by the ``method_of_moments`` command. Stores the scalar
value of the quadratic moment's distance objective function.
.. matvar:: oo_.mom.model_moments_params_derivs
Variable set by the ``method_of_moments`` command. Stores the analytically
computed Jacobian matrix of the derivatives of the model moments with
respect to the estimated parameters. Only for GMM with :opt:`analytic_standard_errors`.
Matrix with dimension equal to the number of orthogonality conditions
times number of estimated parameters.
.. matvar:: oo_.mom.gmm_stage_1_mode, oo_.mom.gmm_stage_2_mode,...
.. matvar:: oo_.mom.smm_stage_1_mode, oo_.mom.smm_stage_2_mode,...
.. matvar:: oo_.mom.verbose_gmm_stage_1_mode, oo_.mom.verbose_gmm_stage_2_mode,...
.. matvar:: oo_.mom.verbose_smm_stage_1_mode, oo_.mom.verbose_smm_stage_2_mode,...
Variables set by the ``method_of_moments`` command when estimating
with GMM or SMM. Stores the estimated values at stages 1, 2,....
The structures contain the following fields:
- ``measurement_errors_corr``: estimated correlation between two measurement errors
- ``measurement_errors_std``: estimated standard deviation of measurement errors
- ``parameters``: estimated model parameters
- ``shocks_corr``: estimated correlation between two structural shocks.
- ``shocks_std``: estimated standard deviation of structural shocks.
If the :opt:`verbose` option is set, additional fields prefixed with
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.
.. matvar:: oo_.mom.gmm_stage_1_std_at_mode, oo_.mom.gmm_stage_2_std_at_mode,...
.. matvar:: oo_.mom.smm_stage_1_std_at_mode, oo_.mom.smm_stage_2_std_at_mode,...
.. matvar:: oo_.mom.verbose_gmm_stage_1_std_at_mode, oo_.mom.verbose_gmm_stage_2_std_at_mode,...
.. matvar:: oo_.mom.verbose_smm_stage_1_std_at_mode, oo_.mom.verbose_smm_stage_2_std_at_mode,...
Variables set by the ``method_of_moments`` command when estimating
with GMM or SMM. Stores the estimated standard errors at stages 1, 2,....
The structures contain the following fields:
- ``measurement_errors_corr``: standard error of estimated correlation between two measurement errors
- ``measurement_errors_std``: standard error of estimated standard deviation of measurement errors
- ``parameters``: standard error of estimated model parameters
- ``shocks_corr``: standard error of estimated correlation between two structural shocks.
- ``shocks_std``: standard error of estimated standard deviation of structural shocks.
If the :opt:`verbose` option is set, additional fields prefixed with
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.