Indirect Inference
==================

Besides full-information (maximum-likelihood / Bayesian) estimation, RISE can
estimate a model by **limited-information** methods: choose the parameters so
that a set of *auxiliary statistics* implied by the model -- moments,
impulse responses, regression coefficients, ... -- matches the same statistics
computed from the data, in a weighted-distance sense. A good reference in the
DSGE literature is :cite:`Ruge-Murcia2012`.

The principle of indirect inference
-----------------------------------

Let :math:`\widehat{a}` collect the auxiliary statistics computed from the data
and :math:`a\left(\theta\right)` the same statistics implied by the model at
parameter vector :math:`\theta`. Indirect inference solves

.. math::

	\widehat{\theta}=\arg\min_{\theta}\;\left[\widehat{a}-a\left(\theta\right)\right]'\,
	W\;\left[\widehat{a}-a\left(\theta\right)\right],

for a positive-definite weighting matrix :math:`W` (often the identity, or the
inverse of the sampling covariance of :math:`\widehat{a}` for the efficient
choice). Different choices of the auxiliary statistics give the generalized
method of moments, the simulated method of moments, impulse-response matching,
and so on.

Indirect inference in RISE
--------------------------

RISE does not prescribe the auxiliary statistics or the distance metric -- you
supply them as a MATLAB function, and RISE handles the optimisation over the
parameters (with the same optimisers, bounds and priors as full-information
estimation; see :doc:`Posterior maximization`)::

	mest = indirect_inference(m, myobjective, 'priors', priors);

Here ``priors`` lists the parameters to estimate and their supports, exactly as
for :doc:`Bayesian / ML estimation <Posterior maximization>`, and
``myobjective`` is a function handle with signature::

	function [critmin, retcode] = myobjective(m)
	    % m : the model object, already parameterised (and, if needed, solved)
	    % critmin : the criterion to minimise
	    % retcode : 0 if critmin was computed successfully, a nonzero RISE
	    %           return code otherwise (decipher(retcode) explains it)

When ``indirect_inference`` runs, RISE skips data loading and filtering -- the
objective does whatever is needed -- and minimises ``critmin`` over the listed
parameters. Any usual model option can be passed through as extra name/value
pairs.

Generalized method of moments
-----------------------------

For (analytical) GMM the auxiliary statistics are population moments. Inside
``myobjective`` you compute the model-implied moments from the solved model --
``theoretical_moments(m)`` and ``theoretical_autocovariances(m)`` /
``theoretical_autocorrelations(m)`` give the variances, covariances and
autocovariances implied by the perturbation solution -- stack them into
:math:`a\left(\theta\right)`, and return the weighted distance to the
empirical moments :math:`\widehat{a}` (computed once from the data and captured
by the function handle, together with the weighting matrix)::

	g       = a_model(m) - a_hat;       % a_model: stacks the model moments
	critmin = g' * W * g;
	retcode = 0;

(``a_hat`` and ``W`` are closed over -- e.g. ``myobjective = @(m) gmm_crit(m, a_hat, W)``.)

Simulated method of moments
---------------------------

When the moments of interest are not available in closed form (nonlinear
models, regime-dependent statistics, ...), simulate the model and use the
*sample* moments instead: inside ``myobjective``, call ``simulate(m, ...)`` (a
long simulation, or several replications), compute the same statistics on the
simulated paths as on the data, and return the weighted distance. Fixing the
simulation seed makes the criterion smooth in :math:`\theta`.

Impulse-response matching
-------------------------

Here the auxiliary statistics are impulse responses -- typically estimated from
the data with an (S)VAR (see :doc:`../SVAR_capabilities/Main Structural VAR Modeling`).
Inside ``myobjective``, compute the model IRFs with ``irf(m, shock_names, H)``
over the matched horizons, stack the entries you want to match, and return the
weighted distance to the empirical IRFs (with :math:`W` typically the inverse
of the covariance of the empirical IRFs).

Indirect inference composes with the rest of the toolbox -- for instance it can
be combined with :doc:`optimal simple rules <../DSGE_capabilities/OSR/OptimizedSimpleRules>`
(choosing rule coefficients to match empirical statistics) by supplying the
corresponding objective.

.. todo::

   Add a full worked example: pick a model, choose a small moment set, build
   the data moments and weighting matrix, write ``myobjective``, run
   ``indirect_inference``, and compare the estimates with the ML ones.