1.7. 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 [Ruge-Murcia, 2012].

1.7.1. The principle of indirect inference

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

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

for a positive-definite weighting matrix \(W\) (often the identity, or the inverse of the sampling covariance of \(\widehat{a}\) 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.

1.7.2. 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 Posterior maximization):

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

Here priors lists the parameters to estimate and their supports, exactly as for Bayesian or maximum-likelihood estimation, 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.

1.7.3. 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 \(a\left(\theta\right)\), and return the weighted distance to the empirical moments \(\widehat{a}\) (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).)

1.7.4. 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 \(\theta\).

1.7.5. Impulse-response matching

Here the auxiliary statistics are impulse responses – typically estimated from the data with an (S)VAR (see 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 \(W\) 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 optimal simple rules (choosing rule coefficients to match empirical statistics) by supplying the corresponding objective.