The Generalized Method of Moments in the Bayesian Framework and a Model and Moment Selection Criterion

While the classical framework has a rich set of limited information procedures such as GMM and other related methods, the situation is not so in the Bayesian framework. We develop a limited information procedure in the Bayesian framework that does not require the knowledge of the full likelihood. The developed procedure is a Bayesian counterpart of the classical GMM but has advantages over the classical GMM for practical applications. The necessary limited information for our approach is a set of moment conditions, instead of the likelihood function, which has a counterpart in the classical GMM. Such moment conditions in the Bayesian framework are obtained from the equality condition of the Bayes' estimator and the GMM estimator. From such moment conditions, a posterior probability measure is derived that forms the basis of our limited information Bayesian procedure. This limited information posterior has some desirable properties for small and large sample analyses. An alternative approach is also provided in this paper for deriving a limited information posterior based on a variant of the empirical likelihood method where an empirical likelihood is obtained from the moment conditions of the classical GMM. This alternative approach yields asymptotically the same result as the approach explained above. Based on our limited information method, we develop a procedure for selecting the moment for GMM. This moment selection procedure is an extension of the Bayesian information criterion to the Bayesian semi-parametric, limited information framework. It is shown that under some conditions the proposed moment selection procedure is a consistent decision rule.