Bayesian Properties of Minimum Divergence and Generalized Empirical Likelihood Methods (PRELIMINARY DRAFT)∗

When the object of the statistical analysis is the estimation of an economic model, the choice of a likelihood function should be coherent with the economic model under investigation. Many econometric models provides the researcher with weak “structural prediction” about the parameter of interest and the data. Econometric models specified through moment conditions and usually estimated by Generalized Method of Moments (GMM) belong to this class. This paper show that valid likelihoods (in a Bayesian sense) can be constructed by substituting the parametric likelihood in the Bayes theorem with some (empirical) likelihoods whose functional forms have a close relationship with the weights generated by Minimum Divergence (MD) and Generalized Empirical Likelihood (GEL). These likelihoods are obtained by eliciting a prior distribution for the nuisance parameters that is maximally uninformative but contains information about the moment conditions. Integrating over the nuisance parameters with respect to this distribution gives likelihoods that have a relationship with the weights that MD/GEL methods assign to a given observation. The likelihoods obtained are proper only when all the possible outcomes of the underlying random vector is observed. This issue is investigated by applying a simply methodology proposed by Monahan and Boos (1992). Higher order properties of the maximum posterior estimators are also discussed.