Bayesian Likelihoods for Moment Condition Models

This paper investigates the viability of conducting Bayesian inference when the only information linking parameters and data is in the form of moment restrictions. Bayesian inference in moment condition models is difficult to implement since the likelihood function is not fully specified. In this paper, we obtain a class of nonparametric likelihoods by formal Bayesian calculations that take into account the semiparametric nature of the problem. These likelihoods are closely connected to Generalized Empirical Likelihood (GEL) methods. The ability of these likelihoods to provide valid probability statements is discussed and examined by studying the coverage properties of the resulting posteriors.