Bayesian estimation and comparison of moment condition models

In this article, we develop a Bayesian semiparametric analysis of moment condition models by casting the problem within the exponentially tilted empirical likelihood (ETEL) framework. We use this framework to develop a fully Bayesian analysis of correctly and misspecified moment condition models. We show that even under misspecification, the Bayesian ETEL posterior distribution satisfies the Bernstein–von Mises (BvM) theorem. We also develop a unified approach based on marginal likelihoods and Bayes factors for comparing different moment-restricted models and for discarding any misspecified moment restrictions. Computation of the marginal likelihoods is by the method of Chib (1995) as extended to Metropolis–Hastings samplers in Chib and Jeliazkov in 2001. We establish the model selection consistency of the marginal likelihood and show that the marginal likelihood favors the model with the minimum number of parameters and the maximum number of valid moment restrictions. When the models are misspecified, the marginal likelihood model selection procedure selects the model that is closer to the (unknown) true data-generating process in terms of the Kullback–Leibler divergence. The ideas and results in this article broaden the theoretical underpinning and value of the Bayesian ETEL framework with many practical applications. The discussion is illuminated through several examples. Supplementary materials for this article are available online.