Bayesian Empirical Likelihood Estimation and Comparison of Moment Condition Models

In this paper we consider the problem of inference in statistical models characterized by moment restrictions by casting the problem within the Exponentially Tilted Empirical Likelihood (ETEL) framework. Because the ETEL function has a well de ned probabilistic interpretation and plays the role of a likelihood, a fully Bayesian framework can be developed. We establish a number of powerful results surrounding the Bayesian ETEL framework in such models. One ma jor concern driving our work is the possibility of misspeci cation. To accommodate this possibility, we show how the moment conditions can be reexpressed in terms of additional nuisance parameters and that, even under misspeci cation, the Bayesian ETEL posterior distribution satis es a Bernstein-von Mises result. A second key contribution of the paper is the development of a framework based on marginal likelihoods (MLs) and Bayes factors to compare models de ned by di erent moment conditions. Computation of the MLs is by Chib (1995)'s method. We establish the consistency of the Bayes factors and show that the ML favors the model with the minimum number of parameters and the maximum number of valid moment restrictions. When the models are misspeci ed, the ML 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 paper provide a further broadening of the theoretical underpinning and value of the Bayesian ETEL framework with likely far-reaching practical consequences. The discussion is illuminated through several examples.