  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 defined probabilistic interpretation and plays the role of
a nonparametric likelihood, a fully Bayesian semiparametric framework can be
developed. We establish a number of powerful results surrounding the Bayesian
ETEL framework in such models. One major concern driving our work is the
possibility of misspecification. To accommodate this possibility, we show how
the moment conditions can be reexpressed in terms of additional nuisance
parameters and that, even under misspecification, the Bayesian ETEL posterior
distribution satisfies a Bernstein-von Mises result. A second key contribution
of the paper is the development of a framework based on marginal likelihoods
and Bayes factors to compare models defined by different moment conditions.
Computation of the marginal likelihoods is by the method of Chib (1995) as
extended to Metropolis-Hastings samplers in Chib and Jeliazkov (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 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.
