Bayesian Estimation and Comparison of Moment Condition Models

TL;DR

This paper develops a Bayesian framework for inference and model comparison in moment condition models using the ETEL approach, addressing misspecification and establishing theoretical properties like Bernstein-von Mises and model selection consistency.

Contribution

It introduces a Bayesian ETEL framework for moment models, including methods for model comparison and robustness under misspecification, with theoretical guarantees.

Findings

Bayesian ETEL posterior satisfies Bernstein-von Mises under misspecification.

Model selection via marginal likelihood favors simpler models with more valid moments.

Marginal likelihood approach is consistent and favors models closer to the true data generating process.

Abstract

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.