Bayesian Estimation and Comparison of Moment Condition Models

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.

Bayesian Empirical Likelihood Inference with Complex Survey Data

“…The basic marginal likelihood was used by Chib et al . () for model selection under the model‐based framework for independent data. For complex survey data, we propose to usewhere is any point in the support of the posterior and the dependence on the model has been made explicit.…”

“…Second, the selection procedure allows models to be defined through non-differentiable estimating functions. The procedure of Chib et al (2018) is developed for independent and identically distributed samples, and direct applications of the method to survey data will lead to invalid results under the design-based framework.…”

“…Recently, within the context of the exponentially tilted empirical likelihood, Chib et al . () studied robust Bayesian semiparametric estimation and model selection criteria for a collection of possibly overidentified differentiable moment condition models.…”

“…Yang and He (2012) considered BEL inferences for quantile regression with independent sample data, which deals with non-differentiable estimating functions. Recently, within the context of the exponentially tilted empirical likelihood, Chib et al (2018) studied robust Bayesian semiparametric estimation and model selection criteria for a collection of possibly overidentified differentiable moment condition models. Rao and Wu (2010b) developed a Bayesian pseudo-empirical-likelihood approach and focused on constructing Bayesian credible intervals for a population mean.…”

“…where {θ .k/ g } are the sampled points from the posterior distribution and {θ .k/ j } are the random draws from q.θ .k/ |θ .k/ / with the givenθ .k/ .The basic marginal likelihood was used byChib et al (2018) for model selection under the model-based framework for independent data. For complex survey data, we propose to uselog{ᑧ B .S, M k /} = log{P.θ .k/ |M k /} + log{L n .θ .k/ |S, M k /} − log{P.θ .k/ |S, M k /}, .16/ whereθ .k/ isany point in the support of the posterior and the dependence on the model M k has been made explicit.…”