Bayesian analysis in moment inequality models

TL;DR

This paper analyzes the asymptotic behavior of Bayesian posteriors in moment inequality models, showing exponential convergence outside the identified region and advantages over frequentist methods, with applications to model selection.

Contribution

It provides a theoretical analysis of Bayesian posterior convergence in moment inequality models and introduces a maximum posterior procedure for model and moment selection.

Findings

Posterior density converges exponentially fast outside the identified region.

Within the identified region, the posterior is bounded below by a positive constant.

The maximum posterior procedure asymptotically selects the true model with the most inequalities and simplest structure.

Abstract

This paper presents a study of the large-sample behavior of the posterior distribution of a structural parameter which is partially identified by moment inequalities. The posterior density is derived based on the limited information likelihood. The posterior distribution converges to zero exponentially fast on any $\delta$-contraction outside the identified region. Inside, it is bounded below by a positive constant if the identified region is assumed to have a nonempty interior. Our simulation evidence indicates that the Bayesian approach has advantages over frequentist methods, in the sense that, with a proper choice of the prior, the posterior provides more information about the true parameter inside the identified region. We also address the problem of moment and model selection. Our optimality criterion is the maximum posterior procedure and we show that, asymptotically, it selects the true moment/model combination with the most moment inequalities and the simplest model.