Posterior consistency of nonparametric conditional moment restricted models

TL;DR

This paper proves the posterior consistency for nonparametric conditional moment models estimated via a quasi-Bayesian approach, accommodating partial identification and non-compact parameter spaces, with applications to instrumental regression.

Contribution

It introduces a quasi-Bayesian estimation framework with various priors for nonparametric models, establishing posterior consistency under broad conditions including partial identification.

Findings

Posterior converges to the identified region with a slowly growing sieve dimension.

Consistency holds for different prior types, including truncated, thin-tail, and normal priors.

Results extend to models with non-compact parameter spaces and partial identification.

Abstract

This paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite-dimensional parameter $g_0$. We estimate it in a quasi-Bayesian way, based on the limited information likelihood, and investigate the impact of three types of priors on the posterior consistency: (i) truncated prior (priors supported on a bounded set), (ii) thin-tail prior (a prior that has very thin tail outside a growing bounded set) and (iii) normal prior with nonshrinking variance. In addition, $g_0$ is allowed to be only partially identified in the frequentist sense, and the parameter space does not need to be compact. The posterior is regularized using a slowly growing sieve dimension, and it is shown that the posterior converges to any small neighborhood of the identified region. We then apply our results to the nonparametric instrumental regression model. Finally, the posterior consistency using a random sieve dimension parameter is studied.