Posterior Consistency of Nonparametric Conditional Moment Restricted Models
30 Pages Posted: 11 Feb 2011 Last revised: 24 May 2011
Date Written: November 9, 2010
Abstract
This paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g0. 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) thintail prior (a prior that has very thin tail outside a growing bounded set), and (iii) normal prior with non-shrinking variance. In addition, g0 is allowed to be only partially identified, 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.
Keywords: Identified Region, Limited Information Likelihood, Sieve Approximation, Nonparametric Instrumental Variable, Ill-Posed Problem, Partial Identification, Bayesian Inference
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