• Abstract: This paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g₀. We estimate g₀in a quasi-Bayesian way, based on the limited information likelihood. We 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 it 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.

• Paper: pdf file

• Supplementary materials
  Proofs: pdf file.
  This document contains all the technical proofs of the main paper.

  Regularization using shrinking normal prior: pdf file.
  The regularization is carried out by a normal prior with shrinking variance, instead of a slowly-growing sieve dimension. Conditions (3.10) and (3.11) are then relaxed.

• Slides: pdf file

• Related literature:

  Chen X. and Pouzo D.(2011).
  Estimation and nonparametric conditional moment models with possibly nonsmooth generalized residuals.
  Econometrica, to appear.

  Hall P. and Horowitz J. (2005).
  Nonparametric methods for inference in the presence of instrumental variables
  Annals of Statistics 33, 2904-2929

  Florens J. and Simoni A. (2009)
  Nonparametric estimation of an instrumental regression: a quasi-Bayesian approach based on regularized posterior
  Manuscript.