Quasi-Bayesian analysis of nonparametric instrumental variables models

TL;DR

This paper develops a quasi-Bayesian framework for nonparametric instrumental variables models, analyzing the asymptotic behavior of quasi-posterior distributions without strict distributional assumptions, and achieves minimax optimal convergence rates.

Contribution

It introduces a quasi-Bayesian approach with priors on finite-dimensional sieves, deriving contraction rates and Bernstein-von Mises results that improve upon previous work.

Findings

Quasi-posterior distribution attains minimax optimal contraction rates.

Derived nonparametric Bernstein-von Mises type theorem.

Established convergence rates for quasi-Bayes estimators.

Abstract

This paper aims at developing a quasi-Bayesian analysis of the nonparametric instrumental variables model, with a focus on the asymptotic properties of quasi-posterior distributions. In this paper, instead of assuming a distributional assumption on the data generating process, we consider a quasi-likelihood induced from the conditional moment restriction, and put priors on the function-valued parameter. We call the resulting posterior quasi-posterior, which corresponds to ``Gibbs posterior'' in the literature. Here we focus on priors constructed on slowly growing finite-dimensional sieves. We derive rates of contraction and a nonparametric Bernstein-von Mises type result for the quasi-posterior distribution, and rates of convergence for the quasi-Bayes estimator defined by the posterior expectation. We show that, with priors suitably chosen, the quasi-posterior distribution (the quasi-Bayes estimator) attains the minimax optimal rate of contraction (convergence, resp.). These results greatly sharpen the previous related work.