Quasi-Bayesian analysis of nonparametric instrumental variables models

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 stringent distributional assumption on the data generating process, we consider a quasi-likelihood induced from the conditional moment restriction, and put priors for the function-valued parameter. We call the resulting posterior quasi-posterior, which corresponds to "Gibbs posterior" in the literature. Here we shall focus on sieve priors, which are priors that concentrate on finite dimensional sieve spaces. The dimension of the sieve space should increase as the sample size. We derive rates of contraction and a non-parametric 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, respectively). These results greatly sharpen the previous related work.