Quasi-Bayesian Estimation and Inference with Control Functions

Abstract

This paper introduces a quasi-Bayesian method that integrates frequentist nonparametric estimation with Bayesian inference in a two-stage process. Applied to an endogenous discrete choice model, the approach first uses kernel or sieve estimators to estimate the control function nonparametrically, followed by Bayesian methods to estimate the structural parameters. This combination leverages the advantages of both frequentist tractability for nonparametric estimation and Bayesian computational efficiency for complicated structural models. We analyze the asymptotic properties of the resulting quasi-posterior distribution, finding that its mean provides a consistent estimator for the parameters of interest, although its quantiles do not yield valid confidence intervals. However, bootstrapping the quasi-posterior mean accounts for the estimation uncertainty from the first stage, thereby producing asymptotically valid confidence intervals.