Semi-parametric Bayesian Partially Identified Models based on Support Function

Bayesian partially identified models have received a growing attention in recent years in the econometric literature, due to their broad applications in empirical studies. Classical Bayesian approach in this literature has been assuming a parametric model, by specifying an ad-hoc parametric likelihood function. However, econometric models usually only identify a set of moment inequalities, and therefore assuming a known likelihood function suffers from the risk of misspecification, and may result in inconsistent estimations of the identified set. On the other hand, moment-condition based likelihoods such as the limited information and exponential tilted empirical likelihood, though guarantee the consistency, lack of probabilistic interpretations. We propose a semi-parametric Bayesian partially identified model, by placing a nonparametric prior on the unknown likelihood function. Our approach thus only requires a set of moment conditions but still possesses a pure Bayesian interpretation. We study the posterior of the support function, which is essential when the object of interest is the identified set. The support function also enables us to construct two-sided Bayesian credible sets (BCS) for the identified set. It is found that, while the BCS of the partially identified parameter is too narrow from the frequentist point of view, that of the identified set has asymptotically correct coverage probability in the frequentist sense. Moreover, we establish the posterior consistency for both the structural parameter and its identified set. We also develop the posterior concentration theory for the support function, and prove the semi-parametric Bernstein von Mises theorem. Finally, the proposed method is applied to analyze a financial asset pricing problem.