Semiparametric Bayesian Inference for a Conditional Moment Equality Model

I propose a semiparametric Bayesian inference framework for conditional moment equalities. The core idea is that these models deterministically map a conditional distribution of data to a structural parameter via the restriction that a conditional expectation equals zero. Consequently, a posterior for the conditional distribution leads to a posterior for the structural parameter by minimizing the distance of the conditional moments to zero. The method has similar flexibility to frequentist semiparametric estimators and does not require converting the conditional moments into unconditional moments. I also establish frequentist asymptotic optimality of my proposal via a semiparametric Bernsteinvon Mises theorem (BvM), which establishes that the posterior for the structural parameter is asymptotically normal and matches the Chamberlain (1987) semiparametric efficiency bound. The BvM conditions are verified for Gaussian process priors and complement the numerical aspects of the paper in which these priors are used to estimate welfare effects.