This paper develops a Bayesian approach to inference in a class of partially identified econometric models. Models in this class are characterized by a known mapping between a point identified reduced-form parameter μ, and the identified set for a partially identified parameter θ. The approach maps posterior inference about μ to various posterior inference statements concerning the identified set for θ, without the specification of a prior for θ. Many posterior inference statements are considered, including the posterior probability that a particular parameter value (or a set of parameter values) is in the identified set. The approach applies also to functions of θ. The paper develops general results on large sample approximations, which illustrate how the posterior probabilities over the identified set are revised by the data, and establishes conditions under which the Bayesian credible sets also are valid frequentist confidence sets. The approach is computationally attractive even in high-dimensional models, in that the approach avoids an exhaustive search over the parameter space. The performance of the approach is illustrated via Monte Carlo experiments and an empirical application to a binary entry game involving airlines.

We address counterfactual analysis in empirical models of games with partially identified parameters, and multiple equilibria and/or randomized strategies, by constructing and analyzing the counterfactual predictive distribution set (CPDS). This framework accommodates various outcomes of interest, including behavioral and welfare outcomes. It allows a variety of changes to the environment to generate the counterfactual, including modifications of the utility functions, the distribution of utility determinants, the number of decision makers, and the solution concept. We use a Bayesian approach to summarize statistical uncertainty. We establish conditions under which the population CPDS is sharp from the point of view of identification. We also establish conditions under which the posterior CPDS is consistent if the posterior distribution for the underlying model parameter is consistent. Consequently, our results can be employed to conduct counterfactual analysis after a preliminary step of identifying and estimating the underlying model parameter based on the existing literature. Our consistency results involve the development of a new general theory for Bayesian consistency of posterior distributions for mappings of sets. Although we primarily focus on a model of a strategic game, our approach is applicable to other structural models with similar features.