Date Published:

Abstract:

We develop a Bayesian approach to inference in a class of partially
identified econometric models. Models in this class have a point identified parameter μ (e.g., characteristics of the distribution of the data) and a partially identified parameter of interest (e.g., parameters of the model); further, if μ is known then the identified set for is known. Many instances of this class are commonly used in empirical work. Our approach maps, via the mapping between μ and , and without the specification of a prior for , the posterior for the point identified parameter μ to posterior probability statements about the identified set for , which is the quantity about which the data are informative. Thus, among other examples, we can report the posterior probability that a particular parameter value (or a set of parameter values, or a function of the parameter) is in the identified set. The paper develops general results on large sample approximations to these posterior probabilities, which illustrate how the posterior probabilities over the identified set are revised by the data. The paper establishes conditions under which the credible sets for the identified set also are valid frequentist confidence sets, providing a connection between Bayesian and frequentist inference in partially identified models (including for functions of the partially identified parameter). The approach is computationally attractive even in high-dimensional models: the approach avoids an exhaustive search over the parameter space (or “guess and verify”), partly by using existing MCMC methods to simulate draws from the posterior for μ. The paper also considers issues related to specification testing and estimation of misspecified models. We illustrate our approach via a set of Monte Carlo experiments and an empirical application to a binary entry game involving airlines. JEL codes: C10, C11. Keywords: partial identification, identified set, criterion function, posterior, Bayesian inference

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