Abstract
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. Partial identification identified set criterion function Bayesian inference C10 C11
Highlights
This paper considers the problem of Bayesian inference in a class of partially identified models
If θ is a parameter of an underlying econometric model and μ are statistics concerning the data, the identified set mapping is the set of θ∗ such that the underlying econometric model evaluated at θ∗ generates μ
By focusing on posterior probability statements concerning the identified set rather than the partially identified parameter, this paper establishes a method for Bayesian inference that results in posterior inference statements that do not depend on the prior asymptotically
Summary
This paper considers the problem of Bayesian inference in a class of partially identified models. The Bayesian approach in this paper can use the developed literature on simulation of posterior distributions for point identified parameters, and can use a variety of analytic and computational simplifications concerning the identified set mapping, implying that it is not necessary to use such an “exhaustive search” grid search. By focusing on posterior probability statements concerning the identified set rather than the partially identified parameter, this paper establishes a method for Bayesian inference that results in posterior inference statements that do not depend on the prior asymptotically. This approach does not even require the specification of any prior for the partially identified parameter, and is a starting point that summarizes the.
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