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Credibility of Confidence Sets in Nonstandard Econometric Problems
Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework to formalize the ''reasonableness'' of confidence intervals and use it for two purposes. First, we quantify the degree of unreasonableness of previously suggested confidence intervals. Second, we derive alternative sets that are reasonable by construction. As was already realized in the literature, any bet-proof set must be a superset of a Bayesian credible set relative to some prior. This suggests that attractive bet-proof confidence sets may be obtained by selecting a prior that induces a given type of credible set (such as HPD) to have frequentist coverage. The main theoretical result of this paper is to show that such a prior exists. Previous results for Bayesian sets with frequentist coverage are either for particular families of distributions, for invariant problems, or for asymptotic equivalence of coverage and credibility in LAN models. In contrast, our existence result is entirely generic as it applies to any (discretized) inference problem under very mild regularity conditions. We apply our framework to several nonstandard problems involving a parameter near a boundary, weak instruments, near unit roots, and moment inequalities. We find that most previously suggested confidence intervals are not reasonable, and numerically determine alternative confidence sets that satisfy our criteria.