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 and a notion of bet‐proofness to formalize the “reasonableness” of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the violations of bet‐proofness for previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are bet‐proof by construction. We apply our framework to several nonstandard problems involving weak instruments, near unit roots, and moment inequalities. We find that previously suggested confidence intervals are not bet‐proof, and numerically determine alternative bet‐proof confidence sets.