Econometrics and Applied Micro Seminar
Confidence intervals are commonly used to describe parameter uncertainty. In non-standard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty with positive probability, even if they are based on inverting powerful tests, or if they are chosen to minimize average expected length. We apply a betting framework to formalize the "reasonableness" of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the degree of unreasonableness of previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are reasonable by construction. 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.