Bonferroni-Based Size-Correction for Nonstandard Testing Problems

Abstract

We develop powerful new size-correction procedures for nonstandard hypothesis testing environments in which the asymptotic distribution of a test statistic is discontinuous in a parameter under the null hypothesis. Examples of this form of testing problem are pervasive in econometrics and complicate inference by making size diffcult to control. This paper introduces two sets of new size-correction methods that correspond to two different general hypothesis testing frameworks. The new methods are designed to maximize the power of the underlying test while maintaining correct asymptotic size uniformly over the parameter space specified by the null hypothesis. They involve the construction of critical values that make use of reasoning derived from Bonferroni bounds. The first set of new methods provides complementary alternatives to existing size-correction methods, entailing substantially higher power for many testing problems. The second set of new methods provides the first available asymptotically size-correct tests for the general class of testing problems to which it applies. This class includes hypothesis tests on parameters after consistent model selection and tests on super-efficient/hard-thresholding estimators. We detail the construction and performance of the new tests in three specific examples: testing after conservative model selection, testing when a nuisance parameter may be on a boundary and testing after consistent model selection.

Keywords