A Two-Step Method for Testing Many Moment Inequalities

This article considers the problem of testing a finite number of moment inequalities. For this problem, Romano, Shaikh, and Wolf proposed a two-step testing procedure. In the first step, the procedure incorporates information about the location of moments using a confidence region. In the second step, the procedure accounts for the use of the confidence region in the first step by adjusting the significance level of the test appropriately. Its justification, however, has so far been limited to settings in which the number of moments is fixed with the sample size. In this article, we provide weak assumptions under which the same procedure remains valid even in settings in which there are "many" moments in the sense that the number of moments grows rapidly with the sample size. We confirm the practical relevance of our theoretical guarantees in a simulation study. We additionally provide both numerical and theoretical evidence that the procedure compares favorably with the method proposed by Chernozhukov, Chetverikov, and Kato, which has also been shown to be valid in such settings.