This paper considers the problem of testing a finite number of moment inequalities. For this problem, Romano et al. (2014) propose 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. An important feature of the proposed method is that it is “practical” in the sense that it remains computationally feasible even if the number of moments is large. Its justification, however, has so far been limited to settings in which the number of moments is fixed with the sample size. In this paper, 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 et al. (2019), which has also been shown to be valid in such settings.

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