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This paper derives asymptotic power functions for Cramer-von Mises (CvM) style tests for inference on a ﬁnite dimensional parameter deﬁned by conditional moment inequalities in the case where the parameter is set identiﬁed. Combined with power results for Kolmogorov-Smirnov (KS) tests, these results can be used to choose the optimal test statistic, weighting function and, for tests based on kernel estimates, kernel bandwidth. The results show that KS tests are preferred to CvM tests, and that a truncated variance weighting is preferred to bounded weightings under a minimax criterion, and for a class of alternatives that arises naturally in these models. The results also provide insight into how moment selection and the choice of instruments aﬀect power. Such considerations have a large eﬀect on power for instrument based approaches when a CvM statistic or an unweighted KS statistic is used and relatively little eﬀect on power with optimally weighted KS tests.
Armstrong, Timothy B., "On the Choice of Test Statistic for Conditional Moment Inequalities" (2014). Cowles Foundation Discussion Papers. 2370.