In this paper, I construct a new test of conditional moment inequalities, which is based on studentized kernel estimates of moment functions with many different values of the bandwidth parameter. The test automatically adapts to the unknown smoothness of moment functions and has uniformly correct asymptotic size. The test has high power in a large class of models with conditional moment inequalities. Some existing tests have nontrivial power against n^{-1/2}-local alternatives in a certain class of these models whereas my method only allows for nontrivial testing against (n/log n)^{-1/2}-local alternatives in this class. There exist, however, other classes of models with conditional moment inequalities where the mentioned tests have much lower power in comparison with the test developed in this paper.
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