  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.
