A simple specification test for models with many conditional moment inequalities

This paper proposes a simple specification test for partially identified models with a large or possibly uncountably infinite number of conditional moment (in)equalities. The approach is valid under weak assumptions, allowing for both weak identification and non-differentiable moment conditions. Computational simplifications are obtained by reusing certain expensive-to-compute components of the test statistic when constructing the critical values. Because of the weak assumptions, the procedure faces a new set of interesting theoretical issues which we show can be addressed by an unconventional sample-splitting procedure that runs multiple tests of the same null hypothesis. The resulting specification test controls size uniformly over a large class of data generating processes, has power tending to 1 for fixed alternatives, and has power against certain local alternatives which we characterize. Finally, the testing procedure is demonstrated in three simulation exercises.