Consistent estimation with many moment inequalities

In this paper, we consider estimation of the identified set when the number of moment inequalities is large relative to sample size, possibly infinite. Many applications in the recent literature on partially identified problems have this feature, including dynamic games, set-identified IV models, and parameters defined by a continuum of moment inequalities, in particular conditional moment inequalities. We provide a generic consistency result for criterion-based estimators using an increasing number of unconditional moment inequalities. We then develop more specific results for set estimation subject to conditional moment inequalities: we first derive the fastest possible rate for estimating the sharp identification region under smoothness conditions on the conditional moment functions. We also give rate conditions for inference under local alternatives.