Essays on set estimation and inference with moment inequalities

This thesis explores power and consistency of estimation and inference procedures with moment inequalities, and applications of the moment inequality framework to estimation of frontiers in finance. In the first chapter, I consider estimation of the identified set and inference on a partially identified parameter when the number of moment inequalities is large relative to sample size. Many applications in the recent literature on set estimation have this feature. Examples discussed in this paper include set-identified instrumental variables models, inference under conditional moment inequalities, and dynamic games. I show that GMM-type test statistics will often be poorly centered when the number of moment inequalities is large. My results establish consistency of the set estimator based on a Wald-type criterion, and I give conditions for uniformly valid inference under many weak moment asymptotics for both plug-in and subsampling procedures. The second chapter evaluates the performance of an Anderson-Rubin (AR) type test for a finite number of moment inequalities, and propose a modified Lagrange Multiplier (LM) and a conditional minimum distance (CMD) statistic. The paper outlines a procedure to construct asymptotically valid critical values for both procedures. All three tests are robust, to weak identification, however in most settings, conservative inference using the LM statistic seems to have greater power against local alternatives than the AR-type test. Furthermore, confidence regions based on the LM statistic will remain non-empty if the model is misspecified.