COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1761R

Inference Based on Conditional Moment Inequalities

Donald W.K. Andrews and Xiaoxia Shi

June 2010
Revised July 2011

In this paper, we propose an instrumental variable approach to constructing confidence sets (CS's) for the true parameter in models defined by conditional moment inequalities/equalities. We show that by properly choosing instrument functions, one can transform conditional moment inequalities/equalities into unconditional ones without losing identification power. Based on the unconditional moment inequalities/equalities, we construct CS's by inverting Cramér-von Mises-type or Kolmogorov-Smirnov-type tests. Critical values are obtained using generalized moment selection (GMS) procedures.

We show that the proposed CS's have correct uniform asymptotic coverage probabilities. New methods are required to establish these results because an infinite-dimensional nuisance parameter affects the asymptotic distributions. We show that the tests considered are consistent against all fixed alternatives and have power against n^{-1/2}-local alternatives to some, but not all, sequences of distributions in the null hypothesis. Monte Carlo simulations for four different models show that the methods perform well in finite samples.

Keywords: Asymptotic size, Asymptotic power, Conditional moment inequalities, Confidence set, Cramér-von Mises, Generalized moment selection, Kolmogorov-Smirnov, Moment inequalities

JEL Classification: C12, C15