Inference Based on Conditional Moment Inequalities


  • Donald W. K. Andrews,

    1. Cowles Foundation for Research in Economics, Yale University, New Haven, CT 06520, U.S.A.;
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  • Xiaoxia Shi

    1. Dept. of Economics, University of Wisconsin, Madison, Madison, WI 53706, U.S.A.;
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    • Andrews gratefully acknowledges the research support of the National Science Foundation via Grants SES-0751517 and SES-1058376. The authors thank the co-editor, four referees, Andrés Aradillas-López, Kees Jan van Garderen, Hidehiko Ichimura, and Konrad Menzel for helpful comments.


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 typically have power against n−1/2-local alternatives to some, but not all, sequences of distributions in the null hypothesis. Monte Carlo simulations for five different models show that the methods perform well in finite samples.