Inference in a Class of Optimization Problems: Confidence Regions and Finite Sample Bounds on Errors in Coverage Probabilities

Abstract

This article describes three methods for carrying out nonasymptotic inference on partially identified parameters that are solutions to a class of optimization problems. Applications in which the optimization problems arise include estimation under shape restrictions, estimation of models of discrete games, and estimation based on grouped data. The partially identified parameters are characterized by restrictions that involve the unknown population means of observed random variables in addition to structural parameters. Inference consists of finding confidence intervals for functions of the structural parameters. Our theory provides finite-sample lower bounds on the coverage probabilities of the confidence intervals under three sets of assumptions of increasing strength. With the moderate sample sizes found in most economics applications, the bounds become tighter as the assumptions strengthen. We discuss estimation of population parameters that the bounds depend on and contrast our methods with alternative methods for obtaining confidence intervals for partially identified parameters. The results of Monte Carlo experiments and empirical examples illustrate the usefulness of our method.

Keywords:

• Finite-sample bounds
• Normal approximation
• Partial identification
• Sub-Gaussian distribution

Supplementary Materials

The supplementary materials consist of five appendices. Appendix A presents the proofs of theorems. Appendix B provides additional technical information about our methods. Appendix C describes Minsker’s method. Appendix D provides an additional empirical example, and Appendix E provides additional Monte Carlo results.

Acknowledgments

An earlier version of this article was presented at the conference “Incomplete Models,” which took place at Northwestern University in November 2018. We thank the conference participants, the associate editor, and two anonymous referees for helpful comments. Part of Joel L. Horowitz’s research was carried out while he was a visitor to the Department of Economics at University College London.

Additional information

Funding

This research was supported in part by the European Research Council (grant no. ERC-2014-CoG-646917-ROMIA) and the UK Economic and Social Research Council (grant no. ES/P008909/1 to the CeMMAP).