Estimation and Confidence Regions for Parameter Sets in Econometric Models1

doi:10.1111/j.1468-0262.2007.00794.x

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Wiley InterScience

Econometrica

Volume 75 Issue 5, Pages 1243 - 1284

Published Online: 3 Aug 2007

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Estimation and Confidence Regions for Parameter Sets in Econometric Models¹

Victor Chernozhukov ^* , Han Hong ^** , and Elie Tamer ^***

^* Dept. of Economics and Operations Research Center, MIT, 50 Memorial Drive, Cambridge, MA 02139, U.S.A.; vchern@mit.edu , ^** Dept. of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305-6072, U.S.A.; han.hong@duke.edu , and ^*** Dept. of Economics, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208, U.S.A.; tamer@northwestern.edu

This is a revision of a previous paper, "Parameter Set Inference in a Class of Econometric Models" (2002), which was circulated as a MIT working paper. We thank A. Abadie, D. Bertsimas, A. Belloni, G. Chamberlain, M. Cohen, A. Galichon, R. Guiteras, J. Hahn, C. Hansen, M. Henry, B. Honoré, G. Imbens, S. Izmalkov, Y. Kitamura, I. Makarov, C. Manski, A. Milkusheva, I. Molchanov, W. Newey, A. Rosen, O. Rytchkov, A. Simsek, and Neshe Yildiz for valuable comments that substantially improved the paper. We also thank four referees and Oliver Linton for valuable comments that also substantially improved the paper. We also thank seminar participants at many institutions where this paper was presented. Chernozhukov gratefully acknowledges research support from the Castle Krob Chair, National Science Foundation, and the Sloan Foundation. Hong gratefully acknowledges research support from the National Science Foundation. Tamer gratefully acknowledges research support from the National Science Foundation and the Sloan Foundation.

KEYWORDS

Set estimator • contour sets • moment inequalities • moment equalities • resampling • bootstrap

ABSTRACT

This paper develops a framework for performing estimation and inference in econometric models with partial identification, focusing particularly on models characterized by moment inequalities and equalities. Applications of this framework include the analysis of game-theoretic models, revealed preference restrictions, regressions with missing and corrupted data, auction models, structural quantile regressions, and asset pricing models.

Specifically, we provide estimators and confidence regions for the set of minimizers Θ_I of an econometric criterion function Q(θ). In applications, the criterion function embodies testable restrictions on economic models. A parameter value θthat describes an economic model satisfies these restrictions if Q(θ) attains its minimum at this value. Interest therefore focuses on the set of minimizers, called the identified set. We use the inversion of the sample analog, Q_n(θ), of the population criterion, Q(θ), to construct estimators and confidence regions for the identified set, and develop consistency, rates of convergence, and inference results for these estimators and regions. To derive these results, we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification.

Manuscript received April, 2004; final revision received March, 2007.

DIGITAL OBJECT IDENTIFIER (DOI)

10.1111/j.1468-0262.2007.00794.x About DOI