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Published online by Cambridge University Press: 07 October 2009
In this paper, we propose nonparametric estimators ofsharp bounds on the distribution of treatmenteffects of a binary treatment and establish theirasymptotic distributions. We note the possiblefailure of the standard bootstrap with the samesample size and apply thefewer-than-n bootstrap to makinginferences on these bounds. The finite sampleperformances of the confidence intervals for thebounds based on normal critical values, the standardbootstrap, and the fewer-than-nbootstrap are investigated via a simulation study.Finally we establish sharp bounds on the treatmenteffect distribution when covariates areavailable.
We thank Jinyong Hahn and two anonymous refereesfor their valuable suggestions that greatlyimproved the paper. We thank Jianqing Fan, JoelHorowitz, Chuck Manski, Per Mykland, BryanShepherd, Elie Tamer, and seminar participants inthe Department of Statistics at the University ofChicago and in the Department of Economics atNorthwestern University, UNC at Chapel Hill, andPrinceton University for helpful discussions. Wealso thank Jeff Smith and Jörg Stoye for providinguseful references. Y. Fan acknowledges financialsupport from the National Science Foundation.