Sharp Bounds on the Distribution of Treatment Effects and Their Statistical Inference

In this paper, we propose nonparametric estimators of sharp bounds on the distribution of treatment e¤ects of a binary treatment and establish their asymptotic distributions. We note the possible failure of the standard bootstrap with the same sample size and apply the fewer-than-n bootstrap to making inferences on these bounds. The …nite sample performances of the con…dence intervals for the bounds based on normal critical values, the standard bootstrap, and the fewer-than-n bootstrap are investigated via a simulation study. Finally we establish sharp bounds on the treatment e¤ect distribution when covariates are available.