Partial Identi cation and Con dence Sets for Functionals of the Joint Distribution of Potential Outcomes

In this paper, we study partial identi cation and inference for a general class of functionals of the joint distribution of potential outcomes of a binary treatment under the strong ignorability assumption or the selection on observables assumption commonly used in evaluating average treatment e ects. Members of this class of functionals include the correlation coe cient between the potential outcomes and many inequality measures of the distribution of treatment e ects. We establish sharp bounds on functionals in this class and characterize conditions under which our lower and upper bounds coincide and thus point identify the true functionals. We propose nonparametric estimators of the sharp bounds, establish their asymptotic distributions, and construct asymptotically valid con dence sets for the true functionals. Another interesting nding of this paper is that under the selection on observables assumption, although the average treatment e ect can be point identi ed from the observable covariates or from the propensity score, in general, the sharp bounds on functionals of the joint distribution of potential outcomes based on the observable covariates are tighter than the corresponding sharp bounds based on the propensity score.