Partial identification of functionals of the joint distribution of “potential outcomes”

In this paper, we present a systematic study of partial identification of two general classes of functionals of the joint distribution of two “potential outcomes” when a bivariate sample from the joint distribution is not available to the econometrician. Assuming the identification of the conditional marginal distributions of potential outcomes and the distribution of the covariate vector, we show that the identified sets for functionals in both classes are intervals and provide conditions under which the identified sets point identify the true value of the functionals. In addition, we establish sufficient and necessary conditions for the covariate information to be informative in the sense of shrinking the identified sets. We focus on the application of our general results to evaluating distributional treatment effects of a binary treatment in two commonly used frameworks in the literature for evaluating average treatment effects: the selection on observables framework and a latent threshold-crossing model. We characterize the role of the propensity score in the selection-on-observables framework and the role of endogenous selection in the latent threshold-crossing model. Examples of policy parameters that our results apply include the correlation coefficient between the potential outcomes, many inequality measures of the distribution of treatment effects, and median of the distribution of the individual treatment effect.