Partial Identifi…cation and Confidence Sets for 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. We establish the identified sets for functionals in both classes under various maintained assumptions and characterize conditions under which our identified sets point identify the true value of the functionals. In addition, we establish sufficient and necessary conditions for the covariate information to tighten the identified sets without the covariate information. Applications include, but not limited to, the evaluation of distributional treatment effects of a binary treatment and pricing options written on two underlying assets when the sample information only contains observations on traded options written on each individual asset. In the former, the class of functionals 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. We focus on two commonly used frameworks for evaluating treatment effects: the selection on observables framework and a latent threshold-crossing model and 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. For the selection-on-observables framework, we construct asymptotically valid confidence sets for the true value of the parameter corresponding to a super-modular functional.

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