Identification and Estimation in a Class of Potential Outcomes Models

Abstract

This paper develops a class of potential outcomes models characterized by three main features: (i) Unobserved heterogeneity can be represented by a vector of potential outcomes and a type describing the manner in which an instrument determines the choice of treatment; (ii) The availability of an instrumental variable that is conditionally independent of unobserved heterogeneity; and (iii) The imposition of convex restrictions on the distribution of unobserved heterogeneity. The proposed class of models encompasses multiple classical and novel research designs, yet possesses a common structure that permits a unifying analysis of identification and estimation. In particular, we establish that these models share a common necessary and sufficient condition for identifying certain causal parameters. Our identification results are constructive in that they yield estimating moment conditions for the parameters of interest. Focusing on a leading special case of our framework, we further show how these estimating moment conditions may be modified to be doubly robust. The corresponding double robust estimators are shown to be asymptotically normally distributed, bootstrap based inference is shown to be asymptotically valid, and the semi-parametric efficiency bound is derived for those parameters that are root-n estimable. We illustrate the usefulness of our results for developing, identifying, and estimating causal models through an empirical evaluation of the role of mental health as a mediating variable in the Moving To Opportunity experiment.