Identification and Estimation in a Class of Potential Outcomes Models

TL;DR

This paper introduces a unified framework for identifying and estimating causal effects in potential outcomes models with unobserved heterogeneity, leveraging instrumental variables and convex restrictions, and provides robust estimation methods with empirical application.

Contribution

It develops a general class of models with a unifying identification approach, including novel and classical designs, and introduces doubly robust estimators with valid inference.

Findings

Identifies a necessary and sufficient condition for causal parameter identification.

Proposes doubly robust estimators with asymptotic normality.

Demonstrates empirical utility through an analysis of mental health in a social experiment.

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.