  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.
