We study instrumental variables models characterized by: (i) Unobserved heterogeneity consisting of potential outcomes and response types that describe how the instrument determines treatment choice; (ii) Conditional independence of the instrument and the unobserved heterogeneity; and (iii) Convex restrictions on the distribution of unobserved heterogeneity. We show certain causal parameters are identified in these models if and only if a version of the kappa of Abadie (2003) exists. Our identification results are constructive in yielding estimating moment conditions. Focusing on a leading special case, we develop asymptotically normal estimators based on a doubly robust version of these moment conditions.

implies and is implied by additive separability of choice of treatment equations in terms of observed and unobserved variables. These results follow from properties of binary matrices developed in this paper. We investigate conditions under which unordered monotonicity arises as a consequence of choice behavior. We characterize IV estimators of counterfactuals as solutions to discrete mixture problems.