Heterogeneous Coefficients, Control Variables, and Identification of Treatment Effects

Multidimensional heterogeneity and endogeneity are important features of models with multiple treatments. We consider a heterogeneous coefficients model where the outcome is a linear combination of dummy treatment variables, with each variable representing a different kind of treatment. We use control variables to give necessary and sufficient conditions for identification of average treatment effects. With mutually exclusive treatments we find that, provided the generalized propensity scores (Imbens, 2000) are bounded away from zero with probability one, a simple identification condition is that their sum be bounded away from one with probability one. These results generalize the classical identification result of Rosenbaum and Rubin (1983) for binary treatments.