  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 heterogeneous coefficients are mean independent from treatments
given the controls, a simple identification condition is that the generalized
propensity scores (Imbens, 2000) be bounded away from zero and that their sum
be bounded away from one, with probability one. Our analysis extends to
distributional and quantile treatment effects, as well as corresponding
treatment effects on the treated. These results generalize the classical
identification result of Rosenbaum and Rubin (1983) for binary treatments.
