Heterogenous coefficients, discrete instruments, and identification of treatment effects

Multidimensional heterogeneity and endogeneity are important features of a wide class of econometric models. We consider heterogenous coefficients models where the outcome is a linear combination of known functions of treatment and heterogenous coefficients. We use control variables to obtain identi cation results for average treatment effects. With discrete instruments in a triangular model we find that average treatment effects cannot be identi ed when the number of support points is less than or equal to the number of coefficients. A sufficient condition for identi fication is that the second moment matrix of the treatment functions given the control is nonsingular with probability one. We relate this condition to identi fication of average treatment effects with multiple treatments.