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Multidimensional heterogeneity and endogeneity are important features of awide class of econometric models. With control variables to correct forendogeneity, nonparametric identification of treatment effects requires strongsupport conditions. To alleviate this requirement, we consider varyingcoefficients specifications for the conditional expectation function of theoutcome given a treatment and control variables. This function is expressed asa linear combination of either known functions of the treatment, with unknowncoefficients varying with the controls, or known functions of the controls,with unknown coefficients varying with the treatment. We use this modelingapproach to give necessary and sufficient conditions for identification ofaverage treatment effects. A sufficient condition for identification isconditional nonsingularity, that the second moment matrix of the knownfunctions given the variable in the varying coefficients is nonsingular withprobability one. For known treatment functions with sufficient variation, wefind that triangular models with discrete instrument cannot identify averagetreatment effects when the number of support points for the instrument is lessthan the number of coefficients. For known functions of the controls, we findthat average treatment effects can be identified in general nonseparabletriangular models with binary or discrete instruments. We extend our analysisto flexible models of increasing dimension and relate conditionalnonsingularity to the full support condition of Imbens and Newey (2009),thereby embedding semi- and non-parametric identification into a commonframework.

Identification of Treatment Effects under Limited Exogenous Variation