Subsample Least Squares Estimator for Heterogeneous Effects of Multiple Treatments with Any Outcome Variable
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For multiple treatments D=0,1,...,J, covariates X and outcome Y, the ordinary least squares estimator (OLS) of Y on (D1,...,DJ,X) is widely applied to a constant-effect linear model, where Dj is the dummy variable for D=j. However, the treatment effects are almost always X-heterogeneous in reality, or Y is noncontinuous, to invalidate such a linear model. The blind hope of practitioners is that the OLS "somehow" estimates a sensible average of the unknown X-heterogeneous effects. This paper shows that, unfortunately, the OLS is inconsistent unless all treatment effects are constant, because the estimand of the Dd-slope involves the X-heterogeneous effects of all treatments, not just Dd. One way to overcome this "contamination" problem is the OLS of Y on Dd-E(Dd|X, D=0,d) using only the subsample D=0,d, and this paper proposes a modified version of the subsample OLS that is robust to misspecifications of E(Dd|X, D=0,d). The robustified subsample OLS is proven to be consistent for an "overlap weight" average of the X-heterogeneous effect of Dd for any form of Y (continuous, binary, count, ...).
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