Distribution and Quantile Structural Functions in Treatment Effect Models: Application to Smoking Effects on Wages

This paper examines the distribution structural functions (DSFs) and quantile structural functions (QSFs) in a semiparametric treatment effect model. The DSF and QSF are defined as the distribution function and quantile function of the counterfactural outcome when covariates are exogenously switched to a fixed value, while the unobserved heterogeneity of the whole population remains unchanged. We show that the DSFs and QSFs are identified under the unconfoundedness assumption, and then propose inverse probability weighted estimators which are n^{-1/2}-consistent and converge weakly to mean zero Gaussian processes. A simulation approach is also proposed to approximate the limiting processes. Finally, we apply the results to construct uniform confidence bands for the structural quantile treatment effect of smoking on wage, and find that smoking does not impose any wage penalty on male workers with low unobserved heterogeneity. JEL Classification: C14, C31, I19