Estimation of counterfactual distributions with a continuous endogenous treatment

In this article, I propose a method to estimate the counterfactual distribution of an outcome variable when the treatment is endogenous, continuous, and its effect is heterogeneous. The types of counterfactuals considered are those in which the change in treatment intensity can be correlated with the individual effects or when some of the structural functions are changed by some other group’s counterparts. I characterize the outcome and the treatment with a triangular system of equations in which the unobservables are related by a copula that captures the endogeneity of the treatment, which is nonparametrically identified by inverting the quantile processes that determine the outcome and the treatment. Both processes are estimated using existing quantile regression methods, and I propose a parametric and a nonparametric estimator of the copula. To illustrate these methods, I estimate several counterfactual distributions of the birth weight of children, had their mothers smoked differently during pregnancy.