Estimation of counterfactual distributions with a continuous endogenous treatment

Policy makers are often interested in the distributional effects that a policy would have. In this paper I propose a method to estimate such effects when the treatment variable is endogenous, continuous, and has a heterogeneous effect. I consider a triangular system of equations in which the unobservables are related by a copula that captures the endogeneity of the model. The copula is nonparametrically identified by inverting the quantile processes conditional on a vector of covariates. I estimate both conditional quantile processes using existing quantile regression methods, and propose a parametric and a nonparametric estimator of the copula, showing the asymptotic properties of the estimators. I consider three kinds of counterfactual experiments: changing the distribution of the treatment, changing the distribution of the instrument, and changing the determination of the treatment, discussing the estimation for each counterfactual. I illustrate these methods by estimating several counterfactuals that affect the distribution of the share of food consumption.