A triangular treatment effect model with random coefficients in the selection equation

This paper considers treatment effects under endogeneity with complex heterogeneity in the selection equation. We model the outcome of an endogenous treatment as a triangular system, where both the outcome and first-stage equations consist symmetrically of a random coefficients model. The first-stage equation allows for nonmonotone selection into treatment. We provide conditions under which, conditional on first-stage unobservables, distributions of potential outcomes, average and quantile treatment effects are identified. Under these conditions, we derive bounds on the conditional joint distribution of potential outcomes and of gains from treatment. We give conditions under which these conditional distributions are identified. The conditional on first-stage unobservables parameters yield unconditional effects as weighted integrals.