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This paper considers treatment effects under endogeneity with complexheterogeneity in the selection equation. We model the outcome of an endogenoustreatment as a triangular system, where both the outcome and first-stageequations consist of a random coefficients model. The first-stage specificallyallows for nonmonotone selection into treatment. We provide conditions underwhich marginal distributions of potential outcomes, average and quantiletreatment effects, all conditional on first-stage random coefficients, areidentified. Under the same conditions, we derive bounds on the (conditional)joint distributions of potential outcomes and gains from treatment, and provideadditional conditions for their point identification. All conditionalquantities yield unconditional effects (\emph{e.g.}, the average treatmenteffect) by weighted integration.

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