  We identify and estimate treatment effects when potential outcomes are weakly
separable with a binary endogenous treatment. Vytlacil and Yildiz (2007)
proposed an identification strategy that exploits the mean of observed
outcomes, but their approach requires a monotonicity condition. In comparison,
we exploit full information in the entire outcome distribution, instead of just
its mean. As a result, our method does not require monotonicity and is also
applicable to general settings with multiple indices. We provide examples where
our approach can identify treatment effect parameters of interest whereas
existing methods would fail. These include models where potential outcomes
depend on multiple unobserved disturbance terms, such as a Roy model, a
multinomial choice model, as well as a model with endogenous random
coefficients. We establish consistency and asymptotic normality of our
estimators.
