Dummy Endogenous Variables in Weakly Separable Multiple Index Models without Monotonicity

We study the identification and estimation of treatment effect parameters in weakly separable models. In their seminal work, Vytlacil and Yildiz (2007) showed how to identify and estimate the average treatment effect of a dummy endogenous variable when the outcome is weakly separable in a single index. Their identification result builds on a monotonicity condition with respect to this single index. In comparison, we consider similar weakly separable models with multiple indices, and relax the monotonicity condition for identification. Unlike Vytlacil and Yildiz (2007), we exploit the full information in the distribution of the outcome variable, instead of just its mean. Indeed, when the outcome distribution function is more informative than the mean, our method is applicable to more general settings than theirs; in particular we do not rely on their monotonicity assumption and at the same time we also allow for multiple indices. To illustrate the advantage of our approach, we provide examples of models where our approach can identify parameters of interest whereas existing methods would fail. These examples include models with multiple unobserved disturbance terms such as the Roy model and multinomial choice models with dummy endogenous variables, as well as potential outcome models with endogenous random coefficients. Our method is easy to implement and can be applied to a wide class of models. We establish standard asymptotic properties such as consistency and asymptotic normality.