Before revision: In this paper, I show that a nonseparable model where the endogenous variable is multivalued can be point-identified even when the instrument (IV) is only binary. Though the order condition generally fails in this case, I show that exogenous covariates are able to generate enough moment equations to restore the order condition as if enlarging the IV's support under very general selection mechanisms for the endogenous variable.
After revision: In this paper, I show that a nonseparable model where the endogenous variable is multivalued can be point-identified even when the instrument takes on too few values. This paper presents a new method that matches pairs of covariates and instruments to restore point identification in this scenario in a triangular model. The model consists of a structural function for a continuous outcome and a selection model for the discrete endogenous variable.