This paper studies models for discrete outcomes which permit explanatory variables to be endogenous. In these models there is a single nonadditive latent variate which is restricted to be locally independent of instruments. The models are silent about the nature of dependence between the latent variate and the endogenous variable and the role of the instrument in this relationship. These single equation IV models which, when an outcome is continuous, can have point identifying power, have only set identifying power when the outcome is discrete. Identi.cation regions shrink as the support of a discrete outcome grows. The paper extends the analysis of structural quantile functions with endogenous arguments to cases in which there are discrete outcomes, cases which have so far been excluded from consideration.