Before revision: We develop a uniform asymptotic expansion for the empirical distribution function of residuals in the nonparametric IV regression. Such expansion opens a door forconstruction of a broad range of residual-based specification tests in nonparametric IV models. Building on obtained result, we develop a  test for the separability of unobservables in econometric models with endogeneity .
After revision: It is common to assume in the empirical practice that the observables are additively separable from the unobservables, especially, when the former are endogenous. This is done because it is widely recognized that identification and estimation challenges arise when interactions between the two are allowed for. Starting from the nonseparable IV model, where the IV is independent of the unobservable, we develop a test for the separability of unobservables in econometric models with endogeneity .