Tests for independence in nonparametric regression

Consider the nonparametric regression model Y = m(X) + E, where the function m is smooth, but unknown. We construct tests for the independence of E and X, based on n independent copies of (X, Y). The testing procedures are based on differences of neighboring Y's. We establish asymptotic results for the proposed tests statistics, investigate their finite sample properties through a simulation study and present an econometric application to household data. The proofs are based on delicate empirical process theory.