Goodness-of-fit tests in nonparametric regression

Consider the nonparametric regression model Y = m(X) + ε, where the function m is smooth, but unknown, and ε is independent of X. We construct omnibus goodness-of-fit tests, based on n independent copies of (X, Y ), for the independence of ε and X and establish asymptotic results for the proposed tests statistics. We investigate their finite sample properties through a simulation study and present an econometric application to household data. One testing procedure is based on differences of neighboring Y ’s, whereas the other one makes use of an estimator of m. The proofs are based on delicate weighted empirical process theory.