Statistica Sinica 18(2008), 601-615

TESTS FOR INDEPENDENCE IN NONPARAMETRIC

REGRESSION

John H.J. Einmahl and Ingrid Van Keilegom

Tilburg University & Université catholique de Louvain

Abstract: Consider the nonparametric regression model $Y=m(X)+\varepsilon$, where the function $m$ is smooth, but unknown. We construct tests for the independence of $\varepsilon$ 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.

Key words and phrases: Empirical process, model diagnostics, nonparametric regression, test for independence, weak convergence.