A procedure for testing the signi cance of a subset of explanatory variables in a nonparametric regression is proposed. Our test statistic uses the kernel method. Under the null hypothesis of no e ect of the variables under test, we show that our test statistic has a nh p 2 =2 standard normal limiting distribution, where p 2 is the dimension of the complete set of regressors. Our test is one-sided, consistent against all alternatives and detect local alternatives approaching the null at rate slower than n ;1=2 h ;p 2 =4 . Our Monte-Carlo experiments indicate that it outperforms the test proposed by F an and Li (1996).Keywords: Hypothesis testing, Kernel estimation, Nested models. JEL classi cation: Primary C52 Secondary C14.R esum e Une proc edure pour tester la signi cativit e d'un sous-ensemble de r egresseurs dans un mod e l e d e r egression non-param etrique est propos ee. Elle s'appuie sur la m ethode du noyau. Sous l'hypoth ese nulle, i.e. lorsque les variables consid er ees ne sont pas pertinentes, la statistique de test a une distribution asymptotique normale en nh p 2 =2 , o u p 2 est le nombre total de r egresseurs. Le test est unilat eral, convergent contre toute alternative e t d etecte des alternatives locales qui s'approchent d e l'hypoth ese nulle a une vitesse inf erieure a n ;1=2 h p 2 =4 . P our des petits echantillons, notre test a de meilleures performances que celui propos e par Fan et Li (1996).Mots-Cl es: Test d'hypoth ese, M ethode du noyau, Mod eles emboit es.

We propose new data-driven smooth tests for a parametric regression function. The smoothing parameter is selected through a new criterion that favors a large smoothing parameter under the null hypothesis. The resulting test is adaptive rate-optimal and consistent against Pitman local alternatives approaching the parametric model at a rate arbitrarily close to 1/\sqrtn. Asymptotic critical values come from the standard normal distribution and the bootstrap can be used in small samples. A general formalization allows one to consider a large class of linear smoothing methods, which can be tailored for detection of additive alternatives.Comment: Published at http://dx.doi.org/10.1214/009053604000001200 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

International audienceFor tests based on nonparametric methods, power crucially depends on the dimension of the conditioning variables, and specifically decreases with this dimension. This is known as the “curse of dimensionality”. We propose a new general approach to nonparametric testing in high dimensional settings and we show how to implement it when testing for a parametric regression. The resulting test behaves against directional local alternatives almost as if the dimension of the regressors was one. It is also almost optimal against classes of one-dimensional alternatives for a suitable choice of the smoothing parameter. The test performs well in small samples compared to several other test

In the context of testing the specification of a nonlinear parametric regression function, we adopt a nonparametric minimax approach to determine the maximum rate at which a set of smooth alternatives can approach the null hypothesis while ensuring that a test can uniformly detect any alternative in this set with some predetermined power. We show that a smooth nonparametric test has optimal asymptotic minimax properties for regular alternatives. As a by-product, we obtain the rate of the smoothing parameter that ensures rate-optimality of the test. We show that, in contrast, a class of nonsmooth tests, which includes the integrated conditional moment test of Bierens (1982, Journal of Econometrics 20, 105–134), has suboptimal asymptotic minimax properties.