  Let X be a d dimensional vector of covariates and Y be the response variable.
Under the nonparametric model Y = m(X) + {\sigma}(X) \in we develop an
ANOVA-type test for the null hypothesis that a particular coordinate of X has
no influence on the regression function. The asymptotic distribution of the
test statistic, using residuals based on Nadaraya-Watson type kernel estimator
and d \leq 4, is established under the null hypothesis and local alternatives.
Simulations suggest that under a sparse model, the applicability of the test
extends to arbitrary d through sufficient dimension reduction. Using p-values
from this test, a variable selection method based on multiple testing ideas is
proposed. The proposed test outperforms existing procedures, while additional
simulations reveal that the proposed variable selection method performs
competitively against well established procedures. A real data set is analyzed.
