This paper proposes a test for selecting explanatory variables in
nonparametric regression. The test does not need to estimate the conditional
expectation function given all the variables, but only those which are
significant under the null hypothesis. This feature is computationally
convenient and solves, in part, the problem of the “curse of
dimensionality” when selecting regressors in a nonparametric context.
The proposed test statistic is based on functionals of a $U$-process.
Contiguous alternatives, converging to the null at a rate $n^{-1/2}$ can be
detected. The asymptotic null distribution of the statistic depends on certain
features of the data generating process,and asymptotic tests are difficult to
implement except in rare circumstances. We justify the consistency of two easy
to implement bootstrap tests which exhibit good level accuracy for fairly small
samples, according to the reported Monte Carlo simulations. These results are
also applicable to test other interesting restrictions on nonparametric curves,
like partial linearity and conditional independence.