Significant Testing in Nonparametric Regression based on the Bootstrap

Abstract

This paper proposes a test for selecting explanatory variables in nonparametric regression. The test does not need to estimate the conditional expectation function given al the variables, butonly 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 seleccting regressors in a nonparametric context. The proposed test statistic is based on functional of a U-process. Contiguous alternatives, converging to the null at a rate n-1/2 can be detected. The asympotetic null distribution of the statistic depends on certain features of the data generating process, and asymptotic tests are difficult to implement except in rare circunstantces. 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 simulation. These result are also applicable to test other interesting restictions on nonparametric curves, like partial linearity and conditional independence.