A nonparametric R2 test for the presence of relevant variables

• A nonparametric test for relevant variables based on a local linear R2R2 estimator.
• Test is asymptotically normal under the null, local alternatives and consistent.
• Wild bootstrap/bootstrap can be used to approximate the null distribution.
• Illustrate finite sample performances with a Monte Carlo study.

We propose a nonparametric test for the presence of relevant variables based on a measure of nonparametric goodness-of-fit (R2)(R2) in a regression model. It does not require correct specifications of the conditional mean function, thus is able to detect presence of relevant variables of unknown form. Our test statistic is based on an appropriately centered and standardized nonparametric R2 estimator, which is obtained from a local linear regression. We establish the asymptotic normality of the test statistic under the null hypothesis that relevant variables are not present and a sequence of Pitman local alternatives. We also prove the consistency of the test, and show that the Wild bootstrap/bootstrap method can be used to approximate the null distribution of the test statistic. Under the alternative hypothesis, we establish the asymptotic normality of the nonparametric R2 estimator at rate n, which facilitates inference using the nonparametric measure of goodness-of-fit. We illustrate the finite sample performance of the tests with a Monte Carlo study and the bootstrap tests perform well relative to other alternatives.