Testing homoscedasticity in nonparametric regression

We consider the nonparametric regression model with random design and derive an asymptotic alpha-test for testing the hypothesis that the conditional variance of the observations is constant against the alternative that it depends on the design. The test statistic is based on a L-2-distance between nonparametric variance estimators in both models-the underlying heteroscedastic model and in the hypothetical homoscedastic model. The proposed test is a consequence of a limit theorem for the weighted Integrated Square Error of nonparametric estimators of the conditional variance. Power considerations complete the approach.