Heteroscedasticity checks for single index models

To test heteroscedasticity in single index models, in this paper two test statistics are proposed via quadratic conditional moments. Without the use of dimension reduction structure, the first test has the usual convergence rate in nonparametric sense. Under the dimension reduction structure of mean and variance functions, the second one has faster convergence rate to its limit under the null hypothesis, and can detect local alternative hypotheses distinct from the null at a much faster rate than the one the first test can achieve. Numerical studies are also carried out to evaluate the performance of the developed tests. Interestingly, the second one works much better than the first one if the variance function does have a dimension reduction structure. However, it is not robust against the violation of dimension reduction structure, in other words, the power performance of the second test may not be encouraging if without the dimension reduction structure.