Pairwise distance-based heteroscedasticity for regressions

We propose a new test for heteroscedasticity in parametric and partial
linear regression models in multidimensional spaces. When the dimension of the
covariates is large, or even moderate, existing tests for heteroscedasticity perform
badly, owing to the “curse of dimensionality.” To address this problem, we
construct a test for heteroscedasticity that uses a projection-based empirical process.
Then, we study the asymptotic properties of the test statistic under the null
and alternative hypotheses. The results show that the test detects the departure
of local alternatives from the null hypothesis at the fastest possible rate during
hypothesis testing. Because the limiting null distribution of the test statistic is
not asymptotically distribution free, we propose a residual-based bootstrap approach
and investigate the validity of its approximations. Simulations verify the
finite-sample performance of the test. Two real-data analyses are conducted to
demonstrate the proposed test.