Abstract: 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.

Key words and phrases: Heteroscedasticity testing, partial linear models, projection, U-process.