Abstract: Testing the constancy of a conditional covariance matrix is a fundamental problem, because deviating from this assumption can result in a severely inefficient estimate. We propose a slice-based procedure to test for constant conditional variance in cases where the data dimension is larger than the sample size. We develop a high-order correction that makes the test statistic robust with respect to high dimensionality, and show that the proposed test statistic is asymptotically normal under some mild conditions. The proposed method allows the dimensionality to increase as the square of the sample size. Furthermore, simulations demonstrate that it exhibits good size and power in a wide range of settings.

Key words and phrases: Asymptotic normality, constant variance condition, high dimensional data, inverse regression, sufficient dimension reduction.