Testing for heteroscedasticity in high-dimensional regressions

Abstract

Testing heteroscedasticity of the errors is a major challenge in high-dimensional regressions where the number of covariates is large compared to the sample size. Traditional procedures such as the White and the Breusch–Pagan tests typically suffer from low sizes and powers. Two new test procedures are proposed based on standard OLS residuals. Using the theory of random Haar orthogonal matrices, the asymptotic normality of both test statistics is obtained under the null when the degrees of freedom tend to infinity. This encompasses both the classical low-dimensional setting where the number of variables is fixed while the sample size tends to infinity, and the proportional high-dimensional setting where these dimensions grow to infinity proportionally. This is the first procedures in the literature for testing heteroscedasticity which are valid for medium and high-dimensional regressions. Notice however that as the procedures are based on the OLS residuals, the number of variables must be smaller than the sample size, although both can grow to infinity. The superiority of our proposed tests over the existing methods are demonstrated by extensive simulations and by several real data analyses as well.