  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. This paper
proposes two new test procedures 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 degree of freedom tends 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. These procedures thus offer a wide coverage of dimensions in
applications. To our best knowledge, this is the first procedures in the
literature for testing heteroscedasticity which are valid for medium and
high-dimensional regressions. The superiority of our proposed tests over the
existing methods are demonstrated by extensive simulations and by several real
data analyses as well.
