Testing for Heteroscedasticity in High-dimensional Regressions

TL;DR

This paper introduces two novel tests for heteroscedasticity in high-dimensional regressions, valid across various dimensions, and demonstrates their effectiveness through simulations and real data applications.

Contribution

It proposes the first heteroscedasticity tests suitable for medium and high-dimensional regressions, leveraging random Haar matrices for asymptotic normality.

Findings

Proposed tests outperform traditional methods in size and power.

Tests are valid for both low and high-dimensional settings.

Extensive simulations and real data analyses confirm effectiveness.

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. 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.