Homoscedastic regression error is a common assumption in many high-dimensional regression models and theories. Although heteroscedastic error commonly exists in real-world datasets, testing heteroscedasticity remains largely underexplored under high-dimensional settings. We consider the heteroscedasticity test proposed in Newey and Powell (1987), whose asymptotic theory has been well-established for the low-dimensional setting. We show that the Newey-Powell test can be developed for high-dimensional data. For asymptotic theory, we consider the setting where the number of dimensions grows with the sample size at a linear rate. The asymptotic analysis for the test statistic utilizes the Approximate Message Passing (AMP) algorithm, from which we obtain the limiting distribution of the test. The numerical performance of the test is investigated through an extensive simulation study. As real-data applications, we present the analysis based on "international economic growth" data (Belloni et al. 2011), which is found to be homoscedastic, and "supermarket" data (Lan et al., 2016), which is found to be heteroscedastic.

翻译：同方差回归误差是许多高维回归模型与理论中的常见假设。尽管异方差误差普遍存在于真实数据集，但在高维环境下异方差性检验的研究仍相对不足。本文研究Newey与Powell（1987）提出的异方差性检验，该检验在低维设定下的渐近理论已臻成熟。我们证明Newey-Powell检验可拓展至高维数据。在渐近理论方面，我们考虑维度随样本量以线性速率增长的情形。通过利用近似消息传递（AMP）算法，我们推导出检验统计量的渐近分布。通过大量仿真研究验证了该检验的数值性能。在真实数据应用中，我们对"国际经济增长"数据（Belloni et al., 2011）和"超市"数据（Lan et al., 2016）进行分析，前者呈现同方差性，后者呈现异方差性。