  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.
