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This paper is to prove the asymptotic normality of a statistic for detectingthe existence of heteroscedasticity for linear regression models withoutassuming randomness of covariates when the sample size $n$ tends to infinityand the number of covariates $p$ is either fixed or tends to infinity. Moreoverour approach indicates that its asymptotic normality holds even withouthomoscedasticity.

Homoscedasticity tests for both low and high-dimensional fixed design regressions