Detecting heteroscedasticity in a simple regression model via quantile regression slopes

Consider the linear regression model Y=β X ₁+α+τ (X)ϵ, where X and ϵ are independent random variables, ϵ has a mean of zero and variance σ², and τ is some unknown function used to model heteroscedasticity. Many methods have been proposed for testing H ₀: τ (X) ≡ 1, the hypothesis that the error term is homoscedastic, with most methods known to be unsatisfactory in terms of controlling the probability of a Type I error. This paper considers several approaches based on a quantile regression estimator, one of which (method N2) is recommended for general use. A minor goal is to report new results related to a method suggested by Koenker. Method N2 does not dominate Koenker’s method in terms of power, but as illustrated, the choice of method can make a considerable difference when testing H ₀. In particular, situations occur where Koenker’s method is highly non-significant, yet method N2 rejects at the 0.01 level.