Tests for Heteroscedasticity Based on Regression Quantiles and Regression Rank Scores

Abstract

We propose a new class of tests for heteroscedasticity in semiparametric linear models with independent errors. The test statistics are a sum of two asymptotically independent parts, one based on regression quantiles, the other on regression ranks. Those components of heteroscedasticity that are correlated to covariates in the model are detected by the regression quantile part of the statistics, whereas components orthogonal to the model are detected by its regression rank part. The proposed tests are a natural application of the concept of regression ranks and regression quantiles. They may be considered as canonical generalizations of rank tests for scale and scale tests based on L-statistics from the two- or p- sample case to a general semiparametric linear model setting. Moreover, the test statistics have desirable invariance properties with respect to the nuisance parameters of the model and do not require symmetry of the error distribution. Their relationship to the likelihood ratio statistic for the corresponding parametric model (with known form of the error distribution) is analogous to that of corresponding rank- and L-statistics to likelihood ratio statistics in the two sample scale testing problem. We also suggest, how generalizations to hypotheses on other departures from i.i.d. errors than classical heteroscedasticity could be obtained.