ARCH test and quantile regressions

The paper analyzes the behavior of a test for heteroskedasticity based on regression quantiles (RQ test). Then it generalizes its applicability to the class of conditional heteroskedastic models. Koenker and Bassett (1982) have presented the first regression quantile based test for heteroskedasticity in a linear model with asymptotically vanishing heteroskedasticity. Their test relies on the idea that, in the presence of heteroskedasticity, different quantile regressions are characterized by different slope coefficients, while in the homoskedastic case these slopes are not significantly different across the various regression quantiles. Not significantly different slopes across quantiles define a group of parallel hyperplanes. By testing the equality of the slope vectors estimated at the different quantile regressions, the regression quantile based test verifies the presence of heteroskedasticity avoiding: i) the definition of distributional assumptions; ii) the specification of a functional form needed to define the heterogeneity in variance which is supposed to characterize the data. Some simple basic models of conditional heteroskedasticity are then estimated by means of the quantile regression estimator. The asymptotic normality of the quantile regression estimators for this class of models is proved, and the Koenker and Bassett (1982) test for heteroskedasticity is thus extended to non-vanishing and conditionally heteroskedastic models.