Flexible Specification Testing in Quantile Regression Models

We propose three novel consistent specification tests for quantile regression models which generalize former tests in three ways. First, we allow the covariate effects to be quantile-dependent and nonlinear. Second, we allow parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. We are hence able to test for functional forms beyond linearity, while retaining the linear effects as special cases. In both cases, the induced class of conditional distribution functions is tested with a Cram\'{e}r-von Mises type test statistic for which we derive the theoretical limit distribution and propose a bootstrap method. Third, to increase the power of the tests, we further suggest a modified test statistic. We highlight the merits of our tests in a detailed MC study and two real data examples. Our first application to conditional income distributions in Germany indicates that there are not only still significant differences between East and West but also across the quantiles of the conditional income distributions, when conditioning on age and year. The second application to data from the Australian national electricity market reveals the importance of using interaction effects for modelling the highly skewed and heavy-tailed distributions of energy prices conditional on day, time of day and demand.