Flexible Specification Testing in Semi-Parametric Quantile Regression Models

We propose several novel consistent specification tests for quantile regression models which generalize former tests by important characteristics. First, we allow the covariate effects to be quantile-dependent and nonlinear, bypassing estimation difficulties such as multicollinearity. Second, we allow for parameterizing the conditional quantile functions by appropriate basis functions, rather than parametrically. In the framework of splines, we can thus test for the order and number of knots. 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 practical bootstrap method. To increase the power of the first test, we further suggest a modified test statistic using the $B$-spline approach from the second test. A detailed Monte Carlo experiment shows that the tests result in reasonable sized testing procedures with large power. Our first application to conditional income disparities between East and West Germany over the period 2001--2010 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.