  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.
