  In this paper the nonparametric quantile regression model is considered in a
location-scale context. The asymptotic properties of the empirical independence
process based on covariates and estimated residuals are investigated. In
particular an asymptotic expansion and weak convergence to a Gaussian process
are proved. The results can, on the one hand, be applied to test for validity
of the location-scale model. On the other hand, they allow to derive various
specification tests in conditional quantile location-scale models. In detail a
test for monotonicity of the conditional quantile curve is investigated. For
the test for validity of the location-scale model as well as for the
monotonicity test smooth residual bootstrap versions of Kolmogorov-Smirnov and
Cramer-von Mises type test statistics are suggested. We give rigorous proofs
for bootstrap versions of the weak convergence results. The performance of the
tests is demonstrated in a simulation study.
