  We discuss nonparametric tests for parametric specifications of regression
quantiles. The test is based on the comparison of parametric and nonparametric
fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile
smoothing estimator.
  An asymptotic treatment of the test statistic requires the development of new
mathematical arguments. An approach that makes only use of plugging in a
Bahadur expansion of the nonparametric estimator is not satisfactory. It
requires too strong conditions on the dimension and the choice of the
bandwidth.
  Our alternative mathematical approach requires the calculation of moments of
Nadaraya-Watson quantile regression estimators. This calculation is done by
application of higher order Edgeworth expansions.
