  Quantile regression provides a framework for modeling statistical quantities
of interest other than the conditional mean. The regression methodology is well
developed for linear models, but less so for nonparametric models. We consider
conditional quantiles with varying coefficients and propose a methodology for
their estimation and assessment using polynomial splines. The proposed
estimators are easy to compute via standard quantile regression algorithms and
a stepwise knot selection algorithm. The proposed Rao-score-type test that
assesses the model against a linear model is also easy to implement. We provide
asymptotic results on the convergence of the estimators and the null
distribution of the test statistic. Empirical results are also provided,
including an application of the methodology to forced expiratory volume (FEV)
data.
