  In ordinary quantile regression, quantiles of different order are estimated
one at a time. An alternative approach, which is referred to as quantile
regression coefficients modeling (QRCM), is to model quantile regression
coefficients as parametric functions of the order of the quantile. In this
paper, we describe how the QRCM paradigm can be applied to longitudinal data.
We introduce a two-level quantile function, in which two different quantile
regression models are used to describe the (conditional) distribution of the
within-subject response and that of the individual effects. We propose a novel
type of penalized fixed-effects estimator, and discuss its advantages over
standard methods based on $\ell_1$ and $\ell_2$ penalization. We provide model
identifiability conditions, derive asymptotic properties, describe
goodness-of-fit measures and model selection criteria, present simulation
results, and discuss an application. The proposed method has been implemented
in the R package qrcm.
