Parametric Modeling of Quantile Regression Coefficient Functions With Longitudinal Data

Abstract

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 article, we describe how the $\mathit{qrcm}$ 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 ${\mathcal{l}}_1$ and ${\mathcal{l}}_2$ 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.

Keywords:

• Longitudinal quantile regression
• Parametric quantile function
• Penalized fixed-effects
• R package qrcm
• Two-level quantile function

Supplementary Materials

Version 3.0 of the qrcm R-package (to be installed as local zip file):

New package version including the iqrL function that implements the estimator.

NGAL dataset: Dataset and R script used in the illustration of the method in Section 8 (.txt file).

Appendix A: Proof of Theorem 1.

Appendix B: Computation.

Appendix C: Extended simulation results.

Notes

Notes

¹ Chernozhukov, Fernández-Val, and Weidner (2018) considered an alternative to quantile regression for estimation of quantile effects in longitudinal data based on distribution regression.

² We index ${\mathit{\alpha }}_N$ by N to emphasize that the dimension grows with the sample size.

³ The expression for $L_1(\mathit{\theta },{\mathit{\alpha }}_N)$ bears some similarity to Koenker’s (2004) loss function for unpenalized fixed-effects quantile regression, which is defined by $L(\mathit{\beta },{\mathit{\alpha }}_N)=\sum _j\sum _i\sum _t{w}_j{\rho }_{u_j}(y_{it}-{\alpha }_i-{\mathit{x}}_{it}\mathit{\beta }(u_j))$ and can be seen as a discretized, nonparameterized, and weighted version of $L_1(\mathit{\theta },{\mathit{\alpha }}_N)$.

⁴ We normalize the mean of the fixed effects. Alternative normalizations on the median or other quantile of the fixed effects are also possible.