QUANTILE REGRESSION: A REVIEW OF METHODOLOGICAL ADVANCES IN LINEAR, BAYESIAN, PANEL DATA, AND MEASUREMENT ERROR MODELS

Abstract

Quantile regression (QR) has become a central tool for modelling heterogeneous covariate effects across the entire conditional distribution of a response, offering clear advantages over mean-based regression when data exhibit skewness, heavy tails, or heteroscedasticity. The methodological literature has grown rapidly and unevenly, spanning linear and functional models, Bayesian formulations, panel and longitudinal structures, and frameworks that accommodate measurement error. Because existing surveys typically treat these strands in isolation, applied researchers find it difficult to form a coherent overview of the field, or to see how advances in one area bear on the others. This paper provides a structured, thematically organized review of recent advances across four interconnected themes: linear and functional QR, Bayesian QR, QR for panel and longitudinal data, and QR under measurement error. Our distinctive contribution is to read these literatures jointly. Rather than cataloguing studies, we organize each theme around the methodological problem being solved, compare competing solutions on common criteria – assumptions, robustness, scalability, interpretability, computational cost, and software availability – and aggregate recurring weaknesses into a single cross-cutting critique. We provide per-theme comparison tables, a master comparison table spanning all four themes and a dedicated analysis of systemic challenges. We conclude that the field is converging toward integrated frameworks that jointly address dimension reduction, sparsity, dependence, and data imperfections, and that the most consequential open problems lie precisely at the intersections of the four themes.