Average quantile regression: a new non-mean regression model and coherent risk measure

TL;DR

This paper introduces Average Quantile Regression (AQR), a novel non-mean regression model that captures distributional information and tail risks, with applications in risk management and portfolio optimization.

Contribution

AQR is a new, flexible regression model that unifies and extends traditional models and risk measures, with rigorous estimators and asymptotic properties.

Findings

AQR effectively models distributional information across all quantile levels.

AQR outperforms traditional models in high-dimensional, large datasets.

AQR provides valuable insights for risk assessment and portfolio optimization.

Abstract

Regression models that go beyond the mean, alongside coherent risk measures, have been important tools in modern data analysis. This paper introduces the innovative concept of Average Quantile Regression (AQR), which is smooth at the quantile-like level, comonotonically additive, and explicitly accounts for the severity of tail losses relative to quantile regression. AQR serves as a versatile regression model capable of describing distributional information across all positions, akin to quantile regression, yet offering enhanced interpretability compared to expectiles. Numerous traditional regression models and coherent risk measures can be regarded as special cases of AQR. As a flexible non-parametric regression model, AQR demonstrates outstanding performance in analyzing high-dimensional and large datasets, particularly those generated by distributed systems, and provides a convenient framework for their statistical analysis. The corresponding estimators are rigorously derived, and their asymptotic properties are thoroughly developed. In a risk management context, the case study confirms AQR's effectiveness in risk assessment and portfolio optimization.