Averaged extreme regression quantile

TL;DR

The paper introduces the averaged extreme regression quantile (AERQ), a robust statistical tool combining regression and extreme value analysis, with applications in economic and market risk assessment.

Contribution

It provides new equivalent forms of AERQ, including a linear programming and residual-based representation, applicable under heteroscedasticity and dependence.

Findings

AERQ is a weighted mean of regression quantile components.

AERQ can be expressed as a maximum residual from a specific R-estimator.

Finite-sample properties hold even with heteroscedasticity and dependent errors.

Abstract

Various events in the nature, economics and in other areas force us to combine the study of extremes with regression and other methods. A useful tool for reducing the role of nuisance regression, while we are interested in the shape or tails of the basic distribution, is provided by the averaged regression quantile and namely by the average extreme regression quantile. Both are weighted means of regression quantile components, with weights depending on the regressors. Our primary interest is the averaged extreme regression quantile (AERQ), its structure, qualities and its applications, e.g. in investigation of a conditional loss given a value exogenous economic and market variables. AERQ has several interesting equivalent forms: While it is originally defined as an optimal solution of a specific linear programming problem, hence is a weighted mean of responses corresponding to the optimal base of the pertaining linear program, we give another equivalent form as a maximum residual of responses from a specific R-estimator of the slope components of regression parameter. The latter form shows that while AERQ equals to the maximum of some residuals of the responses, it has minimal possible perturbation by the regressors. Notice that these finite-sample results are true even for non-identically distributed model errors, e.g. under heteroscedasticity. Moreover, the representations are formally true even when the errors are dependent - this all provokes a question of the right interpretation and of other possible applications.