Comparative e-backtests for general risk measures

TL;DR

This paper introduces a non-parametric, anytime-valid framework for comparative backtesting of various risk measures using e-values, enhancing robustness and informativeness in financial risk assessment.

Contribution

It develops a novel sequential, non-parametric approach for comparative backtests of elicitable risk measures, applicable to many common risk metrics, with improved interpretability.

Findings

The proposed methods are robust under dependence and model misspecification.

Simulation and empirical results demonstrate the effectiveness of the approach.

The framework applies to risk measures like VaR, ES, mean, and variance.

Abstract

Backtesting risk measures is a central task in financial regulation. While standard backtests evaluate whether a forecasting model is statistically consistent with observed losses, regulatory practice often requires assessing the performance of an internal model relative to benchmark models. We develop a non-parametric sequential framework for comparative backtests of general elicitable risk measures using e-values and e-processes. The proposed methods provide anytime-valid inference and remain robust under dependence and model misspecification. In particular, we propose a modified three-zone approach based on weak dominance, which yields more informative conclusions in comparative backtesting. As a technical building block, we also construct general standard e-backtests for identifiable risk measures and characterize the associated e-values and e-processes. The resulting procedures apply to a broad class of commonly used risk measures, including the mean, variance, Value-at-Risk, Expected Shortfall, and expectiles. Simulation studies and empirical analyses illustrate the effectiveness of the proposed approach.