E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models
Abstract
Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR structure. Under the stated assumptions, both control false rejections at any stopping time. For the time-series method, validity requires a fixed parameter--quantile pair representing the conditional VaR path and a positive report margin. The time-series statistic takes an infimum over possible parameter and VaR values. We choose the betting process (λt) by maximizing an empirical growth rate averaged across candidate parameters. We compute the statistic by continuous VaR profiling on a dense parameter grid. For grid approximations, we prove convergence to the continuous infimum, together with convergence of the probability of false rejection, as the grid is refined over a fixed monitoring period. Lower and upper bounds for the VaR infimum on the parameter grid control the accuracy of the numerical profiling. Simulations, including ES reports that change from over-reporting to under-reporting, and an application to daily NASDAQ losses provide numerical comparisons with VaR--ES e-processes and fixed-sample ES backtests, assessing detection frequencies and the timing of rejections.
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BibTeX
@misc{wang2026ebacktesting,
title = {{E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models}},
author = {Ruodu Wang and Jingcheng Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17681},
eprint = {2610.17681},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17681},
}
RIS
TY - UNPB
TI - E-Backtesting Expected Shortfall without Value-at-Risk Forecasts under Time-series Models
AU - Ruodu Wang
AU - Jingcheng Yu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17681
UR - https://arxiv.org/abs/2610.17681
AB - Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR structure. Under the stated assumptions, both control false rejections at any stopping time. For the time-series method, validity requires a fixed parameter--quantile pair representing the conditional VaR path and a positive report margin. The time-series statistic takes an infimum over possible parameter and VaR values. We choose the betting process (λ_t) by maximizing an empirical growth rate averaged across candidate parameters. We compute the statistic by continuous VaR profiling on a dense parameter grid. For grid approximations, we prove convergence to the continuous infimum, together with convergence of the probability of false rejection, as the grid is refined over a fixed monitoring period. Lower and upper bounds for the VaR infimum on the parameter grid control the accuracy of the numerical profiling. Simulations, including ES reports that change from over-reporting to under-reporting, and an application to daily NASDAQ losses provide numerical comparisons with VaR--ES e-processes and fixed-sample ES backtests, assessing detection frequencies and the timing of rejections.
ER -