Joint extreme value-at-risk and expected shortfall dynamics with a single integrated tail shape parameter

André Lucas, Bernd Schwaab, Xin Zhang and D’Innocenzo, Enzo

No 3166, Working Paper Series from European Central Bank

Abstract: We propose a robust semi-parametric framework for persistent time-varying extreme tail behavior, including extreme Value-at-Risk (VaR) and Expected Shortfall (ES). The framework builds on Extreme Value Theory and uses a conditional version of the Generalized Pareto Distribution (GPD) for peaks-over-threshold (POT) dynamics. Unlike earlier approaches, our model (i) has unit root-like, i.e., integrated autoregressive dynamics for the GPD tail shape, and (ii) re-scales POTs by their thresholds to obtain a more parsimonious model with only one time-varying parameter to describe the entire tail. We establish parameter regions for stationarity, ergodicity, and invertibility for the integrated time-varying parameter model and its filter, and formulate conditions for consistency and asymptotic normality of the maximum likelihood estimator. Using two cryptocurrency exchange rates, we illustrate how the simple single-parameter model is competitive in capturing the dynamics of VaR and ES, particularly in the extreme tail. JEL Classification: C22, G11

Keywords: dynamic tail risk; extreme value theory; integrated score-driven models (search for similar items in EconPapers)
Date: 2026-01
Note: 955417
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Persistent link: https://EconPapers.repec.org/RePEc:ecb:ecbwps:20263166

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