Modeling Extreme Events: Time-Varying Extreme Tail Shape

Abstract

We propose a dynamic semiparametric framework to study time variation in tail parameters. The framework builds on the Generalized Pareto Distribution (GPD) for modeling peaks over thresholds as in Extreme Value Theory, but casts the model in a conditional framework to allow for time-variation in the tail parameters. We establish parameter regions for stationarity and ergodicity and for the existence of (unconditional) moments and consider conditions for consistency and asymptotic normality of the maximum likelihood estimator for the deterministic parameters in the model. Two empirical datasets illustrate the usefulness of the approach: daily U.S. equity returns, and 15-min euro area sovereign bond yield changes.

Keywords:

• Dynamic tail risk
• Extreme value theory
• Observation-driven models
• Securities Markets Programme (SMP)
• Stock return tails

Supplementary Materials

Supplementary materials contain proofs and technical details, additional simulation and empirical results, and computer code (language: Ox) for the implementation of the method. The latter is also obtainable via www.gasmodel.com.

Acknowledgments

The views expressed in this article are those of the authors and do not necessarily reflect the views or policies of the BIS, the European Central Bank or Sveriges Riksbank.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 The web appendix considers additional applications to exchange rates and commodity prices.

2 Alternatively, one could opt to not update $f_t$ at all until a new $x_t>0$ arrives. Empirically, both approaches seem to work equally well.

3 See for instance Boswijk et al. (2021).

4 Web Appendix I provides two additional illustrations to other asset classes: exchange rates (GBP/USD) and commodities (Brent crude oil).