Tail Risk Inference via Expectiles in Heavy-Tailed Time Series

Abstract

Expectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to establish that extreme expectiles can be applied with the kind of dependent time series models relevant to finance. In this article we provide a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general β-mixing context that encompasses ARMA and GARCH models with heavy-tailed innovations. Our methods allow the estimation of marginal (pertaining to the stationary distribution) and dynamic (conditional on the past) extreme expectile-based risk measures. Simulations and applications to financial returns show that the new estimators and confidence intervals greatly improve on existing ones when the data are dependent.

Keywords:

• Asymmetric least squares estimation
• Marginal expected shortfall
• Mixing
• Tail copula
• Weak dependence

Supplementary Materials

Supplementary material available at Journal of Business & Economic Statistics online provides a discussion of the technical conditions, further numerical results and all proofs.

Additional information

Funding

A. C. Davison is supported by the Swiss National Science Foundation. S. A. Padoan is supported by the Bocconi Institute for Data Science and Analytics. G. Stupfler gratefully acknowledges support of the Nottingham PEF Fund, of the French National Research Agency (grant ANR-19-CE40-0013) and of the AXA Research Fund.