Time-Varying Tails: Estimation, Inference, and Risk Measurement

Abstract

Motivated by evidence that the tails of economic and financial time series need not be stable over time, we develop a framework for studying time-varying tails. We model the extreme value index as a smooth function of time in the locally stationary framework of Zhou and Wu (2009). We estimate it by a ratio-form kernel Hill estimator in which a single intermediate quantile thresholds the whole local window. We establish consistency and asymptotic normality, and we construct uniform confi- dence bands. We also propose a consistent test of tail stability under extremal clustering. Combined with the Weissman extrapolation method, the estimator of the extreme value index yields time- varying extreme Value-at-Risk and Expected Shortfall. In an empirical application, we find the tail of daily loss of S&P 500 is compatible with a constant shape, whereas that of volatility-filtered loss varies significantly.

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BibTeX RIS

BibTeX

@misc{li2026timevarying,
  title = {{Time-Varying Tails: Estimation, Inference, and Risk Measurement}},
  author = {Shuo Li and Liuhua Peng},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06005},
  eprint = {2611.06005},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06005},
}

RIS

TY  - UNPB
TI  - Time-Varying Tails: Estimation, Inference, and Risk Measurement
AU  - Shuo Li
AU  - Liuhua Peng
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06005
UR  - https://arxiv.org/abs/2611.06005
AB  - Motivated by evidence that the tails of economic and financial time series need not be stable over time, we develop a framework for studying time-varying tails. We model the extreme value index as a smooth function of time in the locally stationary framework of Zhou and Wu (2009). We estimate it by a ratio-form kernel Hill estimator in which a single intermediate quantile thresholds the whole local window. We establish consistency and asymptotic normality, and we construct uniform confi- dence bands. We also propose a consistent test of tail stability under extremal clustering. Combined with the Weissman extrapolation method, the estimator of the extreme value index yields time- varying extreme Value-at-Risk and Expected Shortfall. In an empirical application, we find the tail of daily loss of S&P 500 is compatible with a constant shape, whereas that of volatility-filtered loss varies significantly.
ER  -

Abstract

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