On studying extreme values and systematic risks with nonlinear time series models and tail dependence measures

ABSTRACT

This review paper discusses advances of statistical inference in modeling extreme observations from multiple sources and heterogeneous populations. The paper starts briefly reviewing classical univariate/multivariate extreme value theory, tail equivalence, and tail (in)dependence. New extreme value theory for heterogeneous populations is then introduced. Time series models for maxima and extreme observations are the focus of the review. These models naturally form a new system with similar structures. They can be used as alternatives to the widely used ARMA models and GARCH models. Applications of these time series models can be in many fields. The paper discusses two important applications: systematic risks and extreme co-movements/large scale contagions.

Keywords:

• Extreme value theory
• tail dependence index
• time series of maxima
• maxima of moving maxima
• autoregressive tail index models

Acknowledgments

The author thank Editor Jun Shao and two referees for their valuable comments. The work was partially supported by NSF-DMS-1505367 and NSF-DMS-2012298.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 If we let $d=d^{\prime }$, then the first three cases in the second column of Table 1 correspond to the serial asymptotic dependence index ${\lambda }_{d{d}_r}$.

Additional information

Funding

This work was partially supported by NSF - DMS-1505367 and NSF - DMS-2012298.

Notes on contributors

Zhengjun Zhang

Zhengjun Zhang is Professor of Statistics at the University of Wisconsin. His main research areas of expertise are in financial time series and rare event modeling, virtual standard currency, risk management, nonlinear dependence, asymmetric dependence, asymmetric and directed causal inference, gene-gene relationship in rare diseases.