### Abstract

We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.

Language | English (US) |
---|---|

Pages | 663-676 |

Number of pages | 14 |

Journal | Journal of Statistical Planning and Inference |

Volume | 141 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2011 |

### Fingerprint

### Keywords

- Convolution tail
- Hill estimator
- Stochastic volatility
- Tail dependence

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

**Extremal memory of stochastic volatility with an application to tail shape inference.** / Hill, Jonathan B.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol 141, no. 2, pp. 663-676. DOI: 10.1016/j.jspi.2010.07.007

}

TY - JOUR

T1 - Extremal memory of stochastic volatility with an application to tail shape inference

AU - Hill,Jonathan B.

PY - 2011/1/1

Y1 - 2011/1/1

N2 - We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.

AB - We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.

KW - Convolution tail

KW - Hill estimator

KW - Stochastic volatility

KW - Tail dependence

UR - http://www.scopus.com/inward/record.url?scp=77954995936&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954995936&partnerID=8YFLogxK

U2 - 10.1016/j.jspi.2010.07.007

DO - 10.1016/j.jspi.2010.07.007

M3 - Article

VL - 141

SP - 663

EP - 676

JO - Journal of Statistical Planning and Inference

T2 - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 2

ER -