Abstract
We consider tail empirical processes of long memory stochastic volatility models with heavy tails and leverage. We study the limiting behaviour of the tail empirical process with both fixed and random levels. We show a dichotomous behaviour for the tail empirical process with fixed levels, according to the interplay between the long memory parameter and the tail index; leverage does not play a role. On the other hand, the tail empirical process with random levels is not affected by either long memory or leverage. The tail empirical process with random levels is used to construct a family of estimators of the tail index, including the famous Hill estimator and harmonic mean estimators. The paper can be viewed as an extension of [21] while the presence of leverage in the model creates additional theoretical problems, the limiting behaviour remains unchanged.
Highlights
The tail empirical process (TEP) is an important tool used in nonparametric estimation of extremal quantities, like the Hill estimator of the index of regular variation or various risk measures
Our goal is to study weak convergence for the tail empirical processes associated with heavy tailed long memory stochastic volatility sequences with leverage
We have considered the heavy-tailed long memory stochastic volatility model with leverage given in (1)
Summary
The tail empirical process (TEP) is an important tool used in nonparametric estimation of extremal quantities, like the Hill estimator of the index of regular variation or various risk measures. From a theoretical point of view, our most important contribution is the proof of weak convergence of the tail empirical process (with fixed and random levels) in the presence of heavy tails, long memory and leverage. From a practical point of view, the key result is that the asymptotic behaviour of the TEP with random levels is unaffected by the presence of long memory and/or leverage in the model, and so in applications the log returns may be handled exactly as if they were i.i.d. heavy-tailed random variables. This greatly enhances the utility of the LMSV model with leverage considered here.
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