Seminar in applied mathematics and statistics
SPEAKER: Anja Janßen (University of Hamburg)
TITLE: Asymptotically independent time series and hidden regular variation
ABSTRACT:
Most models for financial time series share the feature of regularly varying marginal distributions in order to reflect the heavy-tailed behavior of log-returns which is commonly accepted as a stylized fact. However, models differ with respect to their joint extremal behavior. Some models for stationary time series $(X_t)_{t \in \mathbb{Z}}$ show asymptotic dependence, meaning that all pairs $(X_t, X_{t+h}), h \in \mathbb{Z},$ satisfy $\lim_{x \to \infty} P(|X_{t+h}|>x | |X_t|>x)>0$. This holds for example true for GARCH$(p,q$) models. These models are well described by multivariate regular variation which gives an exhaustive description of the extremal behavior of the complete process.
On the other hand, there exist time series models which show asymptotic independence, i.e. $\lim_{x \to \infty} P(|X_{t+h}|>x | |X_t|>x)=0$ for all lags $h \neq 0$. Many common stochastic volatility models are examples for this class. For these models, the limiting measure of consecutive observations $(X_t,X_{t+h})$ is concentrated on the axes in $\mathbb{R}^2$ and it does not allow to give useful approximations of exceedance probabilities of the form $P(|X_t|>x,|X_{t+h}|>x)$. To overcome this, it is useful to look at these exceedance probabilities in the framework of so-called hidden regular variation on $(0,\infty]^2$, c.f. Resnick (2002).
We analyze common specifications for standard stochastic volatility models in this framework and show that these models show a very strong form of asymptotic independence, i.e., similar to an i.i.d.\ sequence. We also investigate a different model which allows a more flexible extremal behavior, c.f. Janßen and Drees (2015). To this end, we make use of a new Breiman-type result for hidden regular variation that enables us to analyze the behavior of the product of a random matrix and a hidden regularly varying random vector.
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Tea and chocolate will be served on the 4th floor after the seminar.
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UPCOMING SEMINARS:
February 20, 13:15 (Friday): J.O. Ramsay, Aud. 8
February 25, 15.15: Piotr Zwiernik, lille UP 1, DIKU
March 4, 15.15: Felix Hentschel, Aud. 10
March 18, 15.15: Jes Frellsen, Aud. 10
March 24: Persi Diaconis (Harald Bohr Lecture)
June 2: Daniel Stroock (Department Colloquium)