Abstract: Heavy tails is a common feature of most financial time series. Multivariate regular variation is often used as a building block for modeling heavy-tailed multivariate phenomena. Interestingly, many time series models, such as GARCH and stochastic volatility (SV) models that are commonly used for modeling financial time series, have finite dimensional distributions which are regularly varying. While GARCH and SV models share many of the same properties-both are martingale differences and exhibit heavy tails and volatility clustering, it turns out that the extremal behavior is quite different. Unlike a SV process, extremes cluster for a GARCH process.
To measure extremal dependence, we define an analog of the autocorrelation function, the extremogram, which only depends on the extreme values in the sequence. We propose a natural estimator for the extremogram and study its asymptotic properties under alpha-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram. Ultimately, we hope the extremogram can be used for model building and discriminating between various models, such as SV and GARCH models, for the data.
(This is joint work with Thomas Mikosch.)