Groupe d’études et de recherche en analyse des décisions

Estimating Extremal Dependence in Time Series via the Extremogram

Richard A. Davis

The extremogram is a flexible quantitative tool that measures various types of extremal dependence in a stationary time series. In many respects, the extremogram can be viewed as an extreme-value analogue of the autocorrelation function (ACF) for a time series. Under mixing conditions, the asymptotic normality of the empirical extremogram was derived in Davis and Mikosch (2009). Unfortunately, the limiting variance is a difficult quantity to estimate. Instead we employ the stationary bootstrap to the empirical extremogram and establish that this resampling procedure provides an asymptotically correct approximation to the central limit theorem. This in turn can be used for constructing credible confidence bounds for the sample extremogram. The use of the stationary bootstrap for the extremogram is illustrated in a variety of real and simulated data sets. The cross-extremogram measures cross-sectional extremal dependence in multivariate time series. A measure of this dependence, especially left tail dependence, is of great importance in the calculation of portfolio risk. We find that after devolatilizing the marginal series, extremal dependence still remains, which suggests that the extremal dependence is not due solely to the heteroskedasticity in the stock returns process. However, for the univariate series, the filtering removes all extremal dependence. Following Geman and Chang (2010), a return time extremogram which measures the waiting time between rare or extreme events in univariate and bivariate stationary time series is calculated.. The return time extremogram suggests the existence of extremal clustering in the return times of extreme events for financial assets. The stationary bootstrap can again provide an asymptotically correct approximation to the central limit theorem and can be used for constructing credible confidence bounds for this return time extremogram. (This is joint work with Thomas Mikosch and Ivor Cribben.)