Extremal clustering in non-stationary random sequences

It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.

Using the Extremal Index for Value-at-Risk Backtesting*

We introduce a set of new Value-at-Risk independence backtests by establishing a connection between the independence property of Value-at-Risk forecasts and the extremal index, a general measure of extremal clustering of stationary sequences. For this purpose, we introduce a sequence of relative excess returns whose extremal index is to be estimated. We compare our backtest to both popular and recent competitors using Monte Carlo simulations and find considerable power in many scenarios. In an applied section, we perform realistic out-of-sample forecasts with common forecasting models and discuss advantages and pitfalls of our approach.