Sequential monitoring of the tail behavior of dependent data

- A sequential monitoring procedure for the tail behavior of time series is proposed.
- An algorithm using extreme quantile estimates performs well in simulations.
- Stock returns exhibit extremal instability during the recent crisis of 2007-2008.

We construct a sequential monitoring procedure for changes in the tail index and extreme quantiles of β-mixing random variables, which can be based on a large class of tail index estimators. The assumptions on the data are general enough to be satisfied in a wide range of applications. In a simulation study empirical sizes and power of the proposed tests are studied for linear and non-linear time series. Finally, we use our results to monitor Bank of America stock log-losses from 2007 to 2012 and detect changes in extreme quantiles without an accompanying detection of a tail index break.