Tail behaviour and extremes of two-state Markov-switching autoregressive models

We examine the tail behaviour and extremal cluster characteristics of two-state Markov-switching autoregressive models where the first regime behaves like a random walk, the second regime is a stationary autoregression, and the generating noise is light-tailed. Under additional technical conditions we prove that the stationary solution has asymptotically exponential tail and the extremal index is smaller than one. The extremal index and the limiting cluster size distribution of the process are calculated explicitly for some noise distributions, and simulated for others. The practical relevance of the results is illustrated by examining extremal properties of a regime-switching autoregressive process with Gamma-distributed noise, already applied successfully in river flow modeling. The limiting aggregate excess distribution is shown to possess Weibull-like tail in this special case.