Multivariate dynamic intensity peaks-over-threshold models

We propose a multivariate dynamic intensity peaks-over-threshold model to capture extreme events in a multivariate time series of returns. The random occurrence of extreme events exceeding a threshold is modeled by means of a multivariate dynamic intensity model allowing for feedback effects between the individual processes. We propose alternative specifications of the multivariate intensity process using autoregressive conditional intensity and Hawkes-type specifications. Likewise, temporal clustering of the size of exceedances is captured by an autoregressive multiplicative error model based on a generalized Pareto distribution. We allow for spillovers between both the intensity processes and the process of marks. The model is applied to jointly model extreme returns in the daily returns of three major stock indexes. We find strong empirical support for a temporal clustering of both the occurrence of extremes and the size of exceedances. Moreover, significant feedback effects between both types of processes are observed. Backtesting Value-at-Risk (VaR) and Expected Shortfall (ES) forecasts show that the proposed model does not only produce a good in-sample fit but also reliable out-of-sample predictions. We show that the inclusion of temporal clustering of the size of exceedances and feedback with the intensity thereof results in better forecasts of VaR and ES.