Modeling multivariate extreme events using self-exciting point processes

Abstract

We propose a model that can capture the typical features of multivariate extreme events observed in financial time series, namely clustering behavior in magnitudes and arrival times of multivariate extreme events, and time-varying dependence. The model is developed in the framework of the peaks-over-threshold approach in extreme value theory and relies on a Poisson process with selfexciting intensity. We discuss the properties of the model, treat its estimation, deal with testing goodness-of-fit, and develop a simulation algorithm. The model is applied to return data of two stock markets and four major European banks.

Specification Testing in Hawkes Models

“…Percentage rejections of the null hypothesis of zero cross-excitation, when performing Kolmogorov-Smirnov test on the transformed times (35) of 1000 simulations from Hawkes models over a time period of 5000 and 10000 time instances using 5% critical values. The parameters and the minimum magnitude of events are set on values estimated from the data set of Grothe et al (2014). In the Hawkes models with the parameter restriction α = 0, the magnitude of events have no influence on the triggering subsequent events.…”

Specification Testing in Hawkes Models

“…For example Grothe et al(2014) combine the intensities, with which events in the margins arrive, in a copula function to derive the joint intensity with which in at least one margin an event arrives. As the joint intensity is smaller than the sum of the marginal intensities, the probability of events occurring simultaneously is larger than zero.…”

“…Parameters and the minimum magnitude of events under consideration are set on the models estimated from the data set of Grothe et al (2014). Hawkes models with the parameter restriction α = 0, the magnitude of events have no influence on the triggering subsequent events.…”

“…For the power of the tests the models are simulated under the alternative, while for the size the models are simulated under the null. The parameters and the minimum magnitude of events are set on values estimated from the data set of Grothe et al (2014). The parameters Γ 12 and Γ 21 control respectively the cross-triggering effect from series 2 to series 1 and the cross-triggering effect from series 1 to series 2 (25).…”

“…Percentage rejections of the null hypothesis of zero cross-excitation, when performing the LR test on 1000 simulations from Hawkes models over a time period of 5000 and 10000 time instances using 5% critical values. The parameters and the minimum magnitude of events are set on values estimated from the data set of Grothe et al (2014). The parameters Γ 12 and Γ 21 control respectively the cross-triggering effect from series 2 to series 1 and the cross-triggering effect from series 1 to series 2 (25).…”