Alternative asymptotic inference theory for a nonstationary Hawkes process

Abstract

The Hawkes process is a popular point process model for events that exhibit a local clustering behaviour. Chen and Hall studied the asymptotic inference theory for a nonstationary Hawkes process where the baseline intensity is proportional to some time-varying function with the proportionality constant n tending to infinity. However, they assumed the excitation kernel to be independent of n, and therefore, as n increases, the waiting times from a baseline event to its excited events are of order OP(1) while the waiting times between baseline events is OP(1/n), suggesting the excitation effect is not local anymore, which defeats the purpose of choosing the exponential excitation kernel in the first place. To avoid this issue, we study the model in a more realistic setting where the excitation kernel also depends on the limit index n, so that the waiting times to excited events are of the same order of magnitude as those between baseline events. We establish consistency and asymptotic normality of the Maximum Likelihood Estimator (MLE). We also propose a score test to assess the constancy of the baseline intensity function and derive the asymptotic properties of the test. We apply the MLE and the score test to ultra-high frequency financial data as well as evaluating their finite sample performance via simulation experiments.