Testing the causality of Hawkes processes with time reversal

TL;DR

This paper demonstrates that Hawkes processes are only weakly causal, as their likelihoods are nearly identical for real and reversed event sequences, challenging the interpretation of their fitted kernels as indicators of causality.

Contribution

It reveals the weak causality property of Hawkes processes and emphasizes the importance of testing for time directionality rather than relying solely on kernel fitting.

Findings

Likelihoods of real and reversed data are nearly equal.

Goodness-of-fit tests can distinguish true from reversed data in ideal conditions.

Flexible kernels may fit both time directions in financial data.

Abstract

We show that univariate and symmetric multivariate Hawkes processes are only weakly causal: the true log-likelihoods of real and reversed event time vectors are almost equal, thus parameter estimation via maximum likelihood only weakly depends on the direction of the arrow of time. In ideal (synthetic) conditions, tests of goodness of parametric fit unambiguously reject backward event times, which implies that inferring kernels from time-symmetric quantities, such as the autocovariance of the event rate, only rarely produce statistically significant fits. Finally, we find that fitting financial data with many-parameter kernels may yield significant fits for both arrows of time for the same event time vector, sometimes favouring the backward time direction. This goes to show that a significant fit of Hawkes processes to real data with flexible kernels does not imply a definite arrow of time unless one tests it.