A non-local estimator for locally stationary Hawkes processes

TL;DR

This paper develops a maximum likelihood estimation method for non-stationary Hawkes processes with time-varying parameters, including a test for constancy of the reproduction rate, and applies it to analyze power market data.

Contribution

It introduces a consistent MLE approach for non-stationary Hawkes processes and a time invariance test applicable to continuous functions, extending existing methods.

Findings

MLE remains consistent and asymptotically normal for large T.

The time invariance test is effective over all continuous functions.

Application reveals fluctuations in endogeneity in power market order flow.

Abstract

We consider the problem of estimating the parameters of a non-stationary Hawkes process with time-dependent reproduction rate and baseline intensity. Our approach relies on the standard maximum likelihood estimator (MLE), coinciding with the conventional approach for stationary point processes characterised by [Ogata, 1978]. In the fully parametric setting, we find that the MLE over a single observation of the process over $[0, T]$ remains consistent and asymptotically normal as $T \to \infty$. Our results extend partially to the semi-nonparametric setting where no specific shape is assumed for the reproduction rate $g \colon [0, 1] \mapsto \mathbb{R}_+$. We construct a time invariance test with null hypothesis that g is constant against the alternative that it is not, and find that it remains consistent over the whole space of continuous functions of [0, 1]. As an application, we employ our procedure in the context of the German intraday power market, where we provide evidence of fluctuations in the endogeneity rate of the order flow.