A sparsity test for multivariate Hawkes processes

TL;DR

This paper introduces a new statistical test for detecting causal interactions in multivariate Hawkes processes, with applications demonstrating its effectiveness in financial and auction data analysis.

Contribution

It presents a novel likelihood ratio test for causal relationships in multivariate Hawkes processes, including asymptotic distribution derivation and real-world applications.

Findings

Test detects deviations from static equilibrium in auction bidders' strategies.

Uncovers factors influencing German intraday power prices.

Demonstrates effectiveness on financial datasets.

Abstract

Multivariate Hawkes processes (MHP) are a class of point processes in which events at different coordinates interact through mutual excitation. The weighted adjacency matrix of the MHP encodes the strength of the relations, and shares its support with the causal graph of interactions of the process. We consider the problem of testing for causal relationships across the dimensions of a marked MHP. The null hypothesis is that a joint group of adjacency coefficients are null, corresponding to the absence of interactions. The alternative is that they are positive, and the associated interactions do exist. To this end, we introduce a novel estimation procedure in the context of a large sample of independent event sequences. We construct the associated likelihood ratio test and derive the asymptotic distribution of the test statistic as a mixture of chi squared laws. We offer two applications on financial datasets to illustrate the performance of our method. In the first one, our test reveals a deviation from a static equilibrium in bidders' strategies on retail online auctions. In the second one, we uncover some factors at play in the dynamics of German intraday power prices.