The statistical significance of multivariate Hawkes processes fitted to limit order book data

TL;DR

This paper evaluates the statistical fit of multivariate Hawkes processes with different kernels to limit order book data, demonstrating improved performance with more complex kernels.

Contribution

It introduces a detailed analysis of multivariate Hawkes processes with sum-of-exponentials kernels applied to limit order book events, assessing goodness-of-fit and stability.

Findings

Three-exponential kernels provide the best fit.

Performance improves with increased kernel complexity.

Goodness-of-fit tests confirm model stability.

Abstract

Hawkes processes have seen a number of applications in finance, due to their ability to capture event clustering behaviour typically observed in financial systems. Given a calibrated Hawkes process, of concern is the statistical fit to empirical data, particularly for the accurate quantification of self- and mutual-excitation effects. We investigate the application of a multivariate Hawkes process with a sum-of-exponentials kernel and piecewise-linear exogeneity factors, fitted to liquidity demand and replenishment events extracted from limit order book data. We consider one-, two- and three-exponential kernels, applying various tests to ascertain goodness-of-fit and stationarity of residuals, as well as stability of the calibration procedure. In line with prior research, it is found that performance across all tests improves as the number of exponentials is increased, with a sum-of-three-exponentials yielding the best fit to the given set of coupled point processes.