  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.
