Hawkes model: significance of coefficients estimated with maximum likelihood

Question

I estimated a Hawkes model using maximum likelihood. How can I estimate the significance of the coefficients? I'm going to use the standard errors, i.e., the square root of the values on the diagonal of the inverse Hessian matrix. The latter is indeed an estimator of the asymptotic covariance matrix. Am I correct? I've read some literature but I haven't found an answer to my question, I assume that, being the method maximum likelihood, it goes without saying that the appropriate method is the one I described above. Another option would be computing the bootstrap standard errors. Is there a most common method?

Answer 1

The classical way, i.e. by using the observed Fisher information, is by far most common and is justified by Ogata, "The Asymptotic Behaviour of Maximum Likelihood Estimators for Stationary Point Processes" (1978). The assumptions required are basically the usual regularity assumptions for MLE, and they apply to the Hawkes process, at least in the univariate case (Example 4).

A bootstrap approach is also possible, but because of the computational cost it can be quite difficult to implement. A "fixed-intensity" bootstrap method was proposed by Cavaliere et al., "Bootstrap inference for Hawkes and general point processes" (2021) which makes the computational cost less prohibitive by considering the same conditional intensity in each bootstrap draw.