A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.