A Tale of Two Time Scales: Applications in Nonparametric Hawkes Processes With Ito Semimartingale Baseline

In view of their tractability, Hawkes processes are widely employed in high-frequency data. However, even in the absence of kernel (i.e. Poisson case), it is well-documented empirically that the baseline is not constant, reproducing in particular seasonalities from the financial market. In this paper, we relax this constancy assumption and consider a more realistic nonparametric framework where Hawkes self-exciting processes baseline follows an Ito semimartingale with possible jumps. When the kernel satisfies the short-range condition, we jointly and consistently estimate aggregated local Poisson estimates and truncated Two Scales Realized Volatility of these estimates, together with its central limit theory and feasible statistics. As a byproduct, we provide estimation and feasible limit theory of the branching ratio (i.e. the L1-norm of the kernel), the integrated baseline, the integrated volatility of the baseline and develop branching ratio related tests, which in particular include a test for the absence of Hawkes term and another for near criticality, for the absence of time-varying baseline, and for the absence of Brownian diffusion in the baseline. Simulation studies corroborate the theory, and document superiority of our branching ratio estimator over several concurrent methods in realistic configurations. Empirical studies illustrate a coherent behavior of our branching ratio estimator during the flash crash and COVID-19 crisis, a branching ratio consistently estimated around 0.7-0.8, tests for the absence of Hawkes term and for near criticality systematically rejected.