We consider Hawkes self-exciting processes with a baseline driven by an Itô semimartingale with possible jumps. Under in-fill asymptotics, we characterize feasible statistics induced by central limit theory for empirical average and variance of local Poisson estimates. As a by-product, we develop a test for the absence of a Hawkes component and a test for baseline constancy. Simulation studies corroborate asymptotic theory. An empirical application on high-frequency data of the E-mini S&P500 future contracts shows that the absence of a Hawkes component and baseline constancy is always rejected.

Keywords: Hawkes tests, in-fill asymptotics, high-frequency data, Itô semimartingale, self-exciting process, time-varying baseline.

JEL: C14, C22, C41, C58, G00, G13.