We introduce Hawkes self-exciting processes with a baseline driven by an Itô semimartingale with possible jumps. Three measures for the intensity of these Hawkes processes, i.e., integrated intensity, integrated baseline, and integrated volatility of the baseline are studied. The statistics are based on empirical averages and preaveraging of local Poisson estimates constructed from high- frequency quote times. We also propose feasible statistics induced by central limit theory with in-fill asymptotic theory and develop a test for the absence of a Hawkes component and a test for constant baseline. An empirical application on high-frequency data of the E-mini S&P500 futures contracts finds rejection of both the null hypothesis of no Hawkes excitation and that of constant baseline contributing to the formal identification of patterns in high-frequency trading activity.

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

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