Testing for jumps and jump intensity path dependence

In this paper, we develop a “jump test” for the null hypothesis that the probability of a jump is zero, building on earlier work by Aït-Sahalia (2002). The test is based on realized third moments, and uses observations over an increasing time span. The test offers an alternative to standard finite time span tests, and is designed to detect jumps in the data generating process rather than detecting realized jumps over a fixed time span. More specifically, we make two contributions. First, we introduce our largely model free jump test for the null hypothesis of zero jump intensity. Second, under the maintained assumption of strictly positive jump intensity, we introduce two “self-excitement” tests for the null of constant jump intensity against the alternative of path dependent intensity. These tests have power against autocorrelation in the jump component, and are direct tests for Hawkes diffusions (see, e.g. Aït-Sahalia et al. (2015)). The limiting distributions of the proposed statistics are analyzed via use of a double asymptotic scheme, wherein the time span goes to infinity and the discrete interval approaches zero; and the distributions of the tests are normal and half normal. The results from a Monte Carlo study indicate that the tests have reasonable finite sample properties. An empirical illustration based on the analysis of 11 stock price series indicates the prevalence of jumps and self-excitation.