Consistent pretesting for jumps

If the intensity parameter in a jump diffusion model is identically zero, then parameters characterizing the jump size density cannot be identified. In general, this lack of identification precludes consistent estimation of identified parameters. Hence, it should be standard practice to consistently pretest for jumps, prior to estimating jump diffusions. Many currently available tests have power against the presence of jumps over a finite time span (typically a day or a week); and, as already noted by various authors, jumps may not be observed over finite time spans, even if the intensity parameter is strictly positive. Such tests cannot be consistent against non-zero intensity. Moreover, sequential application of finite time span tests usually leads to sequential testing bias, which in turn leads to jump discovery with probability one, in the limit, even if the true intensity is identically zero. This paper introduces tests for jump intensity, based on both in-fill and long-span asymptotics, which solve both the test consistency and the sequential testing bias problems discussed above, in turn facilitating consistent estimation of jump diffusion models. A "self excitement" test is also introduced, which is designed to have power against path dependent intensity, thus providing a direct test for the Hawkes diffusion model of Ait-Sahalia, Cacho-Diaz and Laeven (2013). In a series of Monte Carlo experiments, the proposed tests are evaluated, and are found to perform adequately in finite samples.