In this thesis, I study the following two problems by using discretely observed high frequency data.

• How active are jumps in the underlying continuous-time dynamic which are modeled by semi-martingales?

• Could we model the underlying continuous-time dynamic by a pure jump process? The thesis consists of three parts.

First, under a general semi-martingale process, I propose a new estimator of the jump activity index (JAI hereafter) which is a natural measure of the relative occurring frequency of jumps defined by Ait-Sahalia and Jacod (2009b). Since small jumps were properly included in the estimator, the newly proposed estimator is more efficient than the one given in Ait-Sahalia and Jacod (2009b). Theoretically, the new estimator is proved to be consistent and a ce...[ Read more ]

In this thesis, I study the following two problems by using discretely observed high frequency data.

• How active are jumps in the underlying continuous-time dynamic which are modeled by semi-martingales?

• Could we model the underlying continuous-time dynamic by a pure jump process? The thesis consists of three parts.

First, under a general semi-martingale process, I propose a new estimator of the jump activity index (JAI hereafter) which is a natural measure of the relative occurring frequency of jumps defined by Ait-Sahalia and Jacod (2009b). Since small jumps were properly included in the estimator, the newly proposed estimator is more efficient than the one given in Ait-Sahalia and Jacod (2009b). Theoretically, the new estimator is proved to be consistent and a central limit theorem is obtained. Simulation studies justifies the good performance of the new estimator. A real example is also presented.

Second, the effect of the microstructure noise on estimation of JAI is investigated. It turns out that the existence of the microstructure noise leads to a significant bias on estimation of JAI. A two-step procedure is proposed to give a consistent estimator of the JAI under noisy observations. In the first step, a local smoothing technique is used to eliminate the effect of the microstructure noise and a modified noise-cleaned data set is obtained. In the second step, based on the noise-cleaned data set, several estimators are presented under three different model settings, Models I-III with increasing generality. Consistency of the estimators is proved under all models. Asymptotic normality is shown for the estimators under Model I and Model II. Simulations and the real data analysis shows that the estimator under Model I is robust to the presence of microstructure noise and performs reasonably well.

Third, to answer whether the underlying dynamic is a pure-jump semi-martingale or not, a test is developed. The test is very simple to use and yet effective. Asymptotic properties of the test statistic are studied. Simulations show that the test could control the type I error probabilities no matter how actively the jumps occur, and that the test is very powerful. Three real data sets are analyzed and all data sets support the pure-jump modeling of the underlying dynamics.