THESIS
2011
xiv, 101 p. : ill. ; 30 cm
Abstract
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...[
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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.
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