Doctor of Philosophy
School of Mathematics and Applied Statistics
Wang, Qiying, Asymptotics on the general fractionally integrated processes with applications to unit root testing, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2001. https://ro.uow.edu.au/theses/2042
As a basic tool, asymptotic theory (in particular, functional limit theorem) plays a key role in characterizing the limit distribution of various statistics arising from statistical inferences in economic time series, such as testing for unit roots, testing for stationarity, and time series regression. Asymptotics on the fractional processes and the summable linear processes have been studied by many people. However, the results in the literature are quite restrictive on both the processes themselves and the conditions used in deriving the results. For example, a functional limit theorem is only available for a general fractional process with innovations being iid N(0, o2) or a simple fractional process under at least fourth moment conditions.
The aim of this work is to investigate systematically asymptotics of the general fractionally integrated processes and the summable linear processes with dependent or independent innovations under quite general conditions. We derive the functional limit theorem on the general fractionally integrated processes with and without "pre-historical influence". We give sufficient conditions so that the partial sum process of a summable linear process converges to a standard Brownian motion, and discuss asymptotics of sample autocovariances and autocorrelations based on nonstationary fractionally integrated processes. In particular, the result for the functional limit theorem on the general fractionally integrated processes provides a unified treatment for previous studies on the functional limit theorem for fractional processes and nonstationary fractionally integrated processes. Additionally, only finite second moments are required for most of the results established in this dissertation. Such a condition is the best possible moment condition in the literature and it is interesting from the theoretical point of view.
Also, we discuss applications of the results established in this dissertation to testing for unit roots, testing for stationarity, and time series regression. The limit distributions of Dickey and Fuller test statistics and KPSS (Kwiatkowski, Phillips, Schmidt and Shin) test statistics are derived for more general models under very weak conditions.