Long Memory Time Series Modelling: Computational and Statistical Efficiency, Nonstationarity/Noninvertibility and Goodness of Fit

鉴于长记忆模型应用的迅速扩展和对高频数据的日益关注，本项目的研究计划强调现代长记忆建模的统计和计算效率。目标是为长记忆时间序列的估计和拟合优度检验开发统计上合理和计算上有效的方法。为了达到这一目标，研究者将重点放在三个方面的研究上。研究的第一个方向是非平稳时间序列记忆参数的半参数估计。在经济、金融和网络流量数据中，非平稳性是一个非常普遍的特征。我们的目标是开发允许序列非平稳的估计器，并在不影响计算效率的情况下保持与平稳情况相同的效率。第二方面的研究重点是全参数长记忆模型的估计，即潜在不可逆的ARFIMA模型。在实践中，时间序列经常被差分一次或多次以诱导平稳性。有时可能会出现过度差异。研究者同时考虑频域和时域MLE，并推导出它们的渐近性质。研究者还提供了一种有效的算法来评估似然函数，以便精确的MLE可以应用于大型数据集。金融资产收益高频数据的可用性引发了对波动率建模的大量研究。在最后的研究中，研究者将以前为线性过程开发的一些统计程序扩展到随机波动模型。研究者提出了一类随机波动模型的拟合优度检验。这些测试是基于周期图的统计，不仅可以避免计算拟合残差，而且可以通过快速傅立叶变换算法进行评估。此外，它们在零下是渐近正态的，并且对大量的备选方案是一致的。最后，研究者研究了两个或多个长记忆随机波动序列之间的关系，这种关系称为分数协整。提出了协整参数的估计量，并推导了其相合性。研究者通过模拟研究研究了所提出程序的有限样本性质。本研究的动机是分析包括经济和金融数据在内的高频时间序列的新兴趋势。所提出的方法不仅为从业者提供了跨多个学科的时间相关数据建模的可行统计程序，而且还为未来时间序列的计算效率研究指明了方向。统计学系将于2006年推出新的网上研究生课程，以满足从事工业统计、生物统计和统计教学的专业人士（包括少数族裔、妇女和有家庭和全职工作的年长学生）在高等教育方面的需要，修订研究计划的最新发展，并将其转化为应用时间序列课程。这些项目可以帮助学生，特别是那些可能无法这样做的学生，实现他们个人丰富和职业发展的目标。

In view of the rapid expansion of the application of long memory models and growing interest in high frequency data, the research plan of this project stresses both statistical and computational efficiency in modern long memory modelling. The goal is to develop statistically sound and computationally efficient methods for long memory time series in estimation and goodness-of-fit testing. To reach this goal, the investigator focuses on the three lines of research. The first line of research concerns the semiparametric estimation of memory parameter of nonstationary time series. Nonstationarity is a very common feature in economics, finance and network traffic data. Our goal is to develop estimators that allow nonstationarity in series and retain the same efficiency as the stationary case without compromising the computational efficiency. The second line of research focuses the estimation of a full parametric long memory model, namely ARFIMA which is potentially noninvertible. In practice, time series are often differenced one or more times to induce stationarity. Sometimes overdifferencing may occur. The investigator considers both frequency and time domains MLE and derives the asymptotic properties for them. The investigator also provides an efficient algorithm to evaluate the likelihood function so that the exact MLE can be applied on a large data set. The availability of high-frequency data on returns of financial assets has intrigued a great amount of research in volatility modelling. In the last line of research, the investigator extends a few statistical procedures which were previously developed for the linear processes to stochastic volatility models. The investigator proposes a class of goodness-of-fit tests for stochastic volatility models. These tests are periodogram-based statistics that not only circumvent the computation of fitted residuals but also can be evaluated via the fast Fourier transform algorithm. Furthermore, they are asymptotically normal under the null and consistent against a wide class of alternatives. Lastly, the investigator studies the relation between two or more long memory stochastic volatility series, a relationship called fractional cointegration. The investigator proposes an estimator for the cointegration parameter and derives its consistency. The investigator studies the finite sample properties of the proposed procedures through simulation studies.This research is motivated by the emerging trend to analyze high frequency time series including economic and financial data. The proposed methods not only provide practitioners feasible statistical procedures in modelling time dependent data across a number of disciplines, but also lead a direction of future research in time series toward computational efficiency. The up-to-date developments in the research plan will be modified and transform into the applied time series course for the new coming online graduate program which will be launched in year 2006 by the Department of Statistics for serving the higher education needs of professionals who work in industrial statistics, biostatistics and statistical teaching, including minorities, women and older students with families and full-time jobs. The programs can help students, especially those who might otherwise not be able to do so, achieve their goals of personal enrichment and career advancement.