A study of Ihigher-ooder moment long-range dependence in economic time-series

在2003-2004年的研究中，建立了一个基本的理论框架来处理平稳时间序列数据高阶矩依赖性的统计推断，并开发了一套数值计算的算法。为此目的的方法是应用二阶多元平稳过程的统计推断理论的联合过程所讨论的系列和它的平方系列。通过这种方法，我们能够建模的二阶，三阶和四阶串行相关。为了适应长程相关性，频域表示中的Whittle似然是适当的。Whittle似然的渐近估计和检验理论在适当的修正下是适用的。为了处理长程相关的非线性多变量过程，研究了Johansen单位根协整模型的推广--分数阶协整模型及其性质。建立了2型分数布朗运动的泛函中心极限定理，得到了协整秩的Whtle似然比统计量的渐近分布.高阶矩依赖性与所关注的过程的非高斯性密切相关。处理这种非高斯性的另一种方法是对过程进行非线性变换。为此，本文研究了一种改进的Box-Cox变换阿尔马模型，并建立了渐近理论。

In the 2003-2004 research, a basic theoretical framework was developed to deal with statistical inference on higher-order moment dependency of stationary time-series data and also developed a set of algorithm for numerical computation. The approach for this purpose is to apply statistical inference theory of second-order multivariate stationary processes to the joint process consisting of the series in question and its squared series. By this method, we are able to model the second, third and fourth-order serial dependency. In order to accommodate long-range dependency, the Whittle likelihood in the frequency-domain representation is appropriate. The asymptotic estimation and testing theory of the Whittle likelihood is applicable under suitable modification. To deal with nonlinear multivariate processes of long range dependency, this research investigated the fractional cointegrated model which extends the Johansen's unit-root cointegration model and its properties. Also a version of functional central limit theorem was established on type 2 fractional Brown motion and the asymptotic distribution of the Whttle likelihood ratio statistic for the cointegration rank was derived. Higher-order moment dependency is closely related to non-Gaussianity of a process in concern. Another approach to deal with such non-Gaussianity is to transform nonlinearly the process. For that purpose, a model of modified Box-Cox transformation ARMA is investigated in this research and an asymptotic theory is developed.