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

在2003-2004年的研究中，建立了处理平稳时间序列数据高阶矩相依的统计推断的基本理论框架，并发展了一套用于数值计算的算法。其方法是将二阶多元平稳过程的统计推断理论应用于由所讨论的序列及其平方序列组成的联合过程。通过这种方法，我们能够对二阶、三阶和四阶串联相关性进行建模。为了适应长期相关性，频域表示中的怀特尔似然是合适的。在适当的修正下，惠特尔似然的渐近估计和检验理论是适用的。为了处理具有长期相关性的非线性多变量过程，研究了分数阶协整模型，该模型是Johansen单位根协整模型的推广，并讨论了它的性质。在第二类分式Brown运动上建立了泛函中心极限定理，得到了协整秩数的Whttle似然比统计量的渐近分布。高阶矩相依性与所关注过程的非高斯性密切相关。处理这种非高斯主义的另一种方法是对这一过程进行非线性转换。为此，研究了一种修正的Box-Cox变换ARMA模型，并给出了一个渐近理论。

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.