19th Ave New York, NY 95822, USA

# 统计代写|时间序列分析代写Time Series Analysis代考|Models utilizing time series structure

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 统计代写|时间序列分析代写Time Series Analysis代考|Fixed effects model

One problem with the model in Eq. (6.23) is that some of the factors may be correlated and become redundant. This leads to the introduction of the following Orthogonal GARCH (shortened to O-GARCH) model, introduced by Alexander and Chibumba (1997),
$$\boldsymbol{\varepsilon}t=\boldsymbol{\Omega} \mathbf{r}_t$$ where $\boldsymbol{\Omega}$ is an $m \times m$ orthogonal matrix often known as the linkage matrix, transformation matrix, or factor loading matrix, $\mathbf{r}_t=\left(r{1, t}, \ldots, r_{m, t}\right)^{\prime}$, and the $r_{i, t}$ ‘s are independent factors and each follows a univariate $\operatorname{GARCH}(p, q)$ such as a $\operatorname{GARCH}(1,1)$ model. That is,
$$\mathbf{r}t=\boldsymbol{\Gamma}_t^{1 / 2} \mathbf{e}_t$$ where the $\mathbf{e}_t$ are i.i.d. $m$-dimensional multivariate random vectors with mean vector $\mathbf{0}$ and covariance matrix $\mathbf{I}$, $$\boldsymbol{\Gamma}_t=\operatorname{Var}{t-1}\left(\mathbf{r}t\right)=\operatorname{diag}\left(\sigma{r_{1, t}}^2, \ldots, \sigma_{r_{n, r}}^2\right),$$
and
$$\sigma_{r, t}^2=\left(1-\alpha_i-\beta_i\right)+\alpha_i \sigma_{r, t-1}^2+\beta_i r_{i, t-1}^2, i=1, \ldots, m$$
To ensure the value in Eq. (6.28) to be positive, we assume that $\alpha_i$ and $\beta_i$ are positive, and $\alpha_i+\beta_i$ $<1$. Thus, the conditional variance of $\varepsilon_t$ and hence that of $\mathbf{Z}t$ becomes $$\boldsymbol{\Sigma}_t=\operatorname{Var}{t-1}\left(\boldsymbol{\varepsilon}t\right)=\operatorname{Var}\left(\boldsymbol{\varepsilon}_t \mid \boldsymbol{\Psi}{t-1}\right)=\boldsymbol{\Omega} \boldsymbol{\Gamma}_t \boldsymbol{\Omega}^{\prime} .$$
The linkage matrix and the independent components in Eq. (6.25) are obtained by performing a principal component analysis (PCA) on the series through the sample covariance matrix. Alexander (2001) further illustrated the use of the O-GARCH model in her book and emphasized that the strength of the model is to choose a small number of principal components from PCA compared to the number of variables (assets).

## 统计代写|时间序列分析代写Time Series Analysis代考|Some common variance–covariance structures

Since a large number of parameters in a variance-covariance matrix will unfavorably affect the estimation efficiency, we should use the correlation pattern of the time series to simplify its form. The following are some commonly used variance-covariance matrices used in repeated measurement studies. Except for the first unstructured matrix, we introduce some simple and useful structures that contain only a small number of parameters.
The unstructured matrix:
$$\mathbf{\Sigma}=\left[\begin{array}{ccccc} \sigma_1^2 & \sigma_{1,2} & \cdots & \cdots & \sigma_{1, p} \ & \sigma_2^2 & \sigma_{2,3} & \cdots & \sigma_{2, p} \ & & \ddots & \vdots & \vdots \ & & & \sigma_{p-1}^2 & \sigma_{(p-1), p} \ & & & & \sigma_p^2 \end{array}\right]$$
The form implies that variances and covariances at different times are not necessarily equal. There are $p(p+1) / 2$ parameters in the matrix.
The identical and independent structure:
$$\boldsymbol{\Sigma}=\left[\begin{array}{ccccc} \sigma^2 & 0 & 0 & \cdots & 0 \ & \sigma^2 & 0 & \cdots & 0 \ & & \ddots & \vdots & \vdots \ & & & \sigma^2 & 0 \ & & & & \sigma^2 \end{array}\right]=\sigma^2 \mathbf{I} .$$
The form in Eq. (7.20) is the simplest one and contains only one parameter. It may be applicable in some applications especially when the repeated measurements are taken far apart such that the correlation between different times is effectively zero relative to the other variation.
The independent but non-identical structure:
$$\mathbf{\Sigma}=\left[\begin{array}{ccccc} \sigma_1^2 & 0 & 0 & \cdots & 0 \ & \sigma_2^2 & 0 & \cdots & 0 \ & & \ddots & \vdots & \vdots \ & & & \sigma_{p-1}^2 & 0 \ & & & 0 & \sigma_p^2 \end{array}\right] .$$
This is a generalized form of Eq. (7.20), where the variances at different times are not necessarily equal. It contains $p$ parameters.
The structure of common symmetry:
$$\boldsymbol{\Sigma}=\left[\begin{array}{ccccc} \sigma^2 & \sigma^2 \rho & \cdots & \cdots & \sigma^2 \rho \ & \sigma^2 & \sigma^2 \rho & \cdots & \sigma^2 \rho \ & \ddots & \vdots & \vdots \ & & \sigma^2 & \sigma^2 \rho \ & & & \sigma^2 \end{array}\right] .$$
The form in Eq. (7.22) assumes that $E\left(e_{i, j, k} e_{i, j, \ell}\right)=\sigma^2$ if $k=\ell$, and $E\left(e_{i, j, k} e_{i, j, \ell}\right)=\sigma^2 \rho$ if $k \neq \ell$. There are only two parameters. However, it implies that (i) variances are equal at all times, and (ii) covariances and hence correlations are equal at all pairs of times. This strong assumption may not hold in many situations.
The structure of heterogeneous common symmetry:
$$\boldsymbol{\Sigma}=\left[\begin{array}{ccccc} \sigma_1^2 & \sigma_1 \sigma_2 \rho & \cdots & \cdots & \sigma_1 \sigma_p \rho \ & \sigma_2^2 & \sigma_2 \sigma_3 \rho & \cdots & \sigma_2 \sigma_p \rho \ & & \ddots & \vdots & \vdots \ & & & \ddots & \sigma_{(p-1)} \sigma_p \rho \ & & & & \sigma_p^2 \end{array}\right] .$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Fixed effects model

$$\boldsymbol{\varepsilon}t=\boldsymbol{\Omega} \mathbf{r}t$$其中$\boldsymbol{\Omega}$是一个$m \times m$正交矩阵，通常称为链接矩阵、变换矩阵或因子加载矩阵，$\mathbf{r}_t=\left(r{1, t}, \ldots, r{m, t}\right)^{\prime}$和$r_{i, t}$是独立的因素，每个因素都遵循一个单变量$\operatorname{GARCH}(p, q)$，如$\operatorname{GARCH}(1,1)$模型。也就是说，
$$\mathbf{r}t=\boldsymbol{\Gamma}t^{1 / 2} \mathbf{e}_t$$其中$\mathbf{e}_t$为i.i.d $m$多维随机向量，具有均值向量$\mathbf{0}$和协方差矩阵$\mathbf{I}$, $$\boldsymbol{\Gamma}_t=\operatorname{Var}{t-1}\left(\mathbf{r}t\right)=\operatorname{diag}\left(\sigma{r{1, t}}^2, \ldots, \sigma_{r_{n, r}}^2\right),$$

$$\sigma_{r, t}^2=\left(1-\alpha_i-\beta_i\right)+\alpha_i \sigma_{r, t-1}^2+\beta_i r_{i, t-1}^2, i=1, \ldots, m$$

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。