## 统计代写|时间序列分析代写Time-Series Analysis代考|S-650

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocorrelation Function

The trouble with covariances is that they generally depend on the units with which $Y_t$ is measured. We can easily get around this problem by working correlations, which are just scaled versions of covariances. We have:
Definition 31 The correlation between $Y_t$ and $Y_{t-k}$ is:
$$\operatorname{Corr}\left[Y_t, Y_{t-k}\right]=\frac{\operatorname{Cov}\left[Y_t, Y_{t-k}\right]}{\operatorname{Var}\left[Y_t\right]^{\frac{1}{2}} \operatorname{Var}\left[Y_{t-k}\right]^{\frac{1}{2}}} .$$
Using stationarity we can simplify this considerably. Since
$$\operatorname{Cov}\left[Y_t, Y_{t-k}\right]=\gamma(k)$$
and by stationarity
$$\operatorname{Var}\left[Y_t\right]^{\frac{1}{2}}=\operatorname{Var}\left[Y_{t-k}\right]^{\frac{1}{2}}=\gamma(0)^{\frac{1}{2}}$$
we have:
$$\operatorname{Corr}\left[Y_t, Y_{t-k}\right]=\frac{\gamma(k)}{\gamma(0)}$$
With this in mind we can define the autocorrelation function
$$\rho(k)=\operatorname{Corr}\left[Y_t, Y_{t-k}\right]$$
as follows:
Definition 32 Autocorrelation Function: The autocorrelation function $\rho(k)$ is defined as:
$$\rho(k)=\frac{\gamma(k)}{\gamma(0)}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocorrelation Function of an AR(1) Process

We have :
Theorem 39 The autocorrelation function of a stationary AR(1) process is:
$$\rho(k)=\phi^{|k|} .$$
Proof. From Theorem 30 it follows that
\begin{aligned} \rho(k) & =\frac{\gamma(k)}{\gamma(0)} \ & =\frac{\phi^{|k|} \frac{\sigma^2}{1-\phi^2}}{\frac{\sigma^2}{1-\phi^2}} \ & =\phi^{|k|} . \end{aligned}

We plot $\rho(k)$ an $\operatorname{AR}(1)$ with $\phi=0.7$ below: ${ }^2$
$$\rho(k) \text { when } \phi=0.7$$
Since $|\phi|<1$ it follows that the autocorrelation function, like the autocovariance function, has the short-memory property so that $\rho(k)=O\left(\tau^k\right)$ as given in Section 2.2 with $A=1$ and $\tau=|\phi|$.

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocorrelation Function

$$\operatorname{Corr}\left[Y_t, Y_{t-k}\right]=\frac{\operatorname{Cov}\left[Y_t, Y_{t-k}\right]}{\operatorname{Var}\left[Y_t\right]^{\frac{1}{2}} \operatorname{Var}\left[Y_{t-k}\right]^{\frac{1}{2}}} .$$

$$\operatorname{Cov}\left[Y_t, Y_{t-k}\right]=\gamma(k)$$

$$\operatorname{Var}\left[Y_t\right]^{\frac{1}{2}}=\operatorname{Var}\left[Y_{t-k}\right]^{\frac{1}{2}}=\gamma(0)^{\frac{1}{2}}$$

$$\operatorname{Corr}\left[Y_t, Y_{t-k}\right]=\frac{\gamma(k)}{\gamma(0)}$$

$$\rho(k)=\operatorname{Corr}\left[Y_t, Y_{t-k}\right]$$

$$\rho(k)=\frac{\gamma(k)}{\gamma(0)}$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocorrelation Function of an AR(1) Process

$$\rho(k)=\phi^{|k|} .$$

\begin{aligned} \rho(k) & =\frac{\gamma(k)}{\gamma(0)} \ & =\frac{\phi^{|k|} \frac{\sigma^2}{1-\phi^2}}{\frac{\sigma^2}{1-\phi^2}} \ & =\phi^{|k|} . \end{aligned}

$$\rho(k) \text { when } \phi=0.7$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STK9060

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocovariance Function

An important implication of stationarity is that the covariance between the business cycle in say the first and third quarters of say 1999 is the same as the covariance between the business cycle the first and third quarters of say 1963 . In general covariances only depend on the number of periods separating $Y_t$ and $Y_s$ so that:

Theorem 20 If $Y_t$ is stationary then $\operatorname{Cov}\left[Y_{t_1}, Y_{t_2}\right]$ depends only on $k=t_1-t_2$; that is the number of periods separating $t_1$ and $t_2$.

Since we will often be focusing on covariances, and since $\operatorname{Cov}\left[Y_t, Y_{t-k}\right]$ only depends on $k$, let us define this as a function of $k$ as: $\gamma(k)$, which we will refer to as the autocovariance function so that:

Definition 21 Autocovariance Function: Let $Y_t$ be a stationary time series with $E\left[Y_t\right]=0$. The autocovariance function for $Y_t$, denoted as $\gamma(k)$, is defined for $k=0, \pm 1, \pm 2, \pm 3, \ldots \pm \infty$ as:
$$\gamma(k) \equiv E\left[Y_t Y_{t-k}\right]=\operatorname{Cov}\left[Y_t, Y_{t-k}\right] .$$
We have the following results for the autocovariance function:
Theorem $22 \gamma(0)=\operatorname{Var}\left[Y_t\right]>0$
Theorem $23 \gamma(k)=E\left[Y_t Y_{t-k}\right]=E\left[Y_s Y_{s-k}\right]$ for any $t$ and $s$.
Theorem $24 \gamma(-k)=\gamma(k)(\gamma(k)$ is an even function $)$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The AR(1) Model

For the AR(1) model we have already shown that:
$$\gamma(0)=\operatorname{Var}\left[Y_t\right]=\frac{\sigma^2}{1-\phi^2}$$
and that for $k>0$ :
\begin{aligned} \gamma(k) & =\phi^k \gamma(0) \ & =\phi^k \frac{\sigma^2}{1-\phi^2} . \end{aligned}
We can make this formula correct for all $k$ by appealing to Theorem 24 and replacing $k$ with $|k|$ to obtain:

Theorem 30 For an $A R(1)$ process the autocovariance function is given by:
$$\gamma(k)=\frac{\phi^{|k|} \sigma^2}{1-\phi^2} .$$

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Autocovariance Function

$$\gamma(k) \equiv E\left[Y_t Y_{t-k}\right]=\operatorname{Cov}\left[Y_t, Y_{t-k}\right] .$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The AR(1) Model

$$\gamma(0)=\operatorname{Var}\left[Y_t\right]=\frac{\sigma^2}{1-\phi^2}$$

\begin{aligned} \gamma(k) & =\phi^k \gamma(0) \ & =\phi^k \frac{\sigma^2}{1-\phi^2} . \end{aligned}

$$\gamma(k)=\frac{\phi^{|k|} \sigma^2}{1-\phi^2} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|MGSC575

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Short-Memory Property

Many of the models that we will be considering will have the property that they quickly forget or quickly become independent of what occurs either in the distant future or the distant past. This forgetting occurs at an exponential rate which represents a very rapid type of decay.

For example if you have a pie in the fridge and you eat one-half of the pie each day, you will quickly have almost no pie. After only ten days you would have:
$$\left(\frac{1}{2}\right)^{10}=\frac{1}{1024}$$
or about one-thousandth of a pie; maybe a couple of crumbs.
We will see that for stationary $\operatorname{ARMA}(\mathrm{p}, \mathrm{q})$ processes, the infinite moving average weights: $\psi_k$, the autocorrelation function $\rho(k)$ and the forecast function $E_t\left[Y_{t+k}\right]$, all functions of the number of periods $k$, all have the short-memory property which we now define:

Definition 12 Short-Memory: Let $P_k$ for $k=0,1,2, \ldots \infty$ be some numerical property of a stationary time series which depends on $k$, the number of periods. We say $P_k$ displays a short-memory or $P_k=O\left(\tau^k\right)$ if
$$\left|P_k\right| \leq A \tau^k$$
where $A \geq 0$ and $0<\tau<1$.

If $P_k=O\left(\tau^k\right)$ or if $P_k$ has a short-memory then $P_k$ decays rapidly in the same, manner that is at least as fast as $\tau^k$ decays to zero as $k \rightarrow \infty$. For example if:
$$P_k=10 \cos (2 k)\left(-\frac{1}{2}\right)^k$$
then $P_k$ decays rapidly in a manner which is bounded by exponential decay since $|\cos (2 k)| \leq 1$ and so we have:
$$\left|P_k\right| \leq 10\left(\frac{1}{2}\right)^k=A \tau^k$$
where $\tau=\frac{1}{2}$ and $A=10$. This is illustrated in the plot below:
$$P_k=10 \cos (2 k)\left(-\frac{1}{2}\right)^k$$
Not everything decays so rapidly. For example if we reverse the $\frac{1}{2}$ and the $k$ in $\left(\frac{1}{2}\right)^k$ we obtain:
$$Q_k=\frac{1}{k^{\frac{1}{2}}}=k^{-\frac{1}{2}} .$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The AR(1) Model

The simplest interesting model for $Y_t$ is a first-order autoregressive process or $\mathrm{AR}(1)$ which can be written as:
$$Y_t=\phi Y_{t-1}+a_t, a_t \sim \text { i.i.n }\left(0, \sigma^2\right),$$
where i.i.n. $\left(0, \sigma^2\right)$ means that $a_t$ is independently and identically distributed (i.i.d.) with a normal distribution with mean 0 and variance $\sigma^2$ so that the density of $a_t$ is:
$$p\left(a_t\right)=\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} a_t^2} .$$
We can attempt to calculate $E\left[Y_t\right]$ by taking expectations of both sides of (2.6) to obtain:
\begin{aligned} E\left[Y_t\right] & =\phi E\left[Y_{t-1}\right]+E\left[a_t\right] \ & =\phi E\left[Y_{t-1}\right] \end{aligned}
since $E\left[a_t\right]=0$. We now need to find $E\left[Y_{t-1}\right]$. We could try the same approach with $E\left[Y_{t-1}\right]$ since $Y_{t-1}=\phi Y_{t-2}+a_{t-1}$ from which we would conclude that: $E\left[Y_{t-1}\right]=\phi E\left[Y_{t-2}\right]$ so that:
$$E\left[Y_t\right]=\phi^2 E\left[Y_{t-2}\right] ;$$
but now we now need to find $E\left[Y_{t-2}\right]$. Clearly this process will never end.
If, however, we assume stationarity then it is possible to break this infinite regress since by the definition of stationarity in Definition 7:
$$E\left[Y_t\right]=E\left[Y_{t-1}\right] .$$
It then follows from (2.8) that:
$$E\left[Y_t\right]=\phi E\left[Y_t\right]$$
or
$$(1-\phi) E\left[Y_t\right]=0 .$$

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|The Short-Memory Property

$$\left(\frac{1}{2}\right)^{10}=\frac{1}{1024}$$

$$\left|P_k\right| \leq A \tau^k$$

$$P_k=10 \cos (2 k)\left(-\frac{1}{2}\right)^k$$

$$\left|P_k\right| \leq 10\left(\frac{1}{2}\right)^k=A \tau^k$$

$$P_k=10 \cos (2 k)\left(-\frac{1}{2}\right)^k$$

$$Q_k=\frac{1}{k^{\frac{1}{2}}}=k^{-\frac{1}{2}} .$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|The AR(1) Model

$$Y_t=\phi Y_{t-1}+a_t, a_t \sim \text { i.i.n }\left(0, \sigma^2\right),$$

$$p\left(a_t\right)=\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{1}{2} a_t^2} .$$

\begin{aligned} E\left[Y_t\right] & =\phi E\left[Y_{t-1}\right]+E\left[a_t\right] \ & =\phi E\left[Y_{t-1}\right] \end{aligned}

$$E\left[Y_t\right]=\phi^2 E\left[Y_{t-2}\right] ;$$

$$E\left[Y_t\right]=E\left[Y_{t-1}\right] .$$

$$E\left[Y_t\right]=\phi E\left[Y_t\right]$$

$$(1-\phi) E\left[Y_t\right]=0 .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|STAT6550

statistics-lab™ 为您的留学生涯保驾护航 在代写时间序列分析Time-Series Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写时间序列分析Time-Series Analysis代写方面经验极为丰富，各种代写时间序列分析Time-Series Analysis相关的作业也就用不着说。

## 统计代写|时间序列分析代写Time-Series Analysis代考|Smoothing spline

The main disadvantage of the kernel smoothing method and the multitaper smoothing method is that they cannot guarantee that the final estimate is positive semidefinite while allowing flexible smoothing for each element of the spectral matrix. Thus, the same bandwidth is often applied to smoothing all the spectral components. However, in many applications, different components of the spectral matrix may need different smoothnesses, and require different smoothing parameters to get optimal estimates. To overcome this difficulty, Dai and Guo (2004) proposed a Cholesky decomposition based smoothing spline method for the spectrum estimation. The method models each Cholesky component separately by using different smoothing parameters. The method first obtains positive-definite and asymptotically unbiased initial spectral estimator $\widetilde{\mathbf{f}}_M(\omega)$ through sine multitapers as shown in Eq. (9.65). Then, it further smooths the Cholesky components of the spectral matrix via the smoothing spline and penalized sum of squares, which allows different degrees of smoothness for different Cholesky elements.

Suppose the spectral matrix $\tilde{\mathbf{f}}(\omega)$ has Cholesky decomposition such that $\tilde{\mathbf{f}}(\omega)=\mathbf{\Gamma} \mathbf{\Gamma}^*$, where $\boldsymbol{\Gamma}$ is $m \times m$ lower triangular matrix. To obtain unique decomposition, the diagonal elements of $\boldsymbol{\Gamma}$ are constrained to be positive. The diagonal elements $\gamma_{j, j}, j=1, \ldots, m$, the real part of $\gamma_{j, k}, \Re\left(\gamma_{j, k}\right)$, and imaginary part of $\gamma_{j, k}, \mathfrak{\Im}\left(\gamma_{j, k}\right), j>k$ are smoothed by spline with different smoothing parameters. Suppose $\gamma \in\left{\gamma_{j, j}, \Re\left(\gamma_{j, k}\right), \mathfrak{\Im}\left(\gamma_{j, k}\right), j>k\right.$, for $\left.j, k=1, \ldots, m\right}$, we have
$$\gamma\left(\omega_{\ell}\right)=a\left(\omega_{\ell}\right)+e\left(\omega_{\ell}\right),$$
where $a\left(\omega_{\ell}\right)=\mathrm{E}\left{\gamma\left(\omega_{\ell}\right)\right}$, and the $e\left(\omega_{\ell}\right), \ell=1, \ldots, n$, are independent errors with zero means and the variances depending on the frequency point $\omega_{\ell} \cdot a(\cdot)$ is periodic and is fitted by periodic smoothing spline (Wahba, 1990) of the form,
$$a(\omega)=c_0+\sum_{\nu=1}^{n / 2-1} c_\nu \sqrt{2} \cos (2 \pi \nu \omega)+\sum_{\nu=1}^{n / 2-1} d_\nu \sqrt{2} \sin (2 \pi \nu \omega)+c_{n / 2} \cos (\pi n \omega)$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Penalized Whittle likelihood

Penalized methods have been one of the most popular research topics for the past decade. In the area of spectrum estimation, Pawitan and O’Sullivan (1994) developed penalized likelihood method for the nonparametric estimation of the power spectrum of a univariate time series. Pawitan (1996) developed a penalized Whittle likelihood estimator for a bivariate time series. However, their approach cannot be easily generalized to higher dimension. More recently, Krafty and Collinge (2013) proposed a penalized Whittle likelihood method to estimate the power spectrum of a vector-valued time series. The method allows for varying levels of smoothness among spectral components while accounting for the positive definiteness of spectral matrices and the Hermitian and periodic structures of power spectra as functions of frequency. In this section, we briefly discuss the method by Krafty and Collinge (2013).
Recall that $\widetilde{\mathbf{Y}}{\ell}=\widetilde{\mathbf{Y}}\left(\omega{\ell}\right)$ are discrete Fourier transform of the time series, and define the negative log Whittle likelihood as
$$L(\mathbf{f})=\sum_{\ell=1}^L\left{\log \left|\mathbf{f}\left(\omega_{\ell}\right)\right|+\widetilde{\mathbf{Y}}{\ell}^\left[\mathbf{f}\left(\omega{\ell}\right)\right]^{-1} \widetilde{\mathbf{Y}}{\ell}\right},$$ where $\omega{\ell}=\ell / n$ and $L=[(n-1) / 2]$. Based on $L$, the method penalizes the roughness of the estimated spectrum. The spectral matrix $\mathbf{f}(\omega)$ is modeled by the Cholesky decomposition such that $\mathbf{f}(\omega)=\left[\mathbf{\Gamma} \boldsymbol{\Gamma}^\right]^{-1}$, where $\boldsymbol{\Gamma}$ is a $m \times m$ lower triangular matrix with real-valued diagonal elements. Further, denote $\boldsymbol{\Gamma}{i, j, R}(\omega ; \mathbf{f})$ and $\boldsymbol{\Gamma}{i, j, l}(\omega ; \mathbf{f})$ as the real and imaginary parts of the $(i, j)$ element of $\boldsymbol{\Gamma}$. The method proposes a measure of roughness of a power spectrum through the integrated squared $k$ th derivatives of the $m(m+1) / 2$ real and $m(m-1) / 2$ imaginary components of the Cholesky decomposition. Suppose the smoothing parameters, $\lambda=\left{\rho_{i, j}\right.$, $\left.\theta_{i, j}: i \leq j=1, \ldots, m\right}$, of $\boldsymbol{\Gamma}{i, j, R}$ and $\boldsymbol{\Gamma}{i, j, I}$ that control the roughness penalty are $\rho_{i, j}>0$ and $\theta_{i, j}$ $>0$, the roughness measure for a spectrum $\tilde{\mathbf{f}}(\omega)$ is
$$J_\lambda(\mathbf{f})=\sum_{i \leq j=1}^m \rho_{i, j} \int_0^{1 / 2}\left{\boldsymbol{\Gamma}{i, j, R}^{(k)}(\omega ; \mathbf{f})\right}^2 d \omega+\sum{i<j=1}^m \theta_{i, j} \int_0^{1 / 2}\left{\boldsymbol{\Gamma}{i, j, I}^{(k)}(\omega ; \mathbf{f})\right}^2 d \omega .$$ The interpretation is that the penalty function $J$ shrinks the estimates of power spectra toward real-valued matrix functions that are constant across frequency. Consequently, we consider minimizing the penalized Whittle negative loglikelihood $$Q\lambda(\mathbf{f})=L(\mathbf{f})+J_\lambda(\mathbf{f}) .$$

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|Smoothing spline

$$\gamma\left(\omega_{\ell}\right)=a\left(\omega_{\ell}\right)+e\left(\omega_{\ell}\right),$$

$$a(\omega)=c_0+\sum_{\nu=1}^{n / 2-1} c_\nu \sqrt{2} \cos (2 \pi \nu \omega)+\sum_{\nu=1}^{n / 2-1} d_\nu \sqrt{2} \sin (2 \pi \nu \omega)+c_{n / 2} \cos (\pi n \omega)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 统计代写|时间序列分析代写Time-Series Analysis代考|Time-varying autoregressive model

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Time-varying autoregressive model

One of the methods used in Dahlhaus (2000) is the time-varying VARMA $(p, q)$ model. For a vector autoregressive model $\operatorname{VAR}(p)$, it is defined by
$$\mathbf{Z}t=\boldsymbol{\mu}(t / n)+\sum{j=1}^p \boldsymbol{\Phi}j(t / n)\left[\mathbf{Z}{t-j}-\boldsymbol{\mu}[(t-j)]+\boldsymbol{\Sigma}(t / n) \boldsymbol{\varepsilon}t, t=1, \ldots, n,\right.$$ where $\boldsymbol{\varepsilon}_t$ are $m$-dimensional independent random variable with mean zero and unit variance $\mathbf{I}$. In addition, we assume some smoothness conditions on $\boldsymbol{\Sigma}(\cdot)$ and $\boldsymbol{\Phi}_j(\cdot)$. In some neighborhood of a fixed time point $u_0=t_0 / n$, the process $\mathbf{Z}_t$ can be approximated by the stationary process $\mathbf{Z}_t\left(u_0\right)$ given by $$\mathbf{Z}_t\left(u_0\right)=\boldsymbol{\mu}\left(u_0\right)+\sum{j=1}^p \boldsymbol{\Phi}j\left(u_0\right)\left[\mathbf{Z}{t-j}\left(u_0\right)-\boldsymbol{\mu}\left(u_0\right)+\boldsymbol{\Sigma}\left(u_0\right) \boldsymbol{\varepsilon}_t, t=1, \ldots, n .\right.$$
$\mathbf{Z}_t$ has a unique time-varying power spectrum, which is locally the same as the power spectrum of $\mathbf{Z}_t(u)$, that is,
$$\mathbf{f}(u, \omega)=\left[\boldsymbol{\Phi}\left(u, e^{-i \omega}\right)\right]^{-1} \boldsymbol{\Sigma}(u)\left{\left[\boldsymbol{\Phi}\left(u, e^{-i \omega}\right)\right]^{-1}\right}^*,$$

where $u=t / n$, and
$$\boldsymbol{\Phi}(u, B)=\left(\mathbf{I}-\boldsymbol{\Phi}1(u, B)-\cdots-\boldsymbol{\Phi}_p\left(u, B^p\right)\right) .$$ Similarly, the locally covariance matrix is $$\boldsymbol{\Gamma}(u, j)=\int{-\pi}^\pi \mathbf{f}(u, \omega) \exp (j i \omega) d \omega .$$
Based on this statement, the time-varying spectrum can be obtained by estimating the timevarying parameters of the VAR model. For more properties of the estimation based on timevarying VARMA $(p, q), \operatorname{VAR}(p)$, and $\operatorname{VMA}(q)$ models, we refer readers to Dahlhaus (2000).

## 统计代写|时间序列分析代写Time-Series Analysis代考|Smoothing spline ANOVA model

Based on the locally stationary process, the time-varying spectrum can also be estimated nonparametrically via the smoothing spline Analysis of Variance (ANOVA) model by Guo and Dai (2006). However, their definition of locally stationary process is slightly different from Dahlhaus (2000). In Section 9.5.2, we mentioned that Dahlhaus (2000) assumes a series of transfer functions $\mathbf{A}^0(t / n, \omega)$ that converge to a large-sample transfer function $\mathbf{A}(u, \omega)$ in order to allow for the fitting of parametric models. Since Guo and Dai (2006) considered a nonparametric estimation, they used $\mathbf{A}(u, \omega)$ directly.

Definition 9.2 Without loss of generality, the $m$-dimensional zero-mean time series of length $n,\left{\mathbf{Z}t: t=1, \ldots, n\right}$, is called locally stationary if $$\mathbf{Z}_t=\int{-\pi}^\pi \mathbf{A}(t / n, \omega) \exp (i \omega t) d \mathbf{U}(\omega),$$
where we assume that the cumulants of $d \mathbf{U}(\omega)$ exists and are bounded for all orders. For the details, please see Brillinger (2002), and Guo and Dai (2006).

Based on this definition, the smoothing ANOVA model takes a two-stage estimation procedure. At the first stage, the locally stationary process is approximated by piecewise stationary time series with small blocks to obtain initial spectrum estimates and the Cholesky decomposition. The initial spectrum estimates are obtained by the multitaper method to reduce variance. At the second stage, each element of the Cholesky decomposition is treated as a bivariate smooth function of time and frequency and is modeled by the smoothing spline ANOVA model by Gu and Wahba (1993). The final estimated time-varying spectrum is reconstructed from the smoothed elements of the Cholesky decomposition. Thus, the method provides a way to ensure the final estimate of the multivariate spectrum is positive-definite while allowing enough flexibility in the smoothness of its elements.

We shall briefly discuss the smoothing spline ANOVA step. Suppose the spectral matrix $\widetilde{\mathbf{f}}(u, \omega)$ has the Cholesky decomposition such that $\widetilde{\mathbf{f}}(u, \omega)=\mathbf{L}(u, \omega) \mathbf{L}(u, \omega)^*$, where $\mathbf{L}(u, \omega)$ is a $m \times m$ lower triangular matrix. The method smooths the diagonal elements $\gamma_{j, j}(u, \omega), j=1, \ldots, m$, the real part of $\gamma_{j, k}(u, \omega), \mathfrak{R}\left{\gamma_{j, k}(u, \omega)\right}$, and the imaginary part of $\gamma_{j, k}(u, \omega), \mathfrak{J}\left{\gamma_{j, k}(u, \omega)\right}$, for $j>k$ separately with their own smoothing parameters. Let $\gamma(u, \omega) \in\left{\gamma_{j, j}(u, \omega), \mathfrak{R}\left(\gamma_{j, k}\right)(u, \omega), \mathfrak{\Im}\left(\gamma_{j, k}\right)\right.$ $(u, \omega), j>k$, for $j, k=1, \ldots, m}$. We have
$$\gamma(u, \omega)=a(u, \omega)+\varepsilon(u, \omega),$$where $a(u, \omega)$ is the corresponding Cholesky decomposition element of the spectrum, such that $a(u, \omega)=E{\gamma(u, \omega)}$, the $\varepsilon(u, \omega)$ are independent errors with zero-mean and the variance depending on the time-frequency point $(u, \omega)$.

# 时间序列分析代考

## 统计代写|时间序列分析代写Time-Series Analysis代考|Time-varying autoregressive model

Dahlhaus(2000)使用的方法之一是时变VARMA $(p, q)$模型。对于向量自回归模型$\operatorname{VAR}(p)$，定义为
$$\mathbf{Z}t=\boldsymbol{\mu}(t / n)+\sum{j=1}^p \boldsymbol{\Phi}j(t / n)\left[\mathbf{Z}{t-j}-\boldsymbol{\mu}[(t-j)]+\boldsymbol{\Sigma}(t / n) \boldsymbol{\varepsilon}t, t=1, \ldots, n,\right.$$式中$\boldsymbol{\varepsilon}_t$为$m$维独立随机变量，均值为零，单位方差为$\mathbf{I}$。此外，我们假设在$\boldsymbol{\Sigma}(\cdot)$和$\boldsymbol{\Phi}_j(\cdot)$上有一些平滑条件。在一个固定时间点$u_0=t_0 / n$的邻域内，过程$\mathbf{Z}_t$可以近似为由$$\mathbf{Z}_t\left(u_0\right)=\boldsymbol{\mu}\left(u_0\right)+\sum{j=1}^p \boldsymbol{\Phi}j\left(u_0\right)\left[\mathbf{Z}{t-j}\left(u_0\right)-\boldsymbol{\mu}\left(u_0\right)+\boldsymbol{\Sigma}\left(u_0\right) \boldsymbol{\varepsilon}_t, t=1, \ldots, n .\right.$$给出的平稳过程$\mathbf{Z}_t\left(u_0\right)$
$\mathbf{Z}_t$具有唯一的时变功率谱，它与$\mathbf{Z}_t(u)$的功率谱局部相同，即:
$$\mathbf{f}(u, \omega)=\left[\boldsymbol{\Phi}\left(u, e^{-i \omega}\right)\right]^{-1} \boldsymbol{\Sigma}(u)\left{\left[\boldsymbol{\Phi}\left(u, e^{-i \omega}\right)\right]^{-1}\right}^*,$$

$$\boldsymbol{\Phi}(u, B)=\left(\mathbf{I}-\boldsymbol{\Phi}1(u, B)-\cdots-\boldsymbol{\Phi}_p\left(u, B^p\right)\right) .$$同样，局部协方差矩阵为$$\boldsymbol{\Gamma}(u, j)=\int{-\pi}^\pi \mathbf{f}(u, \omega) \exp (j i \omega) d \omega .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。