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# 统计代写|时间序列分析代写Time Series Analysis代考|Vector moving average processes

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## 统计代写|时间序列分析代写Time Series Analysis代考|Vector moving average processes

The $m$-dimensional vector moving average process or model in the order of $q$, shortened to $\operatorname{VMA}(q)$, is given by
\begin{aligned} \mathbf{Z}t & =\boldsymbol{\mu}+\mathbf{a}_t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q} \ & =\boldsymbol{\mu}+\boldsymbol{\Theta}q(B) \mathbf{a}_t, \end{aligned} where $\boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\cdots-\boldsymbol{\Theta}_q B^q, \mathbf{a}_t$ is a sequence of the $m$-dimensional vector white noise process, VWN $(\mathbf{0}, \mathbf{\Sigma})$, with mean vector, $\mathbf{0}$, and covariance matrix function $$E\left(\mathbf{a}{\mathbf{t}} \mathbf{a}_{t+k}^{\prime}\right)=\left{\begin{array}{l} \boldsymbol{\Sigma}, \text { if } k=0 \ \mathbf{0}, \text { if } k \neq 0 \end{array}\right.$$

and $\boldsymbol{\Sigma}$ is a $m \times m$ symmetric positive-definite matrix. The $\operatorname{VMA}(q)$ model is clearly stationary with the mean vector,
$$E\left(\mathbf{Z}t\right)=\boldsymbol{\mu},$$ and covariance matrix function, \begin{aligned} \boldsymbol{\Gamma}(k) & =E\left[\left(\mathbf{Z}_t-\boldsymbol{\mu}\right)\left(\mathbf{Z}{t+k}-\boldsymbol{\mu}\right)^{\prime}\right] \ & =E\left[\left(\mathbf{a}t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q}\right)\left(\mathbf{a}{t+k}-\boldsymbol{\Theta}_1 \mathbf{a}{t+k-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t+k-q}\right)^{\prime}\right] \ & =\left{\begin{array}{cc} \sum_{j=0}^{q-k} \boldsymbol{\Theta}j \mathbf{\Sigma} \mathbf{\Theta}{j+k}^{\prime}, & \text { for } k=0,1, \ldots, q, \ \mathrm{O}, & k>q, \end{array}\right. \end{aligned}
where $\boldsymbol{\Theta}_0=\mathbf{I}$ and $\boldsymbol{\Gamma}(-k)=\boldsymbol{\Gamma}^{\prime}(k)$. Thus, $\boldsymbol{\Gamma}(k)$ cuts off after lag $q$.

## 统计代写|时间序列分析代写Time Series Analysis代考|Vector autoregressive processes

The $m$-dimensional vector autoregressive process or model of order $p$, $\operatorname{shortened}$ to $\operatorname{VAR}(p)$, is given by
$$\mathbf{Z}t=\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}t$$ or $$\boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\mathbf{a}_t$$ where $\mathbf{a}_t$ is a sequence of $m$-dimensional vector white noise process, $\operatorname{VWN}(\mathbf{0}, \boldsymbol{\Sigma})$, and $$\boldsymbol{\Phi}_p(\boldsymbol{B})=\mathbf{I}-\boldsymbol{\Phi}_1 B-\cdots-\boldsymbol{\Phi}_p B^p .$$ The model is clearly invertible. It will be stationary if the zeros of $\left|\mathbf{I}-\mathbf{\Phi}_1 B-\cdots-\mathbf{\Phi}_p B^p\right|$ lie outside of the unit circle or equivalently, the roots of $$\left|\lambda^p \mathbf{I}-\lambda^{p-1} \boldsymbol{\Phi}_1-\cdots-\boldsymbol{\Phi}_p\right|=0$$ are all inside the unit circle. In this case, its mean is a constant vector, $E\left(\mathbf{Z}_t\right)=\boldsymbol{\mu}$, which can be found by noting that \begin{aligned} \boldsymbol{\mu} & =E\left(\mathbf{Z}_t\right)=E\left(\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}_t\right) \ & =\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \boldsymbol{\mu}+\cdots+\boldsymbol{\Phi}_p \boldsymbol{\mu} \end{aligned}
and hence
$$\boldsymbol{\mu}=\left(\mathbf{I}-\boldsymbol{\Phi}_1-\cdots-\boldsymbol{\Phi}_p\right)^{-1} \boldsymbol{\theta}_0$$

## 统计代写|时间序列分析代写Time Series Analysis代考|Vector moving average processes

$m$维向量移动平均过程或$q$顺序的模型，简称为$\operatorname{VMA}(q)$，由
\begin{aligned} \mathbf{Z}t & =\boldsymbol{\mu}+\mathbf{a}_t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q} \ & =\boldsymbol{\mu}+\boldsymbol{\Theta}q(B) \mathbf{a}_t, \end{aligned}给出，其中$\boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\cdots-\boldsymbol{\Theta}_q B^q, \mathbf{a}_t$是$m$维向量白噪声过程VWN $(\mathbf{0}, \mathbf{\Sigma})$的序列，具有均值向量$\mathbf{0}$和协方差矩阵函数$$E\left(\mathbf{a}{\mathbf{t}} \mathbf{a}_{t+k}^{\prime}\right)=\left{\begin{array}{l} \boldsymbol{\Sigma}, \text { if } k=0 \ \mathbf{0}, \text { if } k \neq 0 \end{array}\right.$$

$$E\left(\mathbf{Z}t\right)=\boldsymbol{\mu},$$和协方差矩阵函数，\begin{aligned} \boldsymbol{\Gamma}(k) & =E\left[\left(\mathbf{Z}_t-\boldsymbol{\mu}\right)\left(\mathbf{Z}{t+k}-\boldsymbol{\mu}\right)^{\prime}\right] \ & =E\left[\left(\mathbf{a}t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q}\right)\left(\mathbf{a}{t+k}-\boldsymbol{\Theta}_1 \mathbf{a}{t+k-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t+k-q}\right)^{\prime}\right] \ & =\left{\begin{array}{cc} \sum_{j=0}^{q-k} \boldsymbol{\Theta}j \mathbf{\Sigma} \mathbf{\Theta}{j+k}^{\prime}, & \text { for } k=0,1, \ldots, q, \ \mathrm{O}, & k>q, \end{array}\right. \end{aligned}

## 统计代写|时间序列分析代写Time Series Analysis代考|Vector autoregressive processes

$m$维向量自回归过程或$p$, $\operatorname{shortened}$到$\operatorname{VAR}(p)$的模型由
$$\mathbf{Z}t=\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}t$$或$$\boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\mathbf{a}_t$$给出，其中$\mathbf{a}_t$是$m$维向量白噪声过程，$\operatorname{VWN}(\mathbf{0}, \boldsymbol{\Sigma})$和$$\boldsymbol{\Phi}_p(\boldsymbol{B})=\mathbf{I}-\boldsymbol{\Phi}_1 B-\cdots-\boldsymbol{\Phi}_p B^p .$$的序列，模型明显可逆。它是平稳的如果$\left|\mathbf{I}-\mathbf{\Phi}_1 B-\cdots-\mathbf{\Phi}_p B^p\right|$的零点在单位圆外或者等价地，$$\left|\lambda^p \mathbf{I}-\lambda^{p-1} \boldsymbol{\Phi}_1-\cdots-\boldsymbol{\Phi}_p\right|=0$$的根都在单位圆内。在这种情况下，它的平均值是一个常数向量$E\left(\mathbf{Z}_t\right)=\boldsymbol{\mu}$，可以通过注意\begin{aligned} \boldsymbol{\mu} & =E\left(\mathbf{Z}_t\right)=E\left(\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}_t\right) \ & =\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \boldsymbol{\mu}+\cdots+\boldsymbol{\Phi}_p \boldsymbol{\mu} \end{aligned}

$$\boldsymbol{\mu}=\left(\mathbf{I}-\boldsymbol{\Phi}_1-\cdots-\boldsymbol{\Phi}_p\right)^{-1} \boldsymbol{\theta}_0$$

## Matlab代写

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