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

The $m$-dimensional vector autoregressive moving average (VARMA) process or model of orders $p$ and $q$, shortened to $\operatorname{VARMA}(p, q)$ is given by
$$\mathbf{Z}t=\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q},$$

or
$$\boldsymbol{\Phi}p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\boldsymbol{\Theta}_q(B) \mathbf{a}_t$$ where $\mathbf{a}_t$ is a sequence of $m$-dimensional vector white noise process, $\operatorname{VWN}(\mathbf{0}, \mathbf{\Sigma})$, and \begin{aligned} & \boldsymbol{\Phi}_p(B)=\mathbf{I}-\boldsymbol{\Phi}_1 B-\cdots-\boldsymbol{\Phi}_p B^p, \ & \boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\cdots-\boldsymbol{\Theta}_q B^q . \end{aligned} The model is stationary if the zeros of the determinant polynomial $\left|\boldsymbol{\Phi}_p(B)\right|$ are all outside of the unit circle so that \begin{aligned} \dot{\mathbf{Z}}_t & =\left[\boldsymbol{\Phi}_p(B)\right]^{-1} \boldsymbol{\Theta}_q(B) \mathbf{a}_t \ & =\sum{j=0}^{\infty} \boldsymbol{\Psi}j \mathbf{a}{t-j} \end{aligned}
The model is invertible if zeros of the determinant polynomial $\left|\boldsymbol{\Theta}q(B)\right|$ are all outside of the unit circle so that $$\boldsymbol{\Pi}(B) \dot{\mathbf{Z}}_t=\mathbf{a}_t$$ where $$\boldsymbol{\Pi}(B)=\left[\boldsymbol{\Theta}_q(B)\right]^{-1} \boldsymbol{\Phi}_p(B)=\mathbf{I}-\sum{j=0}^{\infty} \boldsymbol{\Pi}_j B^j$$

## 统计代写|时间序列分析代写Time-Series Analysis代考|Nonstationary vector autoregressive moving average processes

In univariate time series analysis, a nonstationary time series is reduced to a stationary time series by proper power transformations and differencing. These can still be used in vector time series analysis. However, it should be noted that these transformations should be applied to a component series individually because not all component series can be reduced to stationary by exactly the same power transformation and the same number of differencing. To be more flexible, after using proper power transformations to a component series, we will use the following presentation for a nonstationary vector time series model
$$\boldsymbol{\Phi}_p(B) \mathbf{D}(B) \mathbf{Z}_t=\boldsymbol{\Theta}_q(B) \mathbf{a}_t$$
where
$$\mathbf{D}(B)=\left[\begin{array}{cccccc} (1-B)^{d_1} & 0 & . & \cdots & 0 & 0 \ 0 & (1-B)^{d_2} & 0 & \cdots & . & 0 \ . & 0 & . & \cdots & . & . \ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \ 0 & \cdots & . & \cdots & . & 0 \ 0 & 0 & . & \cdots & 0 & (1-B)^{d_m} \end{array}\right],$$
and the zeros of $\left|\boldsymbol{\Phi}_p(B)\right|$ and $\left|\Theta_q(B)\right|$ are outside of the unit circle. The unit root test introduced by Dickey and Fuller (1979) can be used to determine the order of $d_i$. For more recent references on a unit root, see Teles et al. (2008), Chambers (2015), Cavaliere et al. (2015), and Hosseinkouchack and Hassler (2016), among others.

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

$$\mathbf{Z}t=\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\cdots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\cdots-\boldsymbol{\Theta}q \mathbf{a}{t-q},$$

$$\boldsymbol{\Phi}p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\boldsymbol{\Theta}_q(B) \mathbf{a}_t$$其中$\mathbf{a}_t$是$m$维矢量白噪声过程的序列，$\operatorname{VWN}(\mathbf{0}, \mathbf{\Sigma})$, \begin{aligned} & \boldsymbol{\Phi}_p(B)=\mathbf{I}-\boldsymbol{\Phi}_1 B-\cdots-\boldsymbol{\Phi}_p B^p, \ & \boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\cdots-\boldsymbol{\Theta}_q B^q . \end{aligned}如果行列式多项式$\left|\boldsymbol{\Phi}_p(B)\right|$的零都在单位圆之外，则模型是平稳的，因此\begin{aligned} \dot{\mathbf{Z}}_t & =\left[\boldsymbol{\Phi}_p(B)\right]^{-1} \boldsymbol{\Theta}_q(B) \mathbf{a}_t \ & =\sum{j=0}^{\infty} \boldsymbol{\Psi}j \mathbf{a}{t-j} \end{aligned}

## 统计代写|时间序列分析代写Time-Series Analysis代考|Nonstationary vector autoregressive moving average processes

$$\boldsymbol{\Phi}_p(B) \mathbf{D}(B) \mathbf{Z}_t=\boldsymbol{\Theta}_q(B) \mathbf{a}_t$$

$$\mathbf{D}(B)=\left[\begin{array}{cccccc} (1-B)^{d_1} & 0 & . & \cdots & 0 & 0 \ 0 & (1-B)^{d_2} & 0 & \cdots & . & 0 \ . & 0 & . & \cdots & . & . \ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \ 0 & \cdots & . & \cdots & . & 0 \ 0 & 0 & . & \cdots & 0 & (1-B)^{d_m} \end{array}\right],$$
$\left|\boldsymbol{\Phi}_p(B)\right|$和$\left|\Theta_q(B)\right|$的零点在单位圆外。Dickey和Fuller(1979)引入的单位根检验可以用来确定$d_i$的顺序。有关单位根的最新参考文献，请参见Teles等人(2008)、Chambers(2015)、Cavaliere等人(2015)、Hosseinkouchack和Hassler(2016)等。

## MATLAB代写

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