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

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

As shown in Chapter 2, the underlying vector time series process is often described by a vector autoregressive moving average (VARMA) model. Specifically, the $m$-dimensional VARMA process of order $p$ and $q, \operatorname{VARMA}(p, q)$ is given by
$$\mathbf{Z}t=\boldsymbol{\theta}_0+\boldsymbol{\Phi}_1 \mathbf{Z}{t-1}+\ldots+\boldsymbol{\Phi}p \mathbf{Z}{t-p}+\mathbf{a}t-\boldsymbol{\Theta}_1 \mathbf{a}{t-1}-\ldots-\boldsymbol{\Theta}q \mathbf{a}{t-q},$$
or
$$\boldsymbol{\Phi}_p(B) \dot{\mathbf{Z}}_t=\boldsymbol{\Theta}_q(B) \mathbf{a}_t$$
where $\dot{\boldsymbol{Z}}_t=\boldsymbol{Z}_t-\boldsymbol{\mu}, \mathbf{a}_t$ is a sequence of $m$-dimensional vector white noise process, VWN $(\mathbf{0}, \mathbf{\Sigma})$, and

\begin{aligned} & \boldsymbol{\Phi}p(B)=\mathbf{I}-\boldsymbol{\Phi}_1 B-\ldots-\boldsymbol{\Phi}_p B^p, \ & \boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\ldots-\boldsymbol{\Theta}_q B^q . \end{aligned} We assume that all zeros of $\left|\boldsymbol{\Phi}_p(B)\right|$ and $\left|\boldsymbol{\Theta}_q(B)\right|$ lie outside of the unit circle, so that we can also represent it as $$\dot{\mathbf{Z}}_t=\boldsymbol{\Psi}(B) \mathbf{a}_t,$$ where $\boldsymbol{\Psi}(B)=\left[\boldsymbol{\Phi}_p(B)\right]^{-1} \boldsymbol{\Theta}_q(B)=\sum{j=0}^{\infty} \boldsymbol{\Psi}j B^j$ and the sequence $\boldsymbol{\Psi}_j$ is square summable. The spectral density matrix of $\operatorname{VARMA}(p, q)$ model is given by $$\mathbf{f}(\omega)=\boldsymbol{\Psi}\left(e^{-i \omega}\right) \mathbf{\Sigma} \boldsymbol{\Psi}^\left(e^{-i \omega}\right),$$ where $\boldsymbol{\Psi}\left(e^{-i \omega}\right)=\sum{j=0}^{\infty} \boldsymbol{\Psi}j e^{-i \omega j}$, and $\boldsymbol{\Psi}^\left(e^{-i \omega}\right)$ is its conjugate transpose.
Given a vector time series of $n$ observations, $\mathbf{Z}=\left(\mathbf{Z}_1, \mathbf{Z}_2, \ldots, \mathbf{Z}_n\right)$, we will first build its $\operatorname{VARMA}(p, q)$ model including the estimation of its parameter matrices. Let $\hat{\boldsymbol{\mu}}=\overline{\mathbf{Z}}$, $\hat{\boldsymbol{\Phi}}_1, \ldots, \hat{\boldsymbol{\Phi}}_p, \hat{\boldsymbol{\Theta}}_1, \ldots, \hat{\boldsymbol{\Theta}}_q$ and $\hat{\boldsymbol{\Sigma}}$ be the corresponding estimates of the parameter matrices. We have
$$\widehat{\dot{\boldsymbol{Z}}}_t=\hat{\boldsymbol{\Psi}}(B) \boldsymbol{a}_t,$$
where $\hat{\dot{Z}}_t=\boldsymbol{Z}_t-\overline{\boldsymbol{Z}}, \hat{\boldsymbol{\Psi}}(B)=\sum{s=0}^{\infty} \hat{\mathbf{\Psi}}s B^s=\left[\mathbf{I}-\hat{\mathbf{\Phi}}_1 B-\ldots-\hat{\boldsymbol{\Phi}}_p B^p\right]^{-1}\left[\mathbf{I}-\hat{\boldsymbol{\Theta}}_1 B-\ldots-\hat{\boldsymbol{\Theta}}_q B^q\right]$, and $\mathbf{a}_t$ is a sequence of $m$-dimensional vector white noise, $\operatorname{VWN}(\mathbf{0}, \hat{\boldsymbol{\Sigma}})$. Then, the spectral density matrix estimation of the underlying process is given by $$\hat{\mathbf{f}}(\omega)=\hat{\boldsymbol{\Psi}}\left(e^{-i \omega}\right) \hat{\mathbf{\Sigma}} \hat{\boldsymbol{\Psi}}^*\left(e^{-i \omega}\right),$$ where \begin{aligned} \hat{\boldsymbol{\Psi}}\left(e^{-i \omega}\right) & =\sum{j=0}^{\infty} \hat{\boldsymbol{\Psi}}_j e^{-i \omega j} \ & =\left[\hat{\boldsymbol{\Phi}}_p\left(e^{-i \omega}\right)\right]^{-1} \hat{\boldsymbol{\Theta}}_q\left(e^{-i \omega}\right) \ & =\left[\left(\mathbf{I}-\hat{\boldsymbol{\Phi}}_1 e^{-i \omega}-\ldots-\hat{\boldsymbol{\Phi}}_p e^{-i \omega p}\right)\right]^{-1}\left[\left(\mathbf{I}-\hat{\boldsymbol{\Theta}}_1 e^{-i \omega}-\ldots-\hat{\boldsymbol{\Theta}}_q e^{-i \omega q}\right)\right] . \end{aligned}

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

Example 9.3(a) Based on the given five-dimensional vector time series $\mathbf{Z}t=\left[Z{1, t}, Z_{2, t}, \ldots\right.$, $\left.Z_{5, t}\right]^{\prime}$ for $t=1,2, \ldots, 89$, we can compute its sample covariance matrix function, $\hat{\boldsymbol{\Gamma}}(k)=\left[\hat{\gamma}{i, j}(k)\right]$. Then, the sample spectral matrix is given by $$\widetilde{\mathbf{f}}\left(\omega_j\right)=\frac{1}{2 \pi} \sum{k=-88}^{88} \hat{\boldsymbol{\Gamma}}(k) e^{-i \omega_j k}=\left[\widetilde{f}{i, j}\left(\omega_j\right)\right],$$ where the $\omega_j=2 \pi j / 89$ are Fourier frequencies. Let us first examine the sample spectrum of each series, which as shown in Wei (2006) and Priestley (1981) can be used to test hidden periodic components. They are displayed in Figure 9.2 and they are noisy. So, we will compute a smoothed kernel sample spectrum matrix $$\hat{\mathbf{f}}(\omega)=\left[\hat{f}{i, j}(\omega)\right]$$
where
$$\hat{f}{i, j}\left(\omega_p\right)=\sum{k=-M}^M W\left(\omega_k\right) \widetilde{f}{i, j}\left(\omega_p-\omega_k\right),$$ where $W(\omega)$ is a spectral window, and $M$ is the corresponding bandwidth. Let us simply consider Daniell’s window, that is, $$\hat{\mathbf{f}}\left(\omega_p\right)=\frac{1}{2 M+1} \sum{k=-M}^M \widetilde{\mathbf{f}}\left(\omega_p-\omega_k\right)=\left[\hat{f}_{i, j}\left(\omega_p\right)\right],$$
where
$$\hat{f}{i, j}\left(\omega_p\right)=\frac{1}{2 M+1} \sum{k=-M}^M \widetilde{f}_{i, j}\left(\omega_p-\omega_k\right) .$$

# 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|VARMA spectral estimation

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

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

\begin{aligned} & \boldsymbol{\Phi}p(B)=\mathbf{I}-\boldsymbol{\Phi}_1 B-\ldots-\boldsymbol{\Phi}_p B^p, \ & \boldsymbol{\Theta}_q(B)=\mathbf{I}-\boldsymbol{\Theta}_1 B-\ldots-\boldsymbol{\Theta}_q B^q . \end{aligned} 我们假设$\left|\boldsymbol{\Phi}_p(B)\right|$和$\left|\boldsymbol{\Theta}_q(B)\right|$的所有零都在单位圆之外，因此我们也可以将其表示为$$\dot{\mathbf{Z}}_t=\boldsymbol{\Psi}(B) \mathbf{a}_t,$$，其中$\boldsymbol{\Psi}(B)=\left[\boldsymbol{\Phi}_p(B)\right]^{-1} \boldsymbol{\Theta}_q(B)=\sum{j=0}^{\infty} \boldsymbol{\Psi}j B^j$和序列$\boldsymbol{\Psi}_j$是平方可和的。$\operatorname{VARMA}(p, q)$模型的谱密度矩阵由$$\mathbf{f}(\omega)=\boldsymbol{\Psi}\left(e^{-i \omega}\right) \mathbf{\Sigma} \boldsymbol{\Psi}^\left(e^{-i \omega}\right),$$给出，其中$\boldsymbol{\Psi}\left(e^{-i \omega}\right)=\sum{j=0}^{\infty} \boldsymbol{\Psi}j e^{-i \omega j}$, $\boldsymbol{\Psi}^\left(e^{-i \omega}\right)$为其共轭转置。

$$\widehat{\dot{\boldsymbol{Z}}}_t=\hat{\boldsymbol{\Psi}}(B) \boldsymbol{a}_t,$$

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

$$\hat{f}{i, j}\left(\omega_p\right)=\sum{k=-M}^M W\left(\omega_k\right) \widetilde{f}{i, j}\left(\omega_p-\omega_k\right),$$ 在哪里 $W(\omega)$ 是光谱窗，又是 $M$ 对应的带宽。让我们简单地考虑丹尼尔的窗口，也就是说， $$\hat{\mathbf{f}}\left(\omega_p\right)=\frac{1}{2 M+1} \sum{k=-M}^M \widetilde{\mathbf{f}}\left(\omega_p-\omega_k\right)=\left[\hat{f}{i, j}\left(\omega_p\right)\right],$$ 在哪里 $$\hat{f}{i, j}\left(\omega_p\right)=\frac{1}{2 M+1} \sum{k=-M}^M \widetilde{f}{i, j}\left(\omega_p-\omega_k\right) .$$

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