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# 统计代写|时间序列分析代写Time-Series Analysis代考|Seasonal vector time series model

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## 统计代写|时间序列分析代写Time-Series Analysis代考|Seasonal vector time series model

The $\operatorname{VARMA}(p, q)$ model in Eq. (2.42) can be extended to a seasonal vector model that contains both seasonal and non-seasonal AR and MA polynomials as follows,
$$\boldsymbol{\alpha}P\left(B^s\right) \boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\boldsymbol{\beta}_Q\left(B^s\right) \boldsymbol{\Theta}_q(B) \mathbf{a}_t$$ where \begin{aligned} & \boldsymbol{\alpha}_P\left(B^s\right)=\mathbf{I}-\boldsymbol{\alpha}_1 B^s-\cdots-\boldsymbol{\alpha}_P B^{P s}=\mathbf{I}-\sum{k=1}^P \boldsymbol{\alpha}k B^{k s}, \ & \boldsymbol{\beta}_Q\left(B^s\right)=\mathbf{I}-\boldsymbol{\beta}_1 B^s-\cdots-\boldsymbol{\beta}_Q B^{Q s}=\mathbf{I}-\sum{k=1}^Q \boldsymbol{\beta}_k B^{k s}, \end{aligned}
and $s$ is a seasonal period. $\boldsymbol{\alpha}_k$ and $\boldsymbol{\beta}_k$ are seasonal AR and seasonal MA parameters, respectively. $P$ is the seasonal AR order, $Q$ is the seasonal MA order. The zeros of $\left|\boldsymbol{\alpha}_P(B)\right|$ and $\left|\boldsymbol{\beta}_Q(B)\right|$ are outside of the unit circle. For simplicity, we will denote the seasonal vector time series model in Eq. (2.78) with a seasonal period $s$ as $\operatorname{VARMA}(p, q) \times(P, Q)_s$.

It is important to point out that for a univariate time series, the polynomials in Eq. (2.78) are scalar and they are commutative. For example, for a univariate seasonal AR model, the two representations,
$$\alpha_P\left(B^s\right) \phi_p(B) Z_t=a_t, \text { and } \phi_p(B) \alpha_P\left(B^s\right) Z_t=a_t,$$
are exactly the same. However, for a vector seasonal VAR model, the two representations,
$$\boldsymbol{\alpha}_P\left(B^s\right) \boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\mathbf{a}_t, \text { and } \boldsymbol{\Phi}_p(B) \boldsymbol{\alpha}_P\left(B^s\right) \mathbf{Z}_t=\mathbf{a}_t$$
are not the same, because the vector multiplications are not commutative. This leads to many complications in terms of model identification, parameter estimation, and forecasting. We refer readers to an interesting and excellent paper by Yozgatligil and Wei (2009) for more details and examples.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Multivariate time series outliers

In time series analysis, it is important to examine the possible outliers and do some proper adjustments because outliers can lead to inaccurate parameter estimation, model misspecification, and poor forecasts. Outlier detection has been studied extensively for univariate time series, including Fox (1972), Abraham and Box (1979), Martin (1980), Hillmer et al. (1983), Chang et al. (1988), Tsay (1986, 1988), Chen and Liu (1993), Lee and Wei (1995), Wang et al. (1995), Sanchez and Pena (2003), and many others. We normally classify outliers in four categories, additive outliers, innovational outliers, level shifts, and temporal changes. For MTS, a natural approach is first to use univariate techniques to the individual component and remove outliers, then treat the adjusted series as outlier-free and model them jointly. However, there are several disadvantages of this approach. First, in MTS an outlier of its univariate component may be induced by an outlier from other component within the multivariate series. Overlooking this situation may lead to overspecification of the number of outliers. Second, an outlier impacting all the components may not be detected by using the univariate outlier detection methods because they do not use the joint information from all time series components in the system at the same time.

To overcome the difficulties, Tsay et al. (2000) extended four types of outliers for univariate time series to MTS and their detections, and Galeano et al. (2006) further proposed a detection method based on projection pursuit, which sometime is more powerful than testing the multivariate series directly. Other references on MTS outliers include Helbing and Cleroux (2009), Martinez-Alvarez et al. (2011), and Cucina et al. (2014), among others.

## 统计代写|时间序列分析代写Time-Series Analysis代考|Seasonal vector time series model

Eq.(2.42)中的$\operatorname{VARMA}(p, q)$模型可以扩展为包含季节性和非季节性AR和MA多项式的季节性向量模型，如下所示:
$$\boldsymbol{\alpha}P\left(B^s\right) \boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\boldsymbol{\theta}_0+\boldsymbol{\beta}_Q\left(B^s\right) \boldsymbol{\Theta}_q(B) \mathbf{a}_t$$ where \begin{aligned} & \boldsymbol{\alpha}_P\left(B^s\right)=\mathbf{I}-\boldsymbol{\alpha}_1 B^s-\cdots-\boldsymbol{\alpha}_P B^{P s}=\mathbf{I}-\sum{k=1}^P \boldsymbol{\alpha}k B^{k s}, \ & \boldsymbol{\beta}_Q\left(B^s\right)=\mathbf{I}-\boldsymbol{\beta}_1 B^s-\cdots-\boldsymbol{\beta}_Q B^{Q s}=\mathbf{I}-\sum{k=1}^Q \boldsymbol{\beta}_k B^{k s}, \end{aligned}
$s$是季节性的。其中$\boldsymbol{\alpha}_k$为季节性AR参数，$\boldsymbol{\beta}_k$为季节性MA参数。$P$为季节性AR订单，$Q$为季节性MA订单。$\left|\boldsymbol{\alpha}_P(B)\right|$和$\left|\boldsymbol{\beta}_Q(B)\right|$的零点在单位圆外。为简单起见，我们将Eq.(2.78)中的季节向量时间序列模型表示为季节性周期$s$为$\operatorname{VARMA}(p, q) \times(P, Q)_s$。

$$\alpha_P\left(B^s\right) \phi_p(B) Z_t=a_t, \text { and } \phi_p(B) \alpha_P\left(B^s\right) Z_t=a_t,$$

$$\boldsymbol{\alpha}_P\left(B^s\right) \boldsymbol{\Phi}_p(B) \mathbf{Z}_t=\mathbf{a}_t, \text { and } \boldsymbol{\Phi}_p(B) \boldsymbol{\alpha}_P\left(B^s\right) \mathbf{Z}_t=\mathbf{a}_t$$

## MATLAB代写

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