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AR
Time Series Analysis
author：zoxiii

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【参考文献】王燕. 应用时间序列分析-第5版[M]. 中国人民大学出版社, 2019.

0-模型

AR(q)

$$\{\begin{array}{l}{x}_t={\varphi }_0+{\varphi }_1{x}_{t-1}+...+{\varphi }_p{x}_{t-p}+{\epsilon }_t\\ {\varphi }_p\ne 0\\ E({\epsilon }_t)=0,Var({\epsilon }_t)={\sigma }_{\epsilon }^2,E({\epsilon }_t{\epsilon }_s)=0,s\ne t\\ E(x_s{\epsilon }_t)=0,\forall s<t\end{array}$$

中心化AR(q)

$$x_t={\varphi }_1{x}_{t-1}+...+{\varphi }_p{x}_{t-p}+{\epsilon }_t$$

引入延迟算子B

$$\begin{gathered} x_t={\varphi }_1{x}_{t-1}+...+{\varphi }_p{x}_{t-p}+{\epsilon }_t \\ ={\varphi }_{1}B{x}_t+...+{\varphi }_p{B}^p{x}_t+{\epsilon }_t \\ =\Phi (B){\epsilon }_t \end{gathered}$$
得到q阶自回归系数多项式：
$$\Phi (B)=1-{\varphi }_{1}B-{\varphi }_2{B}^2-...-{\varphi }_p{B}^p$$

1-均值

$$\mu =\frac{{\varphi }_0}{1-{\varphi }_1-...-{\varphi }_p}$$

2-Green函数

$$\{\begin{array}{c}{G}_0=1\\ G_j={\sum }_{k=1}^j{\varphi }_k^{\prime }{G}_{j-k}\end{array}$$
其中：
$${\varphi }_k^{\prime }=\{\begin{array}{l}{\varphi }_k,k\le p\\ 0,k>p\end{array}$$

Green推导公式过程

$$x_t=\frac{{\epsilon }_t}{\Phi \left(B\right)}=G(B){\epsilon }_t$$

$$\Phi \left(B\right)G\left(B\right){\epsilon }_t={\epsilon }_t$$

$$\left(1-\sum _{k=1}^p\left({\varphi }_k{B}^k\right)\right)\left(\sum _{j=0}^{\infty }\left(G_j{B}^j\right)\right){\epsilon }_t={\epsilon }_t$$

$$\left(\sum _{j=0}^{\infty }{G}_j{B}^j-\sum _{k=1}^p\sum _{j=0}^{\infty }{\varphi }_k{B}^k{G}_j{B}^j\right){\epsilon }_t={\epsilon }_t$$

$$\left(G_0+\sum _{j=1}^{\infty }\left(G_j-\sum _{k=1}^j{{\varphi }_k}^{\prime }{G}_{j-k}\right)B_j\right){\epsilon }_t={\epsilon }_t$$

3-方差

$$Var(x_t)=\sum _{j=0}^{\infty }{G}_j^{2}Var({\epsilon }_{t-j})=\sum _{j=0}^{\infty }{G}_j^2{\sigma }_{\epsilon }^2$$
或者
$$Var(x_t)={\gamma }_0$$

4-延迟k协方差函数

AR(1)

$${\gamma }_k={\varphi }_1^k\frac{{\sigma }_{\epsilon }^2}{1-{\varphi }_1^2}$$

AR(2)

$$\{\begin{array}{c}{\gamma }_0=\frac{1-{\varphi }_2}{(1+{\varphi }_2)(1-{\varphi }_1-{\varphi }_2)(1+{\varphi }_1-{\varphi }_2)}{\sigma }_{\epsilon }^2\\ {\gamma }_1=\frac{{\varphi }_1}{1-{\varphi }_2}{\gamma }_0\\ {\gamma }_k={\varphi }_1{\gamma }_{k-1}+{\varphi }_2{\gamma }_{k-2}\end{array}$$

5-延迟k自相关系数

AR(1)

$${\rho }_k=\frac{{\gamma }_k}{{\gamma }_0}={\varphi }_1^k$$

AR(2)

$$\{\begin{array}{c}{\rho }_0=\frac{{\gamma }_0}{{\gamma }_0}=1\\ {\rho }_1=\frac{{\gamma }_1}{{\gamma }_0}=\frac{{\varphi }_1}{1-{\varphi }_2}\\ {\rho }_k=\frac{{\gamma }_k}{{\gamma }_0}={\varphi }_1{\rho }_{k-1}+{\varphi }_2{\rho }_{k-2}\end{array}$$

6-延迟k偏自相关系数

AR(1)

$$\{\begin{array}{c}{\varphi }_{11}=\frac{{\rho }_1}{{\rho }_0}\\ {\varphi }_{kk}=0,\forall k>1\end{array}$$

AR(2)

$$\{\begin{array}{c}{\varphi }_{11}=\frac{{\rho }_1}{{\rho }_0}=\frac{{\varphi }_1}{1-{\varphi }_2}\\ {\varphi }_{22}={\varphi }_2\\ {\varphi }_{kk}=0,\forall k>2\end{array}$$

7-AR模型平稳性判别(特征根+平稳域)

AR(1)

$$x_t={\varphi }_1{x}_{t-1}+{\epsilon }_t$$
特征方程$$\lambda -{\varphi }_1=0$$
特征根$$\lambda ={\varphi }_1$$
平稳充要条件：特征根在单位圆内，即$$|{\varphi }_1|<1$$
平稳域为 { ϕ 1 ∣ − 1 < ϕ 1 < 1 } \{\phi_1|-1<\phi_1<1\} {ϕ1​∣−1<ϕ1​<1}

AR(2)

$$x_t={\varphi }_1{x}_{t-1}+{\varphi }_2{x}_{t-2}+{\epsilon }_t$$
特征方程$${\lambda }^2-{\varphi }_1\lambda -{\varphi }_2=0$$
特征根$${\lambda }_1=\frac{{\varphi }_1+\sqrt{{\varphi }_1^2+4{\varphi }_2}}{2},{\lambda }_2=\frac{{\varphi }_1-\sqrt{{\varphi }_1^2+4{\varphi }_2}}{2}$$
平稳充要条件：特征根在单位圆内，即$$|{\lambda }_1|<1且|{\lambda }_2|<1$$
平稳域为 { ϕ 1 , ϕ 2 ∣ ∣ ϕ 2 ∣ < 1 且 ϕ 2 ± ϕ 1 < 1 } \{\phi_1,\phi_2||\phi_2|<1且\phi_2\pm\phi_1<1\} {ϕ1​,ϕ2​∣∣ϕ2​∣<1且ϕ2​±ϕ1​<1}