【时间序列分析】AR模型公式总结

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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)

{ x t = ϕ 0 + ϕ 1 x t − 1 + . . . + ϕ p x t − p + ε t ϕ p ≠ 0 E ( ε t ) = 0 , V a r ( ε t ) = σ ε 2 , E ( ε t ε s ) = 0 , s ≠ t E ( x s ε t ) = 0 , ∀ s < t

$$\begin{cases} x_t=\phi_0+\phi_1x_{t-1}+...+\phi_px_{t-p}+\varepsilon_t \\ \phi_p \neq 0\\ E(\varepsilon_t)=0,Var(\varepsilon_t)=\sigma_\varepsilon^2,E(\varepsilon_t\varepsilon_s)=0,s \neq t \\ E(x_s\varepsilon_t)=0,\forall s \lt t \end{cases}$$ ⎩⎪⎪⎪⎨⎪⎪⎪⎧​xt​=ϕ0​+ϕ1​xt−1​+...+ϕp​xt−p​+εt​ϕp​​=0E(εt​)=0,Var(εt​)=σε2​,E(εt​εs​)=0,s​=tE(xs​εt​)=0,∀s<t​

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

{ G 0 = 1 G j = ∑ k = 1 j ϕ k ′ G j − k \left \{

$$\begin{array}{c} G_0=1 \\ G_j=\sum_{k=1}^{j}{\phi_k'G_{j-k}} \end{array}$$ \right. {G0​=1Gj​=∑k=1j​ϕk′​Gj−k​​
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)

{ γ 0 = 1 − ϕ 2 ( 1 + ϕ 2 ) ( 1 − ϕ 1 − ϕ 2 ) ( 1 + ϕ 1 − ϕ 2 ) σ ε 2 γ 1 = ϕ 1 1 − ϕ 2 γ 0 γ k = ϕ 1 γ k − 1 + ϕ 2 γ k − 2 \left \{

$$\begin{array}{c} \gamma_0=\frac{1-\phi_2}{(1+\phi_2)(1-\phi_1-\phi_2)(1+\phi_1-\phi_2)}{\sigma_\varepsilon^2} \\ \gamma_1=\frac{\phi_1}{1-\phi_2}{\gamma_0} \\ \gamma_k=\phi_1\gamma_{k-1}+\phi_2\gamma_{k-2} \end{array}$$ \right. ⎩⎪⎨⎪⎧​γ0​=(1+ϕ2​)(1−ϕ1​−ϕ2​)(1+ϕ1​−ϕ2​)1−ϕ2​​σε2​γ1​=1−ϕ2​ϕ1​​γ0​γk​=ϕ1​γk−1​+ϕ2​γk−2​​

5-延迟k自相关系数

AR(1)

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

AR(2)

{ ρ 0 = γ 0 γ 0 = 1 ρ 1 = γ 1 γ 0 = ϕ 1 1 − ϕ 2 ρ k = γ k γ 0 = ϕ 1 ρ k − 1 + ϕ 2 ρ k − 2 \left \{

$$\begin{array}{c} \rho_0=\frac{\gamma_0}{\gamma_0}=1\\ \rho_1=\frac{\gamma_1}{\gamma_0}=\frac{\phi_1}{1-\phi_2}\\ \rho_k=\frac{\gamma_k}{\gamma_0}=\phi_1\rho_{k-1}+\phi_2\rho_{k-2} \end{array}$$ \right. ⎩⎪⎨⎪⎧​ρ0​=γ0​γ0​​=1ρ1​=γ0​γ1​​=1−ϕ2​ϕ1​​ρk​=γ0​γk​​=ϕ1​ρk−1​+ϕ2​ρk−2​​

6-延迟k偏自相关系数

AR(1)

{ ϕ 11 = ρ 1 ρ 0 ϕ k k = 0 , ∀ k > 1 \left \{

$$\begin{array}{c} \phi_{11}=\frac{\rho_1}{\rho_0} \\ \phi_{kk}=0,\forall k\gt 1 \end{array}$$ \right. {ϕ11​=ρ0​ρ1​​ϕkk​=0,∀k>1​

AR(2)

{ ϕ 11 = ρ 1 ρ 0 = ϕ 1 1 − ϕ 2 ϕ 22 = ϕ 2 ϕ k k = 0 , ∀ k > 2 \left \{

$$\begin{array}{c} \phi_{11}=\frac{\rho_1}{\rho_0}=\frac{\phi_1}{1-\phi_2} \\ \phi_{22}=\phi_2\\ \phi_{kk}=0,\forall k\gt 2 \end{array}$$ \right. ⎩⎨⎧​ϕ11​=ρ0​ρ1​​=1−ϕ2​ϕ1​​ϕ22​=ϕ2​ϕkk​=0,∀k>2​

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}