【时间序列分析】AR模型公式总结

AR
Time Series Analysis
author：zoxiii

AR

0-模型
- AR(q)
- 中心化AR(q)
- 引入延迟算子B
1-均值
2-Green函数
- Green推导公式过程
3-方差
4-延迟k协方差函数
- AR(1)
- AR(2)
5-延迟k自相关系数
- AR(1)
- AR(2)
6-延迟k偏自相关系数
- AR(1)
- AR(2)
7-AR模型平稳性判别(特征根+平稳域)
- AR(1)
- AR(2)

【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)

{xt=ϕ0+ϕ1xt−1+...+ϕpxt−p+εtϕp≠0E(εt)=0,Var(εt)=σε2,E(εtεs)=0,s≠tE(xsεt)=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+ϕ1xt−1+...+ϕpxt−p+εtϕp=0E(εt)=0,Var(εt)=σε2,E(εtεs)=0,s=tE(xsεt)=0,∀s<t

中心化AR(q)

xt=ϕ1xt−1+...+ϕpxt−p+εtx_t=\phi_1x_{t-1}+...+\phi_px_{t-p}+\varepsilon_t xt=ϕ1xt−1+...+ϕpxt−p+εt

引入延迟算子B

xt=ϕ1xt−1+...+ϕpxt−p+εt=ϕ1Bxt+...+ϕpBpxt+εt=Φ(B)εtx_t=\phi_1x_{t-1}+...+\phi_px_{t-p}+\varepsilon_t \\ ~=\phi_1Bx_t+...+\phi_pB^px_t+\varepsilon_t\\ =\Phi(B)\varepsilon_t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ xt=ϕ1xt−1+...+ϕpxt−p+εt =ϕ1Bxt+...+ϕpBpxt+εt=Φ(B)εt
得到q阶自回归系数多项式：
Φ(B)=1−ϕ1B−ϕ2B2−...−ϕpBp\Phi(B)=1-\phi_1B-\phi_2B^2-...-\phi_pB^p Φ(B)=1−ϕ1B−ϕ2B2−...−ϕpBp

1-均值

μ=ϕ01−ϕ1−...−ϕp\mu=\frac{\phi_0}{1-\phi_1-...-\phi_p} μ=1−ϕ1−...−ϕpϕ0

2-Green函数

{G0=1Gj=∑k=1jϕk′Gj−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=1jϕk′Gj−k
其中：
ϕk′={ϕk,k≤p0,k>p\phi_k' =\begin{cases} \phi_k, k\le p \\ 0, k\gt p \end{cases} ϕk′={ϕk,k≤p0,k>p

Green推导公式过程

xt=εtΦ(B)=G(B)εtx_t=\frac{\varepsilon_t}{\Phi\left(B\right)}=G(B)\varepsilon_t xt=Φ(B)εt=G(B)εt

Φ(B)G(B)εt=εt\Phi\left(B\right)G\left(B\right)\varepsilon_t=\varepsilon_t Φ(B)G(B)εt=εt

(1−∑k=1p(ϕkBk))(∑j=0∞(GjBj))εt=εt\left(1-\sum_{k=1}^{p}\left(\phi_kB^k\right)\right)\left(\sum_{j=0}^{\infty}\left(G_jB^j\right)\right)\varepsilon_t=\varepsilon_t (1−k=1∑p(ϕkBk))(j=0∑∞(GjBj))εt=εt

(∑j=0∞GjBj−∑k=1p∑j=0∞ϕkBkGjBj)εt=εt\left(\sum_{j=0}^{\infty}{G_jB^j}-\sum_{k=1}^{p}\sum_{j=0}^{\infty}{\phi_kB^kG_jB^j}\right)\varepsilon_t=\varepsilon_t (j=0∑∞GjBj−k=1∑pj=0∑∞ϕkBkGjBj)εt=εt

(G0+∑j=1∞(Gj−∑k=1jϕk′Gj−k)Bj)εt=εt\left(G_0+\sum_{j=1}^{\infty}\left(G_j-\sum_{k=1}^{j}{{\phi_k}^\prime G_{j-k}}\right)B_j\right)\varepsilon_t=\varepsilon_t (G0+j=1∑∞(Gj−k=1∑jϕk′Gj−k)Bj)εt=εt

3-方差

Var(xt)=∑j=0∞Gj2Var(εt−j)=∑j=0∞Gj2σε2Var(x_t)=\sum_{j=0}^{\infty}{G_j^2Var(\varepsilon_{t-j})}=\sum_{j=0}^{\infty}{G_j^2\sigma_\varepsilon^2} Var(xt)=j=0∑∞Gj2Var(εt−j)=j=0∑∞Gj2σε2
或者
Var(xt)=γ0Var(x_t)=\gamma_0 Var(xt)=γ0

4-延迟k协方差函数

AR(1)

γk=ϕ1kσε21−ϕ12\gamma_k=\phi_1^k\frac{\sigma_\varepsilon^2}{1-\phi_1^2} γk=ϕ1k1−ϕ12σε2

AR(2)

{γ0=1−ϕ2(1+ϕ2)(1−ϕ1−ϕ2)(1+ϕ1−ϕ2)σε2γ1=ϕ11−ϕ2γ0γk=ϕ1γk−1+ϕ2γk−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−ϕ2σε2γ1=1−ϕ2ϕ1γ0γk=ϕ1γk−1+ϕ2γk−2

5-延迟k自相关系数

AR(1)

ρk=γkγ0=ϕ1k\rho_k=\frac{\gamma_k}{\gamma_0}=\phi_1^k ρk=γ0γk=ϕ1k

AR(2)

{ρ0=γ0γ0=1ρ1=γ1γ0=ϕ11−ϕ2ρk=γkγ0=ϕ1ρk−1+ϕ2ρk−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=γ0γ0=1ρ1=γ0γ1=1−ϕ2ϕ1ρk=γ0γk=ϕ1ρk−1+ϕ2ρk−2

6-延迟k偏自相关系数

AR(1)

{ϕ11=ρ1ρ0ϕkk=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=ρ0ρ1ϕkk=0,∀k>1

AR(2)

{ϕ11=ρ1ρ0=ϕ11−ϕ2ϕ22=ϕ2ϕkk=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=ρ0ρ1=1−ϕ2ϕ1ϕ22=ϕ2ϕkk=0,∀k>2

7-AR模型平稳性判别(特征根+平稳域)

AR(1)

xt=ϕ1xt−1+εtx_t=\phi_1x_{t-1}+\varepsilon_t xt=ϕ1xt−1+εt
特征方程λ−ϕ1=0\lambda-\phi_1=0λ−ϕ1=0
特征根λ=ϕ1\lambda=\phi_1λ=ϕ1
平稳充要条件：特征根在单位圆内，即∣ϕ1∣<1|\phi_1|<1∣ϕ1∣<1
平稳域为{ϕ1∣−1<ϕ1<1}\{\phi_1|-1<\phi_1<1\}{ϕ1∣−1<ϕ1<1}

AR(2)

xt=ϕ1xt−1+ϕ2xt−2+εtx_t=\phi_1x_{t-1}+\phi_2x_{t-2}+\varepsilon_t xt=ϕ1xt−1+ϕ2xt−2+εt
特征方程λ2−ϕ1λ−ϕ2=0\lambda^2-\phi_1\lambda-\phi_2=0λ2−ϕ1λ−ϕ2=0
特征根λ1=ϕ1+ϕ12+4ϕ22,λ2=ϕ1−ϕ12+4ϕ22\lambda_1=\frac{\phi_1+\sqrt{\phi_1^2+4\phi_2}}{2},\lambda_2=\frac{\phi_1-\sqrt{\phi_1^2+4\phi_2}}{2}λ1=2ϕ1+ϕ12+4ϕ2,λ2=2ϕ1−ϕ12+4ϕ2
平稳充要条件：特征根在单位圆内，即∣λ1∣<1且∣λ2∣<1|\lambda_1|<1且|\lambda_2|<1∣λ1∣<1且∣λ2∣<1
平稳域为{ϕ1,ϕ2∣∣ϕ2∣<1且ϕ2±ϕ1<1}\{\phi_1,\phi_2||\phi_2|<1且\phi_2\pm\phi_1<1\}{ϕ1,ϕ2∣∣ϕ2∣<1且ϕ2±ϕ1<1}