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【时间序列分析】AR模型公式总结

ar模型公式

AR
Time Series Analysis
author:zoxiii


【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

{xt=ϕ0+ϕ1xt1+...+ϕpxtp+εtϕp0E(εt)=0,Var(εt)=σε2,E(εtεs)=0,stE(xsεt)=0,s<t
xt=ϕ0+ϕ1xt1+...+ϕpxtp+εtϕp=0E(εt)=0,Var(εt)=σε2,E(εtεs)=0,s=tE(xsεt)=0,s<t

中心化AR(q)

x t = ϕ 1 x t − 1 + . . . + ϕ p x t − p + ε t x_t=\phi_1x_{t-1}+...+\phi_px_{t-p}+\varepsilon_t xt=ϕ1xt1+...+ϕpxtp+εt

引入延迟算子B

x t = ϕ 1 x t − 1 + . . . + ϕ p x t − p + ε t   = ϕ 1 B x t + . . . + ϕ p B p x t + ε t = Φ ( B ) ε t                                  x_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=ϕ1xt1+...+ϕpxtp+εt =ϕ1Bxt+...+ϕpBpxt+εt=Φ(B)εt                                
得到q阶自回归系数多项式:
Φ ( B ) = 1 − ϕ 1 B − ϕ 2 B 2 − . . . − ϕ p B p \Phi(B)=1-\phi_1B-\phi_2B^2-...-\phi_pB^p Φ(B)=1ϕ1Bϕ2B2...ϕpBp

1-均值

μ = ϕ 0 1 − ϕ 1 − . . . − ϕ p \mu=\frac{\phi_0}{1-\phi_1-...-\phi_p} μ=1ϕ1...ϕpϕ0

2-Green函数

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

G0=1Gj=k=1jϕkGjk
\right. {G0=1Gj=k=1jϕkGjk
其中:
ϕ k ′ = { ϕ k , k ≤ p 0 , k > p \phi_k' =
{ϕk,kp0,k>p
ϕk={ϕk,kp0,k>p

Green推导公式过程

x t = ε t Φ ( B ) = G ( B ) ε t x_t=\frac{\varepsilon_t}{\Phi\left(B\right)}=G(B)\varepsilon_t xt=Φ(B)εt=G(B)εt

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

( 1 − ∑ k = 1 p ( ϕ k B k ) ) ( ∑ j = 0 ∞ ( G j B j ) ) ε t = ε t \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 (1k=1p(ϕkBk))(j=0(GjBj))εt=εt

( ∑ j = 0 ∞ G j B j − ∑ k = 1 p ∑ j = 0 ∞ ϕ k B k G j B j ) ε t = ε t \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=0GjBjk=1pj=0ϕkBkGjBj)εt=εt

( G 0 + ∑ j = 1 ∞ ( G j − ∑ k = 1 j ϕ k ′ G j − k ) B j ) ε t = ε t \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(Gjk=1jϕkGjk)Bj)εt=εt

3-方差

V a r ( x t ) = ∑ j = 0 ∞ G j 2 V a r ( ε t − j ) = ∑ j = 0 ∞ G j 2 σ ε 2 Var(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=0Gj2Var(εtj)=j=0Gj2σε2
或者
V a r ( x t ) = γ 0 Var(x_t)=\gamma_0 Var(xt)=γ0

4-延迟k协方差函数

AR(1)

γ k = ϕ 1 k σ ε 2 1 − ϕ 1 2 \gamma_k=\phi_1^k\frac{\sigma_\varepsilon^2}{1-\phi_1^2} γk=ϕ1k1ϕ12σε2

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 \{

γ0=1ϕ2(1+ϕ2)(1ϕ1ϕ2)(1+ϕ1ϕ2)σε2γ1=ϕ11ϕ2γ0γk=ϕ1γk1+ϕ2γk2
\right. γ0=(1+ϕ2)(1ϕ1ϕ2)(1+ϕ1ϕ2)1ϕ2σε2γ1=1ϕ2ϕ1γ0γk=ϕ1γk1+ϕ2γk2

5-延迟k自相关系数

AR(1)

ρ k = γ k γ 0 = ϕ 1 k \rho_k=\frac{\gamma_k}{\gamma_0}=\phi_1^k ρk=γ0γk=ϕ1k

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 \{

ρ0=γ0γ0=1ρ1=γ1γ0=ϕ11ϕ2ρk=γkγ0=ϕ1ρk1+ϕ2ρk2
\right. ρ0=γ0γ0=1ρ1=γ0γ1=1ϕ2ϕ1ρk=γ0γk=ϕ1ρk1+ϕ2ρk2

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 \{

ϕ11=ρ1ρ0ϕkk=0,k>1
\right. {ϕ11=ρ0ρ1ϕkk=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 \{

ϕ11=ρ1ρ0=ϕ11ϕ2ϕ22=ϕ2ϕkk=0,k>2
\right. ϕ11=ρ0ρ1=1ϕ2ϕ1ϕ22=ϕ2ϕkk=0,k>2

7-AR模型平稳性判别(特征根+平稳域)

AR(1)

x t = ϕ 1 x t − 1 + ε t x_t=\phi_1x_{t-1}+\varepsilon_t xt=ϕ1xt1+εt
特征方程 λ − ϕ 1 = 0 \lambda-\phi_1=0 λϕ1=0
特征根 λ = ϕ 1 \lambda=\phi_1 λ=ϕ1
平稳充要条件:特征根在单位圆内,即 ∣ ϕ 1 ∣ < 1 |\phi_1|<1 ϕ1<1
平稳域为 { ϕ 1 ∣ − 1 < ϕ 1 < 1 } \{\phi_1|-1<\phi_1<1\} {ϕ11<ϕ1<1}

AR(2)

x t = ϕ 1 x t − 1 + ϕ 2 x t − 2 + ε t x_t=\phi_1x_{t-1}+\phi_2x_{t-2}+\varepsilon_t xt=ϕ1xt1+ϕ2xt2+εt
特征方程 λ 2 − ϕ 1 λ − ϕ 2 = 0 \lambda^2-\phi_1\lambda-\phi_2=0 λ2ϕ1λϕ2=0
特征根 λ 1 = ϕ 1 + ϕ 1 2 + 4 ϕ 2 2 , λ 2 = ϕ 1 − ϕ 1 2 + 4 ϕ 2 2 \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=2ϕ1ϕ12+4ϕ2
平稳充要条件:特征根在单位圆内,即 ∣ λ 1 ∣ < 1 且 ∣ λ 2 ∣ < 1 |\lambda_1|<1且|\lambda_2|<1 λ1<1λ2<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,ϕ2ϕ2<1ϕ2±ϕ1<1}

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