第7講確率過程・時系列解析

自己回帰モデル

次のようなモデルを $p$ 次自己回帰モデルと呼び， $AR(p)$ と書く．

X_{t} = ϕ_{0} + ϕ_{1} X_{t - 1} + \dots + ϕ_{p} X_{t - p} + ϵ_{t}

$\begin{equation} X_t = \phi_0 + \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + \epsilon_t \end{equation}$

ϵ_{t} \overset{i . i . d .}{\sim} N (0, σ)

$\begin{equation} \epsilon_t \stackrel{i.i.d.}{\sim} N(0, \sigma) \end{equation}$

A R (1)

$AR(1)$ の場合を考える．

\begin{aligned} X_{t} & = ϕ_{0} + ϕ_{1} X_{t - 1} + ϵ_{t} \\ = ϕ_{0} + ϕ_{1} (ϕ_{0} + ϕ_{1} X_{t - 2} + ϵ_{t - 1}) + ϵ_{t} \\ = ϕ_{0} + ϕ_{1} ϕ_{0} + {ϕ_{1}}^{2} X_{t - 2} + ϕ_{1} ϵ_{t - 1} + ϵ_{t} \\ = ϕ_{0} (1 + ϕ_{1}) + {ϕ_{1}}^{2} X_{t - 2} + (ϵ_{t} + ϕ_{1} ϵ_{t - 1}) \\ = ϕ_{0} (1 + ϕ_{1}) + {ϕ_{1}}^{2} (ϕ_{0} + ϕ_{1} X_{t - 3} + ϵ_{t - 2}) + (ϵ_{t} + ϕ_{1} ϵ_{t - 1}) \\ = ϕ_{0} (1 + ϕ_{1} + {ϕ_{1}}^{2}) + {ϕ_{1}}^{3} X_{t - 3} + (ϵ_{t} + ϕ_{1} ϵ_{t - 1} + {ϕ_{1}}^{2} ϵ_{t - 2}) \\ ⋮ \\ = ϕ_{0} (1 + ϕ_{1} + \dots + {ϕ_{1}}^{n}) + {ϕ_{1}}^{n + 1} X_{t - n - 1} + (ϵ_{t} + ϕ_{1} ϵ_{t - 1} + \dots + {ϕ_{1}}^{n} ϵ_{t - n}) \end{aligned}

$\begin{align} X_t &= \phi_0 + \phi_1 X_{t-1} + \epsilon_t \\ &= \phi_0 + \phi_1(\phi_0 + \phi_1 X_{t-2} + \epsilon_{t-1}) + \epsilon_t \\ &= \phi_0 + \phi_1 \phi_0 + {\phi_1}^2 X_{t-2} + \phi_1 \epsilon_{t-1} + \epsilon_t \\ &= \phi_0(1 + \phi_1) + {\phi_1}^2 X_{t-2} + (\epsilon_t + \phi_1 \epsilon_{t-1}) \\ &= \phi_0(1 + \phi_1) + {\phi_1}^2 (\phi_0 + \phi_1 X_{t-3} + \epsilon_{t-2}) + (\epsilon_t + \phi_1 \epsilon_{t-1}) \\ &= \phi_0(1 + \phi_1 + {\phi_1}^2) + {\phi_1}^3 X_{t-3} + (\epsilon_t + \phi_1 \epsilon_{t-1} + {\phi_1}^2 \epsilon_{t-2}) \\ &~\vdots \\ &= \phi_0(1 + \phi_1 + \cdots + {\phi_1}^n) + {\phi_1}^{n+1} X_{t-n-1} + (\epsilon_t + \phi_1 \epsilon_{t-1} + \cdots + {\phi_1}^{n} \epsilon_{t-n}) \end{align}$

ここで，

- 1 < | ϕ_{1} | < 1

$-1 < |\phi_1| < 1$ という条件を課し，

n \to \infty

$n \to \infty$ とすると，

\begin{aligned} X_{t} & = ϕ_{0} (1 + ϕ_{1} + {ϕ_{1}}^{2} + \dots) + {ϕ_{1}}^{n + 1} X_{t - n - 1} + (ϵ_{t} + ϕ_{1} ϵ_{t - 1} + {ϕ_{1}}^{2} ϵ_{t - 2} + \dots) \\ = \frac{ϕ_{0}}{1 - ϕ_{1}} + \sum_{i = 0}^{\infty} {ϕ_{1}}^{i} ϵ_{t - i} \end{aligned}

$\begin{align} X_t &= \phi_0(1 + \phi_1 + {\phi_1}^2 + \cdots) + {\phi_1}^{n+1} X_{t-n-1} + (\epsilon_t + \phi_1 \epsilon_{t-1} + {\phi_1}^{2} \epsilon_{t-2} + \cdots) \\ &= \frac{\phi_0}{1 - \phi_1} + \sum_{i=0}^{\infty} {\phi_1}^{i} \epsilon_{t-i} \end{align}$

これを

A R (1)

$AR(1)$ の移動平均表現 (MA(

\infty

$\infty$ ) 表現) という．これより，

\begin{aligned} E [X_{t}] & = E [\frac{ϕ_{0}}{1 - ϕ_{1}} + \sum_{i = 0}^{\infty} {ϕ_{1}}^{i} ϵ_{t - i}] \\ = \frac{ϕ_{0}}{1 - ϕ_{1}} + \sum_{i = 0}^{\infty} {ϕ_{1}}^{i} E [ϵ_{t - i}] \\ = \frac{ϕ_{0}}{1 - ϕ_{1}} \end{aligned}

$\begin{align} E[X_t] &= E\left[\frac{\phi_0}{1 - \phi_1} + \sum_{i=0}^{\infty} {\phi_1}^{i} \epsilon_{t-i}\right] \\ &= \frac{\phi_0}{1 - \phi_1} + \sum_{i=0}^{\infty} {\phi_1}^{i} E\left[\epsilon_{t-i}\right] \\ &= \frac{\phi_0}{1 - \phi_1} \end{align}$

\begin{aligned} C o v [X_{t}, X_{t - h}] & = C o v [\frac{ϕ_{0}}{1 - ϕ_{1}} + \sum_{i = 0}^{\infty} {ϕ_{1}}^{i} ϵ_{t - i}, \frac{ϕ_{0}}{1 - ϕ_{1}} + \sum_{j = 0}^{\infty} {ϕ_{1}}^{j} ϵ_{t - h - j}] \\ = C o v [\sum_{i = 0}^{\infty} {ϕ_{1}}^{i} ϵ_{t - i}, \sum_{j = 0}^{\infty} {ϕ_{1}}^{j} ϵ_{t - h - j}] \\ = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} {ϕ_{1}}^{i} {ϕ_{1}}^{j} C o v [ϵ_{t - i}, ϵ_{t - j - h}] \\ = \sum_{j = 0}^{\infty} {ϕ_{1}}^{j + h} {ϕ_{1}}^{j} σ^{2} \\ = σ^{2} \sum_{j = 1}^{\infty} {ϕ_{1}}^{h} {({ϕ_{1}}^{2})}^{j - 1} \\ = \frac{{ϕ_{1}}^{h}}{1 - {ϕ_{1}}^{2}} σ^{2} \end{aligned}

$\begin{align} Cov[X_t, X_{t-h}] &= Cov\left[\frac{\phi_0}{1 - \phi_1} + \sum_{i=0}^{\infty} {\phi_1}^{i} \epsilon_{t-i},~\frac{\phi_0}{1 - \phi_1} + \sum_{j=0}^{\infty} {\phi_1}^{j} \epsilon_{t-h-j}\right] \\ &= Cov\left[\sum_{i=0}^{\infty} {\phi_1}^{i} \epsilon_{t-i},~\sum_{j=0}^{\infty} {\phi_1}^{j} \epsilon_{t-h-j}\right] \\ &= \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}{\phi_1}^{i} {\phi_1}^{j}\,Cov[\epsilon_{t-i}, \epsilon_{t-j-h}] \\ &= \sum_{j=0}^{\infty}{\phi_1}^{j+h} {\phi_1}^{j} \sigma^2 \\ &= \sigma^2 \sum_{j=1}^{\infty}{\phi_1}^{h} \left({\phi_1}^{2}\right)^{j-1} \\ &= \frac{{\phi_1}^h}{1 - {\phi_1}^2}\sigma^2 \end{align}$

第7講 確率過程・時系列解析

自己回帰モデル

第7講確率過程・時系列解析