AR過程
AR(1)過程
AR過程の一番単純なモデル. \[ X_t = c + \phi_1 X_{t-1} + W_t \\ \mathrm{E}[W_t] = 0 \\ \mathrm{V}[W_t] = \sigma^2 \\ \mathrm{Cov}[W_t,W_{t-s}] = 0 \\ \mathrm{Cov}[X_t,W_s] = 0 \\ \] について考察する. \[ \begin{align} X_t &= c + \phi_1 X_{t-1} + W_t\\ &= \phi_1(c + \phi_1 X_{t-2} + W_{t-1}) + W_t + c\\ &= {\phi_1}^2 X_{t-2} + \phi_1(c+W_{t-1}) + W_t + c\\ &= {\phi_1}^2 (c + \phi_1 X_{t-3} + W_{t-2}) + \phi_1 (W_{t-1}+c) + W_t + c\\ &= {\phi_1}^3 X_{t-3} + {\phi_1}^2 (W_{t-2}+c) + \phi_1 (W_{t-1}+c) + W_t + c \\ &= {\phi_1}^t X_{0} + \sum^{t}_{s=1} {\phi_1}^{t-s} (W_{s} + c) \end{align} \] なので,\(|\phi_1| < 1\)の時は,過去遡るほどその影響は小さくなり, \(\phi_1 = 1\)の場合は,同じように影響し, \(|\phi_1| > 1\)の場合は,過去遡るほど現在に影響する.
\[ \begin{align} \mathrm{E}[X_t] &= {\phi_1}^t \mathrm{E}[X_{0}] + \sum^{t}_{s=1} {\phi_1}^{t-s} \mathrm{E}[W_{s}+c] \\ &= {\phi_1}^t \mathrm{E}[X_{0}] + \sum^{t}_{s=1} {\phi_1}^{t-s}c \end{align} \]
定常過程の場合,\(\mathrm{E}[X_t] = \mu\)なので, \[ \begin{align} \mu &= c + \phi_1 \mu \\ \mu &= \frac{c}{1-\phi_1} \end{align} \]
弱定常過程の場合,\(\mathrm{V}[X_t] = \gamma(0), \mathrm{Cov}[X_t,X_{t-s}] = \gamma_{|s|} = \gamma(s)\)なので, \[ \begin{align} \mathrm{V}[X_t] &= \mathrm{V}[c + \phi_1 X_{t-1} + W_t] \\ &= {\phi_1}^2\mathrm{V}[X_{t-1}] + \sigma^2 \\ \gamma(0) &= {\phi_1}^2\gamma(0) + \sigma^2 \\ \gamma(0) &= \frac{\sigma^2}{1-{\phi_1}^2} \end{align} \] \(\gamma(0) > 0 \)なので,AR(1)過程が弱定常過程ならば,\({\phi_1}^2 < 1,|\phi_1| < 1,\) \[ \begin{align} \mathrm{Cov}[X_t,X_{t-1}] &= \mathrm{Cov}[c + \phi_1 X_{t-1} + W_t,X_{t-1}] \\ &= \mathrm{Cov}[\phi_1 X_{t-1} + W_t,X_{t-1}] \\ &= \mathrm{E}[(\phi_1 X_{t-1} + W_t - \mathrm{E}[\phi_1 X_{t-1} + W_t])(X_{t-1} - \mu)] \\ &= \mathrm{E}[\phi_1 X_{t-1}(X_{t-1} - \mu)] \\ &= \mathrm{E}[(\phi_1 X_{t-1} - \phi_1\mu + \phi_1 \mu)(X_{t-1} - \mu)] \\ &= \phi_1\mathrm{V}[X_{t-1}] \\ &= \phi_1\frac{\sigma^2}{1-{\phi_1}^2} \end{align} \] \[ \begin{align} \mathrm{Cov}[X_t,X_{t-s}] &= \mathrm{Cov}[ X_{t-s} + \sum^{s}_{u=1} {\phi_1}^{t-u} (W_{u} + c),X_{t-s}] \\ &= {\phi_1}^s \mathrm{V}[X_{t-s}] \\ &= {\phi_1}^s \frac{\sigma^2}{1-{\phi_1}^2} \\ \end{align} \] なので, \[\gamma(s) = {\phi_1}^s \frac{\sigma^2}{1-{\phi_1}^2}\] \[\rho(s) = \frac{\gamma(s)}{\gamma(0)} = \frac{{\phi_1}^s \frac{\sigma^2}{1-{\phi_1}^2}}{\frac{\sigma^2}{1-{\phi_1}^2}} = {\phi_1}^s \].
AR(2)過程
2次のAR過程のを考察する. \[ X_t = c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t \]
定常過程の場合,\(\mathrm{E}[X_t] = \mu\)なので,
\[
\begin{align}
\mathrm{E}[X_t] &= \mathrm{E}[c + \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t] \\
&= c + \phi_1\mathrm{E}[X_{t-1}] + \phi_2\mathrm{E}[X_{t-2}] \\
\mu &= c + \phi_1\mu + \phi_2\mu \\
\mu &= \frac{c}{1 - \phi_1 - \phi_2}
\end{align}
\]
ここで,\(c=0\)とすると,\(\mu=0\). なので,これは,\(X’_t = X_t-\mu\)として,
\[
X_t-\mu = \phi_1 (X_{t-1}-\mu) + \phi_2 (X_{t-1}-\mu) + W_t \\
X’_t = \phi_1 X’_{t-1} + \phi_2 X’_{t-2} + W_t \\
\]
とすることをできる.\(X’_t\)は\(X_t\)の平均を0に標準化したものとみなせる.
共分散の時間発展を考察するのは,平均を0に標準化でも同じなので,
\[
X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + W_t \\
\]
で考察を続ける.
行列で表すと, \[ \begin{pmatrix} X_t \\ X_{t-1} \end{pmatrix} = \begin{pmatrix} \phi_1 & \phi_2 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} X_{t-1} \\ X_{t-2} \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix} W_t \] \[ \boldsymbol{X}_t = \begin{pmatrix} X_t \\ X_{t-1} \end{pmatrix},\; \mathbf{A} = \begin{pmatrix} \phi_1 & \phi_2 \\ 1 & 0 \end{pmatrix},\; \mathbf{B} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \] として, \[ \mathbf{X}_t = \mathbf{A}\mathbf{X}_{t-1} + \mathbf{B}W_t \] と表せる.