第7講 確率過程・時系列解析

$$\begin{equation} X_t = \theta_0 + \epsilon_t - \theta_1 \epsilon_{t-1} - \cdots - \theta_q \epsilon_{t-q} \end{equation}$$

$$\begin{equation} E[\epsilon_t] = 0 \end{equation}$$

$$\begin{equation} V[\epsilon_t] = \sigma^2 \end{equation}$$

$$\begin{equation} Cov[\epsilon_s,\,\epsilon_t] = 0~~~~(s \ne t) \end{equation}$$

$$\begin{align} E[X_t] &= E[\theta_0 + \epsilon_t - \theta_1 \epsilon_{t-1} - \cdots - \theta_q \epsilon_{t-q}] \\ &= \theta_0 \end{align}$$

$$\begin{align} V[X_t] &= V[\theta_0 + \epsilon_t - \theta_1 \epsilon_{t-1} - \cdots - \theta_q \epsilon_{t-q}] \\ &= (1 + {\theta_1}^2 + \cdots + {\theta_q}^2)\sigma^2 \end{align}$$

$$\begin{align} \gamma_{0} = V[X_{t}] = (1 + {\theta_1}^2 + \cdots + {\theta_q}^2)\sigma^2 \end{align}$$

$$\begin{align} \gamma_{h} &= Cov[Y_{t}, Y_{t-h}] \\ &= Cov[\theta_0 + \epsilon_t - \theta_1 \epsilon_{t-1} - \cdots - \theta_q \epsilon_{t-q},~\theta_0 + \epsilon_{t-h} - \theta_1 \epsilon_{t-h-1} - \cdots - \theta_q \epsilon_{t-h-q}] \\ &= Cov\left[\theta_0 + \epsilon_t - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i},~\theta_0 + \epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right] \\ &= E\left[\left(\theta_0 + \epsilon_t - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i} - E\left[\theta_0 + \epsilon_{t} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right]\right)\left(\theta_0 + \epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i} - E\left[\theta_0 + \epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right]\right)\right] \\ &= E\left[\left(\theta_0 + \epsilon_t - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i} - \theta_0\right)\left(\theta_0 + \epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i} - \theta_0\right)\right] \\ &= E\left[\left(\epsilon_t - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] \\ &= E\left[\epsilon_t \left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right) - \left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] \\ &= E\left[\epsilon_t \left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] + E\left[-\left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] \\ &= E\left[-\left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\left(\epsilon_{t-h} - \sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] \\ &= E\left[-\left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\epsilon_{t-h}\right] + E\left[\left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-i}\right)\left(\sum_{i=1}^{q} \theta_{i} \epsilon_{t-h-i}\right)\right] \\ &= E\left[-\theta_{h}{\epsilon_{t-h}}^2\right] + E\left[\sum_{i=1}^{q-h} \theta_{i+h}\theta_{i}{\epsilon_{t-h-i}}^2\right] \\ &= -\theta_{h}E\left[{\epsilon_{t-h}}^2\right] + \sum_{i=1}^{q-h} \theta_{i+h}\theta_{i}E\left[{\epsilon_{t-h-i}}^2\right] \\ &= -\theta_{h}\sigma^2 + \sigma^2\sum_{i=1}^{q-h} \theta_{i+h}\theta_{i} \\ &= \left(-\theta_{h} + \sum_{i=1}^{q-h} \theta_{i+h}\theta_{i}\right)\sigma^2 \\ \end{align}$$

$$\begin{equation} \gamma_{q} = Cov[X_{t}, X_{t-q}] = -\theta_{q}\sigma^2 \\ \end{equation}$$