Part (a): Derive an equation similar to Equation (5.1.3)
Misal .
Mulai dengan dan . Didapati,
- , jadi \begin{align} Y_t &= \phi (\phi Y_{t-2} + e_{t-1}) + e_t \\ &= \phi^2 Y_{t-2} + \phi e_{t-1} + e_t \end{align}
- jadi \begin{align} Y_t &= \phi^2 (\phi Y_{t-3} + e_{t-2}) + \phi e_{t-1} + e_t \\ &= \phi^3 Y_{t-3} + \phi^2 e_{t-2} + \phi e_{t-1} + e_t \end{align}
Selanjutnya saat dengan dan . Didapati:
Y_t &= e_t + \phi e_{t-1} + \phi^2 e_{t-2} + \cdots + \phi^{t-1} e_1 + \phi^t Y_0\\ &= e_t + \phi e_{t-1} + \phi^2 e_{t-2} + \cdots + \phi^{t-1} e_1 \\ &= \sum_{j=0}^{t-1} \phi^j e_{t-j} \end{align}$$ ## Part (b): Derive an equation similar to Equation (5.1.4) Misal $|\phi| > 1$. Dengan $Y_t = \sum_{j=0}^{t-1}\phi^j e_{t-j}$ dari (a), dan karena $\{e_t\}$ adalah white noise, didapati:\begin{align} \operatorname{Var}(Y_t) &= \operatorname{Var}\left( \sum_{j=0}^{t-1} \phi^j e_{t-j} \right) \ &= \sum_{j=0}^{t-1} \operatorname{Var}(\phi^j e_{t-j}) \ &= \sum_{j=0}^{t-1} (\phi^j)^2 \operatorname{Var}(e_{t-j}) \ &= \sum_{j=0}^{t-1} \phi^{2j} \sigma_e^2 \ &= \frac{1 - \phi^{2t}}{1 - \phi^2} \quad \text{(Geometric Series)} \ &= \frac{\phi^{2t} - 1}{\phi^2 - 1} \end{align}
## Part (c): Derive an equation similar to Equation (5.1.5) Misal $|\phi| > 1$. Untuk $k \geq 0$ berlaku: - $Y_t = e_t + \phi e_{t-1} + \cdots + \phi^{t-1} e_1$ - $Y_{t-k} = e_{t-k} + \phi e_{t-k-1} + \cdots + \phi^{t-k-1} e_1$ Karena $E(Y_t) = 0$, maka $\operatorname{Cov}(Y_t, Y_{t-k}) = E(Y_t Y_{t-k})$. Sehingga didapati: $$\begin{align} E(Y_t Y_{t-k}) &= E\left( \sum_{j=0}^{t-1} \phi^j e_{t-j} \sum_{i=0}^{t-k-1} \phi^i e_{t-k-i} \right) \\ \operatorname{Var}(Y_{t}, Y_{t-k})&= \sum_{j=k}^{t-1} \phi^j \phi^{j-k} \sigma_e^2 \\ &= \phi^k \sum_{j=k}^{t-1} \phi^{2(j-k)} \sigma_e^2 \\ &= \frac{\phi^{2(t-k) + k} - \phi^k}{\phi^2 - 1} \sigma_e^2 \\ &= \frac{\phi^{2t - k} - \phi^k}{\phi^2 - 1} \sigma_e^2 \end{align}$$ ## Part (d): Is $\operatorname{Corr}(Y_t, Y_{t-k}) \approx 1$ for large $t$ and moderate $k$? $$\begin{align} \operatorname{Corr}(Y_t, Y_{t-k}) &= \frac{\operatorname{Cov}(Y_t, Y_{t-k})}{\sqrt{\operatorname{Var}(Y_t) \operatorname{Var}(Y_{t-k})}} \\ &= \frac{\frac{\phi^{2t - k} - \phi^k}{\phi^2 - 1}}{\sqrt{\frac{\phi^{2t} - 1}{\phi^2 - 1} \cdot \frac{\phi^{2(t-k)} - 1}{\phi^2 - 1}}} \\ &= \frac{\phi^{2t - k} - \phi^k}{\sqrt{(\phi^{2t} - 1)(\phi^{2(t-k)} - 1)}} \end{align}$$ Untuk $t$ besar dan $k$ moderate (tetap), karena $|\phi| > 1$, maka didapati: - $\phi^{2t} \gg 1$, $\phi^{2t-k} \gg \phi^k$, dan $\phi^{2(t-k)} \gg 1$. - Pembilang $\approx \phi^{2t-k}$ - Penyebut $\approx \sqrt{\phi^{2t} \phi^{2(t-k)}} = \phi^{2t - k}$. - $\operatorname{Corr}(Y_t, Y_{t-k}) \approx \frac{\phi^{2t - k}}{\phi^{2t - k}} = 1$.