# Time Series Analysis

- 1 min

For AR(1) process, $$X_{t} = - \sum_{j=1}^{\infty}\phi^{-j}Z_{t+j}$$ where $$\{Z_{t}\} \sim WN(0,\sigma^{2})$$, $$\mid{\phi}\mid > 1$$.

First we can derive the $$\gamma_{X}(0)$$ since $${Z_{t}}$$ are uncorrelated to each other.

\begin{align} \gamma_{x}(0) & = Var(X_{t}) = Var( - \sum_{j=1}^{\infty}\phi^{-j}Z_{t+j}) = Var( \sum_{j=1}^{\infty}\phi^{-j}Z_{t+j}) \\\\ & = \sum_{j=1}^{\infty}Var(\phi^{-j}Z_{t+j}) = \sum_{j=1}^{\infty}(\phi^{-j})^{2}Var(Z_{t+j}) = \sum_{j=1}^{\infty}(\phi^{-j})^{2}\sigma^{2} \\\\ & = \sum_{j=1}^{\infty}\phi^{-2j}\sigma^{2} = \sigma^{2}\sum_{j=1}^{\infty}\phi^{-2j} = \sigma^{2}\frac{\phi^{-2}}{1-\phi^{-2}} =\frac{\sigma^{2}}{\phi^{2} - 1} \end{align}

Then AR(1) model can also be written as

$X_{t} = Z_{t} + \phi X_{t-1}$

But since $$\mid\phi\mid > 1$$, so we interpret it in another way

$X_{t+1}= Z_{t+1} + \phi X_{t}$ $\phi^{-1}X_{t+1} = \phi^{-1}Z_{t+1} + X_{t}$ $X_{t} = \phi^{-1}X_{t+1} - \phi^{-1}Z_{t+1}$

So we plug it in $\gamma_{x}(h)$, $(h > 0)$

\begin{align} \gamma_{x}(h) & = \gamma_{x}(X_{t+h},X_{t}) \\\\ & =Cov(X_{t+h},\phi^{-1}X_{t+1} - \phi^{-1}Z_{t+1}) \\\\ & = Cov(X_{t+h},\phi^{-1}X_{t+1}) - Cov(X_{t+h},\phi^{-1}Z_{t+1}) \\\\ & = \phi^{-1}Cov(X_{t+h},X_{t+1}) - \phi^{-1}Cov(X_{t+h},Z_{t+1}) \\\\ & = \phi^{-1}\gamma_{x}(h-1) - \phi^{-1}Cov(X_{t+h},Z_{t+1}) \end{align}

We also know that $Cov(X_{s},Z_{t}) = 0$ if $s \geq t$ since

$X_{t} = - \sum_{j=1}^{\infty}\phi^{-j}Z_{t+j}$ $Cov(X_{s},Z_{t}) = Cov(- \sum_{j=1}^{\infty}\phi^{-j}Z_{s+j},Z_{t})$

where $s \geq t$. So $s + j > t$ for j = 1, 2, …

Hence $Cov(X_{s},Z_{t}) = 0$ if $s \geq t$

Therefore,

$\gamma_{x}(h) = \phi^{-1}\gamma_{x}(h-1)$

By induction, we know that

$\gamma_{x}(h) = \phi^{-h}\gamma_{X}(0)$

Then plug $$\gamma_{x}(0)$$ in equation we can get

$\gamma_{x}(h) = \phi^{-h}\frac{\sigma^{2}}{\phi^{2} - 1} = \frac{\sigma^{2}}{\phi^{h}(\phi^{2} - 1)}$