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

\[\gamma_{x}(h) = \phi^{-h}\frac{\sigma^{2}}{\phi^{2} - 1} = \frac{\sigma^{2}}{\phi^{h}(\phi^{2} - 1)}\]
Ying Zhang

Ying Zhang

A statistician who gets lost in analysis.

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