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For a GARCH(1,1) model, how does one write $X_t^2$ in terms of $X_{t-1}^2, \sigma_{t-1}^2,$ and $Z_{t}^2$?

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  • $\begingroup$ What do you think about my answer? If it is clear, you may accept it by clicking on the tick mark to the left, otherwise you may ask for further clarification. This is how Cross Validated works. $\endgroup$ – Richard Hardy Jun 1 '18 at 20:06
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Given a GARCH(1,1) model

\begin{aligned} X_{t} &= \sigma_{t}Z_{t} \\ \sigma^{2}_{t} &= \omega + \alpha_{1}X^{2}_{t-1} + \beta_{1}\sigma^{2}_{t-1} \end{aligned}

where $Z_{t} \sim i.i.d(0,1)$, square the first equation and substitute for the $\sigma_t^2$ as follows:

\begin{aligned} X_{t}^2 &= \sigma_{t}^2 Z_{t}^2 \\ &= (\omega + \alpha_{1}X^{2}_{t-1} + \beta_{1}\sigma^{2}_{t-1}) Z_t^2. \end{aligned}

This is $X_t^2$ expressed in terms of $X_{t-1}^2$, $\sigma_{t-1}^2$, $Z_t^2$ and $\omega$. I doubt it is possible to express $\omega$ in terms of $X_{t-1}^2$, $\sigma_{t-1}^2$ and $Z_t^2$, so it has to be included in the expression for $X_t^2$.

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