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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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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