The GARCH model is a model for the heteroskedastic variations where the changes in variance is assumed to be auto correlated: that, though the variance changes, it changes in a predictable manner.
It is especially useful to
GARCH 1,1
Conditional mean:
\begin{equation} y_{t} = x’_{t} \theta + \epsilon_{t} \end{equation}
Then, the epsilon parameter:
\begin{equation} \epsilon_{t} = \sigma_{t}z_{t} \end{equation}
where:
\begin{equation} z_{t} \sim \mathcal{N}(0,1) \end{equation}
and:
conditional variance
\begin{equation} {\sigma_{t}}^{2} = \omega + \lambda {\sigma_{t-1}}^{2} + \beta {\sigma_{t-1}}^{2} \end{equation}
Finally, with initial conditions:
\begin{equation} w>0; \alpha >0; \beta >0 \end{equation}