Hi Quantitative Fiance Stack Exchange,
It's my first go at GARCH models so please give me a chance with my phrasing.
I understand that GARCH models are used to forecast volatility. The GARCH(1,1) takes the form:
$$\sigma^2_t=\alpha+\beta_1\epsilon_{t-1}+\beta_2\sigma^2_{t-1}$$
I understand the lagged term $\sigma^2_{t-1}$ makes up the AR part of GARCH. However, I also understand the error term $\epsilon_{t-1}$ is dependent on the forecasting model. Consider, forecasting returns using one of the two models:
$$\hat{y_t}=\gamma\cdot y_{t-1}+\epsilon_t$$
and
$$\hat{y_t}=\theta\cdot x_{t-1}+\epsilon_t$$
Each model gives a different error term, which I believe is calculated as $\epsilon_t=y_t-\hat{y_t}$. So for the above models, error terms are $\epsilon_t=y_t-\gamma\cdot y_{t-1}$ and $\epsilon_t=y_t-\theta\cdot x_{t-1}$
Hence, is my understanding correct that calculating $\beta_1$ and $\beta_2$ of the GARCH(1,1) model depends on which forecasting model we're using?
Thank you for the help, Donny