# Are GARCH models dependent on the returns forecasting model?

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

I also understand the error term $\varepsilon_{t-1}$ is dependent on the forecasting model.
Yes, it is. The error term $\varepsilon_t$ in the GARCH model is coming from the full distributional model of $y_t$. The full model is \begin{aligned} y_t &= \mu_t + \varepsilon_t, \\ \varepsilon_t &= \sigma_t \xi_t, \\ \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \xi_t &\sim i.i.d(0,1), \end{aligned} where $\mu_t$ is the conditional mean of $y_t$, $\sigma_t^2$ is the conditional variance of $y_t$ and $d$ is some probability distribution with zero mean and unit variance.
If you are not sure which conditional mean model is best for $y_t$, you may end up with a few alternative models characterized by the conditional means $\mu_{1,t}, \mu_{2,t}, \dots$. The the corresponding error terms will differ across the models and will be $\varepsilon_{1,t} = y_t-\mu_{1,t}, \varepsilon_{2,t} = y_t-\mu_{2,t}, \dots$. This will affect the parameter estimates of the conditional variance model, just as you said.