Suppose we have a GARCH (1,1) equation as follows:
$$\sigma_t^2 = \alpha_0 + \alpha\epsilon_{t-1}^2 + \beta\sigma_{t-1}^2$$
- How are the lagged variance terms calculated?
- Suppose I am also implementing an ARMA-GARCH process sequentially, so I am using the residuals from ARMA to optimize GARCH. (I believe) Then it is assumed that the $\epsilon_i$ in both ARMA and GARCH follow the same distribution. If this is the case, since $\epsilon_i$ and $\sigma_i$ are known, why can't this parameter identification be down through OLS?