From what I know, the GARCH(p,q) model is estimated via MLE and through an iterative process. Let's say if i wanted to recreate a GARCH(1,1) parameter estimation with excel solver (through maximizing the log-likelihood), how are my initial GARCH terms $ \sigma_t^2$ set?
More specifically, given $ \epsilon_t = v_t\sqrt{\sigma_t^2}$ where $$\sigma_t^2 = \alpha_0 +\alpha_1\epsilon^2_{t-1}+\alpha_2\sigma_{t-1}^2 $$
how does the process of parameter estimation start since we do not know what $\sigma_{t-1}$ is?
One answer I've read from here shows that the program set the initial GARCH term to be the sample variance or its expected value. Is this how we approach it?
Best Answer
I know of at least five ways of initializing the volatility process:
1) Set it equal to $\varepsilon_{t-1}^2$,
2) The sample variance,
3) Unconditional variance of the model ($\alpha_0/(1-\alpha_1 - \alpha_2)$),
4) Allow it it to be an parameter to be estimated,
5) Backcasting with an exponential filter.
The topic is discussed in further detail here