# Is my understanding on how to estimate the parameters in a GARCH model correct?

Assume (for the sake of simplicity) we have observed only $$X_1,X_2$$ and we want to estimate the parameters of a GARCH(1,1) that tells us the variance of $$X_t$$ (that is normally distributed) evolves through time as

$$\sigma_t^2 = \omega + \alpha\sigma_{t-1}^2 + \beta X_{t-1}^2$$

where $$\omega, \alpha, \beta$$ are parameters to be estimated and the starting value of the variance $$\sigma_1^2$$ is a known constant. To estimate the parameters of this model my understanding is that we maximize the likelihood of the observations:

$$\frac{1}{\sqrt{2 \pi} \sigma_1} \exp(-X_1^2 / 2\sigma^2_1 ) \frac{1}{\sqrt{2 \pi (\omega +\alpha\sigma_1^2 + \beta X_1^2 )} } \exp(-X_2^2 / 2( \omega +\alpha\sigma_1^2 + \beta X_1^2 ))$$

with respect to $$\omega , \alpha, \beta$$.

Is this procedure correct? How does one usually chose the initial value $$\sigma^2_1$$?

Typically, we will assume that $$X_t = \sigma_t Z_t$$ where $$Z_t$$ is iid(0,1).
Regarding initial values of $$\sigma_1^2$$, I have seen the approaches in Initial value of the conditional variance in the GARCH process