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

Question

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 $\sigma^2_1$?

Answer 1

Typically, we will assume that $X_t = \sigma_t Z_t$ where $Z_t$ is iid(0,1).

How to find the log-likelihood is described in Maximum likelihood in the GJR-GARCH(1,1) model

and how to implement the procedure is described in Fitting a GARCH(1, 1) model

Regarding initial values of $\sigma_1^2$, I have seen the approaches in Initial value of the conditional variance in the GARCH process