# 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).

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

• Thabk you for the answer! So the likelihood I write is indeed the correct one to maximize? if I understand well from your links. Apr 15, 2019 at 10:14
• Yes, it looks correct - it is just the product of two normal pdfs. I guess you need more than one data point, if you want to estimate 3 parameters, but that is more a question regarding implementation. Apr 15, 2019 at 10:29
• You mean my estimates will likely have a high variance but it can be done even with two datapoints? Apr 15, 2019 at 14:56
• I'm believe you will run into problems if you try to estimate a GARCH model with two data points. Typically, one needs to have a lot of data when estimating GARCH model - often many years of daily returns are used in finance. Apr 15, 2019 at 17:37
• Is it possible to fit GARCH without making an assumption about the form of the distribution function of the innovations? Maybe something based on moments and autocorrelation of squared observations? Thank you Sep 15, 2022 at 17:49