I am modelling an univariate GARCH and I would like to know if I am operating in the correct way.
First of all, I suppose that the performance of my asset can be described as:
$Y_t=\mu+\epsilon_t $
in other words, I suppose the conditional mean of the process is constant over time, thus $\text{Var}(Y_t)=\text{Var}(\epsilon_t)=\sigma^2_t$. Later I develop the following GARCH(1,1) model:
$\sigma^2_t=\omega+\alpha Y_{t-1}^2+\beta\sigma^2_{t-1} $
At the implementation level, therefore, I fix $\mu_0 = \bar{Y}, \omega_0,\alpha_0,\beta_0$, I get the value of
$\epsilon_t =Y_t - \mu_0$
and finally I set the log likelihood function of the GARCH model
$-\frac{1}{2}\sum_{t=1}^{T}\big{[}\ln(\sigma^2_t)+\frac{Y^2_t}{\sigma^2_t}\big{]}$
My question is: is it correct that the parameter $\mu$ is estimated together as the parameters $\omega,\alpha,\beta$?