GARCH Model Estimation I am analysing a GARCH(1,1) model under the assumption of t-Student distribution. 
In particular, I set the problem in the following way. I have a series ${y_t}, t \in{1,2,...,T}$ and I assume that:
1) $y_t = \sigma_t \epsilon_t$ where $\epsilon_t\sim t_{\nu}$ where $t_{\nu}$ is a 
   t-Student distribution with $\nu$ degrees of freedom, to be estimated by the 
   model
2) $\sigma_t^2=\omega + \alpha y_{t-1} + \beta \sigma_{t-1}^2$ is the equation of variance
My question is: when creating $\sigma_t^2$ I have to consider the real values of the series $y_t$ or I have to generate a random number $\epsilon_t$ distributed as a t-Student distribution, calculate $y_t = \sigma_t \epsilon_t$ and then evaluate $\sigma_t^2=\omega + \alpha y_{t-1} + \beta \sigma_{t-1}^2$?
Another question is, assuming that $\epsilon_t\sim t_{\nu}$ means that the degrees of freedom of the distribution have to estimated by the model or I have to set these degrees of freedom before the GARCH(1,1) estimation problem?
 A: The answer provided in this post may be of help to you. The code is in MATLAB but that should be irrelevant, because the transformations are algebraic anyway, not code-specific. The main idea is that you build your model in such a way that you transform your original parameters $\vec{\alpha}$ (which have certain constraints on them) to a different set of parameters $\vec{\theta}$ which do not have constraints. You do the transformation using a specific, "1-to-1" function for each parameter, perform the unconstrained optimisation on $\vec{\theta}$, and then transform the resulting $\vec{\hat{\theta}}$ into the original $\vec{\alpha}$ by using the inverse of the "1-to-1" function used in the first place.
You should check the book "Quantitative Risk Management" by Embrechts, McNeil, Frey. Its section called "GARCH models for Changing Volatility" covers pretty much everything you are looking for, including the topic of QMLE.
A: Ad 1) Yes, when creating $\sigma_t^2$ you use $y_t$ in the equation 
$$
\sigma_t^2 = \omega + \beta \sigma_{t-1}^2 + \alpha y_{t-1}^2
$$
You have made the assumption $E[y_t] = 0$. You have an error in the GARCH equation - it should be $y_{t-1}^2$. 
Ad 2) You decide. You can estimate the degree of freedom or simply assume a value. If the goal is to understand how fat tailed distribution you need, then I would estimate the degree of freedoms. 
