GARCH Model Estimation

by Tommaso Ferrari   Last Updated September 11, 2019 15:19 PM

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?

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