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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