[R-SIG-Finance] Parameterization of the GED distribution in rugarch package

[Copula Guy]

Hi,
I am wondering about the following:

When I use the rugarch package I can get the ugarchroll output, showing me the mean, sigma, shape, skew shape(GIG) and realized values. I understand, that these are the 1 step ahead forecasted values. So I can then apply the plot function applied to it and choose the first plot, which shows me the one step ahead forecasted densities. The forecasted density is determined by the values give in the output, i.e. it uses the mean, the variance and the other estimates (shape parameter and so) to draw the different densities. 


I used the generalized error distribution in my model specification. According to the rugarch manual page 15 you an see, that this distribution is determined by three parameters: The location, scale and shape parameter. So I need three values to draw the density. Now my question is the following:

From the output table (of ugrachroll) I can see, that I have estimates for the mu sigma and shape. I think, that the mu estimate is exactly the location parameter. The shape parameter is also clear to me. Now I am not getting the following: What is the sigma?

Is this the estimate for the scale parameter beta or is it an estimate of the standard deviation?

If it is an estimate of the standard deviation, that would mean according to fomula 55, that I have to plug in the squared value to the left and the shape esimtate to the right and then solve for the beta parameter to get the right parameter and to be able to draw the density? Is this right?



Thanks a lot for your help to clarify this!