If you have a question about Tsay's book (which I don't have), you should ask him about it.
The gamma functions are just the integrating constants. If you have t density and inverse CDF functions, they already incorporate those.
UV MAX garch model forecasts
Re: UV MAX garch model forecasts
The Generalized is
comp ES = ((sqrt(hhat) * ((%shape+(%invtcdf(.01,%shape))^2)/(%shape-1.0)) * %tdensity(%invtcdf(.01,%shape),%shape)/.01) - %beta(1))
And I think I divide by sqrt(%shape/(%shape-2.0)) to Standardize as in VaR
comp ES = ((sqrt(hhat) * ((%shape+(%invtcdf(.01,%shape))^2)/(%shape-1.0)) * %tdensity(%invtcdf(.01,%shape),%shape)/.01 / sqrt(%shape/(%shape-2.0))) - %beta(1))
comp ES = ((sqrt(hhat) * ((%shape+(%invtcdf(.01,%shape))^2)/(%shape-1.0)) * %tdensity(%invtcdf(.01,%shape),%shape)/.01) - %beta(1))
And I think I divide by sqrt(%shape/(%shape-2.0)) to Standardize as in VaR
comp ES = ((sqrt(hhat) * ((%shape+(%invtcdf(.01,%shape))^2)/(%shape-1.0)) * %tdensity(%invtcdf(.01,%shape),%shape)/.01 / sqrt(%shape/(%shape-2.0))) - %beta(1))
Re: UV MAX garch model forecasts
Your formula is correct (actually more correct than the wikipedia formula was---I had to correct the divisor from 1-alpha to alpha.
I think you need to understand the point of the correction for the degrees, as it's not just a "divide by this thing". The sigma in the formula described in the wikipedia page isn't the variance of the distribution; it's the scale for the t distribution. The density given is for a random variable mu+sigma*t(nu). That has mean mu and variance sigma^2*nu/(nu-2). If, instead, you have a scaled t with mean mu and *variance* h, then sigma (in the formulas) needs to be sqrt(h*(nu-2)/nu).
I think you need to understand the point of the correction for the degrees, as it's not just a "divide by this thing". The sigma in the formula described in the wikipedia page isn't the variance of the distribution; it's the scale for the t distribution. The density given is for a random variable mu+sigma*t(nu). That has mean mu and variance sigma^2*nu/(nu-2). If, instead, you have a scaled t with mean mu and *variance* h, then sigma (in the formulas) needs to be sqrt(h*(nu-2)/nu).