Solved – Does shape parameter = degrees of freedom for Student’s $t$ distribution in “rugarch”

Question

When I specify a GARCH-model (using "rugarch" package in R) and choose "std" as conditional distribution then the fit gives me a shape parameter. I read both, some say this is equal to DF (a thread on R-SIG-Finance), some say not.

If I use rdist function and plot the density, the higher the shape the fatter the tails, this is not very "std" like. The question is, is the shape equal to "v" in the LogLikelihood formula of Engle & Bollerslev (1986) p. 35 or is, these days, a completely different formula used for maximum likelihood estimation of GARCH models with "std" error distribution where a shape parameter is included?

Reference:

• Engle, Robert F., and Tim Bollerslev. "Modelling the persistence of conditional variances." Econometric Reviews 5.1 (1986): 1-50.

Answer 1

There are two different objects that can be named degrees of freedom in a GARCH model with Student's $t$ errors:

1. The degrees of freedom needed for calculating information criteria such as AIC or BIC is the number of model parameters being estimated, which will be the $1+s+r$ parameters from the conditional variance equation in a GARCH($s,r$) model plus the parameter determining the shape of the Student's $t$ distribution, so $1+s+r+1$ in total.
2. The degrees of freedom of the Student's $t$ distribution.

Obviously, 1. does not give the shape parameter of the Student's $t$ distribution.
Meanwhile, 2. does. This is confirmed by the answer in R-SIG-Finance list by the author of "rugarch" package (Alexios Ghalanos), and he certainly knows what he is talking about.