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?


  • $\begingroup$ some say not - could you cite any source? $\endgroup$ – Richard Hardy Dec 22 '16 at 8:37
  • $\begingroup$ Down there in the comments! My Interpretation of it is that it isn't equal... $\endgroup$ – jürgen Dec 22 '16 at 8:42
  • $\begingroup$ <stats.stackexchange.com/questions/201699/…> $\endgroup$ – jürgen Dec 22 '16 at 8:42
  • $\begingroup$ The d.f. for calculating the AIC or BIC will be simply p + q if the intercept is not included in the model (e.g. because differencing is used) -- can someone confirm this please? $\endgroup$ – David Jul 15 '17 at 6:42

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.

  • $\begingroup$ Okay, thank you for that! Really! But I have one further question, I hope it's not too specific, if I model the density of the std (with "rdist"), then the density's peak is very high for a low shape parameter and it comes down for a higher parameters ( it approaches the normal from above) , but shouldn't a student distribution approach the normal from below, the peak rises with the DF $\endgroup$ – jürgen Dec 22 '16 at 9:05
  • $\begingroup$ @jürgen, Student's $t$ distribution converges to the normal distribution when the degrees of freedom go to infinity. From around d.f.=30 and higher Student's $t$ is really close to normal, but for low d.f. it has much heavier tails than normal. Regarding the peak, what you see from rdist is correct. $\endgroup$ – Richard Hardy Dec 22 '16 at 9:20

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