# Error term/Innovation process in ARCH/GARCH processes?

I am wondering about the distribution of the error term/innovation process in a ARCH/GARCH process and its implementation, I am not sure about some points. The basic assumption is

$r_t=\sigma_t*\epsilon_t$

where the $\sigma_t$ is the volatility, modeled by ARCH/GARCH and the $\epsilon_t$ are mostly assumed to be N(0,1).

Now my questions are:

1. More sophisticated models drop this assumption. So I can say, e.g. $\epsilon_t$ follows a generalized hyperbolic distribution. So the mean does not need to be zero and the variance does not need to be equal to 1. This is correct, right?

2. If I use the rugarch package: It supports different distributional assumptions. But I am not getting the following: So they also drop the assumption of mean zero and variance one? Or are they using something like a "standardized" version?

3. Suppose I want to fit a GARCH(1,1) assuming, that the $\epsilon_t$ follow a generalized hyperbolic distribution, but the mean does not have to be zero and the variance does not need to be one. Is rugarch doing a jointly parameter estimation? So in my final output, do I get the parameters of the GARCH process and the parameters of my generalized hyperbolic distribution?

My last question is, how can I implement this?

I guess I have to use the following command:

ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1),
submodel = NULL, external.regressors = NULL, variance.targeting = FALSE),
mean.model = list(armaOrder = c(1, 1), include.mean = TRUE, archm = FALSE,
archpow = 1, arfima = FALSE, external.regressors = NULL, archex = FALSE),
distribution.model = "norm", start.pars = list(), fixed.pars = list(), ...)


the distribution.model has to be set to ghyp. Is this assuming a mean of zero and a variance of one?

I think no, right?

How can I use the hyperbolic distribution for distribution.model?

• What do you think about my answer? If it is helpful and clear, you may accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. This is how Quantitative Finance SE works. Dec 13, 2020 at 18:25

4. "How can I implement this?" You can implement this using the function ugarchspec just as you wrote.
6. "How can I use the hyperbolic distribution for distribution.model?" I don't know, but see below.