# Confidence Intervals for ARMA+GARCH forecasts

I have fitted an ARMA(1,1)+GARCH(1,1) model to my logreturns series. When it comes to my standarized error's distribution however, I have opted for a Skewed Generalized Error Distribution, because of the much better fit.

My model is therefore:

$$r_t \cdot (1 - \phi_1 \cdot B) = \epsilon_t \cdot (1 + \theta_1 \cdot B)$$

where $${B}$$ is the lag-operator and:

$$\epsilon_t = \sigma_t \cdot e_t,$$

$$\sigma_t = \sqrt{\omega + \alpha_1 \cdot \epsilon^2_{t-1} + \beta_1 \cdot \sigma^2_{t-1}}$$

Finally:

$$e_t \stackrel{iid}{\sim} SKED (mean=0, variance =1, skew, shape)$$

At the moment of forecasting (I use R and the rugarch package), I have a point-estimate and a sigma estimate.

Instead of having "one-sigma estimates" I would like to have actual confidence intervals. To calculate this I would of coarse have to know the distribution of $${\epsilon_t}$$.

My actual question is: how are these distributed? I suppose that they are also SGED but with other parameters. Could someone confirm this? Or maybe show some convenient function to extract this intervals.