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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.

Thanks in advance.

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If you just want a confidence interval for the sigma squared produced by the garch notice that the most popular approach is to assume that the distribution of the estimator is unknown and, as such, use non-parametric methods like bootstrapping. For more info read this answer which is also consistent with other sources like this . On bootstrapping meaning I also suggest this and this

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