Say we have a GARCH(1,1) equation such as
$h_t = \omega + \alpha\varepsilon^2_{t-1} + \beta h_{t-1}$
and $\varepsilon_{t-1}$ follows a standardized skew normal distribution. Using MLE we get the parameter estimates of $\omega$, $\alpha$ and $\beta$. We also get the shape parameter $\gamma$ of the skew normal distribution. If this $\gamma$ equals one we get the symmetric normal distribution.
Say, we want to estimate a DCC-GARCH model using this skew normal distributional assumption, where does this shape parameter fit? Should the GARCH equation be adjusted, for example, or should one use this parameter in the DCC equation (i.e. $Q_t = (1-a-b) + au_{t-1}u_{t-1}' +b\bar{Q}$).
The equations I stated are the well-known equations used when $\varepsilon_{t-1}$ has the normal distribution.
Basically, my question is, what should I do with this $\gamma$ shape parameter of the skew normal distribution in the context of GARCH(1,1) or DCC(1,1)-GARCH(1,1)?