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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)?

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The distributional assumption for a DCC-GARCH model considers standardized model residuals (a multivariate time series). Standardization is done by scaling the raw residuals (from the conditional mean model, if any) by premultiplying them by the square root of inverted conditional variance matrix, $\hat\Sigma^{-1/2}$, for each time period, so as to make the residuals roughly uncorrelated with approximately unit variances.

In theory, each univariate series could have a different marginal distribution, and different copulas could be used to obtain different joint distributions. In practice, software implementations usually have a more limited choice of distributions; for example, you may choose between multivariate normal, multivariate skew normal and multivariate $t$-distribution. The main thing is to achieve an empirical distribution that is not too far from the assumed one. However, perhaps the idea of quasi maximum likelihood estimation could be used to defend a mismatch if the assumed distribution is multivariate normal.

The DCC-GARCH model is estimated in two stages; first, the univariate GARCH models, and second, the DCC part. In the first stage, the relevant distributional assumptions will be those of the marginal distributions. So if the assumed multivariate distribution is multivariate skew normal, I guess the marginal distributions will be univariate skew normal, and the skew parameter(s) will be important in the first stage. I am not sure how one builds a multivariate skew normal distribution from univariate skew normal distributions; if the skew parameter(s) have a role there, then they will be important also in the second stage of the estimation (the DCC part). I do not have full control of the details here, but the idea should be clear.

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  • $\begingroup$ I know how Multivariate GARCH models are estimated. The only question I had was how to incorporate the shape parameter $\gamma$ in the GARCH equations (if the shape parameter has a value different from one then we have a right or left skewed distribution). I could not find anything on the internet about what one should do with this shape parameter. $\endgroup$
    – Eren
    Jan 5 '16 at 13:53
  • $\begingroup$ @Eren, $\gamma$ does not appear in the equations because it characterizes the error distribution, not the mechanics of how $h_t$ evolves given $h_{t-1}$ and $\varepsilon_{t-1}^2$. Think about it: the GARCH equation does not change regardless of whether we assume Student $t$ errors, normal errors or whatever other errors. $\endgroup$ Jan 5 '16 at 13:57
  • $\begingroup$ True, I was confused with the mechanics of the returns which will change as in this question stats.stackexchange.com/questions/152300/… Thanks for your answers! $\endgroup$
    – Eren
    Jan 5 '16 at 14:02

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