# Gaussian QMLE in estimating CCC-GARCH model

I am having some troubles understanding the estimation of a CCC-GARCH model (where the univariate GARCH models are GJR-GARCH(1,1)) by the means of Gaussian QMLE with the likelihood function of multivariate normal distribution. All the necessary formulas are given here where $r_t$ stand for returns, $\Omega$ for the correlation matrix, $\hat{\mu_i}$ and $\bar{\sigma_i}$ are the unconditional mean and volatility used in volatility targeting and as it can be seen at the very end $\varepsilon_t$ stands for standardized residuals, but I think it is a mistake and the residuals should be used instead of the standardized residuals. $n$ stands for the number of assets and $T$ for the number of time periods. Is this formula for likelihood maximization correct? There are already some mistakes in the paper and I would be grateful if you could give your opinion on this formula. From my understanding the additional term in the likelihood function, i.e., $\sum_{t=1}^{T} \sum_{i=1}^{n} log |\sigma_{it}|$ comes from the fact that standardized residuals are used instead of regular ones in the formula. However, using standardized residuals as $\varepsilon$ makes no sense in the GARCH equation.

Also, I understand that by $|\Omega|$ I should understand the determinant of the correlation matrix, but what should I think about $|\sigma_{it}|$ since that value is a scalar? Is it simply the same value?

• Could $|\sigma_{it}|$ simple be the absolute value of $\sigma_{it}$ (since $\sigma_{it}$ can be negative)? Mar 27 '16 at 9:37
• In not a single one of my series have I encountered negative volatility. But the main question is whether this form of likelihood function is correctly specified, since there is an extra term that does not appear in other sources. Mar 27 '16 at 13:29
• Sorry, my bad, I was too quick and mistook it for a covariance. Of course, variance must not be negative. Mar 27 '16 at 19:25