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

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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?

  • $\begingroup$ Could $|\sigma_{it}|$ simple be the absolute value of $\sigma_{it}$ (since $\sigma_{it}$ can be negative)? $\endgroup$ Mar 27 '16 at 9:37
  • $\begingroup$ 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. $\endgroup$
    – Masher
    Mar 27 '16 at 13:29
  • $\begingroup$ Sorry, my bad, I was too quick and mistook it for a covariance. Of course, variance must not be negative. $\endgroup$ Mar 27 '16 at 19:25

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