Constant conditional correlation (CCC) shows high spikes compared to Dynamic CC (DCC), why? I have calculated 5 beta factors (beta is defined as the ratio between covariance between two time series and the variance between the one series).
When I used CCC with GJR dynamics I get this

When calculating with DCC the picture looks very similar just without this extreme spikes of the blue line. 
I am searching for an explanation. My first guess: The constant correlation matrix is not able to capture breaks in variance and covariance adequately?
But this is very high level. Do you have a more detailed idea for this phenomen?
 A: The diagonal elements of the conditional covariance matrix are modelled separately by univariate GJRGARCH models in both CCC-GJRGARCH and DCC-GJRGARCH. Thus the fitted variances will be the same in both cases. Therefore, the differences in betas (which are ratios of covariances to variances) will be due to differences in cond. covariances, or more precisely cond. correlations. 
When the spikes in betas are more pronounced in CCC than in DCC, it means cond. correlations at these spike periods are higher in CCC than in DCC. Since cond. correlations are constant in CCC, it means the fitted cond. correlations do not dip in those periods, while they do dip in DCC. (You can inspect the fitted cond. correlations to see if this is indeed the case.)
Whether CCC or DCC fits the data better can be seen by comparing the AIC or BIC values of the models. 
Regarding breaks in variance and covariance, neither CCC nor DCC produces them by default. There is dynamics, but not breaks in the traditional sense. You would need to include dummy variables or fit the models on subsamples to properly account for breaks.
