Is there any research on whether the correlations among stocks rise when stock indices decline? Which model could account and test for that effect ? Maybe GARCH-BEKK, or some models using copulas?
You will probably be interested in the following papers:
Ang & Chen: Asymmetric Correlations of Equity Portfolios
Longin & Solnik: Extreme Correlation of International Equity Markets
The last paper goes further into exploring the implications of asymmetric correlation on portfolio selection and investor utility.
As you mention, GARCH-based methods like BEKK or DCC can address time-varying correlation and copulas are a good way to capture asymmetric dependence more generally.