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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?

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Conditional correlations do rise (this can be show via a simple R simulation) when overall volatility increases. It would be interesting to see if research if the unconditional correlations rise as well –  Quant Guy Apr 25 '12 at 2:07
    
I believe I've read that unconditional correlation increases also. –  Qbik Apr 26 '12 at 13:21
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You will probably be interested in the following papers:

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.

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