Measure of tail dependency; how dependency changes as you move to extremes of the distribution. Between two normal distributed variables you expect /-shape (for positively correlated variables).
Correlation for subset of data, shrinking quadrants
Threshold correlations of GARCH residuals.
Why residuals? Because GARCH filters whatever it can capture, remaining is what GARCH essentially cannot model.
The presence of threshold correlations in GARCH residuals is evidence that they are not independent of each other (that's not a particularly deep insight in itself tho).
Threshold correlations are never on the unconditional correlation line, i.e. conditional on being in the quadrant, assuming you're operating under such and such correlation is wrong.
Concrete example: Mkt.RF-HML. If you're in a week when both are within their 25% worst, the correlation is most certainly not -0.30, but actually positive.
We want a model that can capture this; you care about what happens in the extremes.
Asymmetry in threshold correlations: What do we see? CMA-Mom displays asymmetry, although not very strong, reflecting higher downside risk than anticipated. RMW-HML and RMW-CMA too. Correlations are not constant as you move to the edges.
CMA-HML remains very high, flat and symmetric by comparison. All others have negative or close to zero unconditional, whereas CMA-HML is above 0.50. That's very high!
Extreme positive very different to extreme negative
Why residuals? Because that's what the copula is modeling
Copula vs GARCH residuals
Can constant capture?
Not great to be honest.
Absolutely no overlap -- expect there note to be. More uncertainty the further out you go.
These models can capture some of the patterns seen in the data.
Student's t appears to be improvement over Gaussian copula.
We do have some asymmetry.
However, still not flexible enough to capture the very strong asymmetries seen between some pairs. Not enough to capture e.g. RMW-HML or RMW-CMA.