2016-11-02

[gustafsoldan]

[victorandree]

Two (three with explanatory) sets of graphs:

• [ ] What is threshold correlation?
• [ ] Residual threshold correlations
• [ ] Residuals with copula models

For writing today:

• What is threshold correlation?
  • 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.
  • Limitation of the parsimonious model