Tail dependence

In probability theory, the tail dependence of a pair of random variables describes their comovements in the tails of the distributions, used in extreme value theory. Random variables that appear to exhibit no correlation can show tail dependence in extreme deviations. For instance, it is a stylized fact of stock returns that they commonly exhibit tail dependence.^[1]

Definition

The lower tail dependence is defined as^[2]

${\displaystyle \lambda _{l}=\lim _{q\rightarrow 0}\operatorname {P} (X_{2}\leq F_{2}^{\leftarrow }(q)\mid X_{1}\leq F_{1}^{\leftarrow }(q)).}$

where ${\displaystyle F^{\leftarrow }(q)=\inf\{x\in \mathbb {R} :F(x)\geq q\}}$, that is, the inverse distribution function for q.

The upper tail dependence is defined analogously as

${\displaystyle \lambda _{u}=\lim _{q\rightarrow 1}\operatorname {P} (X_{2}>F_{2}^{\leftarrow }(q)\mid X_{1}>F_{1}^{\leftarrow }(q)).}$

See also

• Correlation
• Dependence

References

1. Hartmann, Philip; Straetmans, Stefan T.M.; De Vries, Casper G. (2004). "Asset Market Linkages in Crisis Periods". Review of Economics and Statistics. 86 (1): 313–326. doi:10.1162/003465304323023831.
2. McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul (2005), Quantitative Risk Management. Concepts, Techniques and Tools, Princeton Series in Finance, Princeton, NJ: Princeton University Press, ISBN 0-691-12255-5, MR 2175089, Zbl 1089.91037