Tail dependence

In probability theory, the tail dependence of a pair of random variables describes their comovements in the tails of the distributions. It is a stylized fact of stock returns that they commonly exhibit tail dependence.^[1]

Definition

The lower tail dependence is defined as^[2]

$\lambda_l = \lim_{q\rightarrow 0} \operatorname{P}(X_2 \le F_2^{\leftarrow}(q) \mid X_1 \le F_1^{\leftarrow}(q)).$

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

The upper tail dependence is defined analogously as

$\lambda_u = \lim_{q\rightarrow 1} \operatorname{P}(X_2 > F_2^{\leftarrow}(q) \mid X_1 > F_1^{\leftarrow}(q)).$

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