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

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
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