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9.1. Les différentes mesures de dépendances extrêmes 257
parametric method exists, which is based upon the mathematical results by (Ledford and Tawn 1996, Ledford
and Tawn 1998) and (Coles et al. 1999) and has recently been applied by (Poon et al. 2001). The method
consists in tranforming the original random variables X and Y into Fréchet random variables denoted by S
and T respectively. Then, considering the variable Z = min{S, T }, its survival distribution is:
Pr{Z > z} = d · z 1/η
as z → ∞, (35)
and ¯ λ = 2 · η − 1, with λ = 0 if ¯ λ < 1 or ¯ λ = 1 and λ = d. The parameters η and d can be estimated
by maximum likelihood, and deriving their asymptotic statistics allows one to test whether the hypothesis
¯λ = 1 can be rejected or not, and consequently, whether the data present tail dependence or not.
We have implemented this procedure and the estimated values of the coefficient of tail dependence are given
in table 1 both for the positive and the negative tails. Our tests show that we cannot reject the hypothesis of
tail dependence between the four considered Latin American markets (Argentina, Brazil, Chile and Mexico).
Notice that the positive tail dependence is almost always slightly smaller than the negative one, which could
be linked with the trend asymmetry of (Longin and Solnik 2001), but it turns out that these differences are
not statistically significant. These results indicate that, according to this analysis of the extreme dependence
coefficient, the propensity of extreme co-movements is almost the same for each pair of stock markets:
even if the transmission mechanisms of a crisis are different from a country to another one, the propagation
occur with the same probability overall. Thus, the subsequent risks are the same. In table 2, we also
give the coefficients of tail dependence estimated under the Student’s copula (or in fact any copula derived
from an elliptical distribution) with three dregrees of freedom, given by expression (26). One can observe
a remarkable agreement between these values and the non-parametric estimates given in table 1. This
is consistent with the results given by the conditional Spearman’s ρ, for which we have shown that the
Student’s copula seems to reasonably account for the extreme dependence.
4 Summary and Discussion
Table 3 summarizes the asymptotic dependences for large v and u of the signed conditional correlation
coefficient ρ + v , the unsigned conditional correlation coefficient ρ s v and the correlation coefficient ρu conditioned
on both variables for the bivariate Gaussian, the Student’s model, the Gaussian factor model and
the Student’s factor model. Our results provide a quantitative proof that conditioning on exceedance leads
to conditional correlation coefficients that may be very different from the unconditional correlation. This
provides a straightforward mechanism for fluctuations or changes of correlations, based on fluctuations of
volatility or changes of trends. In other words, the many reported variations of correlation structure might
be in large part attributed to changes in volatility (and statisical uncertainty).
We also suggest that the distinct dependences as a function of exceedance v and u of the conditional correlation
coefficients may offer novel tools for characterizing the statistical multivariate distributions of extreme
events. Since their direct characterization is in general restricted by the curse of dimensionality and the scarsity
of data, the conditional correlation coefficients provide reduced robust statistics which can be estimated
with reasonable accuracy and reliability. In this respect, our empirical results encourage us to assert that
a Student’s copula, or more generally and elliptical copula, with a tail index of about three seems able to
account for the main extreme dependence properties investigated here.
Table 4 gives the asymptotic values of ρ + v , ρ s v and ρu for v → +∞ and u → ∞ in order to compare them
with the tail-dependence λ.
These two tables only scratch the surface of the rich sets of measures of tail and extreme dependences. We
have shown that complete independence implies the absence of tail dependence: λ = 0. But λ = 0 does not
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