statistique, théorie et gestion de portefeuille - Docs at ISFA

258 9. Mesure de la dépendance extrême entre deux actifs financiers
implies independence, at least in the intermediate range, since it is only an asymptotic property. Conversely,
a non-zero tail dependence λ implies the absence of asymptotic independence. Nonetheless, it does not
imply necessarily that the conditional correlation coefficients ρ + v=∞ and ρ s v=∞ are non-zero, as one would
have expected naively.
Note that the examples of Table 4 are such that λ = 0 seems to go hand-in-hand with ρ + v→∞ = 0. However,
the logical implication (λ = 0) ⇒ (ρ + v→∞ = 0) does not hold in general. A counter example is offered by
the Student’s factor model in the case where νY > νɛ (the tail of the distribution of the idiosynchratic noise
is fatter than that of the distribution of the factor). In this case, X and Y have the same tail-dependence
as ɛ and Y , which is 0 by construction. But, ρ + v=∞ and ρ s v=∞ are both 1 because a large Y almost always
gives a large X and the simultaneous occurrence of a large Y and a large ɛ can be neglected. The reason for
this absence of tail dependence (in the sense of λ) coming together with asymptotically strong conditional
correlation coefficients stems from two facts:
• first, the conditional correlation coefficients put much less weight on the extreme tails that the taildependence
parameter λ. In other words, ρ + v=∞ and ρ s v=∞ are sensitive to the marginals, i.e., there are
determined by the full bivariate distribution, while, as we said, λ is a pure copula property independent
of the marginals. Since ρ + v=∞ and ρ s v=∞ are measures of tail dependence weighted by the specific
shapes of the marginals, it is natural that they may behave differently.
• Secondly, the tail dependence λ probes the extreme dependence property of the original copula of the
random variables X and Y . On the contrary, when conditioning on Y , one changes the copula of X
and Y , so that the extreme dependence properties investigated by the conditional correlations are not
exactly those of the original copula. This last remark explains clearly why we observe what (Boyer et
al. 1997) call a “bias” in the conditional correlations. Indeed, changing the dependence between two
random variables obviously leads to changing their correlations.
The consequences are of importance. In such a situation, one measure (λ) would conclude on asymptotic
tail-independence while the other measures ρ + v=∞ and ρ s v=∞ would conclude the opposite. Thus, before
concluding on a change in the dependence structure with respect to a given parameter - the volatility or
the trend, for instance - one should check that this change does not result from the tool used to probe the
dependence. In this respect, our study allows us to shed new lights on recent controversial results about the
occurrence or abscence of contagion during the Latin American crises. As in every previous works, we find
no contagion evidence between Chile and Mexico, but contrarily to (Forbes and Rigobon 2002), we think it
is difficult to ignore the possibility of contagion towards Argentina and Brazil, and in this respect we agree
with (Calvo and Reinhart 1996).
In fact, we think that most of the discrepancies between these different studies stem from the fact that
the conditional correlation coefficient does not provide an accurate tool for probing the potential changes
of dependence. Indeed, even when the bias have been accounted for, the fat-tailness of the distributions
of returns are such that the Pearson’s coefficient is subjected to very strong statistical fluctuations which
forbid an accurate estimation of the correlation. Moreover, when studying the dependence properties, it is
interesting to free oneself from the marginal behavior of each random variable. This is why the conditional
Spearman’s rho seems a good tool: it only depends on the copula and is statistically well-behaved.
The conditional Spearman’s rho has allowed us to identify a change in the dependence structure during
downward trends in Latin American markets, similar to that found by (Longin and Solnik 2001) in their
study of the contagion across five major equity markets. It has also enabled us to put in light the asymmetry
in the contagion effects: Mexico and Chile can be potential source sof contagion toward Argentina and
Brazil, while reverse does not seem to hold. This phenomenom has been observed during the 1994 Mexican
20