statistique, théorie et gestion de portefeuille - Docs at ISFA

9.1. Les différentes mesures de dépendances extrêmes 247
constraining conditioning. To this aim, we consider two random variables X and Y and define their conditional
correlation coefficient ρA,B, conditioned upon X ∈ A and Y ∈ B, where A and B are two subsets of
R such that Pr{X ∈ A, Y ∈ B} > 0, by
ρA,B =
Cov(X, Y | X ∈ A, Y ∈ B)
Var(X | X ∈ A, Y ∈ B) · Var(Y | X ∈ A, Y ∈ B) . (12)
In this case, it is much more difficult to obtain general results for any specified class of distributions compared
to the previous case of conditioning on a single variable. We have only been able to give the asymptotic
behavior for a Gaussian distribution in the situation detailed below, using the expressions in (Johnson and
Kotz 1972, p 113) or proposition A.1 of (Ang and Chen 2001).
Let us assume that the pair of random variables (X,Y) has a Normal distribution with unit unconditional variance
and unconditional correlation coefficient ρ. The subsets A and B are both choosen equal to [u, +∞),
with u ∈ R+, so that we focus on the correlation coefficient conditional on the returns of both X and Y
larger than the threshold u. Denoting by ρu the correlation coefficient conditional on this particular choice
for the subsets A and B, we are able to show (see eppendix A.2) that, for large u:
ρu ∼u→∞ ρ
1 + ρ
1 − ρ
1
· , (13)
u2 which goes to zero. This decay is faster than in the case governed by (3) resulting from the conditioning on
a single variable leading to ρ + v ∼v→+∞ 1/v, but, unfortunately, we do not observe a qualitative change.
Thus, the correlation coefficient conditioned on both variables does not yield new significant information
and does not provide any special improvement with respect to the correlation coefficient conditioned on a
single variable.
1.5 Empirical evidence
We consider four national stock markets in Latin America, namely Argentina (MERVAL index), Brazil
(IBOV index), Chile (IPSA index) and Mexico (MEXBOL index). We are particularly interested in the
contagion effects which may have occurred across these markets. We will study this question for the market
indexes expressed in US Dollar to emphasize the effect of the devaluations of local currencies and so to
account for monetary crises. Doing so, we follow the same methodology as in most contagion papers (see
(Forbes and Rigobon 2002), for instance). Our sample contains the daily (log) returns of each stock in
local currency and US dollar during the time interval from January 15, 1992 to June 15, 2002 and thus
encompasses both the Mexican crisis as well as the current Argentina crisis.
Before applying the theoretical results derived above, we first need to check whether we are allowed to do
so. Namely, we have to test whether the index returns distributions are not too fat tailed. Indeed, it its well
known that the correlation coefficient exists if and only if the tail of the distribution decays faster than a
power law with tail index α = 2, and its estimator given by the Pearson’s coefficient is well behaved if at
least the fourth moment of the distribution is finite.
Figure 1 represents the complementary distribution of the positive and negative tails of the index returns in
US dollar. We observe that the positive tail clearly decays faster than a power law with tail index µ = 2.
In fact, Hill’s estimator provides a value ranging between 3 and 4 for the four indexes. The case of the
negative tail is slightly different, particularly for the Brazilian index. Indeed, for the Argentina, the Chilean
and the Mexican indexes, the negative tail behaves almost like the positive one, but for the Brazilian index,
the negative tail exponent is hardly larger than two, as confirmed by Hill’s estimator. This means that, in
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