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312 9. Mesure de la dépendance extrême entre deux actifs financiers
them, but in every case, we found that the relevant range for the tail index estimation was between
the 1% and 5% quantiles. Tables 3 and 4 give the estimated tail index for each asset and residues
at the 1%, 2.5% and 5% quantile, for both the positive and the negative tails for the two time
sub-intervals. The second time interval from January 1980 to December 2000 is characterized by
values of the tail indexes that are homogeneous over the various quantiles and range between 3 and
4 for the negative tails and between 3 and 5 for the positive tails. There is slightly more dispersions
in the first time interval from July 1962 to December 1979.
For each asset and their residue of the regression on the market factor, we tested whether the
hypothesis, according to which the tail index measured for each asset and each residue is the same
as the tail index of the Standard & Poor’s 500 index, can be rejected at the 95% confidence level,
for a given quantile. Before proceding with the presentation of our tests, two caveats have to
be accounted for. First, due to the phenomenon of volatility clustering in financial time series,
extremes are more likely to occur together. In this situation, Hill’s estimator is no more normally
distributed with variance α 2 /k. In fact, for weakly dependent time series, it can only be asserted
that the estimator remains consistent (see Rootzén and de Haan (1998)). Moreover, as shown by
Kearns and Pagan (1997) for heteroskedastic time series, the variance of the estimated tail index
can be seven times larger than the variance given by the asymptic normality assumption. Second,
the idiosyncratic noise is estimated by substracting β times the factor from the asset return. Thus,
even when the factor and the error-term are independent, the empirically estimated residues depend
on the realizations of the factor. As a consequence, the tail index estimators for the factor and for
the idiosyncratic noise are correlated. This correlation obviously depends on the exact form of the
distributions of the factor and the indiosyncratic noise. Even without the knowledge of the true
test statistics, for both problematic points, we can assert that the fluctuations of the estimators
are larger than those given by the asymptotically normal statisics for i.i.d realizations. Thus,
performing the test under the asymptotic normality assumption is more constraining than under
the true (but unknown) test statistics, so that the non-rejection of the equality hypothesis under
the assumption of a normally distributed estimator ensures that we would not be able to reject this
hypothesis under the real statistics of the estimator.
The values which reject the equality hypothesis are indicated by a star in the tables 3 and 4. During
the second time interval from January 1980 to December 2000, only four residues have a tail index
significantly different from that of that Standard & Poor’s 500, and only in the negative tail. The
situation is not as good during the first time interval, especially for the negative tail, for which
no less than 13 assets and 10 residues out of 20 have a tail index significantly different from the
Standard & Poor’s 500 ones, for the 5% quantile. Recall that the the equality tests have been
performed under the assumption of a normally distributed estimator with variance α 2 /k which, as
explained above, is too strong an hypothesis. As a consequence, a rejection under the normality
hypothesis does not imply necessarily that the equality hypothesis would have been rejected under
the true statistics. While providing a note of caution, this statement is nevertheless not very useful
from a practical point of view. More importantly, we stress that the equality of the tail indices of
the distribution of the factor and of the idiosyncratic noise is not crucial. Indeed, we shall propose
below two different estimators for the coefficient of tail dependence. One of them does not rely on
the equality of these two tail indices and thus remains operational even when they are different
and in particular when the tail index of the idiosyncratic noise appears larger than the tail of the
factor.
To summarize, our tests confirm that the tail indexes of most stock return distributions range
between three and four, even though no better precision can be given with good significance.
Moreover, in most cases, we can assume that both the asset, the factor and the residue have the
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