statistique, théorie et gestion de portefeuille - Docs at ISFA

concerning tail fatness can be formulated as follows: tails of the distribution of returns are heavier than a
Gaussian tail and heavier than an exponential tail; they certainly admit the existence of a finite variance
(b > 2), whereas the existence of the third (skewness) and the fourth (kurtosis) moments is questionable.
These apparent contradictory results actually do not apply to the same quantiles of the distributions of
returns. Indeed, Mantegna and Stanley (1995) have shown that the distribution of returns can be described
accurately by a Lévy law only within a range of approximately nine standard deviations, while a faster decay
of the distribution is observed beyond. This almost-but-not-quite Lévy stable description explains (in part)
the slow convergence of the returns distribution to the Gaussian law under time aggregation (Sornette 2000).
And it is precisely outside this range where the Lévy law applies that a tail index of about three have been
estimated. This can be seen from the fact that most authors who have reported a tail index b ∼ = 3 have used
some optimality criteria for choosing the sample fractions (i.e., the largest values) for the estimation of the
tail index. Thus, unlike the authors supporting stable laws, they have used only a fraction of the largest
(positive tail) and smallest (negative tail) sample values.
It would thus seem that all has been said on the distributions of returns. However, there are dissenting
views in the literature. Indeed, the class of regularly varying distributions is not the sole one able to account
for the large kurtosis and fat-tailness of the distributions of returns. Some recent works suggest alternative
descriptions for the distributions of returns. For instance, Gouriéroux and Jasiak (1998) claim that the
distribution of returns on the French stock market decays faster than any power law. Cont et al. (1997)
have proposed to use exponentially truncated stable distributions (Cont et al. 1997) while Laherrère and
Sornette (1999) suggest to fit the distributions of stock returns by the Stretched-Exponential (SE) law. These
results, challenging the traditional hypothesis of power-like tail, offer a new representation of the returns
distributions and need to be tested rigorously on a statistical ground.
A priori, one could assert that Longin (1996)’s results should rule out the exponential and Stretched-
Exponential hypotheses. Indeed, his results, based on extreme value theory, show that the distributions
of log-returns belong to the maximum domain of attraction of the Fréchet distribution, so that they are necessarily
regularly varying power-like laws. However, his study, like almost all others on this subject, has
been performed under the assumption that (1) financial time series are made of independent and identically
distributed returns and (2) the corresponding distributions of returns belong to one of only three possible
maximum domains of attraction. However, these assumptions are not fulfilled in general. While Smith
(1985)’s results indicate that the dependence of the data does not constitute a major problem in the limit
of large samples, we shall see that it can significantly bias standard statistical methods for samples of size
commonly used in extreme tails studies. Moreover, Longin’s conclusions are essentially based on an aggregation
procedure which stresses the central part of the distribution while smoothing the characteristics of
the tail, which are essential in characterizing the tail behavior.
In addition, real financial time series exhibit GARCH effects (Bollerslev 1986, Bollerslev et al. 1994) leading
to heteroskedasticity and to clustering of high threshold exceedances due to a long memory of the
volatility. These rather complex dependent structures make difficult if not questionable the blind application
of standard statistical tools for data analysis. In particular, the existence of significant dependence in the
return volatility leads to dramatic underestimates of the true standard deviation of the statistical estimators
of tail indices. Indeed, there are now many examples showing that dependences and long memories as well
as nonlinearities mislead standard statistical tests (Andersson et al. 1999, Granger and Teräsvirta 1999, for
instance). Consider the Hill’s and Pickand’s estimators, which play an important role in the study of the
tails of distributions. It is often overlooked that, for dependent time series, Hill’s estimator remains only
consistent but not asymptotically efficient (Rootzen et al. 1998). Moreover, for financial time series with a
dependence structure described by a GARCH process, Kearns and Pagan (1997) have shown that the standard
deviation of Hill’s estimator obtained by a bootstrap method can be seven to eight time larger than the
standard deviation derived under the asymptotic normality assumption. These figures are even worse for
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