statistique, théorie et gestion de portefeuille - Docs at ISFA

68 3. Distributions exponentielles étirées contre distributions régulièrement variables
Pickand’s estimator.
The question then arises whether the many results and seemingly almost consensus obtained by ignoring
the limitations of usual statistical tools could have led to erroneous conclusions about the tail behavior of
the distributions of returns. Here, we propose to investigate once more this delicate problem of the tail
behavior of distributions of returns in order to shed new lights. To this aim, we investigate two time series:
the daily returns of the Dow Jones Industrial Average (DJ) Index over a century and the five-minutes returns
of the Nasdaq Composite index (ND) over one year from April 1997 to May 1998. These two sets of data
have been chosen since they are typical of the data sets used in most previous studies. Their size (about
20,000 data points), while significant compared with those used in investment and portfolio analysis, is
however much smaller than recent data-intensive studies using ten of millions of data points (Gopikrishnan
et al. 1998, Matia et al. 2002, Mizuno et al. 2002).
Our first conclusion is that none of the standard parametric family distributions (Pareto, exponential, stretchedexponential
and incomplete Gamma) fits satisfactorily the DJ and ND data on the whole range of either
positive or negative returns. While this is also true for the family of stretched exponential distribution, this
family appears to be the best among the four considered parametric families, in so far as it is able to fit the
data over the largest interval. Our second and main conclusion comes from the discovery that the Pareto
distribution is a limit case of the Stretched-Exponential distribution. This allows us to test the encompassing
of these two models with respect to the true data, from which it seems that the Stretched-Exponential distribution
is the most relevant model of returns distributions. The regular decay of the fractional exponent of
the SE model together with the regular increase of the tail index of the Pareto model lead us to think that the
extreme tail of true distribution of returns is fatter that any stretched-exponential, strictly speaking -i.e., with
a strickly positive fractional exponent- but thinner than any power law. Notwithstanding our best efforts, we
cannot conclude on the exact nature of the far-tail of distributions of returns. As already mentioned, other
works have proposed the so-called inverse-cubic law (b = 3) based on the analysis of distributions of returns
of high-frequency data aggregated over hundreds up to thousands of stocks. This aggregating procedure
leads to novel problems of interpretation. We think that the relevant question for most practical applications
is not to determine what is the true asymptotic tail but what is the best effective description of the tails in
the domain of useful applications. As we shall show below, it may be that the extreme asymptotic tail is
a regularly varying function with tail index b = 3 for daily returns, but this is not very useful if this tail
describes events whose recurrence time is a century or more. Our present work must thus be gauged as
an attempt to provide a simple efficient effective description of the tails of distribution of returns covering
most of the range of interest for practical applications. We feel that the efforts requested to go beyond the
tails analyzed here, while of great interest from a scientific point of view to potentially help unravel market
mechanisms, may be too artificial and unreachable to have significant applications.
The paper is organized as follows.
The next section is devoted to the presentation of our two data sets and to some of their basic statistical
properties, emphasizing their fat tailed behavior. We discuss, in particular, the importance of the so-called
“lunch effect” for the tail properties of intraday returns. We then demonstrate the presence of a significant
temporal dependence structure and study the possible non-stationary character of these time series.
Section 3 attempts to account for the temporal dependence of our time series and investigates its effect on
the determination of the extreme behavior of the tails of the distribution of returns. In this goal, we build
two simple long memory stochastic volatility processes whose stationary distributions are by construction
either asymptotically regularly varying or exponential. We show that, due to the long range dependence
on the volatility, the estimation with standard statistical estimators is severely biased. This leads to a very
unreliable estimation of the parameters. These results justify our re-examination of previous claims of
regularly varying tails.
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