statistique, théorie et gestion de portefeuille - Docs at ISFA

ased on the fact that the quantity
ˆηT = ĉ
ĉ
u
+ 1
dˆ
goes to b under the null hypothesis, whatever c being positive or negative. This can be seen from the
asymptotic correspondence given by (26) and (27). Moreover, it is proved in appendix D that the variable
ˆηT
ζT = T − 1 , (36)
ˆb
asymptoticaly follows a χ 2 -distribution with one degree of freedom.
The results of this test are given in table 14. They show that H0 is more often rejected for the Dow Jones
than for the Nasdaq. Indeed, beyond quantile q12 = 95%, H0 cannot be rejected at the 95% confidence level
for the Nasdaq data. For the Dow Jones, we must consider quantiles higher than q18 = 99% in order not to
reject H0 at the 95% significance level. These results are in agreement with the central limit theorem: the
power-law regime (if it really exists) is pushed back to higher quantiles due to time agregation (recall that
our Dow Jones data is at the daily scale while our Nasdaq data is at the 5 minutes time scale).
In summary, the (SE) model encompasses the Pareto model as soon as one considers quantiles higher than
q6 = 50%. On the other hand, the null hypothesis that the true distribution is the Pareto distribution is
strongly rejected until quantiles 90% − 95% or so. Thus, within this range, the (SE) model seems the best.
But, for the very highest quantiles (above 95% − 98%) we cannot any more reject the hypothesis that the
Pareto model is the right one. Thus, for these extreme quantiles, this last model seems to slightly outperform
the (SE) model.
However, recall that section 4.1 has shown that the Pareto model is retrieved as a limiting case of the (SE)
model when the fractional exponent c and the scale factor d go to zero at an appropriated rate. Thus, all the
results above are compatible with the (SE) model in a generalized version. Indeed, defining a generalized
Stretched-Exponential model as
¯Fu(x) = exp − xc−uc dc
, c > 0
¯Fu(x) =
u b
x , c = 0 and b = limc→0 c
u c (37)
d = b,
our tests show the relevance of this representation and its superiority over all the models considered here.
Indeed, we have shown that it is the best (i.e, the most parcimonious) representation of the data for all
quantiles above q9 = 80%.
6 Discussion and Conclusions
We have presented a statistical analysis of the tail behavior of the distributions of the daily log-returns of
the Dow Jones Industrial Average and of the 5-minutes log-returns of the Nasdaq Composite index. We
have emphasized practical aspects of the application of statistical methods to this problem. Although the
application of statistical methods to the study of empirical distributions of returns seems to be an obvious
approach, it is necessary to keep in mind the existence of necessary conditions that the empirical data must
obey for the conclusions of the statistical study to be valid. Maybe the most important condition in order to
speak meaningfully about distribution functions is the stationarity of the data, a difficult issue that we have
not considered here. In particular, the importance of regime switching is now well established (Ramcham
and Susmel 1998, Ang and Bekeart 2001) and should be accounted for.
23
87
(35)