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70 3. Distributions exponentielles étirées contre distributions régulièrement variables
Table 1 summarizes the main statistical properties of these two time series (both for the raw and for the
corrected Nasdaq returns) in terms of the average returns, their standard deviations, the skewness and the
excess kurtosis for four time scales of five minutes, an hour, one day and one month. The Dow Jones
exhibits a significaltly negative skewness, which can be ascribed to the impact of the market crashes. The
raw Nasdaq returns are significantly positively skewed while the returns corrected for the “lunch effect” are
negatively skewed, showing that the lunch effect plays an important role in the shaping of the distribution
of the intra-day returns. Note also the important decrease of the kurtosis after correction of the Nasdaq
returns for lunch effect, confirming the strong impact of the lunch effect. In all cases, the excess-kurtosis are
high and remains significant even after a time aggregation of one month. Jarque-Bera’s test (Cromwell et
al. 1994), a joint statistic using skewness and kurtosis coefficients, is used to reject the normality assumption
for these time series.
2.2 Existence of time dependence
It is well-known that financial time series exhibit complex dependence structures like heteroskedasticity
or non-linearities. These properties are clearly observed in our two times series. For instance, we have
estimated the statistical characteristic V (for positive random variables) called coefficient of variation
V = Std(X)
E(X)
, (1)
which is often used as a testing statistic of the randomness property of a time series. It can be applied to
a sequence of points (or, intervals generated by these points on the line). If these points are “absolutely
random,” that is, generated by a Poissonian flow, then the intervals between them are distributed according
to an exponential distribution for which V = 1. If V > 1 are associated with a clustering phenomenon. We estimated V = V (u) for extrema X > u
and X < −u as function of threshold u (both for positive and for negative extrema). The results are shown
in figure 2 for the Dow Jones daily returns. As the results are essentially the same for the Nasdaq, we do not
show them. Figure 2 shows that, in the main range |X| < 0.02, containing ∼ 95% of the sample, V increases
with u, indicating that the “clustering” property becomes stronger as the threshold u increases.
We have then applied several formal statistical tests of independence. We have first performed the Lagrange
multiplier test proposed by Engle (1984) which leads to the T · R 2 test statistic, where T denotes the sample
size and R 2 is the determination coefficient of the regression of the squared centered returns xt on a constant
and on q of their lags xt−1,xt−2,··· ,xt−q. Under the null hypothesis of homoskedastic time series, T · R 2
follows a χ 2 -statistic with q degrees of freedom. The test have been performed up to q = 10 and, in every
case, the null hypothesis is strongly rejected, at any usual significance level. Thus, the time series are
heteroskedastics and exhibit volatility clustering. We have also performed a BDS test (Brock et al. 1987)
which allows us to detect not only volatility clustering, like in the previous test, but also departure from
iid-ness due to non-linearities. Again, we strongly rejects the null-hypothesis of iid data, at any usual
significance level, confirming the Lagrange multiplier test.
3 Can long memory processes lead to misleading measures of extreme properties?
Since the descriptive statistics given in the previous section have clearly shown the existence of a significant
temporal dependence structure, it is important to consider the possibility that it can lead to erroneous conclusions
on estimated parameters. We first briefly recall the standard procedures used to investigate extremal
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