statistique, théorie et gestion de portefeuille - Docs at ISFA

To fit our two data sets, section 4 proposes a general parametric representation of the distribution of returns
encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions
of the class of stretched exponential distributions in another limit. The use of regularly varying
distributions have been justified above. From a theoretical view point, the class of stretched exponentials is
motivated in part by the fact that the large deviations of multiplicative processes are generically distributed
with stretched exponential distributions (Frisch and Sornette 1997). Stretched exponential distributions are
also parsimonious examples of the important subset of sub-exponentials, that is, of the general class of
distributions decaying slower than an exponential. This class of sub-exponentials share several important
properties of heavy-tailed distributions (Embrechts et al. 1997), not shared by exponentials or distributions
decreasing faster than exponentials.
The descriptive power of these different hypotheses are compared in section 5. We first consider nested
hypotheses and use Wilk’s test to this aim. It appears that both the stretched-exponential and the Pareto
distributions are the most parsimonous models compatible with the data with a slight advantage in favor
of the stretched exponential model. Then, in order to directly compare the descriptive power of these two
models, we perform encompassing tests, which prove the validity of the two representations, but for different
quantile ranges. Finally we show that these two distributions can be set within a single model.
Section 7 summarizes our results and concludes.
2 Some basic statistical features
2.1 The data
We use two sets of data. The first sample consists in the daily returns 2 of the Dow Jones Industrial Average
Index (DJ) over the time interval from May 27, 1896 to May 31, 2000, which represents a sample size
n = 28415. The second data set contains the high-frequency (5 minutes) returns of Nasdaq Composite (ND)
index for the period from April 8, 1997 to May 29, 1998 which represents n=22123 data points. The choice
of these two data sets is justified by their similarity with (1) the data set of daily returns used by Longin
(1996) particularly and (2) the high frequency data used by Guillaume et al. (1997), Lux (2000), Müller et
al. (1998) among others.
For the intra-day Nasdaq data, there are two caveats that must be addressed. First, in order to remove the
effect of overnight price jumps, we have determined the returns separately for each of 289 days contained in
the Nasdaq data and have taken the union of all these 289 return data sets to obtain a global return data set.
Second, the volatility of intra-day data are known to exhibit a U-shape, also called “lunch-effect”, that is, an
abnormally high volatility at the begining and the end of the trading day compared with a low volatility at
the approximate time of lunch. Such effect is present in our data, as depicted on figure 1, where the average
absolute returns are shown as a function of the time within a trading day. It is desirable to correct the data
from this systematic effect. This has been performed by renormalizing the 5 minutes-returns at a given
moment of the trading day by the corresponding average absolute return at the same moment. We shall refer
to this time series as the corrected Nasdaq returns in constrast with the raw (incorrect) Nasdaq returns and
we shall examine both data sets for comparison.
Although the distributions of positive and negative returns are known to be very similar (Jondeau and
Rockinger 2001, for instance), we have chosen to treat them separately. For the Dow Jones, this gives
us 14949 positive and 13464 negative data points while, for the Nasdaq, we have 11241 positive and 10751
negative data points.
2 Throughout the paper, we will use compound returns, i.e., log-returns.
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