98 M. Corazza, A. Ellero and A. Zorzi
We deepen our analysis taking into account also a confidence levelαequal to 1% (we
recall thatχ 82 , 0. 99 = 20 .09); it results that the null is rejected 890 times, i.e., in about
29.02% of cases.
In order to check if such rejection percentages are reasonable, we perform the
following computational experiment:
- First, for each of the considered 361 stocks we generate a simulated time series of
its logarithmic returns which has the same length as the original time series, and
whose probability distribution follows a Gaussian one with mean and variance
equal to the real ones estimated for the stock^5 (the Gaussian probability distribu-
tion is chosen for coherence with the classical theory of financial markets); - Second, we perform a day-by-day analysis on the generated financial market in
the same way as for the true financial market.
Repeating the experiment 50 times, we obtain the following mean values of the
rejection percentages: 57.92% ifα=5% (about 1776 cases) and 33.50% ifα=1%
(about 1027 cases). This results have not to be considered particularly surprising. In
fact, to each of the considered stocks we associate always the same kind of probability
distribution, the Gaussian one, instead of selecting it at random as would be required
to obtain a Benford distribution (see section 2).
The fact that the rejection percentages in a classical-like market are greater than the
corresponding percentages in the true one denotes that a certain number of deviations
from Benford’s law, i.e., a certain number of days in which the financial market is
not ordinary working, is physiological. Moreover, the significant differences between
rejection percentages concerning the classical-like market and the true one can be
interpreted as a symptom of the fact that, at least from a distributional point of view,
the true financial market does not always follow what is prescribed by the classical
theory.
In Table 2 we report the 45 most rejected days at a 5% significance level with
the corresponding values of the chi-square goodness-fit-of tests with 8 degrees of
freedom.
We notice that some of the days and periods reported in Table 2 are characterised
by well known events. For instance, the Wall Street crash on February, 2007 (the most
rejected day) and the troubles of important hedge funds since 2003 (24.44% of the first
45 most rejected days falls in 2003). Nevertheless, in other rejection days/periods the
link with analogous events cannot generally be observed. In such cases the day-by-
day analysis can be profitaby used to detect hidden anomalous behaviours in financial
markets. On the other hand, the most accepted day is September 5,1995, whose value
of the chi-square goodness-fit-of test is 0.91. In Figure 3 we graphically compare the
empirical probability distributions of the most rejected and of the mostaccepted days
against Benford’s law and the uniform probability distribution.
(^5) In generating this simulated financial market, we do not consider the correlation structure
existing among the returns of the various stocks because, during the investigation period,
such a structure does not appear particularly relevant. So, the simulated financial market
can be reckoned as a reasonable approximation of the true one.