In this section we describe several classical characteristics of
financial time series using daily returns of German stocks from
1974 to 1996. We will concentrate, on the one hand, on the linear,
chronological (in)dependence of the returns, and on the other
hand, on the distribution characteristics. Table 10.1
displays the summarized descriptive statistics. The
autocorrelation of first order is for all stock returns close to
zero. The largest positive autocorrelation is with PREUSSAG
(0.08), the largest negative autocorrelation is with ALLIANZ
(-0.06). The majority of autocorrelations are positive (14 as
compared to 6 negative). This is an empirical phenomenon which is
also documented for the American market.
While the first order autocorrelation of the returns of all stock
returns are all close to zero, the autocorrelations of the squared
and absolute returns of all stocks are positive and significantly
larger than zero. Obviously there is a linear relationship in the
absolute values of the chronologically sequential returns. Since
the autocorrelation is positive, it can be concluded, that small
(positive or negative) returns are followed by small returns and
large returns follow large ones again. In other words, there are
quiet periods with small prices changes and turbulent periods with
large oscillations. Indeed one can further conclude that these
periods are of relatively longer duration, i.e., the
autocorrelations of squared returns from mainly very large orders
are still positive. These effects have already been examined by
Mandelbrot and Fama in the sixties. They can be modelled using,
among others, the ARCH models studied in Chapter
12. Furthermore we will consider estimates for the
skewness and kurtosis. Whereas the skewness in most cases is close
to zero and is sometimes positive, sometimes negative, the
kurtosis is in every case significantly larger than 3. The
smallest estimated kurtosis is by THYSSEN (
),
the largest by ALLIANZ (
). Under the null
hypothesis of the normal distribution, the estimates in
(3.2) and (3.3) are independent
and asymptotically normally distributed with
Table 10.1:
First order autocorrelation of the returns
, the squared returns
and the
absolute returns
as well as skewness (
),
kurtosis (
) and the Bera-Jarque test statistic (
) for the daily returns of German stocks
1974-1996.
|
 |
 |
. |
 |
. |  |
. |  |
. |  |
. | |
ALLIANZ |
-0 |
. | 0632 |
0 |
. | 3699 |
0 |
. | 3349 |
0 |
. | 0781 |
32 |
. | 409 |
207116 |
. | 0 |
BASF |
-0 |
. | 0280 |
0 |
. | 2461 |
0 |
. | 2284 |
-0 |
. | 1727 |
8 |
. | 658 |
7693 |
. | 5 |
BAYER |
-0 |
. | 0333 |
0 |
. | 3356 |
0 |
. | 2487 |
0 |
. | 0499 |
9 |
. | 604 |
10447 |
. | 0 |
BMW |
-0 |
. | 0134 |
0 |
. | 3449 |
0 |
. | 2560 |
-0 |
. | 0107 |
17 |
. | 029 |
47128 |
. | 0 |
COMMERZBANK |
0 |
. | 0483 |
0 |
. | 1310 |
0 |
. | 2141 |
-0 |
. | 2449 |
10 |
. | 033 |
11902 |
. | 0 |
DAIMLER |
-0 |
. | 0273 |
0 |
. | 4050 |
0 |
. | 3195 |
0 |
. | 0381 |
26 |
. | 673 |
134201 |
. | 0 |
DEUTSCHE BANK |
0 |
. | 0304 |
0 |
. | 2881 |
0 |
. | 2408 |
-0 |
. | 3099 |
13 |
. | 773 |
27881 |
. | 0 |
DEGUSSA |
0 |
. | 0250 |
0 |
. | 3149 |
0 |
. | 2349 |
-0 |
. | 3949 |
19 |
. | 127 |
62427 |
. | 0 |
DRESDNER |
0 |
. | 0636 |
0 |
. | 1846 |
0 |
. | 2214 |
0 |
. | 1223 |
8 |
. | 829 |
8150 |
. | 2 |
HOECHST |
0 |
. | 0118 |
0 |
. | 2028 |
0 |
. | 1977 |
-0 |
. | 1205 |
9 |
. | 988 |
11708 |
. | 0 |
KARSTADT |
0 |
. | 0060 |
0 |
. | 2963 |
0 |
. | 1964 |
-0 |
. | 4042 |
20 |
. | 436 |
72958 |
. | 0 |
LINDE |
-0 |
. | 0340 |
0 |
. | 1907 |
0 |
. | 2308 |
-0 |
. | 2433 |
14 |
. | 565 |
32086 |
. | 0 |
MAN |
0 |
. | 0280 |
0 |
. | 2824 |
0 |
. | 2507 |
-0 |
. | 5911 |
18 |
. | 034 |
54454 |
. | 0 |
MANNESMANN |
0 |
. | 0582 |
0 |
. | 1737 |
0 |
. | 2048 |
-0 |
. | 2702 |
13 |
. | 692 |
27442 |
. | 0 |
PREUSSAG |
0 |
. | 0827 |
0 |
. | 1419 |
0 |
. | 1932 |
0 |
. | 1386 |
10 |
. | 341 |
12923 |
. | 0 |
RWE |
0 |
. | 0408 |
0 |
. | 1642 |
0 |
. | 2385 |
-0 |
. | 1926 |
16 |
. | 727 |
45154 |
. | 0 |
SCHERING |
0 |
. | 0696 |
0 |
. | 2493 |
0 |
. | 2217 |
-0 |
. | 0359 |
9 |
. | 577 |
10360 |
. | 0 |
SIEMENS |
0 |
. | 0648 |
0 |
. | 1575 |
0 |
. | 1803 |
-0 |
. | 5474 |
10 |
. | 306 |
13070 |
. | 0 |
THYSSEN |
0 |
. | 0426 |
0 |
. | 1590 |
0 |
. | 1553 |
-0 |
. | 0501 |
6 |
. | 103 |
2308 |
. | 0 |
VOLKSWAGEN |
0 |
. | 0596 |
0 |
. | 1890 |
0 |
. | 1687 |
-0 |
. | 3275 |
10 |
. |
235 |
12637 |
. | 0 |
SFEReturns.xpl
|
and
From this the combined test of the normal distribution from Bera
and Jarque (
) can be derived:
is asymptotically
distribution with two degrees of
freedom. The last column in Table 10.1 shows, that in
all cases the normal distribution hypothesis is clearly rejected
by a significance level of 1% (critical value 9.21). This is
above all caused by the value of the kurtosis. Typically in
financial time series, the kurtosis is significantly larger than
3, which is caused by the frequent appearance of outliers.
Furthermore, there are more frequent appearances of very small
returns than what one would expect under the normal distribution
hypothesis. One says that the empirical distribution of the
returns is leptokurtic, which means that
the distribution is more mass around the center and in the tails
than the normal distribution. The opposite, a weaker asymmetry or
platykurtic distribution rarely appears
in financial markets.