11.2 Statistical Analysis of German Stock Returns

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 ( $ \widehat{\mathop{\text{\rm Kurt}}}=6.1$), the largest by ALLIANZ ( $ \widehat{\mathop{\text{\rm Kurt}}}=32.4$). 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 $ \rho _1(r_t)$, the squared returns $ \rho _1(r_t^2)$ and the absolute returns $ \rho _1(\vert r_t\vert)$ as well as skewness ($ S$), kurtosis ($ K$) and the Bera-Jarque test statistic ($ BJ$) for the daily returns of German stocks 1974-1996.
$ \rho _1(r_t)$ $ \rho _1(r_t^2)$ . $ \rho _1(\vert r_t\vert)$ . $ S$ . $ K$ . $ BJ$ .
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
14288 SFEReturns.xpl


$\displaystyle \sqrt{n}\hat{S} \stackrel{{\mathcal{L}}}{\longrightarrow} {\text{\rm N}}(0,6)
$

and

$\displaystyle \sqrt{n}(\widehat{\mathop{\text{\rm Kurt}}}-3) \stackrel{{\mathcal{L}}}{\longrightarrow}
{\text{\rm N}}(0,24).
$

From this the combined test of the normal distribution from Bera and Jarque ($ BJ$) can be derived:

$\displaystyle BJ = n \left(\frac{\hat{S}^2}{6} + \frac{(\widehat{\mathop{\text{\rm Kurt}}}-3)^2}{24}\right).
$

$ BJ$ is asymptotically $ \chi^2$ 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.