14.4 Data Analysis

A set of four pairs of voting and non-voting German stocks will be subject to our empirical analysis. More precisely, our data sample retrieved from the data information service Thompson Financial Datastream, consists of $ 7290$ daily closing prices of stocks of WMF, Dyckerhoff, KSB and RWE from January $ 01$, $ 1973$, to December $ 12$, $ 2000$.

Figure 14.6 shows the performance of WMF stocks in our data period. The plot indicates an intimate relationship of both assets. Since the performance of both kinds of stocks are influenced by the same economic underlyings, their relative value should be stable over time. If this holds, the log-difference $ X_t$ of the pairs of voting ($ S_t^v$) and non-voting stocks ($ S_t^{nv}$),

$\displaystyle X_t \stackrel{\mathrm{def}}{=}\log S_t^v -\log S_t^{nv}$ (14.6)

should exhibit a reverting behavior and therefore an R/S analysis should yield estimates of the Hurst coefficient smaller than $ 0.5$. In order to reduce the number of plots we show only the plot of WMF stocks. One may start the quantlet 28910 XFGStocksPlots.xpl to see the time series for the other companies as well. First, we perform R/S analysis on both individual stocks and the voting/non-voting log-differences. In a second step, a trading strategy is applied to all four voting/non-voting log-differences.

Figure: Time series of voting and non-voting WMF stocks. 28914 XFGStocksPlots.xpl
\includegraphics[width=1.4\defpicwidth]{gTSWMF.ps}

Table 14.1 gives the R/S statistic of each individual stock and of the log-difference process of voting and non-voting stocks. While $ \hat{H}$ is close to $ 0.5$ for each time series taken separately, we find for the log differences a Hurst coefficient indicating negative persistence, i.e.  $ H<0.5$.

Table 14.1: Estimated Hurst coefficients of each stock and of log-differences.
  WMF Dyck. KSB RWE
  nv v nv v nv v nv v
Stock $ 0.51$ $ 0.53$ $ 0.57$ $ 0.52$ $ 0.53$ $ 0.51$ $ 0.50$ $ 0.51$
Differences 0.33 0.37 0.33 0.41


To test for the significance of the estimated Hurst coefficients we need to know the finite sample distribution of the R/S statistic. Usually, if the probabilistic behavior of a test statistic is unknown, it is approximated by its asymptotic distribution when the number of observations is large. Unfortunately, as, for example, Lo (1991) shows, such an asymptotic approximation is inaccurate in the case of the R/S statistic. This problem may be solved by means of bootstrap and simulation methods. A semiparametric bootstrap approach to hypothesis testing for the Hurst coefficient has been introduced by Hall et al. (2000), In the spirit of this chapter we use Brownian motion ($ H=0.5$) to simulate under the null hypothesis. Under the null hypothesis the log-difference process follows a standard Brownian motion and by Monte Carlo simulation we compute $ 99$%, $ 95$% and $ 90$% confidence intervals of the R/S statistic. The results are given in Table 14.2. While the estimated Hurst coefficients for each individual stock are at least contained in the $ 99$% confidence interval, we consider the R/S statistic for voting/non-voting log differences as significant.

Table 14.2: Simulated confidence intervals for R/S statistic for Brownian motion.
$ N$ Mean $ 90\%$ $ 95\%$ $ 99\%$
$ 7289$ $ 0.543$ $ \left[0.510,0.576\right]$ $ \left[0.504,0.582\right]$ $ \left[0.491,0.595\right]$