Is the German Stock Market Chaotic ? Some NEGM- and BDS-test results for the DAX

The NEGM (Nychka, Ellner, Gallant, MacCaffrey)-test for a positive largest Lyapunov-Exponent via neural nets (NN) led to new interest in testing economic time series for chaos. This is mainly due to two reasons: First, their different concept of chaos, admitting stochastic nonlinear systems to be chaotic too, is more applicable to real economic time series. Second, neural nets seem to bear reliable results even for very noisy systems. The drawback of the algorithm is, that it is very time consuming because of multiple local minima and the ill-conditioned nature of NN-fitting. Using a sparser net topologie, replacing one hidden unit by so called shortcut connections, can improve the accuracy in many cases. We use this topologie to estimate the dominant Lyapunov-Exponent for the DAX (Deutscher Aktienindex) from 1988 to 1996. In this period the german stock market didn't seem to be chaotic - even in the week stochastic sense. Furthermore an inspection of the residuals with the BDS (Brock, Dechert, Scheinkmann)-test shows that the apparent nonlinearity in the data can't be explained by a nonlinear model with additive noise.