2.4 Short summaries of Papers [i]

This paper presents estimates for the largest Lyapunov exponents for some selected exchange rate series, thus revealing the local stability properties of the dynamical system generating these exchange rate series. Estimates of the correlation dimension are also given which means that the number of variables sufficient to mimic a system's behavior are determined.

The correlation dimension, originally proposed by Grassberger and Procaccia (1983), provides a measure of the spatial correlation of states on an attractor generated by a dynamical system. The exchange rate series that were examined are the Swedish Krona versus Deutsche Mark, ECU, U.S. Dollar
and Yen exchange rates. Daily data from January, 1986 to August, 1995 were used.

Each exchange rate series includes 2 409 data points which, in the field of economics, represent rather long time series. The estimated values of the largest Lyapunov exponents were positive in all
exchange rate series suggesting that the behavior of these time series is chaotic.

The correlation dimension estimates revealed a low-dimensional dynamical system in the case of the Swedish Krona ECU and possibly also in the case of the Swedish Krona U.S. Dollar. In the other exchange rate series, the correlation dimension estimates continued to increase as the embedding dimension was increased.

Thus, the attractor was never completely unfolded in the reconstructed phase space which is an indication of a high noise to signal ratio. One of the shortcomings of this paper is the absence of a distributional theory that provides a framework for statistical inference of the estimates.

Prof. Mikael Bask