Empirical asset return distributions: is chaos the culprit?

Abstract

This study employs Rescaled-range analysis; the Correlation Dimension test, and the BDS test, to analyse lengthy daily time series of financial data. Two equity and two commodity indices are examined. The results reject the hypothesis that the series are purely random, independent and identically distributed. Rather, they suggest consistency with the Pareto-Levy family of processes. Motivated by the capacity of certain chaotic models to generate data consistent with these processes, evidence is accumulated consistent with a strange attractor, a long-term memory effect, and a-periodic motion. The evidence is consistent with insights derived from the theory of non-linear dynamics.

Acknowledgements

The author thanks without implication Brock, Dechert and Scheinkman for their excellent Grassberger-Procaccia correlation dimension estimating code and Colm Kearney, Brian Lucey and, Ray Dunne for their help with certain technical aspects of this short paper. Financial support from Anglo Irish Bank Corporation plc is gratefully acknowledged.

Notes

* This study grew from some research the author commenced at Dublin City University.

¹ Lo's R/S analysis (1991) improves the precision with which we can define the alternative hypothesis.

² The author complements this analysis with a graphical inquiry: firstly, the observation-standardized P.D.F.s (over the full sample and across resolutions for each of our empirical series) are contrasted with the normal distribution. Secondly, the differentials of the empirical data P.D.F.s across resolutions with the normal distribution are demonstrably self-similar, and finally, the behaviour of the recursive first and second moments for the full sample period are assessed. As with Gaussian data, the first moment converges almost immediately, but the second moment refuses to converge throughout (i.e. infinite variance). Also, the standard deviations measured scale at a faster pace than the square root of time, for all series under analysis except wheat. These illustrative figures are available from the author upon request.

³ Given the potentially spurious nature of visual inspection alone [of the plots] – we regress C_m on ϵ for each embedding dimension, thus statistically estimating the average embedding dimension-specific slope. Then we regress these slopes on increasing embedding dimension and note the dimension when the correlation coefficient, β, falls below 0.2. Wagner et al. (1989) use a bootstrapping technique on sub samples of their data and find this cut-off point to be highly conservative.