Nonlinearity and Complexity of Stock Returns

M.A. Kaboudan
Penn State University
mak7@psuvm.psu.edu

Abstract

This paper applies a measure of complexity of time series to three frequencies of ten different stock returns to determine their relative complexity and hence predictability. The complexity measure is a ratio. It is designed such that data generated by a pseudo-random process (i.e. computer-generated random numbers) yields a ratio of approximately one, while data from a purely deterministic process yields a ratio that approaches zero. Among the ten stocks, five trade on the NYSE, while the others trade on the NASDAQ. Each stock is represented by three series of returns computed at different frequencies. Two of the frequencies are time stamped every five minutes and every minute, while the third is returns computed only when there is a price change. Three conclusions are obtained from the ratios computed: (1) Price returns taken every five minutes demonstrated random behavior. (2) Price returns taken every minute and every price change are more predictable. (3) Complexity or predictability differs between stocks traded on the NYSE and those traded on the NASDAQ.

The computation of the complexity ratio finds its origin in the literature on chaos theory. It is based on the correlation dimension or exponent. First the exponent of a time series data set is computed. Second, that series is randomly scrambled and the correlation dimension is re-computed. The effect of scrambling the observed sequence on the correlation dimension estimate will depend on the complexity or predictability of the data generating process. Clearly, the more complex it is the lesser the effect. Given that pseudo random data are highly complex and unpredictable, the ratio of the correlation dimension estimate after to the dimension before shoud be approximately one. For deterministic processes, the ratio of before to after dimension estimates will be less than one. The measure seems to work well with a minimum of 1000 observations.

Software and instructions:

• theta.exe
• theta.ps