Estimating the Complexity Function of Financial Time Series: An Estimation Based on Predictive Stochastic Complexity

Using a measure of predictive stochastic complexity, this paper examines the complexity of two types of financial time series of several Pacific Rim countries, including 11 series on stock returns and 9 series on exchange-rate returns. Motivated by Chaitin's application of Kolmogorov complexity to the definition of "Life," we examine complexity as a function of sample size and call it the complexity function. According to Chaitin (1979), if a time series is truly random, then its complexity should increase at the same rate as the sample size, which means one would not gain or lose any information by fine tuning the sample size. Geometrically, this means that the complexity function is a 45 degree line. Based on this criterion, we estimate the complexity function for 20 financial time series and their iid normal surrogates. It is found that, while the complexity functions of all surrogates lie near to the 45 degree line, those of the financial time series are above it, except for the Indonesian stock return. Therefore, while the complexity of most financial time series is initially low compared to pseudo random time series, it gradually catches up as sample size increases. The catching-up effect indicates a short-lived property of financial signals. This property may lend support to the hypothesis that financial time series are not random but are composed of a sequence of structures whose birth and death can be characterized by a jump process with an embedded Markov chain. The significance of this empirical finding is also discussed in light of the recent progress in financial econometrics. Further exploration of this property may help data miners to select moderate sample sizes in either their data-preprocessing procedures or active-learning designs.