A Random Walk or Color Chaos on the Stock Market? Time-Frequency Analysis of S&P Indexes

The random-walk (white-noise) model and the harmonic model are two polar models in linear systems. A model in between is color chaos, which generates irregular oscillations with a narrow frequency (color) band. Time-frequency analysis is introduced for evolutionary time-series analysis. The deterministic component from noisy data can be recovered by a time-variant filter in Gabor space. The characteristic frequency is calculated from the Wigner decomposed distribution series. It is found that about 70 percent of fluctuations in Standard & Poor stock price indexes, such as the FSPCOM and FSDXP monthly series, detrended by the Hodrick-Prescott (HP) filter, can be explained by deterministic color chaos. The characteristic period of persistent cycles is around three to four years. Their correlation dimension is about 2.5. The existence of persistent chaotic cycles reveals a new perspective of market resilience and new sources of economic uncertainties. The nonlinear pattern in the stock market may not be wiped out by market competition under nonequilibrium situations with trend evolution and frequency shifts. The color-chaos model of stock-market movements may establish a potential link between business-cycle theory and asset-pricing theory. Acknowledgments. The algorithms of the time-frequency distribution were developed and modified by Shie Qian and Dapang Chen. The Hodrick-Prescott algorithm was suggested by Victor Zarnowitz, and provided by Finn Kydland. Their help has been indispensable to our progress in empirical analysis. The author also thanks Michael Woodford, William Barnett, Paul Samuelson, Richard Day, William Brock, Arnold Zellner, Clive Granger, and Kehong Wen for their stimulating discussions on various issues in testing economic chaos, and Philip Rothman and anonymous referees for their valuable criticisms of the early manuscript. Our interdisciplinary research in nonlinear economic dynamics is a long-term effort in the studies of complex systems supported by Professor Ilya Prigogine. Financial support from the Welch Foundation and IC2 Institute is gratefully acknowledged.