A Random Walk or Color Chaos on the Stock Market? TimeFrequency
Analysis of S&P Indexes
Ping Chen
Ilya Prigogine Center for Studies in Statistical Mechanics & Complex
Systems
The University of Texas at Austin
Pages 87103
Abstract
The randomwalk (whitenoise) 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. Timefrequency analysis is introduced for evolutionary
timeseries analysis. The deterministic component from noisy data can
be recovered by a timevariant 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 HodrickPrescott
(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 colorchaos model of stockmarket movements may
establish a potential link between businesscycle theory and
assetpricing theory.
Bibliography
 Barnett, W. A., and P. Chen (1988).
"The AggregationTheoretic Monetary
Aggregates Are Chaotic and Have
Strange Attractors: An Econometric Application
of Mathematical Chaos." In: W. Barnett, E. Berndt, and H. White, eds.,
Dynamic Econometric Modeling. Cambridge: Cambridge University Press.
 Barnett, W., R. Gallant, M. Hinich, J. Jungeilges, D. Kaplan,
and M. Jensen (1994). "A SingleBlind Controlled Competition among Tests for
Nonlinearity and Chaos." Working Paper 190, Washington University, St. Louis, Missouri.
 Baba, Y., D. F. Hendry, and R. M. Starr (1992). "The Demand for M1 in the U.S.A."
Review of Economic Studies, 59:2561.
 Benhabib, J., ed. (1992). Cycle and Chaos in Economic Equilibrium.
Princeton, New Jersey: Princeton University Press.
 Bierens, H. J. (1995). "Testing the Unit
Root Hypothesis Against Nonlinear Trend Stationary, with an Application to the
U.S. Price Level and Interest Rate." Working Paper 9507, Southern Methodist University, Dallas,
Texas.
 Black, F. (1990). "Living Up to the Model." RISK, vol. 3, March.
 Bollerslev, T. (1986). "Generalized Autoregressive
Conditional Heteroscedicity." Journal of Econometrics, 31:307327.
 Brock, W. A., and C. Sayers (1988). "Is the Business
Cycle Characterized by Deterministic Chaos?" Journal of Monetary
Economics, 22:7180.
 Chen, P. (1988). "Empirical and Theoretical Evidence of
Monetary Chaos." System Dynamics Review, 4:81108.
 Chen, P. (1993). "Searching for Economic Chaos: A Challenge to Econometric Practice
and Nonlinear Tests." In: R. Day and P. Chen, eds., Nonlinear
Dynamics and Evolutionary Economics. Oxford: Oxford University Press.
 Chen, P. (1994). "Study of Chaotic
Dynamical Systems via TimeFrequency Analysis." Proceedings of IEEESP
International Symposium on TimeFrequency and TimeScale Analysis. Piscataway, New Jersey: IEEE,
pp. 357360.
 Chen, P. (1995). "Deterministic Cycles in Evolving Economy:
TimeFrequency Analysis of Business Cycles." In:
N. Aoki, K. Shiraiwa, and Y. Takahashi, eds.,
Dynamical Systems and Chaos. Singapore: World Scientific.
 Chen, P. (1996). "Trends, Shocks,
Persistent Cycles in Evolving Economy: Business Cycle Measurement in
TimeFrequency Representation." In: W. A. Barnett, A. P. Kirman, and M. Salmon, eds.,
Nonlinear Dynamics and Economics, chapter 13. Cambridge: Cambridge University Press.

Day, R., and P. Chen (1993). Nonlinear Dynamics and Evolutionary Economics.
Oxford: Oxford University Press.
 DeCoster, G. P., and D. W. Mitchell (1991a). "Nonlinear Monetary
Dynamics." Journal of Business and Economic Statistics, 9:455462.
 DeCoster, G. P., and D. W. Mitchell (1991b). "Reply." Journal of Business and
Economic Statistics, 9:455462.
 Engle, R. (1982). "Autoregressive Conditional
Heteroscedasticity with Estimates of the Variance of the United Kingdom
Inflations." Econometrica, 50:9871008.
 Fleming, J., B. Ostdiek, and R. E. Whaley (1994). "Predicting Stock Market Volatility:
A New Measure." Working Paper, Futures and Options Research Center at the Fuqua School of Business.
Durham, North Carolina: Duke University.
 Friedman, M. (1953). "The Case of Flexible Exchange Rates."
In: Essays in Positive Economics. Chicago: University of Chicago Press.
 Friedman, M. (1969). The Optimum Quantity of Money
and Other Essays. Chicago: Aldine.
 Frisch, R. (1933). "Propagation Problems and Impulse Problems in Dynamic
Economics." In: Economic Essays in Honour of Gustav Cassel. London: George Allen
& Unwin.
 Gabor, D. (1946). "Theory of Communication." J.I.E.E..(London),
93(3):429457.
 Granger, C. W. J., and T. Teräsvirta (1993).
Modeling Nonlinear Economic Relationships. Oxford: Oxford University Press.
 Hodrick, R. J., and E. C. Prescott (1981). "PostWar U.S. Business
Cycles: An Empirical Investigation." Discussion Paper 451, CarnegieMellon
University.
 King, R. G., and S. T. Rebelo (1993). "Low Frequency Filtering and
Real Business Cycles." Journal of Economic Dynamics and Control, 17:207231.
 Knight, F. H. (1921). Risk, Uncertainty and Profit.
New York: Sentry Press.
 Keynes, J. M. (1925). A Treatise on Probability. Chicago: University of Chicago
Press.
 Mehra, R., and E. C. Prescott (1985). "The Equity Premium
Puzzle." Journal of Monetary Economics, 15:145161.
 Merton, R. C. (1990). ContinuousTime Finance. Cambridge: Blackwell.
 Nelson, C. R., and C. I. Plosser (1982). "Trends and Random Walks in
Macroeconomic Time Series, Some Evidence and Implications." Journal of
Monetary Economics, 10:139162.

Prigogine, I. (1980). From Being to Becoming:
Time and Complexity in the Physical Sciences.
San Francisco: Freeman.
 Prigogine, I. (1993). "Bounded Rationality: From Dynamical
Systems to Socioeconomic Models." In: R. Day and P. Chen, eds.,
Nonlinear Dynamics and Evolutionary Economics. Oxford: Oxford University Press.
 Qian, S., and D. Chen (1994a). "Discrete Gabor
Transform." IEEE Transaction: Signal Processing, 41:24292439.
 Qian, S., and D. Chen (1994b). "Decomposition of the Wigner
Distribution and TimeFrequency Distribution
Series." IEEE Transaction: Signal Processing, 42:28362842.
 Qian, S., and
D. Chen (1996). Joint TimeFrequency Analysis.
New Jersey: PrenticeHall.
 Ramsey, J. B., C. L. Sayers, and P. Rothman (1990). "The
Statistical Properties of Dimension Calculations using Small Data Sets: Some
Economic Applications." International Economic Review, 31(4):9911020.
 Shleifer, A., and L. H. Summers (1990). "The
Noise Trader Approach in Finance." Journal of Economic Perspectives,
4(2):1933.
 Sun, M., S. Qian, X. Yan, S. B. Baumman, X. G. Xia, R. E. Dahl,
N. D. Ryan, and R. J. Sclabassi (forthcoming). "TimeFrequency Analysis and Synthesis for
Localizing Functional Activity in the Brain."
Proceedings of the IEEE on TimeFrequency Analysis, 84:9.
 Wen, K. H. (1993). "Complex Dynamics
in Nonequilibrium Economics and Chemistry." Ph.D. Dissertation,
University of Texas, Austin.

Wigner, E. P. (1932). "On the Quantum Correction for Thermodynamic Equilibrium."
Physical Review, 40:749759.
 Zarnowitz, V. (1992). Business Cycles,
Theory, History, Indicators, and Forecasting. Chicago: University of Chicago Press.