Chaotic Oscillations in Real Economic Time Series Data: Evaluation of Logistic Model Fit and Forecasting Performance

  • John Dimoticalis
  • Sotiris Zontos
  • Christos H. Skiadas
Part of the Applied Optimization book series (APOP, volume 19)


In this article, we investigate the existence of chaotic oscillations in real economic time series data. To achieve this, we use the well known logistic model equation in discrete form. We apply the logistic model to the M3 Competition time series data. At first, the data transformed in order to vary in the close interval [0,1]. Then, we search for those of the parameter values of the logistic equation that best fit them. By this approach we find a number of time series which best fitted by logistic model for values of control parameter b>3.57 (the chaotic limit). Thus, the data evolution seems to follow chaotic paths. Some implications are presented, especially the expected sensitive dependence on initial conditions (model parameter values) for chaotic paths. This issue is investigated further performing an evaluation of Logistic model forecasting performance. Finally some concluding remarks and future directions are presented.


Logistic Model M3 Competition Forecasting Time Series Analysis Chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baumol, W.J. and Quandt, R. (1985), ‘Chaos models and their implications for forecasting’, Eastern Econ. J. 11, 3–15,.Google Scholar
  2. Baumol, W.J. and Benhabib, J. (1989), ‘Chaos: Significance, Mechanism, and Economic Applications’, Journal of Economic Prospectives 3, 77–105.CrossRefGoogle Scholar
  3. Brock, W.A. (1986),’Distinguishing random and deterministic systems: a bridged version’, J. Econ. Theory 40, 168–195.CrossRefGoogle Scholar
  4. Brock, W.A. and Sayers, C.L. (1988), ‘Is the business cycle characterized by deterministic chaos?’, J. Mon. Econ. 22, 71–90.CrossRefGoogle Scholar
  5. Brock, W.A. and Malliaris, A.G. (1990), ‘Stability and chaos in dynamic economics : Differential Equations* Advanced Textbooks in Economics 77, North-Holland. Google Scholar
  6. Brock, W.A., Hsieh, D.A., and LeBaron, B. (1992), ‘Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence’, MIT Press, Second Printing. Google Scholar
  7. Casdagli, Martin. (1989), ‘Nonlinear prediction of chaotic time series’, Physica D 35, 335–356.CrossRefGoogle Scholar
  8. Cecen, A.A. and Erkal, C. (1996), ‘Distinguishing between stochastic and deterministic behavior in high frequency foreign exchange rate returns: Can non-linear dynamics help forecasting?’, Int. J. of Forecasting, 465–473.Google Scholar
  9. Ditto, W.L. and Pecora, L.M. (1993), ‘Mastering Chaos’, Scientific American, 62–68.Google Scholar
  10. Farmer, J.D. and Sidorowich, J.J. (1987), ‘Predicting chaotic time series’, Physical Review Letters, Vol 59, No 8, 845–848.CrossRefGoogle Scholar
  11. Feigenbaum, M.J. (1983), ‘Universal Behavior in nonlinear systems’, Physica D 7, 16–39.CrossRefGoogle Scholar
  12. Gerr, N.L. and Allen, J.C. (1993), ‘Stochastic versions of chaotic time series: Generalized Logistic and Henon time series models’, Physica D 68, 232–249.Google Scholar
  13. Gordon, T.J. and Greenspan, D. (1988), ‘Chaos and Fractals: New Tools for Technological and Social Forecasting’, Techn. For. and Soc. Change 34, 1–25.CrossRefGoogle Scholar
  14. Gordon, T.J. (1991), ‘Notes on forecasting a chaotic series using regression’, Techn. For. and Soc. Change 39, 337–348.CrossRefGoogle Scholar
  15. Grassberger, P. and Procaccia, I. (1983), ‘Measuring the strangeness of strange attractors’, Physica D 9, 189–208.CrossRefGoogle Scholar
  16. Harvey, A.C., (1984) ‘Time series forecasting based on the logistic curve’, J. Oper. Res. Soc., Vol 35, No 7, 641–646.Google Scholar
  17. Hinich, M.J. and Patterson, D.M. (1985), ‘Evidence of nonlinearity in daily stock returns’, J. of Business & Econ. Stat., Vol 3, No 1,69–77.Google Scholar
  18. Hsieh, D.A. (1993), ‘Implications of Nonlinear Dynamics for Financial Risk Management’, Journal of Financial and Quantitative Analysis, Vol 28, No 1,41–64.CrossRefGoogle Scholar
  19. LeBaron, Blake. (1992), ‘Nonlinear Forecasting for S&P Stock Index, in Nonlinear Modeling and Forecasting’, SFI Studies in the Sciences of Complexity, Proc. Vol XIII, Eds Casdagli M. and Eybank S, Addison-Wesley, 381–393.Google Scholar
  20. Li, Tien-Yien and Yorke, J. A. (1975), ‘Period three implies chaos’, American Math. Monthly 82, 985–992.CrossRefGoogle Scholar
  21. Lorenz, H.W. (1987), ‘Strange attractors in a multisector business cycle model’, J. of Economic Behavior and Organization 8, 397–411.CrossRefGoogle Scholar
  22. Mahajan, Vijay, Mason, C.H., and Srinivasan, V. (1986), ‘An Evaluation of estimation procedures for new product diffusion models, in: Innovation diffusion models of new product acceptance’, Mahajan V and Wind Y. (Eds), Ballinger, Cambridge MA, 202–232.Google Scholar
  23. May, R.M. (1976), ‘Simple mathematical models with very complicated dynamics’, Nature, 261, 459–467.CrossRefGoogle Scholar
  24. Meade, Nigel. (1984), ‘The use of growth curves in forecasting development: A review and appraisal’, Journal of Forecasting, Vol 3, 429–451.CrossRefGoogle Scholar
  25. Mosekilde, E. and Larsen, E.R. (1988), ‘Deterministic chaos in the beer production-distribution model’, Sys. Dyn. Rev. 4, 131–147.CrossRefGoogle Scholar
  26. Mulhern, F.J. and Caprara, R.J. (1994), ‘A nearest neighbor model for forecasting market response’, Int. J. of Forecasting 10, 181–189.CrossRefGoogle Scholar
  27. Osbom, A.R. and Provenzale, A. (1989), ‘Finite Correlation Dimension for stochastic systems with Power low Spectra’, Physica D 35, 357–381.Google Scholar
  28. Phillips, F. and Kim, N. (1996), ‘Implications of Chaos Research for New product Forecasting’, Tech. For. And Soc. Change 53, 239–261.CrossRefGoogle Scholar
  29. Scheinkman, P.A. and LeBaron, B. (1989), ‘Nonlinear dynamics and stock returns’, J. of Business, Vol 62, No 3, 313–337.CrossRefGoogle Scholar
  30. Skiadas, C.H. (1986), ‘Innovation Diffusion Models expressing asymmetry and/or positively influencing forces’, Techn. For. and Soc. Change 30, 319–330.Google Scholar
  31. Skiadas, C.H., Dimoticalis, J., and Zontos, S. (1997), ‘Chaotic Delay Models and Related Simulations’, in Proceedings of VIII International Symposium on Applied Stochastic Models and Data Analysis, Anacapri, Italy, June 11–14, Eds J. Janssen and C.N. Lauro.Google Scholar
  32. Srinivasan, V. and Mason, C.H. (1986), ‘Nonlinear least squares estimation of new product diffusion models’, Marketing Science 5, 169–178.CrossRefGoogle Scholar
  33. Sterman, J.D. (1988), ‘Deterministic chaos in models of human behavior: methodological issues and experimental results’, Sys. Dyn. Rev. 4, 148–178.CrossRefGoogle Scholar
  34. Sterman, J.D. (1989a), ‘Deterministic chaos in an experimental economic system’, J. Econ. Behavior and Organiz. 12, 1–28.CrossRefGoogle Scholar
  35. Sterman, J.D. (1989b), ‘Modeling managerial behavior: Misperceptions of feedback in a dynamic decision making experiment’, Management Science 35, 321–339.CrossRefGoogle Scholar
  36. Sugihara, G. and May, R.M. (1990), ‘Nonlinear forecasting as a way of distinguishing chaos from measurement error in times series’, Nature 344, 734–741.CrossRefGoogle Scholar
  37. Takala, K. and Viren, M. (1996), ‘Chaos and nonlinear dynamics in financial and nonfinancial time series: Evidence from Finland’, Eur. J. of Oper. Res. 93, 155–172.CrossRefGoogle Scholar
  38. Tong, Howell. (1990), ‘Non-Linear time Series: A dynamical system approach’, Clarendon Press Oxford, New York.Google Scholar
  39. Tsonis, A.A. and Eisner, J.B. (1992), ‘Nonlinear prediction as a way of distinguishing chaos from random fractal sequences’, Nature, Vol 358.Google Scholar
  40. Wolf, A., Swift, J., Swinney, H., and Vastano, J. (1985), ‘Determining Lyapounov exponents from a time series’, Physica D 16, 285–317.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • John Dimoticalis
    • 1
  • Sotiris Zontos
    • 1
  • Christos H. Skiadas
    • 1
  1. 1.Dept. of Production Engineering and ManagementTechnical University of CreteGreece

Personalised recommendations