Applications of Methods and Algorithms of Nonlinear Dynamics in Economics and Finance

  • Abdol S. Soofi
  • Andreas Galka
  • Zhe Li
  • Yuqin Zhang
  • Xiaofeng Hui
Chapter
Part of the New Economic Windows book series (NEW)

Abstract

The traditional financial econometric studies presume the underlying data generating processes (DGP) of the time series observations to be linear and stochastic. These assumptions were taken face value for a long time; however, recent advances in dynamical systems theory and algorithms have enabled researchers to observe complicated dynamics of time series data, and test for validity of these assumptions. These developments include theory of time delay embedding and state space reconstruction of the dynamical system from a scalar time series, methods in detecting chaotic dynamics by computation of invariants such as Lyapunov exponents and correlation dimension, surrogate data analysis as well as the other methods of testing for nonlinearity, and mutual prediction as a method of testing for synchronization of oscillating systems. In this chapter, we will discuss the methods, and review the empirical results of the studies the authors of this chapter have undertaken over the last decade and half. Given the methodological and computational advances of the recent decades, the authors of this chapter have explored the possibility of detecting nonlinear, deterministic dynamics in the data generating processes of the financial time series that were examined. We have conjectured that the presence of nonlinear deterministic dynamics may have been blurred by strong noise in the time series, which could give the appearance of the randomness of the series. Accordingly, by using methods of nonlinear dynamics, we have aimed to tackle a set of lingering problems that the traditional linear, stochastic time series approaches to financial econometrics were unable to address successfully. We believe our methods have successfully addressed some, if not all, such lingering issues. We present our methods and empirical results of many of our studies in this chapter.

Keywords

Nonlinear deterministic dynamics Financial integration Nonlinear prediction Synchronization of stock markets Correlation dimension Time-delay embedding 

References

  1. Afraimovich, V., Verichev, N., Rabinovich, M.: Stochastic synchronization of oscillation in dissipative systems. Radiophys. Quantum Electron. 29, 795–803 (1986)CrossRefGoogle Scholar
  2. Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716–723 (1974)CrossRefGoogle Scholar
  3. Albano, A.M., Passamante, A., Hediger, T., Farrell, M.E.: Using neural nets to look for chaos. Physica D 58, 1–9 (1992)CrossRefGoogle Scholar
  4. Bajo-Rubio, O., Fernndez-Rodrguez, F.: Chaotic behaviour in exchange-rate series: first results for the Peseta-U.S. dollar case. Econ. Lett. 39, 207–211 (1992)CrossRefGoogle Scholar
  5. Balanov, A., Janson, N., Postnov, D., Sosnovtseva, O.: Synchronization: From Simple to Complex. Springer, Heildelberg (2009)Google Scholar
  6. Breakspear, M., Terry, J.R.: Topographic organization of nonlinear interdependence in multichannel human EEG. NeuroImage 16, 822–835 (2002)CrossRefGoogle Scholar
  7. Brock, W.A., Dechert, W.: A test for independence based on the correlation dimension. Econom. Rev. 15, 197–235 (1996)CrossRefGoogle Scholar
  8. Broomhead, D.S., King, Gregory P.: Extracting qualitative dynamics from experimental data. Physica D 20, 217–236 (1986)CrossRefGoogle Scholar
  9. Cao, L.: Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50 (1997)CrossRefGoogle Scholar
  10. Cao, L., Mees, A.: Dynamics from multivariate time series. Physica D 121, 75–88 (1998)Google Scholar
  11. Cao, L., Hong, Y., Zhao, H., Deng, S.: Predicting economic time series using a nonlinear deterministic technique. Comput. Econ. 9, 149–178 (1996)CrossRefGoogle Scholar
  12. Cao, L., Kim, B.G., Kurths, J., Kim, S.: Detecting determinism in human posture control data. Int. J. Bifurcat. Chaos 8, 179–188 (1998a)CrossRefGoogle Scholar
  13. Cao, L., Mees, A., Judd, K., Froyland, G.: Determining the minimum embedding dimensions of input-output time series data. Int. J. Bifurcat. Chaos 8, 1491–1504 (1998b)CrossRefGoogle Scholar
  14. Cao, L., Soofi, A.: Nonlinear deterministic forecasting of daily dollar exchange rates. Int. J. Forecast. 15, 421–430 (1999)CrossRefGoogle Scholar
  15. Das, A., Das, P.: Chaotic analysis of the foreign exchange rates. Appl. Math. Comput. 185, 388–396 (2007)CrossRefGoogle Scholar
  16. Devaney, R.L.: An Introduction to Nonlinear Dynamical Systems, 2nd edn. Addison-Wesley, Redwood City (1989)Google Scholar
  17. Efron, B.: The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia (1982)CrossRefGoogle Scholar
  18. Fraser, A., Swinney, H.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140 (1986)CrossRefGoogle Scholar
  19. Fujisaka, H., Yamada, T.: Stability theory of synchoronised motion in coupled-oscillator system. Prog. Theor. Phys. 69, 32–47 (1983)CrossRefGoogle Scholar
  20. Galka, A.: Topics in Nonlinear Time Series Analysis: With Implications for EEG Analysis. World Scientific Publishing Company, Singapore (2000)Google Scholar
  21. Galka, A., Ozaki, T.: Testing for nonlinearity in high-dimensional time series from continuous dynamics. Physica D 158, 32–44 (2001)CrossRefGoogle Scholar
  22. Grassberger, P., Procaccia, I.: Measuring the strangeness of the strange attractors. Physica D 9, 189–208 (1983)Google Scholar
  23. Grassberger, P., Hegger, R., Kanz, H., Schaffrath, C., Screiber, T.: On noise reduction methods for chaotic data. Chaos 3, 127–141 (1993)Google Scholar
  24. Harvey, D., Leybourne, S., Newbold, P.: Testing the equality of prediction mean squared errors. Int. J. Forecast. 13, 281–291 (1997)CrossRefGoogle Scholar
  25. Henon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976)CrossRefGoogle Scholar
  26. Hilborn, R.C.: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press, Oxford (1994)Google Scholar
  27. Hinich, M.J.: Testing for gaussianity and linearity of a stationary time series. J. Time Ser. Anal. 3, 169–176 (1982)CrossRefGoogle Scholar
  28. Hsieh, D.A.: Chaos and nonlinear dynamics: application to financial markets. J. Financ. 46, 1839–1877 (1991)Google Scholar
  29. Kantz, H., Schreiber, T.: Nonlinear Time series analysis. Cambridge University Press, Cambridge (1997)Google Scholar
  30. Kaplan, D.: Math for biomedical engineering: activities for nonlinear dynamics signal processing. http://www.macalester.edu/kaplan/ (2004)
  31. Kennel, M., Brown, R., Abarbanel, H.: Determining embedding dimension for phase space reconstruction using a geometrical reconstruction. Phys. Rev. A 45, 3403–3411 (1992)CrossRefGoogle Scholar
  32. Kohzadi, N., Boyd, M.S.: Testing for chaos and nonlinear dynamics in cattle prices. Can. J. Agr. Econ. 43, 475–484 (1995)CrossRefGoogle Scholar
  33. Kugiumtzis, D.: Surrogate data test on time series. In: Soofi, A., Cao, L. (eds.) Modelling and Forecasting Financial Data: Techniques of Non- linear Dynamics. Kluwer Academic Publishers, Boston (2002)Google Scholar
  34. Larsen, C., Lam, L.: Chaos and the foreign exchange market. In: Lam, L., Naroditsky, V. (eds.) Modeling Complex Phenomena. Springer, New York (1992)Google Scholar
  35. Lisi, F., Medio, A.: Is a random walk the best exchange rate predictor? Int. J. Forecast. 13, 255–267 (1997)CrossRefGoogle Scholar
  36. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  37. May, R.: Simple mathematical models with very complicated dynamics. Nature 261(5560), 459–467 (1976)CrossRefGoogle Scholar
  38. McLeod, A.I., Li, W.K.: Diagnostic checking ARMA time series models using squared-residual autocorrelations. J. Time Ser. Anal. 4, 269–73 (1983)CrossRefGoogle Scholar
  39. Osborne, A., Provencale, A.: Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35, 357–381 (1989)CrossRefGoogle Scholar
  40. Ott, E., Sauer, T., Yorke, J.A.: Coping with chaos: analysis of chaotic data and the exploitation of chaotic systems. Wiley, New York (1994)Google Scholar
  41. Pecora, L., Carroll, T.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)CrossRefGoogle Scholar
  42. Pesin, Y.: Characteristic exponents and smooth ergodic theory. Russ. Math. Surv. 32, 55–114 (1977)CrossRefGoogle Scholar
  43. Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995)CrossRefGoogle Scholar
  44. Sauer, T., Yorke, J.A., Casdagli, M.: Embodology. J. Stat. Phys. 65, 579–616 (1991)CrossRefGoogle Scholar
  45. Scheinkman, J.A., LeBaron, B.: Nonlinear dynamics and stock returns. J. Bus. 62, 311–337 (1989)CrossRefGoogle Scholar
  46. Schiff, S., So, P., Chang, T., Burke, R., Sauer, T.: Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble. Phys. Rev. E 54, 6708–6724 (1996)CrossRefGoogle Scholar
  47. Schreiber, T.: Extremely simple nonlinear noise reduction method. Phys. Rev. E 47, 2401–2404 (1993)CrossRefGoogle Scholar
  48. Schreiber, T.: Interdisciplinary application of nonlinear time series methods. Phys. Rep. 320, 1–86 (1998)Google Scholar
  49. Screiber, T.: Constrained randomization of time series data. Phys. Rev. Lett. 80, 2105 (1999)CrossRefGoogle Scholar
  50. Schreiber, T., Schmitz, A.: Improved surrogate data for nonlinearity tests. Phys. Rev. Lett. 77, 635–638 (1996)CrossRefGoogle Scholar
  51. Soofi, A., Cao, L.: Nonlinear deterministic forecasting of daily peseta-dollar exchange rate. Econ. Lett. 62, 175–180 (1999)CrossRefGoogle Scholar
  52. Soofi, A., Cao, L. (eds.): Modelling and forecasting financial data: techniques of nonlinear dynamics. Kluwer Academic Publisher, Boston (2002a)Google Scholar
  53. Soofi, A., Cao, L.: Nonlinear forecasting of noisy financial data. In: Soofi, A., Cao, L. (eds.) Modelling and Forecasting Financial Data: Techniques of Nonlinear Dynamics. Kluwer Academic Publisher, Boston (2002b)Google Scholar
  54. Soofi, A., Galka, A.: Measuring the complexity of currency markets by fractal dimensional analysis. Int. J. Theor. Appl. Financ. 6, 553–563 (2003)CrossRefGoogle Scholar
  55. Soofi, A., Li, Z., Hui, X.: Nonlinear interdependence of Chinese stock markets. Quant. Financ. 12, 397–410 (2012)CrossRefGoogle Scholar
  56. Strogatz, S.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Cambridge (1994)Google Scholar
  57. Takens, F.: Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence. Lecture notes in mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981)Google Scholar
  58. Theiler, J.: Spurious diemnsions from correlation algorithms applied to limited time series data. Phys. Rev. A 34, 2427–2432 (1986)CrossRefGoogle Scholar
  59. Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., Farmer, J.D.: Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77–94 (1992)CrossRefGoogle Scholar
  60. Tsay, R.S.: Nonlinearity tests for time series. Biometrika 73, 461–466 (1986)CrossRefGoogle Scholar
  61. Vautard, R., Ghil, M.: Singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series. Physica D 35, 395–424 (1989)CrossRefGoogle Scholar
  62. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)CrossRefGoogle Scholar
  63. Yang, S., Brorsen, B.W.: Nonlinear dynamics of daily futures prices: conditional heteroscedasticity or chaos? J. Futures Markets 13, 175–191 (1993)CrossRefGoogle Scholar
  64. Zhang, Y., Soofi, A., Wang, S.: Testing for nonlinearity of exchange rates: an information theoretic approach. J. Econ. Stud. 38, 637–657 (2011)CrossRefGoogle Scholar
  65. Zhu, Q., Lu, L., Wang, S., Soofi, A.: Causal linkages among Shanghai, Shenzhen, and Hong Kong stock markets. Int. J. Theor. Appl. Financ. 7, 135–149 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Abdol S. Soofi
    • 1
  • Andreas Galka
    • 2
  • Zhe Li
    • 3
  • Yuqin Zhang
    • 4
  • Xiaofeng Hui
    • 5
  1. 1.Department of EconomicsUniversity of Wisconsin-PlattevillePlattevilleUSA
  2. 2.Department of NeuropediatricsChristian-Albrechts-University of KielKielGermany
  3. 3.School of Mathematics and StatisticsNortheastern University at QinhuangdaoQinhuangdaoPeople’s Republic of China
  4. 4.School of Public AdministrationUniversity of International Business and EconomicsBeijingChina
  5. 5.School of ManagementHarbin Institute of TechnologyHarbinPeople’s Republic of China

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