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Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos

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Author Info
Oliver Linton
Mototsugu Shintani

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Abstract

This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. A simulation study to evaluate small sample performance is reported. We also apply our procedures to daily stock return data. In most cases, the hypothesis of chaos in the stock return series is rejected at the 1% level with an exception in some higher power transformed absolute returns.

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Paper provided by Suntory and Toyota International Centres for Economics and Related Disciplines, LSE in its series STICERD - Econometrics Paper Series with number /2003/455.

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Date of creation: May 2003
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Handle: RePEc:cep:stiecm:/2003/455

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Keywords: Artificial neural networks; nonlinear dynamics; nonlinear time series; nonparametric regression; sieve estimation;

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This paper has been announced in the following NEP Reports: References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Brock, William A. & Hommes, Cars H., 1998. "Heterogeneous beliefs and routes to chaos in a simple asset pricing model," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1235-1274, August. [Downloadable!] (restricted)
  2. Barnett, William A. & Gallant, A. Ronald & Hinich, Melvin J. & Jungeilges, Jochen A. & Kaplan, Daniel T. & Jensen, Mark J., 1997. "A single-blind controlled competition among tests for nonlinearity and chaos," Journal of Econometrics, Elsevier, vol. 82(1), pages 157-192. [Downloadable!] (restricted)
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  3. Xiaohong Chen & Xiaotong Shen, 1998. "Sieve Extremum Estimates for Weakly Dependent Data," Econometrica, Econometric Society, vol. 66(2), pages 289-314, March.
  4. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-58, May. [Downloadable!] (restricted)
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  5. Barnett, William A. & Ronald Gallant, A. & Hinich, Melvin J. & Jungeilges, Jochen A. & Kaplan, Daniel T. & Jensen, Mark J., 1995. "Robustness of nonlinearity and chaos tests to measurement error, inference method, and sample size," Journal of Economic Behavior & Organization, Elsevier, vol. 27(2), pages 301-320, July. [Downloadable!] (restricted)
  6. Mototsugu Shintani & Oliver Linton, 2003. "Is There Chaos in the World Economy? A Nonparametric Test Using Consistent Standard Errors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 44(1), pages 331-357, February. [Downloadable!] (restricted)
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  7. Dechert, W D & Gencay, R, 1992. "Lyapunov Exponents as a Nonparametric Diagnostic for Stability Analysis," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 7(S), pages S41-60, Suppl. De. [Downloadable!] (restricted)
  8. Mikael Bask & Xavier de Luna, 2002. "Characterizing the Degree of Stability of Non-linear Dynamic Models," Studies in Nonlinear Dynamics & Econometrics, Berkeley Electronic Press, vol. 6(1), pages 1002-1002. [Downloadable!] (restricted)
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  9. Ramazan Gencay & W. Dechert, 1996. "The Identification of Spurious Lyapunov Exponents in Jacobian Algorithms," Studies in Nonlinear Dynamics & Econometrics, Berkeley Electronic Press, vol. 1(3), pages 145-154. [Downloadable!] (restricted)
  10. Abhyankar, A & Copeland, L S & Wong, W, 1997. "Uncovering Nonlinear Structure in Real-Time Stock-Market Indexes: The S&P 500, the DAX, the Nikkei 225, and the FTSE-100," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 1-14, January.
  11. Serletis, Apostolos, 1995. "Random Walks, Breaking Trend Functions, and the Chaotic Structure of the Velocity of Money," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(4), pages 453-58, October.
  12. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June. [Downloadable!] (restricted)
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  13. W. A. Broock & J. A. Scheinkman & W. D. Dechert & B. LeBaron, 1996. "A test for independence based on the correlation dimension," Econometric Reviews, Taylor and Francis Journals, vol. 15(3), pages 197-235. [Downloadable!] (restricted)
  14. Chung-Ming Kuan & Halbert White, 1994. "Artificial neural networks: an econometric perspective," Econometric Reviews, Taylor and Francis Journals, vol. 13(1), pages 1-91. [Downloadable!] (restricted)
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  15. Whang, Yoon-Jae & Linton, Oliver, 1999. "The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series," Journal of Econometrics, Elsevier, vol. 91(1), pages 1-42, July. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Elena Olmedo & Ricardo Gimeno & Lorenzo Escot & Ruth Mateos, 2007. "Convergencia y Estabilidad de los Tipos de Cambio Europeos: Una Aplicación de Exponentes de Lyapunov," Cuadernos de Economía (Latin American Journal of Economics), Instituto de Economía. Pontificia Universidad Católica de Chile., vol. 44(129), pages 91-108. [Downloadable!]
  2. Rodney Wolff & Qiwei Yao & Howell Tong, 2003. "Statistical Tests for Lyapunov Exponents of Deterministic Systems," School of Economics and Finance Discussion Papers and Working Papers Series 167, School of Economics and Finance, Queensland University of Technology. [Downloadable!]
  3. William Barnett, 2005. "Comment on 'Chaotic Monetary Dynamics with Confidence'," Macroeconomics 0505017, EconWPA. [Downloadable!]
    Other versions:
  4. Dominique Guégan & Justin Leroux, 2008. "Local Lyapunov exponents: Zero plays no role in Forecasting chaotic systems," Cahiers de recherche 08-10, HEC Montréal, Institut d'économie appliquée. [Downloadable!]
  5. Mikael Bask & Tung Liu & Anna Widerberg, 2006. "The Stability of Electricity Prices: Estimation and Inference of the Lyapunov Exponent," Working Papers 200603, Ball State University, Department of Economics, revised Apr 2006. [Downloadable!]
    Other versions:
  6. Hommes, C.H. & Manzan, S., 2005. "Testing for Nonlinear Structure and Chaos in Economic Time Series: A Comment," CeNDEF Working Papers 05-14, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance. [Downloadable!]
  7. Rodney C Wolff & Qiwei Yao & Howell Tong, 2006. "Statistical tests for Lyapunov exponents of deterministic systems," Rodney Wolff Papers 2006-8, School of Economics and Finance, Queensland University of Technology. [Downloadable!]
  8. Cars Hommes & Sebastiano Manzan, 2006. "Testing for Nonlinear Structure and Chaos in Economic Time. A Comment," Tinbergen Institute Discussion Papers 06-030/1, Tinbergen Institute. [Downloadable!]
  9. Vitaliy Vandrovych, 2005. "Study of Nonlinearities in the Dynamics of Exchange Rates: Is There Any Evidence of Chaos?," Computing in Economics and Finance 2005 234, Society for Computational Economics. [Downloadable!]
  10. Mototsugu Shintani & Oliver Linton, 2000. "Is There Chaos in the World Economy? A Nonparametric Test Using Consistent Standard Errors," Working Papers 0111, Department of Economics, Vanderbilt University, revised Jun 2001. [Downloadable!]
    Other versions:
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