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

Author

Listed:
  • Mototsugu Shintani

    () (Department of Economics, Vanderbilt University)

  • Oliver Linton

    () (Department of Economics, London School of Economics)

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.

Suggested Citation

  • Mototsugu Shintani & Oliver Linton, 2003. "Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos," Vanderbilt University Department of Economics Working Papers 0309, Vanderbilt University Department of Economics.
  • Handle: RePEc:van:wpaper:0309
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    References listed on IDEAS

    as
    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.
    2. 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.
    3. Gencay Ramazan & Dechert W. Davis, 1996. "The Identification of Spurious Lyapunov Exponents in Jacobian Algorithms," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 1(3), pages 1-12, October.
    4. 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.
    5. 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.
    6. Bask Mikael & de Luna Xavier, 2002. "Characterizing the Degree of Stability of Non-linear Dynamic Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 6(1), pages 1-19, April.
    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 41-60, Suppl. De.
    8. 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.
    9. 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 453-458, October.
    10. Andrews, Donald W K, 1991. "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, Econometric Society, vol. 59(3), pages 817-858, May.
    11. 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.
    12. 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.
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    More about this item

    Keywords

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

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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