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Random Walk or Chaos: A Formal Test on the Lyapunov Exponent

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  • Joon Y. Park

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  • Yoon-Jae Whang

Abstract

A formal test on the Lyapunov exponent is developed to distinguish a random walk model from a chaotic system. The test is based on the Nadaraya-Watson kernel estimate of the Lyapunov exponent. We show that the estimator is consistent: The estimated Lyapunov exponent converges to zero under the random walk hypothesis, while it converges to a positive constant for the chaotic system. The test is thus expected to have discriminatory powers. We derive the asymptotic distribution of the estimator, and make it possible to formally test for the null hypothesis of random walk against chaos. The proposed test statistic is a simple normalization of the estimated Lyapunov exponent. It is shown that the null distribution of the test statistic is given by the range of standard Brownian motion on the unit interval. We confirm through simulation that our test performs reasonably well in finite samples. We also apply out test to some of the standard macro and financial time series. For various stock price indices, the random walk hypothesis is rather strongly rejected in favor of the presence of a chaotic behavior. Contrarily, we find little evidence of chaos for most exchange rates and interest rates.

Suggested Citation

  • Joon Y. Park & Yoon-Jae Whang, 1999. "Random Walk or Chaos: A Formal Test on the Lyapunov Exponent," Working Paper Series no9, Institute of Economic Research, Seoul National University.
  • Handle: RePEc:snu:ioerwp:no9
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    File URL: http://econ.snu.ac.kr/~ecores/activity/paper/no9.pdf
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    1. Shintani, Mototsugu & Linton, Oliver, 2004. "Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos," Journal of Econometrics, Elsevier, vol. 120(1), pages 1-33, May.
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    8. Linton, O. & Whang, Yoon-Jae, 2007. "The quantilogram: With an application to evaluating directional predictability," Journal of Econometrics, Elsevier, vol. 141(1), pages 250-282, November.
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    Cited by:

    1. Arturo Lorenzo Valdés, 2002. "Pruebas de no linealidad de los rendimientos del mercado mexicano accionario: coeficientes de Lyapunov," Estudios Económicos, El Colegio de México, Centro de Estudios Económicos, vol. 17(2), pages 305-322.
    2. Domowitz, Ian & El-Gamal, Mahmoud A., 2001. "A consistent nonparametric test of ergodicity for time series with applications," Journal of Econometrics, Elsevier, vol. 102(2), pages 365-398, June.
    3. 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.
    4. Giannerini Simone & Rosa Rodolfo, 2004. "Assessing Chaos in Time Series: Statistical Aspects and Perspectives," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 8(2), pages 1-25, May.

    More about this item

    Keywords

    Lyapunov exponent; chaos; random walk; unit root; kernel regression; Brownian motion; local time; stochastic integrals;

    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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