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Random walk or chaos: A formal test on the Lyapunov exponent

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  • Park, Joon Y.
  • Whang, Yoon-Jae

Abstract

A formal test on the Lyapunov exponent is developed to distinguish a random walk model from a chaotic system, which is based on the Nadaraya–Watson kernel estimator of the Lyapunov exponent. The asymptotic null distribution of our test statistic is free of nuisance parameter, and simply given by the range of standard Brownian motion on the unit interval. The test is consistent against the chaotic alternatives. A simulation study shows that the test performs reasonably well in finite samples. We apply our test to some of the standard macro and financial time series, finding no significant empirical evidence of chaos.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 169 (2012)
Issue (Month): 1 ()
Pages: 61-74

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Handle: RePEc:eee:econom:v:169:y:2012:i:1:p:61-74

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Web page: http://www.elsevier.com/locate/jeconom

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Keywords: Lyapunov exponent; Chaos; Random walk; Unit root; Kernel regression; Brownian motion; Local time; Stochastic integrals;

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References

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  1. 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.
  2. Oliver Linton & Mototsugu Shintani, 2002. "Nonparametric Neutral Network Estimation of Lyapunov Exponents and a Direct Test for Chaos," STICERD - Econometrics Paper Series /2002/434, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  3. Peter C.B. Phillips & Tassos Magdalinos, 2004. "Limit Theory for Moderate Deviations from a Unit Root," Cowles Foundation Discussion Papers 1471, Cowles Foundation for Research in Economics, Yale University.
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  5. Peter C.B. Phillips & Joon Y. Park, 1998. "Asymptotics for Nonlinear Transformations of Integrated Time Series," Cowles Foundation Discussion Papers 1182, Cowles Foundation for Research in Economics, Yale University.
  6. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(03), pages 560-586, June.
  7. 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.
  8. Diba, Behzad T & Grossman, Herschel I, 1988. "Explosive Rational Bubbles in Stock Prices?," American Economic Review, American Economic Association, vol. 78(3), pages 520-30, June.
  9. Brock, W. A., 1986. "Distinguishing random and deterministic systems: Abridged version," Journal of Economic Theory, Elsevier, vol. 40(1), pages 168-195, October.
  10. Yoon-Jae Whang & Oliver Linton, 1997. "The Asymptotic Distribution of Nonparametric Estimates of the Lyapunov Exponent for Stochastic Time Series," Cowles Foundation Discussion Papers 1130R, Cowles Foundation for Research in Economics, Yale University.
  11. Simón Sosvilla-Rivero & Fernando Fernández-Rodriguez & Julián Andrada-Félix, 2005. "Testing chaotic dynamics via Lyapunov exponents," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(7), pages 911-930.
  12. Chi-Young Choi & Young-Kyu Moh, 2007. "How useful are tests for unit-root in distinguishing unit-root processes from stationary but non-linear processes?," Econometrics Journal, Royal Economic Society, vol. 10(1), pages 82-112, 03.
  13. 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.
  14. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
  15. Rodney C Wolff & Qiwei Yao & Howell Tong, 2006. "Statistical tests for Lyapunov exponents of deterministic systems," School of Economics and Finance Discussion Papers and Working Papers Series 208i, School of Economics and Finance, Queensland University of Technology.
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Cited by:
  1. repec:att:wimass:9716 is not listed on IDEAS
  2. Oliver Linton & Mototsugu Shintani, 2001. "Is There Chaos in the World Economy? A Nonparametric Test Using Consistent Standard Errors," FMG Discussion Papers dp383, Financial Markets Group.
  3. 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.
  4. 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.

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