Skip to main content
Article thumbnail
Location of Repository

Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos

By Mototsugu Shintani and Oliver Linton


This paper derives the asymptotic distribution of nonparametric neural network estimator of the Lyapunov exponent in a noisy system proposed by Nychka et al (1992) and others. 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 datasets. In most cases we strongly reject the hypothesis of chaos; one mild exception is in some higher power transformed absolute returns, where we still find evidence against the hypothesis but it is somewhat weaker

Topics: HB Economic Theory
Publisher: Suntory and Toyota International Centres for Economics and Related Disciplines, London School of Economics and Political Science
Year: 2002
OAI identifier:
Provided by: LSE Research Online

Suggested articles


  1. (1997). 12The data is downloaded from the archive given in
  2. (2001). 14Other economic theories predict chaos in real aggregate series. The method proposed in this paper is also applied to international real output series by Shintani and
  3. (1993). 17to the return series
  4. (1998). 2;2 · E [l(Zt; µ) ¡l(Zt; µ0)] · c2 kµ ¡µ0k 2 2;2 which is required for Theorem 1 in Chen and Shen
  5. (1999). 3.3 Lyapunov exponents when d ¸ 2 10It should also be noted that Whang and Linton
  6. (1991). 35The number of hidden units (r) are selected based on BIC. QS kernel with optimal bandwidth (Andrews,
  7. (1991). 3See also McCa¤rey
  8. (1991). 4.2 Full sample estimation As discussed by Ellner
  9. (1996). 4The bootstrap may be an alternative way to conduct a statistical test for the Lyapunov exponent. This line of research is pursued by Gençay
  10. (1996). 5The well-known BDS test proposed by
  11. (1998). 6 Application to …nancial data Over the past decades, numerous models that can generate chaos in economic variables have been developed. For example, Brock and Hommes
  12. (1992). 7 Conclusion This paper derived the asymptotic distribution of the neural network Lyapunov exponent estimator proposed by Nychka etal.
  13. (1993). A long memory property of stock market returns and a new model.” doi
  14. (1997). A single-blind controlled competition among tests for nonlinearity and chaos.” doi
  15. (1996). A statistical framework for testing chaotic dynamics via Lyapunov exponents.” doi
  16. (1996). A test for independence based on the correlation dimension.” Econometric Reviews doi
  17. (1981). Abstract Inference. doi
  18. (1999). Ámin1·t·M jDµ ¤(Xt Op(1) from A4¤, respectively. The latter can be veri…ed by using the argument given in the proof of Theorem 1 in Whang and Linton
  19. (1992). An algorithm for the n Lyapunov exponents of an ndimensional unknown dynamical system.” doi
  20. (1984). An invariance principle forweakly dependentsequencesofrandomvariables.”
  21. (1984). and A5¤. For the …rst term,
  22. (1960). and H.Kesten doi
  23. (1994). Approximation rate in Sobolev norm is derived in Hornik
  24. (1993). Assumption A3 is a slightly modi…ed version of the smoothness condition …rst introduced by Barron
  25. (1997). competition data Powerful properties of the neural network approach were con…rmed by the successful results in the single-blind controlled competition conducted by William Barnett. Detail of the competition design and the results can be found in
  26. (1994). Convergence rate of sieve estimates.” doi
  27. (1991). Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data.” doi
  28. (1994). Degree of approximation results for feedforward networks approximating unknown mappings and their derivatives.” doi
  29. (1985). Ergodic theory of chaos and strange attractors.” doi
  30. (1991). Estimating Lyapunov exponents with nonparametric regression and convergence rates for feedforward single hidden layer networks.
  31. (1978). Estimating the dimension of a model.” doi
  32. (1992). Estimating the Lyapunov exponent of a chaotic systemwith nonparametric regression.” doi
  33. (1992). Finding chaos in noisy system.”
  34. (1996). For the neural network estimation, we use FUNFITS programdeveloped by Nychka
  35. (1998). FT) is the L2 metric entropy with bracketing which controls the size of the space of criterion di¤erences induced by µ 2
  36. (1996). FUNFITS: data analysis and statistical tools forestimating functions” North Carolina Institute ofStatisticsMimeoseries No.
  37. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation.” doi
  38. (1998). Hommes doi
  39. (1999). Improved rates and asymptotic normality for nonparametric neural network estimators.” doi
  40. (1967). Inequalities for the norms of a function and its derivatives in metric doi
  41. (2001). Is there chaos in the world economy? A nonparametric test using consistent standard errors.” International Economic Review, doi
  42. (1986). Liapunov exponents from time series.” Physical Review A doi
  43. (1996). Local Lyapunov exponents: Predictability depends on where you are. Nonlinear dynamics and economics, edited by
  44. (1992). Lyapunov exponents as a nonparametric diagnostic for stability analysis.” doi
  45. (1992). M rate of convergence and CLT for this term were derived by McCa¤rey
  46. (1999). minft:Xt¡12ÂT g jDµ ¤(Xt¡1)j ¢¡2 = Op(1) by A4 ¤. The third equality follows from stochastic equicontinuity argument employed in Whang and Linton
  47. (1986). Modelling Financial Time Series, doi
  48. (1992). Necessary conditions for Assumption A4 have been discussed in the literature [For example, see Nychka
  49. (1992). On learning the derivatives of an unknown mapping with multilayer feedforward networks.” doi
  50. (1997). On methods of sieves and penalization.” doi
  51. (1996). Random approximants and neural networks.” doi
  52. (1995). Random walks, breaking trend functions, and the chaotic structure of the velocity of money.” doi
  53. (1995). Robustness of nonlinearity and chaos tests to measurement error, inference method, and sample size.” doi
  54. (2000). See also note for Table 1. 31Table 3 Barnett Competition Data T = 380 T =
  55. (1998). Sieve extremum estimates for weakly dependent data.” doi
  56. (1999). Since the proof is similar to the one for Theorem 1(a) in Whang and Linton
  57. (1996). Suppose that assumptions A1 to A4 and B1 to B3 hold, BT ¸ const: £kµk3, CT = const: and r(T) satis…es r2(1+®=d¤) logr = O(T); where d¤ = d if à is homogeneous (Ã(cz) = cÃ(z)), 9See Makovoz
  58. (1999). t=1 ´t # is positive and …nite, where ´t = lnjDµ0(Xt¡1)j ¡¸: Assumption A5¤ is acondition on the properties ofthe dataaround zeroderivatives …rst employed by Whang and Linton
  59. (1984). t=1 [»it ¡¸i] ) N(0;©i) by the CLT of Herrndorf
  60. (1991). t=jjj+1 ´t´t¡jjj where ´t = flnjDµ(Xt¡1)j ¡¸g. From Proposition 1 of Andrews
  61. (1992). The (largest)Lyapunov exponent is known tobe positive (¸ = 0:408)in this model. Thisexample is almost identical to the one used by Dechert and Gençay
  62. (1994). The activation function à is a possibly nonsigmoid function satisfying
  63. (1999). The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series.” doi
  64. (1992). The authors thank Doug Nychka and Barb Bailey for providing their computer program. We also thank
  65. (1991). the covariance matrix estimation, we employ the following class of kernel functions similar to that given in Andrews
  66. (1991). The lag length (d) and the number of hidden units (r) are jointly selected based on BIC. QS kernel with optimal bandwidth (Andrews,
  67. (2002). The Suntory Centre Suntory and Toyota International Centres for Economics and Related Disciplines London doi
  68. (1999). then S("; N) is called a bracketing "-covering of FT with respect to k¢k. We de…ne H("; FT) by log(minfN : S("; N)g).) Using the result in the proof of Theorem 3.1 in Chen and White
  69. (1993). These results are very similar to those of Ding, Granger and Engle
  70. (1993). Universal approximation bounds for superpositions of a sigmoidal function.” doi
  71. (1991). we consider consistent estimation of ©. Since ´t’s are serially dependent and not identically distributed, we need to employ a heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimator (see, e.g. Andrews,
  72. (1998). we consider the approximate maximization problem where exact maximization is included as a special case when "T = 0. Similar to the case shown in Chen and Shen
  73. (1997). where the elements of J¤ t¡1 lie between those of b Jt¡1 and Jt¡1. The result follows from the argument similar (but simpler) to the one used in the proof of Theorem 2.
  74. (1992). which was also employed by Nychka
  75. (1997). µ0[b µ ¡ µ0; Xt¡1] and inner product h:; :i used in Shen

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.