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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:
oai:eprints.lse.ac.uk:2093

Provided by:
LSE Research Online

- (1997). 12The data is downloaded from the archive given in
- (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
- (1993). 17to the return series
- (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
- (1999). 3.3 Lyapunov exponents when d ¸ 2 10It should also be noted that Whang and Linton
- (1991). 35The number of hidden units (r) are selected based on BIC. QS kernel with optimal bandwidth (Andrews,
- (1991). 3See also McCa¤rey
- (1991). 4.2 Full sample estimation As discussed by Ellner
- (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
- (1996). 5The well-known BDS test proposed by
- (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
- (1992). 7 Conclusion This paper derived the asymptotic distribution of the neural network Lyapunov exponent estimator proposed by Nychka etal.
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- (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
- (1992). An algorithm for the n Lyapunov exponents of an ndimensional unknown dynamical system.”
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- (1984). and A5¤. For the …rst term,
- (1960). and H.Kesten
- (1994). Approximation rate in Sobolev norm is derived in Hornik
- (1993). Assumption A3 is a slightly modi…ed version of the smoothness condition …rst introduced by Barron
- (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
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- (1992). Finding chaos in noisy system.”
- (1996). For the neural network estimation, we use FUNFITS programdeveloped by Nychka
- (1998). FT) is the L2 metric entropy with bracketing which controls the size of the space of criterion di¤erences induced by µ 2
- (1996). FUNFITS: data analysis and statistical tools forestimating functions” North Carolina Institute ofStatisticsMimeoseries No.
- (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation.”
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- (1967). Inequalities for the norms of a function and its derivatives in metric
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- (1986). Liapunov exponents from time series.” Physical Review A
- (1996). Local Lyapunov exponents: Predictability depends on where you are. Nonlinear dynamics and economics, edited by
- (1992). Lyapunov exponents as a nonparametric diagnostic for stability analysis.”
- (1992). M rate of convergence and CLT for this term were derived by McCa¤rey
- (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
- (1986). Modelling Financial Time Series,
- (1992). Necessary conditions for Assumption A4 have been discussed in the literature [For example, see Nychka
- (1992). On learning the derivatives of an unknown mapping with multilayer feedforward networks.”
- (1997). On methods of sieves and penalization.”
- (1996). Random approximants and neural networks.”
- (1995). Random walks, breaking trend functions, and the chaotic structure of the velocity of money.”
- (1995). Robustness of nonlinearity and chaos tests to measurement error, inference method, and sample size.”
- (2000). See also note for Table 1. 31Table 3 Barnett Competition Data T = 380 T =
- (1998). Sieve extremum estimates for weakly dependent data.”
- (1999). Since the proof is similar to the one for Theorem 1(a) in Whang and Linton
- (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 Ã is homogeneous (Ã(cz) = cÃ(z)), 9See Makovoz
- (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
- (1984). t=1 [»it ¡¸i] ) N(0;©i) by the CLT of Herrndorf
- (1991). t=jjj+1 ´t´t¡jjj where ´t = flnjDµ(Xt¡1)j ¡¸g. From Proposition 1 of Andrews
- (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
- (1994). The activation function Ã is a possibly nonsigmoid function satisfying
- (1999). The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series.”
- (1992). The authors thank Doug Nychka and Barb Bailey for providing their computer program. We also thank
- (1991). the covariance matrix estimation, we employ the following class of kernel functions similar to that given in Andrews
- (1991). The lag length (d) and the number of hidden units (r) are jointly selected based on BIC. QS kernel with optimal bandwidth (Andrews,
- (2002). The Suntory Centre Suntory and Toyota International Centres for Economics and Related Disciplines London
- (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
- (1993). These results are very similar to those of Ding, Granger and Engle
- (1993). Universal approximation bounds for superpositions of a sigmoidal function.”
- (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,
- (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
- (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.
- (1992). which was also employed by Nychka
- (1997). µ0[b µ ¡ µ0; Xt¡1] and inner product h:; :i used in Shen

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