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Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos
, 2000
"... 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 est ..."
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Cited by 16 (2 self)
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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.
A Nonparametric Measure of Convergence Toward Purchasing Power Parity
, 2004
"... It has been claimed that the deviations from purchasing power parity are highly persistent and have quite long halflives under the assumption of a linear adjustment of real exchange rates. However, inspired by trade cost models, nonlinear adjustment has been widely employed in recent empirical stud ..."
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Cited by 7 (0 self)
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It has been claimed that the deviations from purchasing power parity are highly persistent and have quite long halflives under the assumption of a linear adjustment of real exchange rates. However, inspired by trade cost models, nonlinear adjustment has been widely employed in recent empirical studies. This paper proposes a simple nonparametric procedure for evaluating the speed of adjustment in the presence of nonlinearity, using the largest Lyapunov exponent of the time series. The empirical result suggests that the speed of convergence to a longrun price level is indeed faster than what was found in previous studies with linear restrictions.
Random Walk or Chaos: A Formal Test on the Lyapunov Exponent,” mimeo
, 1999
"... 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 NadarayaWatson kernel estimate of the Lyapunov exponent. We show that the estimator is consistent: The estimated Lyapunov exponent converges to zero under the rando ..."
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Cited by 2 (0 self)
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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 NadarayaWatson 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 our test to some of the standard macro and financial time series. For most of the series we considered, however, we find no significant empirical evidence of chaos. We also discuss some of the limitations of our empirical findings.
A Nonlinear Factor Analysis of Business Cycles with A Large Data Set: Evidence from Japan and the U.S. ¤
, 2002
"... This paper …rst constructs a business cycle index of Japanese economic activity based on a dynamic factor model with a large data set, using a principal components method employed by Stock and Watson (1998) in their analysis of the U.S. di¤usion index. As in the U.S., the factordi¤usion index in Ja ..."
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This paper …rst constructs a business cycle index of Japanese economic activity based on a dynamic factor model with a large data set, using a principal components method employed by Stock and Watson (1998) in their analysis of the U.S. di¤usion index. As in the U.S., the factordi¤usion index in Japan is found to be useful in the context of outofsample forecasting. Secondly, the business cycle characteristics of the U.S. and Japan are further investigated by the twostep estimation of the dynamic factor structure. The evidence suggests the possibility of nonlinearity in both the U.S. and Japan while it excludes the class of nonlinearity that can generate endogenous ‡uctuation or chaos.
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, 2009
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Estimating the predictability of economic and financial time series
, 2014
"... Abstract: The predictability of a time series is determined by the sensitivity to initial conditions of its data generating process. In this paper our goal is to characterize this sensitivity from a finite sample by assuming few hypotheses on the data generating model structure. In order to measure ..."
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Abstract: The predictability of a time series is determined by the sensitivity to initial conditions of its data generating process. In this paper our goal is to characterize this sensitivity from a finite sample by assuming few hypotheses on the data generating model structure. In order to measure the distance between two trajectories induced by a same noisy chaotic dynamic from two close initial conditions, a symmetric KullbackLeiber divergence measure is used. Our approach allows to take into account the dependence of the residual variance on initial conditions. We show it is linked to a Fisher information matrix and we investigated its expressions in the cases of covariancestationary processes and ARCH(∞) processes. Moreover, we propose a consistent nonparametric estimator of this sensitivity matrix in the case of conditionally heteroscedastic autoregressive nonlinear processes. Various statistical hypotheses can so be tested as for instance the hypothesis that the data generating process is “almost ” independently distributed at a given moment. Applications to simulated data and to the S&P500 index illustrate our findings. More particularly, we highlight a significant relationship between the sensitivity to initial conditions of the daily returns of the S&P500 and their volatility.