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.

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 factor-di¤usion index in Japan is found to be useful in the context of out-of-sample forecasting. Secondly, the business cycle characteristics of the U.S. and Japan are further investigated by the two-step 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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