Random Walk or Chaos: A Formal Test on the Lyapunov Exponent

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 Nadaraya-Watson 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 out test to some of the standard macro and financial time series. For various stock price indices, the random walk hypothesis is rather strongly rejected in favor of the presence of a chaotic behavior. Contrarily, we find little evidence of chaos for most exchange rates and interest rates.