Introduction
ecently, a good deal of evidence has been accumulated which suggests that prices R in certain markets and over certain time horizons may be correlated in subtle ways that elude even common statistical tests. Although there are exceptions, such correlations may be induced by an underlying nonlinear process, and in particular, one that is chaotic, or nearly chaotic. Such nonlinear processes are expected to occur, generically, in self-regulating systems. Indeed, studies in a wide range of fields have indicated the important role that such processes play in systems as diverse as the human heart, the heating of water in a teapot, animal populations, and the availability of production parts in manufacturing. Insofar as the financial and commodities markets are self-regulating systems with intricate feedback and feedforward loops, one may expect to find effects of these nonlinearities in the prices generated by those markets.'
In one important development, Brock, Dechert, and Scheinkman' have developed a new statistical test (the BDS statistic), based on the idea of the correlation dimension, a notion developed in the physical sciences in the study of nonlinear dynamical systems. The BDS statistic, and related methods of analysis3 are explicitly constructed to be sensitive to the kinds of higher dimensional correlations that are typically induced by nonlinear processes, and against which many common statistical tests have little power. Using the BDS statistic, various detrended macroeconomic data sets4 as well as more market specific data5 including the price movements of certain fixed income instru-men& have been shown to have a significant nonrandom structure, common statistical tests to the contrary, notwithstanding. Specifically, the BDS test is a test against the null hypothesis that a sequence of numbers is IID. The failure of detrended economic and financial data to pass this test, despite their consistency with other more common tests of randomness, strongly suggests that such data may be produced by processes in which there are underlying dynamics with at least a partially deterministic character. Understanding the nature of the subtle deterministic correlations in this kind of data is cer-I am grateful to the Center for the Studies of Futures Markets at Columbia University for partial support for this work. I also thank Mark Powers for his encouragement, David Hirschfeld for comments on the manuscript, and Matthew Green for help with the numerical simulations.
'For a pedagogical review of nonlinear dynamics and chaos, and its applicability to finance see R. Savit (1988a).