Independence in time series: another look at the BDS test

Abstract

This paper examines a statistical method derived from chaos theory. The correlation integral was proposed over a decade ago as a way of detecting chaos in a possibly partial realization of a dynamical system, because it depends on the spatial arrangement of the reconstructed attractor of the system. We exploit geometrical properties of an embedded time series to establish a test of independence in the original time series. Earlier efforts here have used the Central Limit Theorem to obtain normality as the null distribution; however, the testing procedure was, to an extent, ad hoc . By making moderately weak assumptions about the marginal distribution of the given series, we obtain a Poisson law for the correlation integral under the null hypothesis of independence, and use nonparametric methods to specify the test precisely. We compare the size and power of the present test with its predecessor and with other non-parametric tests for serial dependence.