Very Fast and Correctly Sized Estimation of the Bds Statistic

This paper is concerned with the study of some fundamental aspects of the BDS test. Brock, Dechert, Scheinkman & LeBaron (Econometric Reviews, 1996) propose this non-parametric tool as a test of the null hypothesis of an independently and identically distributed (i.i.d.) time series, with power against virtually all linear and non-linear, stochastic and deterministic ("chaotic") alternatives. Unfortunately, it is extremely processing-intensive and requires an efficient computer algorithm to be viably run even on relatively small data sets. An algorithm is presented which is very fast, rather simple and easily implemented in common programming environments; a version for MATLAB is part of the paper. The algorithm overcomes a number of deficiencies of the two most widely used BDS packages by Dechert and LeBaron. Extensive Monte-Carlo simulations are conducted which show that the properties of the BDS statistic are sensitive to the choice of embedding dimension and dimensional distance, and to sample size. Unless the choice parameters are set in accordance with the recommendations emerging from the simulations, a statistical test for "iidness" is bound to be badly sized on small samples and thus yield misleading conclusions. Tables of the small-sample distribution are offered for correctly sized testing. The recommendations, tabulated quantile values and computer algorithms put forward will hopefully help render the BDS test part of the standard econometric toolbox