Nonparametric correlation integral–based tests for linear and nonlinear stochastic processes

The BDS test is the best-known correlation integral–based test, and it is now an important part of most standard econometric data analysis software packages. This test depends on the proximity (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\varepsilon )$$</EquationSource> </InlineEquation> and the embedding dimension (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$m)$$</EquationSource> </InlineEquation> parameters both of which are chosen by the researcher. Although different studies (e.g., Kanzler in Very fast and correctly sized estimation of the BDS statistic. Department of Economics, Oxford University, Oxford, <CitationRef CitationID="CR10">1999</CitationRef>) have been carried out to provide an adequate selection of the proximity parameter, no relevant research has yet been done on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$m$$</EquationSource> </InlineEquation>. In practice, researchers usually compute the BDS statistic for different values of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$m$$</EquationSource> </InlineEquation>, but sometimes these results are contradictory because some of them accept the null and others reject it. This paper aims to fill this gap. To that end, we propose a new simple, yet powerful, aggregate test for independence, based on BDS outputs from a given data set, that allows the consideration of all of the information contained in several embedding dimensions without the ambiguity of the well-known BDS tests. Copyright Springer-Verlag Italia 2014