Abstract
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 (\(\varepsilon )\) and the embedding dimension (\(m)\) 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, 1999) have been carried out to provide an adequate selection of the proximity parameter, no relevant research has yet been done on \(m\). In practice, researchers usually compute the BDS statistic for different values of \(m\), 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.
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Notes
The symbol \(^{\circ } \) refers to function composition.
Usually, embedding dimensions from 2 to 6 are considered because larger dimensions will require a huge amount of data points, a characteristic that is hardly ever found in economic time series.
Another possible general solution is to conduct each individual test with the significance level determined by the Bonferroni inequality.
This is an interesting property since it means that the confidence intervals for the population location have a 95 % chance of covering the population location regardless of what the underlying distribution is.
In fact, 50,000 BDS tests were calculated with 10,000 replications, since BDS test is applied over five embedding dimensions.
Equality holds for the limiting case of \(m = 2\), where only one \(m\) is used.
A table with two panels as in Table 3 has been elaborated; however, for the sake of space it is not presented. It is available upon request.
Despite the fact that MAX-bds test indirectly uses the available information due to its own concept.
We have used the correct critical values of Table 2.
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This paper was financed by a research grant ECO2012-36032-C03-03 from Ministerio de Economía y Competitividad of Spain and from the Social Sciences and Humanities Research Council of Canada.
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Matilla-García, M., Ruiz Marín, M., Dore, M.I. et al. Nonparametric correlation integral–based tests for linear and nonlinear stochastic processes. Decisions Econ Finan 37, 181–193 (2014). https://doi.org/10.1007/s10203-013-0143-0
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DOI: https://doi.org/10.1007/s10203-013-0143-0