# BDS Statistic

## BDS Statistic

A statistic based upon the correlation integral which examines the probability that a purely random system could have the same scaling properties as the system under study. See: Correlation Integral.
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For large samples, the BDS statistic has a standard normal limiting distribution under the null of i.i.d.
The BDS statistic is proposed by Brock, Dechert and Scheinkman (1987), which is based on the correlation integral that has been quite robust in discerning various types of nonlinearity as well as deterministic chaos.
And then, we perform the noise reduction analysis for the noise added chaotic series by using two filtering techniques and investigate the noise cancellation capabilities of the techniques by the attractor of the series and by the BDS statistic [29-39].
The BDS statistic, which can be denoted as [W.sub.m,T([member of])] is given by
The BDS statistic applied to the standardized re siduals of exponential generalized auto regressive conditional heteroskedasticity (EGARCH) models strongly rejects the null of independent and identically distributed, indicating that conditional heteroskedasticity is not responsible for the presence of the nonlinear structures in the data.
One of the more popular statistical procedures that has evolved from recent progress in nonlinear dynamics is the BDS statistic, developed by Brock et al (1991), which tests whether a data series is independently and identically distributed (IID).
There are three tests that we employ here: the Correlation Dimension of Grassberger and Procaccia (1983), and the BDS statistic of Brock, Deckert, and Scheinkman (1987), and a measure of entropy termed Kolmogorov-Sinai invariant, also known as Kolmogorov entropy.
One of the more popular statistical procedures that has evolved from recent progress in chaos theory is the BDS statistic, developed by Brock et al.
We also used the BDS statistic devised by Brock, Dechert and Scheinkman (1987).
The critical values for the BDS statistic of the standardized residuals are developed by bootstrapping the null distribution and reported in Appendix 1.
Next, each of these series is tested for the null of IID using the BDS statistic. The above process is then repeated for four shorter time periods, which subdivides the sample period into four subperiods of equal lengths, to rule out the possibility that any observed rejection of the IID assumption in the previous step was due to nonstationarity of the data series.
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