The BDS test, also called the Brock-Dechert-Scheinkman test, is a nonparametric test of the null hypothesis that a scalar
time series is independent
and identically distributed. For observations 
, ..., 
,
define the overlapping 
-dimensional
vectors
 |
(1)
|
where 
ranges from 1 to 
and 
.
Their sample correlation integral at distance
is
 |
(2)
|
Here 
is the indicator
function of an event 
:
it equals 1 when the event occurs and 0 otherwise. Under the null
hypothesis, the population correlation integrals satisfy 
. The BDS statistic is therefore
of the form
![W_(m,n)(epsilon)=(sqrt(n)[C_(m,n)(epsilon)-C_(1,n)(epsilon)^m])/(sigma_(m,n)(epsilon)),](/images/equations/BDSTest/NumberedEquation3.svg) |
(3)
|
where 
consistently estimates
the null standard deviation. For fixed 
and suitable regularity conditions,
has an asymptotic standard normal distribution under the
null hypothesis.
Large absolute values of the statistic provide evidence against the null hypothesis of independent and identically distributed observations. The test is often applied to residuals from a fitted time series model, where rejection indicates remaining dependence or distributional structure. Rejection by itself does not identify a particular alternative and does not establish that the data are chaotic.
