A Note on the Distribution of BDS Statistics for a Real Exchange Rate Series
| Author | Joanne Padmore,Catherine Ellis,David Chappell |
| Date | 01 August 1996 |
| DOI | http://doi.org/10.1111/j.1468-0084.1996.mp58003009.x |
| Published date | 01 August 1996 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 58,3 (1996)
0305-9049
A Note on the Distribution of BDS Statistics for a
Real Exchange Rate Series
David Chappell, Joanne Padmore and Catherine Ellis
I. INTRODUCTION
The use of BDS statistics for testing financial and other time-series data
for non-linear structure is now a fairly well established practice. The BDS
statistics may be defined in the following way: Suppose we have a scalar
time series X(t); t = 1, 2, 3, ..., N and define the set of m-histories:-
{(X(1), X(2), ..., X(m)), (X(2), ..., X(m + 1)), ..., (X(Nm + 1),
Let Cm.N() be the fraction of all m-histories which are a distance apart of
no more than i. Then the BDS statistics are:-
Wm,N(t) - Ñ{CmN()_CTN(c)}
m, N(e) (1)
where JmN(8) is an estimate of the standard deviation under the null
hypothesis that the data are independently and identically distributed
(lID). Brock, Dechert and Scheinkman (1987) show that for a time series
which is lID, Wm,N(E) is asymptotically N(O, 1).
To detect non-linearity in a scalar time series one can fit a linear
(ARIMA) model with sufficient dynamic structure to ensure that the
residuals are serially uncorrelated and then test the residuals for lID by
calculating BDS statistics for a range of embedding dimensions, m, and
distances, a. A rejection of the lID null hypothesis means that the model
(data generating process) is non-linear. One potential pitfall in this
approach, however, is that for 'small' values of N the probability distribu-
tions of the BDS statistics may deviate quite substantially from the
standard normal distribution. Brock, Hseih and LeBaron (1991) use
Monte Carlo methods to demonstrate that, for N =100, 500 and 1,000,
the size of the BDS statistics for standardized residuals from ARCH and
GARCH models differ quite markedly from the size of the BDS statistics
for lID data. This is rather unfortunate since ARCH and GARCH
models have been quite successfully used in modelling a range of finan-
cial time series.
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