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1
CCF
Notation
The following notation is used throughout this chapter unless otherwise stated:
X, Y Any two series of length n
rxy 1 6k Sample cross correlation coefficient at lag^ k
S (^) x Standard deviation of series^ X
S (^) y Standard deviation of series^ Y
C xy 1 6k Sample cross covariance at lag^ k
Cross Correlation
The cross correlation coefficient at lag k is estimated by
r k
C k xy S S
xy x y
where
C k
n
x x y y k
n
y y x x k
xy
t t k t
n k
t t k t
1 6 n k
K
KK
K
K
K
=
โ
โ
1
1
K
K
2 CCF
S
n x x^ t x t
n = โ =
1
S
n y y^ t y t
n = โ =
1
The cross correlation function is not symmetric about k = 0.
Approximate standard error of r xy 1 6k is
se r k n k
3 xy 1 6 8 โ
^ k
, 0 , 1 , 2 ,K
The standard error is based on the assumption that the series are not cross correlated and one of the series is white noise. (The general formula for the standard error can be found in Box and Jenkins, 1976, p. 376, 11.1.7.)
References
Box, G. E. P., and Jenkins, G. M. 1976. Time series analysis: Forecasting and control. San Francisco: Holden-Day.
