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Xj. causes Yt where current Y is better predicts by including
current X, and
(4) No causality.
Pierce (1977) discusses other causal patterns and these will be men
tioned later. Each time series Xfc and Y^. is assumed stationary. Though
the above definitions are not in testable form, definition (1) implies a
recursive relationship between Xt and Yt, while (3) implies simul
taneity. The "strength" of causality and the existence of a lead/lag
relationship lose any meaning if (2) exists (Bishop, 1979). Testable
forms of these definitions regarding the null hypothesis of no causality
are given below.
Granger Method
The Granger test for unidirectional and instantaneous causality
between two stationary time series X^ and Yt involves the estimation via
ordinary least squares of a four-equation regression model given as
A.l
n
1
A.2 X E a X + u
t j., J t-3 t
3-1 Tt + + t
n
E
i-1
2
B-2 V-i T
where n is the maximum number of lags used. To test the null hypothesis
that Y does not cause X, an F-test is performed using the residuals from
A.l and A.2 to see if the c^ are different from zero. The F statistic
with q and T-t degrees of freedom is defined as