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(ESS ESS )/(q)
F r u
q,T-t (ESSu)/(T-t)
where t is the number of parameters estimated in the unrestricted model
(A. 1), q is the number of parameters estimated in the restricted model
(A.2), T is total number of observations, and ESSr and ESSU are error
sums of squares for the restricted and unrestricted model,
respectively. If the F statistic for A.l and A.2 is significant then
the null hypothesis is rejected, suggesting that Y causes X. A test of
the d^ can be performed testing causality in the opposite direction to
support this result (Colclough and Lange, 1982) or check for the
existence of feedback. To check for either instantaneous or
unidirectional causality, the index i in equations A.l and B.l is
initialized to zero* The present study, however, will use the Granger
method to test hypotheses regarding strictly unidirectional causality.
These tests assume the error terms are uncorrelated white noise, such
that E(utug) = E(vtvs) 0 for s*t, for every t and s. Rejecting the
null hypothesis that Y does not cause X suggests that X should be
specified as some function of lagged Y.
Sims Method
Another method of testing for unidirectional causality has been
proposed by Sims (1972) where the test involves a system of two regres
sion equations
In this case,
- jLVm <
X ib.Y + e
c J-0 J C-J *
a test
of the hypothesis that X does not cause Y is