On the Granger Condition for Non-Causality

ON THE GRANGER CONDITION FOR NON-CAUSALITY
C. A. SIMS [3]SHOWED that the Granger condition for non-causality can be characterized
by simple conditions on the parameters of the moving-average or the distributed-lag
representation of bivariate second-order stationary processes. Though he is only
concerned with non-deterministic stationary processes, some of his results turn out to be
generalized so as to apply to more general bivariate processes which may possibly contain
deterministic components.
This paper exclusively considers bivariate processes with finite second-order moments.
Given a bivariate process {(x,,y,): t~ I} (I is the set of all integers), denote by
H(x,), H(y,), H(x,, y,) the completions with respect to the mean-square norm of the linear
hulls of the subsets {xi: i s t},{yi: i s t}, {xi, yj: i s t,j S s} in the Hilbert space H of the
random variables with finite mean-square (with the understanding that two random
variables which differ on a set of probability measure zero only are to be identified). If
random variables x and y are orthogonal, write x I y ; when a random variable x is
orthogonal to any random variables belonging to a subspace K of H,write x I K. Denote by
KLthe subspace of the random variables orthogonal to a subspace K.
The Granger condition for non-causality is generalized as this: For a bivariate process
{(x,,y,): t~ I}, {y,} does not cause {x,}if the projection of x, on H(x,-', y,-') belong to
H(x,-') for all t (c.f. Granger [I]).Let the projection of x, on H(x,-') be h, and let
E, =X , -h,;then that condition may be interpreted as follows:LEMMA: The above Granger condition is equivalent to that E, I H ( X , - ~y,,-, ) for all t.
PROOF: Suppose that E 1H ( ,y ) Because of the relations h,E
H(x,-') cH(x,-', ytWl)t,h e projection of h, on H(x,-', y,-,) is h, itself. Since the projection
of E, on H(x,-', y,-') is 0,the projection of x, on H(x,-,, y,-,) is h, which belongs to H(x,-').
The reverse implication of the assertion is obvious.
In general, the decomposition y, = g,+q, where g, E H(x,)and q,1H(x,)may be called
the distributed-lag representation of the process y,. In terms of this generalized sense, the
Sims condition for {y,} not causing{x,} is that q,in the above decomposition is orthogonal to
H(x)for all t, where H(x)denotes the completion of the linear hull of the set {x,: t E I}.
THEOREM: The Granger condition is equivalent to the Sims condition.