Granger causality between vectors of time series: A puzzling property

Let us consider a discrete-time n-dimensional stochastic process z, with components x=(x1,…,xm1)′and y=(y1,…,ym2)′, m1+m2=n. We want to study causality relationships between the variables in x andy. Suppose that we find that y Granger causes x. Then we would expect to be able to pick out at least one of these variables, say yj, having a causal impact on x. It turns out that, when we consider the conditioning information set defined by the past observations of x and all the yi, i≠j, it may be that yjhas no causal impact on x, irrespective of the particular j=1,2,…,m2that we tried to pick out. This is a puzzling property. The paper provides a condition under which this property cannot hold.