Granger causality between vectors of time series: A puzzling property

Let us consider a discrete-time n-dimensional stochastic process z, with components x = (x(1),...,x(m1))' and y = (y(1),...,y(m2))', m(1) + m(2) = 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 y(j), 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 y(i), i not equal j, it may be that y(j) has no causal impact on x, irrespective of the particular j = 1, 2,..., m(2) that we tried to pick out. This is a puzzling property. The paper provides a condition under which this property cannot hold. (C) 2018 Elsevier B.V. All rights reserved.