 where \(d = b + ac\) and \(W = cU + V\). Since (3) has been obtained by substituting (1) into (2), the system (1)–(2) has exactly the same solutions in \(X, Y\), and \(Z\) as the system (1)–(3). Since \(X, Y\) and \(Z\) are the only measured variables, (1)–(2) and (1)–(3) are “observationally equivalent” in the sense that they imply or represent exactly the same facts about the patterns of correlations that obtain among the measured variables. Nonetheless, the two systems correspond to different causal structures. (1)–(2) says that \(X\) is a direct cause of \(Y\) and that \(X\) and \(Y\) are direct causes of \(Z\). By contrast, (1)–(3) says that \(X\) is a direct cause of \(Y\) and that \(X\) is a direct cause of \(Z\) but says nothing about a causal relation between \(Y\) and \(Z\). We can cash this difference out within the interventionist/manipulationist framework described above—(2) claims that an intervention on \(Y\) will change \(Z\) while (1)–(3) denies this. (Recall that an intervention on \(Y\) with respect to \(Z\) must not be correlated with any other cause of \(Z\) such as \(X\), and will break any causal connection between \(X\) and \(Y\).) Thus while the two systems of equations agree about the correlations so far observed, they disagree about what would happen under an intervention on \(Y\). According to an interventionist/manipulationist account of causation, it is the system that gets such counterfactuals right that correctly represents the causal facts.