Instrumental variables

Problem

Suppose that \(x\) is correlated with \(ε\) and that \(z\) is correlated with \(x\) . Will it then follow that \(z\) is correlated with \(ε\) ?

Solution

No, correlation is not transitive. If this was true, instrumental variables could never work. A simple example: \(z\) and \(ε\) are independent random variables and \(x=z+ε\) . Then clearly \(x\) is correlated with \(ε\) and that \(z\) is correlated with \(x\) but \(z\) is not correlated with \(ε\) .