Reverse Causality on a "exogenous" instrument Z

Question

Suppose we want to do an instrumental variable regression. We have an endogenous variable X, exogenous instrument Z, dependent variable Y and an error term U.

In order for the approach to be valid, we need to convincingly rule out that the exogenous instrument is not related to the error term. But I read that furthermore we have also to rule out that there is reverse causality of Y on Z.

If I suspect that there is reverse causality, what assumption of IV would be violated? Is it also the exogeneity assumption? Does reverse causality of Y on Z lead to correlation between Z and the error term and can therefore be considered as a special case of the general assumption of uncorrelatedness between Z and U?

Answer 1

For the instrumental variable regression estimator to be unbiased, you need the instruments $Z$ to be uncorrelated with the hidden variables. If there is a causal link $Y \rightarrow Z$, this is no longer the case via the path $H \rightarrow Y \rightarrow Z$. Yes, "reverse causality" leads to correlation between the instruments and the error term. The exogeneity assumption is violated.