Instrumental variable: Indirect effect of Z on Y I have found an instrument variable ($Z$) for my econometric model. The relevance constraint holds, however I still have a question about the instrument exogeneity and that is:
It is said that it needs to be convincingly ruled out any direct effect of the instrument on the dependent variable or any effect running through omitted variables. But what if $Z$ influences another variable, $V,$ by which $Y$ is influenced ($Z\to V\to Y$)? Is then my exclusion restriction violated? I have data on $V,$ such that I could control for it.
 A: Having data on $V$ is typically not enough to "control for" the fact that $Z$ fails the exclusion restriction. Specifically, we can decompose the change in $Y$ due to an (exogenous) change in $Z$ as
$$\underbrace{\frac{\Delta Y}{\Delta Z}}_{\text{''Reduced Form"}} = \underbrace{\frac{\Delta Y}{\Delta X}}_{\text{''Treatment effect"}}\times\underbrace{\frac{\Delta X}{\Delta Z}}_{\text{''First stage"}} + \frac{\Delta Y}{\Delta V} \times \frac{\Delta V}{\Delta Z}$$
The idea behind the IV estimator to assume the exclusion restriction that the last term on the RHS is 0, so that the treatment effect is the reduced form divided by the first stage.
So if you had data on $V$, would it suffice to restore identification? Not without very strong assumptions. To see why, consider trying to estimate the two factors in that last term. The factor $\frac{\Delta V}{\Delta Z}$ is directly estimable in the same way that the first stage effect of $Z$ on $X$ is. The trouble arises because we also need to know $\frac{\Delta Y}{\Delta V}$. The trouble is that this is itself a treatment effect, this time of $V$ on $Y$. Thus, in order to be able to control account for the influence of $V$, we need an assumption that amounts to being able to know the effect of $V$ on $Y$. But given that the whole point of using causal inference techniques is that we typically have to get very clever to understand these treatment effects, it is typically not the case that we will have a good estimate of the causal effect of $V$ on $Y$, which is what we would need to know to make use of the fact that we know $V$.
