# 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.

• My knee-jerk reaction would be to say that if $Z\to V\to Y,$ then $Z$ is by definition not an instrument. An instrument has to have all its causal effect on $Y$ mediated through your main cause of investigation, $X.$ Commented Jun 17, 2022 at 15:17

Building on the excellent answer by @stats_model:

If you have some idea of the range of values the treatment effect of $$V$$ on $$Y$$ could reasonably take, you could plug in those bounds and solve the equation in the @stats_model answer to get a range of treatment effects of $$X$$ on $$Y$$. YMMV.

$$\frac{\frac{\Delta Y}{\Delta Z}-\text{[lower bound]}\cdot\frac{\Delta V}{\Delta Z}}{\frac{\Delta X}{\Delta Z}}\text{ and }\frac{\frac{\Delta Y}{\Delta Z}-\text{[upper bound]}\cdot\frac{\Delta V}{\Delta Z}}{\frac{\Delta X}{\Delta Z}}$$ Which one of those is higher would depend on the sign of $$\frac{\Delta V}{\Delta Z}$$.

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 $$V$$.

• Thank you very much! @stats_model Commented Jun 17, 2022 at 17:08
• Could you explain what is meant by "convincingly rule out...any effect running through omitted variables" (page 7 of schmidheiny.name/teaching/iv.pdf)? Commented Jun 18, 2022 at 15:16
• I'm afraid the answer to this sort of language is more art than science. The basic idea is that in order to justify your IV, you need to make some sort of argument about why there is no some alternative channel through which the instrument affects outcomes other than the variable which it is instrumenting for Commented Jun 18, 2022 at 17:14
• This is often quite difficult to do, as it is very hard to prove a negative. An example where this is maybe plausible could be the original Imbens and Angrist (1994), who argue that the mere draft lottery number itself should not affect people's outcomes directly, since it's just a number. It's not like people are going to attach any special significance to it except that it changes the likelihood that they join the military. Commented Jun 18, 2022 at 17:15
• Of course, even here, you could imagine a counterargument that finding out that you are have to serve in the military, even if you eventually get out of it, might itself be such a negative surprise that it leaves you traumatizing. It ends up being a matter of judgement whether one finds this example contrived or plausible, but it is your responsibility as a researcher to try to lay out as many of these considerations as you can anticipate. The ultimately provisional nature of these sorts of arguments is a big part of what makes social science research difficult. Commented Jun 18, 2022 at 17:17