# ‘Metrics Monday: One IV for Two Endogenous Variable, and Testing for Mechanisms

A few months ago, a post in this series discussed a recently published article in the American Political Science Review by Acharya et al. (2016, ungated version here) in which the authors developed a method to test whether a mediator variable $M$ is a mechanism whereby treatment variable $D$ causes outcome variable $Y$.

At the time, I suggested to one of my PhD students that she should use that method to test for a presumed mechanism in her job-market paper, but since her identification strategy was based on an IV, it really wasn’t clear that Acharya et al.’s method could be applied to her research question.

A few weeks ago, a new working paper by Dippel et al. (2017) was released titled “Instrumental Variables and Causal Mechanisms: Unpacking the Effect of Trade on Workers and Voters.” Although Dippel et al.’s application is really timely–Do trade shocks cause people to vote for populist parties by turning them into disgruntled workers?–I’ll focus in this post on their methodological innovation.

The title of this post gets to the idea behind that innovation: when a mediator variable $M$ is a presumed mechanism whereby treatment variable $D$ causes outcome variable $Y$, it is possible to use the same instrument $Z$ to estimate:

1. The total effect of $D$ on $Y$. This is the usual 2SLS estimate.
2. The indirect effect of $D$ on $Y$. This is the effect that the treatment variable has on the outcome variable through the mediator variable $M$.
3. The direct effect of $D$ on $Y$. This is the effect that the treatment variable has on the outcome variable net of the effect of mediator variable $M$.

If you have taken an econometrics class, it has almost surely been drilled into your mind that every endogenous variable needs its own instrumental variable (IV). How can Dippel et al. use the same IV for two endogenous variable? Quite simply, it is possible to use the same IV for two endogenous variable when one of those endogenous variables is on the path (in a directed acyclic graph sense) between the treatment and outcome variables.

The three estimands above rely on three easy to perform 2SLS estimates, but Dippel et al.’s method relies on a crucial assumption, viz. that the unobserved confounders are “separable” between (i) those that affect the treatment and mediator variables, and (ii) those that affect the mediator and the outcome variables. The nice thing here is that Dippel et al. provide the reader with a simple statistical test of that hypothesis, which only relies on the three estimands listed above.

In sum, Dippel et al. present a simple yet powerful (where applicable, that is) method to test whether a proximate variable is a mechanism for another, more distal variable when your identification strategy relies on a valid IV, and they give the reader a method to test the core hypothesis behind that method.

The only quibble I have with the paper is that I wish it would discuss what is being estimated in the context of the indirect and direct effects in 2 and 3 above. We know that what’s estimated in 1 is a local average treatment effect (LATE). Presumably, that is also what is being estimated in 2 and 3, but it isn’t clear to me what that looks like in 2.

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