Lecture 24 Instrumental Variables

Nick Huntington-Klein

We’ve covered quite a few methods for isolating causal effects!
Controlling for variables to close back doors (explain X and Y with the control, remove what’s explained)
Matching on variables to close back doors (find treated and non-treated observations with )
Using a control group to control for time (before/after difference for treated and untreated, then difference them)
Using a cutoff to construct a very good control group (treated/untreated difference near a cutoff)

We’ve got ONE LAST METHOD!
Today we’ll be covering instrumental variables
The basic idea is that we have some variable - the instrumental variable - that causes X but has no other back doors!

This calls back to our idea of trying to mimic an experiment without having an experiment. In fact, let’s think about an actual randomized experiment.
We have some random assignment R that determines your X. So even though we have back doors between X and Y, we can identify X -> Y

The idea of instrumental variables is this:
What if we can find a variable that can take the place of R in the diagram despite not actually being something we randomized in an experiment?
If we can do that, we’ve clearly got a “natural experiment”
When we find a variable that can do that, we call it an “instrument” or “instrumental variable”
Let’s call it Z

So, for Z take the place of R in the diagram, what do we need?

Z must be related to X (typically Z -> X but not always)
There must be no open paths from Z to Y except for ones that go through X

In other words “Z is related to X, and all the effect of Z on Y goes THROUGH X”

How?

Explain X with Z, and keep only what is explained, X'
Explain Y with Z, and keep only what is explained, Y'
[If Z is logical/binary] Divide the difference in Y' between Z values by the difference in X' between Z values
[If Z is not logical/binary] Get the correlation between X' and Y'