Lecture 24 Instrumental Variables
Nick Huntington-Klein
March 20, 2019
Recap
- 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)
Today
- 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!
Natural Experiments
- 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
Natural Experiments
- 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
Instrumental Variable
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
”
Instrumental Variable
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'
Graphically