[MUSIC] The Local Average Treatment Effect is defined when the instrument and the treatment
are both binary variables.
That means they can take on two different values.
For example, suppose we're looking at the effect of education on wages, and in our dataset
we have a variable that says whether someone went to college or not, and another variable
that says whether there was a college in that person's home town or not.
That's the case where the local average treatment effect is defined.
And this is very common in practice - having a binary instrument and a binary treatment.
In general, when we have treatments or instruments that take on more than just two values, instrumental
variables analysis will still apply, but things get a little bit more complicated.
So we're not going to look at that.
Instead, we'll focus on just the binary treatment and the binary instrument case where we're
going to be able to learn about all the main ideas and all the intuition.
So in particular, I'm going to look at the application to randomize experiments with
non-compliance.
That's a very helpful setting for understanding how the local average treatment effect is
defined.
Now remember that in this case, our instrument is treatment assignment: whether you were
assigned to receive treatment or not.
That treatment assignment is an instrument for actually receiving treatment, because
there's non-compliance - people who are assigned to get treated don't necessarily get treated.
So that's the setup we're in.
Now what we're going to do is take our population of people, and split them into four distinct
types, depending on how they react to their treatment assignment.
The first group is called Always Takers.
This people always receive treatment, regardless of whether they were assigned to.
So that is, regardless of their instrument value, they always get treatment.
The second group are called Never Takers.
These people never get treated, regardless of their treatment assignment.
The third group are called Compliers.
These people receive treatment if and only if they're assigned to the treatment group.
So they comply with their treatment assignment, that's why we call them compliers.
The fourth and final group are called Defiers.
They get treated if and only if they are not assigned to the treatment group.
So they do not comply with the treatment assignment.
They instead defy their treatment assignment.
So every single person can be put in one of this four groups.
So now we can define the Local Average Treatment Effect.
It's just the average of the unit level causal effects for the Compliers.
So we've got our population, we take our Compliers and we put them over here, then our Defiers,
our Never Takers, and Always Takers - will just put them over here and then ignore them
for now.
Among our Compliers, each person has a unit level causal effect, so we just take the average
of all of those numbers for those people.
That's all that the Local Average Treatment Effect is.
So it's very similar to the Average Treatment on the Treated and the Conditional Average
Treatment Effect.
Those also were just average treatment effects for a subset of units from the whole population.
The average treatment on the treated was just the average unit of our causal effect for
the people who were treated, and the condition average treatment effect for example for men,
was just the average unit level causal effect only for the men, so the local average treatment
effect is very similar: it’s just the average of the unit level causal effect for the Compliers.
Now one thing that makes this a little bit more complicated than CATE and ATT, (treatment
on the treated), is that from our data we don't actually know who the Compliers are.
So for the Conditional Average Treatment Effect, we know who the men are in our dataset.
And for Average Treatment on the Treated, we know who the people who are treated are
in our dataset.
But Compliers, we don't know who they are and that's because the definition of what
makes somebody a Complier depends on two potential outcomes.
It depends on what they would do if they were assigned to be treated, and what they would
do if they were assigned not to be treated.
And because of the Fundamental Problem of Causal Inference, we can never observe both
of these potential outcomes at the same time.
Because for each person we either assign them to be treated, or assign them not to be treated.
We can't have it both.
So we just don't know who is and who isn't a Complier.
So that makes sort of conceptually thinking about the Local Average Treatment Effect a
little bit harder because these groups, this particular subset of people we're looking
at, is a little more complicated than just thinking about this set of men, or this set
of people who were treated.
But it's still the same fundamental idea, an average unit level of causal effect for
a particular kind of people.
And another place that it makes it more difficult is how do we even learn the Local Average
Treatment Effect from the data, if we don't know who is a Complier in our data set?
Well, to answer that, we'll do that in another module.
[MUSIC].