the local average treatment effect is defined when the instrument
and the treatment are both binary variables.
That means that they can take on two different variables.
For example, suppose we're looking at the effect of education on
wages.
And in our data set we have a variable that says whether someone
went to college or not and another variable that says whether there's
a college in that person's hometown or not.
Well 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.
And 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 binary instrument
case, where we are going to be able to learn all the main ideas
and all the intuition.
So in particular
I'm gonna look at the application to randomized 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 are 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 set up we're in.
Now what we're gonna 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.
These people always receive treatment regardless of whether they were assigned to, 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 are assigned
to the treatment group.
So they comply with their treatment assignments, 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 their assignment they instead defy their
treatment assignments.
So every single person can be put in one of this four groups.
So now we can define the local average treatment effect.
I's just the average of the unit level causal effect 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 we
just put them over here and 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 under the condition
average treatment effect, those also were just average treatment
effects for a subset of units from the whole population, the average
treatment under treated was just the average unit level causal
effect for the people who were treated, and the condition average
treatment in fact, 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 just the average of
the unit level causal effect for the compliers, now one thing that
makes this a little bit more complicated than KATE and ATT, treatment
only treated, is that from our data we don't actually know who the 71 00:03:55,887 --> 00:00:00,000 compliers are, so for
the conditional average treatment effect, we know who the men are
in our data set, and for average treatment on the treated, we know
who the people who were treated are in our data set, but compilers,
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 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 thus don't don't know who is and
who isn't a compiler.
So that makes sort of conceptually thinking about the local average
treatment effect a little bit harder because these groups, these
particular sub-set of people we're looking at is a lot more complicated
than just thinking about the set of men
or this set of people who are treated, but it's still the same
fundamental idea, an average uni-level cost of a particular kind
of people.
And another place that makes it more difficult is how do we even
learn the local average treatment effect from data if we don't
know who is the complier in our data set.
Well, to answer that we will do that in another module.
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