remember CATE, the conditional average treatment effect.
This is just the average effect for a substance of units.
Here I'm gonna introduce a new kind of average effect, and to do
that I've got a table with eight data points.
I've got eight people.
One, two, three, four, five, six, seven, eight.
Four of this people are male, and four of them are female.
And over here I've got a column telling me in my data was this person
treated, yes, or not treated, no.
So the first three people are treated, the second, the fourth person
is not and for the females, the first person is treated and the
other three people are not.
Now over here I have my columns of potential outcomes.
So the first column is what this person would get, what outcome
would they have, if they were treated, so the first person was 80,
75, 85, 70 and then.
And then similarly for the females the next column tells me the
potential outcome of what would happen to them if they weren't treated.
60, 70, 80, 60 etc.
Now we're never ever, ever going to see both of this numbers simultaneously
for the same person in a data set.
That's the fundamental problem of causal inference is that you can't
observe the counterfactual outcome, what didn't happen.
So in real data, for say these people we will only ever observe this
number, what happened to them when they were treated because they
were treated, we would not observe these numbers for this person,
this person was not treated so we were to observe this number what
happens to them when they are not treated but not this number.
But for the purpose of defining what effect we wanna go for I've
just written out the full potential outcomes even though we can't
actually observe them in the data.
Now once I got this outcomes that you get with the policy and without
I can compute the unit level causal effect which is just the difference
in these two numbers.
So for the first person the unit level causal effect is 20.
80 minus 60.
So this is the effect that the person would get if weren't in the
policy a 20 point increase in their outcome and similarly for everybody
else.
So remember back from earlier?
The average treatment effect is just the average of these eight
numbers, the average unit causal effect for everybody in the population.
For all of these people.
The conditional average treatment effect for men is just the average
of these four numbers just the average unit level causal effects
only for the men and similarly the conditional average treatment
effect for females is just the average of these four numbers.
The average of the unit level causal effects only for females.
So that's what we did before now what's new
we can think about the treated people as a particular subset of
people, the people who were treated.
And we can define the average treatment effect for those people.
So the average treatment on the treated, ATT, is just going to be the
average unit causal effects for the people who are treated.
So this person, this person, this person, that's 20 plus five,
plus five and then this person here, this one female who is treated
plus five divided by four people.
So that's the average treatment on the treated, the average of this
three people and this person.
We are just thinking of this variable telling us whether someone
was treated or not is just another variable on our data set just
like sex is here, male or female so that's average treatment on
on the treated, now under the unconfoundedness assumption or other
assumptions, even though you don't observe both of this columns
you're still gonna be able to compute this number ATT from your
data, and as well as a TE or K or other perimeters of interest.
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