- [Instructor] The path from cause
to effect is dark and dangerous.
But the weapons
of Econometrics are strong.
[Attack] with fierce
and flexible instrumental variables
when nature blesses you
with fortuitous random assignment.
[gong rings]
Randomized trials are the surest
path to ceteris parabus comparisons.
Alas, this powerful tool
is often unavailable.
But sometimes, randomization
happens by accident.
That's when we turn
to instrumental variables,
IV for short.
- [Voice whispers] Instrumental
variables.
- [Instructor] Today's lesson
is the first of two on IV.
Our first IV lesson begins
with a story of schools.
- [Josh] Charter schools
are public schools
freed from daily district oversight
and teacher union contracts.
The question of whether charters
boost achievement
is one of the most important
in the history
of American education reform.
- The most popular charter schools
have more applicants
so the luck of the lottery draw
decides who's offered a seat.
A lot is at stake for the students
vying for their chance.
Waiting for the lottery results
brings up lots of emotions
as was captured
in the award-winning documentary
"Waiting For Superman."
- [Mother] Don't cry.
You're gonna make Mommy cry.
Okay?
- Do charters really provide
a better education?
Critics most definitely say no,
arguing that charters enroll
better students to begin with,
smarter or more motivated
so differences in later outcomes
reflects selection bias.
- [Kamal] Wait, this one seems easy.
In a lottery, winners
are chosen randomly,
so just compare winners and losers.
- [Student] Obviously.
- On the right track, Kamal,
but charter lotteries
don't force kids into
or out of a particular school.
They randomize offers
of a charter seat.
Some kids get lucky.
Some kids don't.
If we just wanted to know
the effect of charter school offers,
we could treat this
as a randomized trial.
But we we're interested
in the effects
of charter school attendance,
not offers.
And not everyone
who is offered, accepts.
IV turns the effect of being offered
a charter seat into the effect
of actually attending
a charter school.
- [Student] Cool.
- Oh nice.
- Let's look at an example,
a charter school from
the Knowledge Is Power
Program, or KIPP for short.
This KIPP school is in Lynn,
a faded industrial town
on the coast of Massachusetts.
The school has
more applicants than seats
and therefore picks its students
using a lottery.
From 2005 to 2008,
371 fourth and fifth graders
put their names
in the KIPP/Lynn lottery,
253 students won a seat at KIPP,
118 students lost.
A year later, lottery winners had
much higher match scores
than lottery losers.
But remember,
we're not trying to figure out
whether winning a lottery
makes you better at math.
We want to know if attending KIPP
makes you better at math.
Of the 253 lottery winners,
only 199 actually went to KIPP.
The others chose
a traditional public school.
Similarly of the 118 lottery losers,
a few actually ended up at KIPP.
They got an offer later.
So what was the effect of test scores
of actually attending KIPP?
- [Student] Why can't we just
measure their math scores?
- [Instructor] Great question.
Who would you compare them to?
- [Student] Those who didn't attend.
- [Instructor] Is attendance random?
- [Camilla] No.
- Selection bias.
- [Instructor] Correct.
- [Otto] What?
- [Instructor] The KIPP offers
are random so we can be confident
of ceteris parabus,
but attendance is not random.
The choice to accept the offer
might be due to characteristics
that are related
to math performance.
Say, for example,
that dedicated parents
are more likely
to accept the offer.
Their kids are also more likely
to do better in math,
regardless of school.
- [Student] Right.
- [Instructor] IV converts
the offer effect
into the effect of KIPP attendance,
adjusting for the fact
that some winners go elsewhere
and some losers manage
to attend KIPP anyway.
Essentially, IV takes
an incomplete randomization
and makes the appropriate
adjustments.
How? IV describes a chain reaction.
Why do offers affect achievement?
Probably because they affect
charter attendance
and charter attendance
improves math scores,
the first link in the chain
called the first stage
is the effect of the lottery
on charter attendance.
The second stage is the length
between attending a charter
and an outcome variable,
in this case, math scores.
The instrumental variable
or instrument for short
is the variable
that initiates the chain reaction.
The effect of the instrument
on the outcome is called
the reduced form.
This chain reaction can be
represented mathematically.
We multiply the first stage,
the effect of winning
on attendance, by the second stage,
the affect of attendance on scores.
And we get the reduced form,
the effect of winning
the lottery on scores.
Reduced form and first stage
are observable and easy to compute.
However, the effect of attendance
on achievement
is not directly observed.
This is the causal effect
we're trying to determine.
Given some important assumptions
we'll discuss shortly,
we can find the effect
of KIPP attendance
by dividing the reduced form
by the first stage.
This will become more clear
as we work through an example.
- [Student] Let's do this.
- A quick note on measurement.
We measure achievement
using standard deviations,
often denoted
by the Greek letter sigma (σ).
One σ is a huge move
from around the bottom 15%
to the middle of most
achievement distributions.
Even a 1/4 or 1/2 σ
difference is big.
- [Instructor] Now we're ready
to plug some numbers
into the equation
we introduced earlier.
First up, what's the effect
of winning the lottery
on math scores?
KIPP applicant math scores
are a third of a standard deviation
below the state average in
the year before they apply to KIPP.
But a year later, lottery winners
score right at the state average
while the lottery losers
are still well behind.
With an average score
around - 0.36 σ.
The effect of winning the lottery
on scores is the difference
between the winners' scores
and the losers' scores.
Take the winners'
average math scores,
subtract the losers'
average math scores,
and you will have 0.36 σ .
Next stop: what's the effect
of winning the lottery on attendance?
In other words,
if you win the lottery,
how much more likely are you
to attend KIPP than if you lose?
First, what percentage
of lottery winners attend KIPP?
Divide the number of winners
who attended KIPP
by the total number
of lottery winners -- that's 78%.
To find the percentage
of lottery losers who attended KIPP,
we divide the number of losers
who attended KIPP
by the total number
of lottery losers -- that's 4%.
Subtract 4 from 78, and we find
that winning the lottery
makes you 74%
more likely to attend KIPP.
Now we can find
what we're really after,
the effect of attendance on scores,
by dividing 0.36 by 0.74.
Attending KIPP raises math scores
by 0.48 standard deviations
on average.
That's an awesome achievement game,
equal to moving
from about the bottom
to the middle
of the achievement distribution.
- [Student] Whoa, half a sig.
- [Instructor] These estimates
are for kids opting in
to the KIPP lottery,
whose enrollment status
is changed by winning.
That's not necessarily
a random sample
of all children in Lynn.
So we can't assume
we'd see the same effect
for other types of students.
- [Students] Huh.
- But this effect
on keen for KIPP kids
is likely to be a good indicator
of the consequences
of adding additional charter seats.
- [Student] Cool.
- [Student] Got it.
- IV eliminates selection bias
but like all of our tools,
the solution builds on a set
of assumptions
not to be taken for granted.
First, there must be
a substantial first stage --
that is the instrumental variable,
winning or losing the lottery,
must really change the variable
whose effect we're interested in --
here, KIPP attendance.
In this case, the first stage
is not really in doubt.
Winning the lottery makes
KIPP attendance much more likely.
Not all IV stories are like that.