- [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?