因果推断之路径既黑暗又危险

但是计量经济学是很厉害的武器

当自然界给你带来偶然的随机分配时

使用气势汹汹与灵活多變的
工具变量进行攻击

[]

随机试验是完成
“其他条件不变”的比较

的最可靠途径

但我们经常无法使用
这个功能强大的工具

但是有时候，随机是偶然发生的

这时候我们转向工具变量

—简称IV

工具变量

今天的课堂是IV两节课的第一节

我们的第一节IV课
从学校的故事开始

[]

特许学校是一些公立学校

不受日常学区监督
与教师工会合同约束

特许学校能否提高成绩

是美国教育改革史上

最重要的问题之一

最受欢迎的特许学校的申请人数
远多于学位

因此抽奖运决定了
谁家孩子可获得学位

在学生争夺机会时需要面对很多风险

正如获奖纪录片“等待超人”中

所描述的那样

等待结果时会产生很多种情绪

别哭，你会让妈妈哭的
好吗？

特许学校真的能提供更好的教育吗？

评论家肯定会说不是的

他们会争辩说特许学校
能夠招募更好

更聪明或更主动的学生
因此以后的结果的差异

反映了选择性偏差

等一下，这个似乎很容易

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

that charter school offers,

we could treat this
as a randomized trial.

But 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 math 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
on test scores

of actually attending KIPP?

- [Kamal] Why can't we just measure
their math scores?

- [Instructor] Great question.

Who would you compare them to?

- [Kamal] 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 paribus,
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 link
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 effect of attendance on scores.

And we get the reduced form,

the effect of winning
the lottery on scores.

The 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 ¼ or ½ 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 applicants' 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 up: 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 gain,

equal to moving
from about the bottom third

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.
- [Student] 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.

Second, the instrument
must be as good

as randomly assigned,

meaning lottery winners and losers
have similar characteristics.

This is the independence assumption.

Of course, KIPP lottery wins
really are randomly assigned.

Still, we should check for balance
and confirm that winners and losers

have similar family backgrounds,

similar aptitudes and so on.

In essence, we're checking
to ensure KIPP lotteries are fair

with no group of applicants
suspiciously likely to win.

Finally, we require
the instrument change outcomes

solely through
the variable of interest --

in this case, attending KIPP.

This assumption is called
the exclusion restriction.

- IV only works if you can satisfy
these three assumptions.

- I don't understand
the exclusion restriction.

How could winning the lottery
affect math scores

other than by attending KIPP?

- [Student] Yeah.
- [Instructor] Great question.

Suppose lottery winners
are just thrilled to win,

and this happiness motivates them
to study more and learn more math,

regardless of where
they go to school.

This would violate
the exclusion restriction

because the motivational effect
of winning is a second channel

whereby lotteries
might affect test scores.

While it's hard
to rule this out entirely,

there's no evidence
of any alternative channels

in the KIPP study.

- IV solves the problem
of selection bias

in scenarios like the KIPP lottery
where treatment offers are random

but some of those offered opt out.

This sort of intentional
yet incomplete random assignment

is surprisingly common.

Even randomized clinical trials
have this feature.

IV solves the problem
of non-random take-up

in lotteries or clinical research.

But lotteries are not the only source
of compelling instruments.

Many causal questions
can be addressed

by naturally occurring

as good as randomly
assigned variation.

Here's a causal question for you:

Do women who have children
early in their careers

suffer a substantial earnings penalty

as a result?

After all, women earn less than men.

We could, of course, simply compare
the earnings of women

with more and fewer children.

But such comparisons are fraught
with selection bias.

If only we could
randomly assign babies

to different households.

Yeah, right,
sounds pretty fanciful.

Our next IV story -- fantastic
and not fanciful --

illustrates an amazing,
naturally occurring instrument

for family size.

♪ [music] ♪

- [Instructor] You're on your way
to mastering econometrics.

Make sure this video sticks

by taking a few
quick practice questions.

Or, if you're ready,
click for the next video.

You can also check out
MRU's website for more courses,

teacher resources, and more.

♪ [music] ♪