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] ♪