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 谁家孩子可获录取 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 因此以后结果的差异 反映了选择性偏差 等一下,这个似乎很容易 在抽奖活动中 我们会随机选择优胜者 因此只比较赢家和输家 - 很明显的 On the right track,卡马尔 但是特许学校的抽签安排 不会强迫孩子们进入 或离开特定的学校 他们随机分配了特许学校的学位 有些孩子很幸运 有些孩子不是 如果我们只是想知道特许学校 所带来的影响 我们可以将其视为随机试验 但是,我们只对特许学校 就学的影响 感兴趣 而对录取不感兴趣 并非所有获录取的学生 都会接受学位 IV将被录取为特许学校学生的影响 转变为实际就读特许学校的影响 - 太酷了 - 哦,太好了 让我们看一个例子 这是一所执行知识就是力量专案 的特许学校,或简称为KIPP 这所KIPP特许学校位于林恩 一座位于麻省海边的 褪色工业城镇 这所学校的申请者多于学位 因此他们要抽签来挑选学生 从2005年到2008年 共有371名四年级以及五年级生 参加了KIPP林恩的抽签 当中253名学生KIPP获录取 118名学生没有录取 一年后,获录取者的数学分数 比未获录取者更高 我们并不是试图弄清楚 获录取后是否会提高 你的数学水平 我们想知道参加KIPP 是否会使你的数学成绩改进 在253位获录取者中 实际上只有199位到KIPP上学 其他学生选择了传统的公立学校 同样,在118名未被录取的学生中 事实上有一些最终参加了KIPP 他们后来也获录取 那么,实际上参加KIPP 对考试成绩有何影响呢? 为什么我们不能只衡量 他们的数学成绩? 这是很好的问题 你将他们与谁进行比较呢? 那些没有参加的学生 上学率是随机的吗? - 不是啊 - 选择性偏差 - 对啊 - 什么? The KIPP offers are random so we can be confident of ceteris paribus, 但上学率不是随机的 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] ♪