Econ 3120 1st Edition Lecture 24 Outline of Last Lecture I Dependent Variable Errors Outline of Current Lecture II Counterfactual Averages Current Lecture 2 We can also define what we don t observe counterfactual averages E Y0i T 1 average outcome for individuals facing the policy in the state where they didn t face the policy E Y1i T 0 average outcome for individuals not facing the policy in the state where they faced the policy Now let s return to the regression Yi 0 1Ti ui I showed above that this estimates 1 E Yi T 1 E Yi T 0 or in our new notation E Y1i T 1 E Y0i T 0 Note that if I omit the state of the world in the subscript for Yi you can assume it s the observed state of the world But the way we ve the treatment effect the causal effect on the policy on the treated group is actually E Y1i T 1 E Y0i T 1 Thus we have to assume that E Y0i T 1 E Y0i T 0 Sometimes I call this the critical assumption of causal inference It turns out that this is just a more detailed way of expressing the zero conditional mean assumption E ui Ti 0 E ui T 0 E ui T 1 0 Why Because it turns out that ui represents the actual outcome for the control group and the counterfactual outcome for the treatment group The easest way to see this is a situation where 0 0 so Yi 1Ti ui For the control group ui Yi This is the outcome for the control group in the absence of the treatment For the treatment group ui Yi 1Ti But think about the hypothetical situation where the treatment goup did not get treated the counterfactual Using the model we take the treatment group and set T 0 yielding ui Yi0 So the zero conditional mean assumption that E ui T 0 E ui T 1 is the same thing as E Yi0 T 0 E Yi0 T 1 One way to think about the critical assumption is that there unobserved characteristics in this case the counterfactual that differ between the two groups 2 3 When can we assume that E Y0i T 1 E Y0i T 0 Let s take a simple example Suppose we are interested in whether a college scholarship program increases attendance An organization e g Gates Foundation gives out scholarships to high school students and we measure subsequent college attendance Does the critical assumption hold It depends on how the scholarships are targeted Let s consider a situation where sholarships have are given to the most qualified applicants to the program In that case the counterfactual outcome for students who receive the scholarships might be higher than the actual outcome for students who do receive them Thus E Y0i T 1 E Y0i T 0 and hence our estimate of 3 E Y1i T 1 E Y0i T 0 will be greater than the true effect of E Y1i T 1 E Y0i T 1 and we will have upwardly biased estimates Exercise What if scholarships are instead targeted towards disadvantaged applicants Finally let s consider random assignment Suppose that within a population These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute scholarships are given to a randomly selected group of students This would occur for example if the program had more qualified applicants than it had scholarships and the scholarships were allocated by lottery If assignment is random then that ensures that for a large enough sample E Y0i T 1 E Y0i T 0 This occurs because on average all characteristics observable and unobservable will be the same for both groups