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# CORNELL ECON 3120 - Counterfactual Averages

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Econ 3120 1st Edition Lecture 24Outline of Last Lecture I. Dependent Variable ErrorsOutline of Current Lecture II. Counterfactual AveragesCurrent Lecture2 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 = β0 +β1Ti +ui I showed above that this estimates β1 = 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 theactual outcome for the control group and the counterfactual outcome for the treatment group. The easest way to see this is a situation where β0 = 0, so Yi = β1Ti +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 − β1Ti . 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” (inthis 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 of3 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 samefor both

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