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Difference in means estimator on Average Treatment Effect

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Difference in Means Estimand: "In this question we will explore how the difference in means estimator differs from causal quantities of interest such as the ATT, ATC, and ATE. Assume we have potential outcomes Y1i,Y0i, treatment assignment Di, and observed outcome Yi. Note that the difference in means estimator is what we use on a given observed sample; the difference in means estimand is the expectation of this."
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Problem Set 2 Question 1. Practice with potential outcomes (18) a) Consider binary treatment Di ∈ {0, 1} and observed outcome variable Yi. What is the meaning of Y1i and Y0i? Describe both in words, and choose an example to illustrate. (3) b) Is Y1i different form “Yi for a unit that actually received the treatment’ ’? Explain the difference using the example you began in section a. (2) c) Define the average treatment effect (ATE) and average treatment effect among the treated (ATT) using potential outcomes notation. Describe in words what each quantity means. (2) Question 2. Practice with the Difference in Means Estimand (17) In this question we will explore how the difference in means estimator differs from causal quantities of interest such as the ATT, ATC, and ATE. Assume we have potential outcomes Y1i, Y0i, treatment assignment Di, and observed outcome Yi. Note that the difference in means estimator is what we use on a given observed sample; the difference in means estimand is the expectation of this. a) Show that the difference in means estimand (the expectation of the difference in means estimator) can be decomposed into the ATT and a bias term. Derive the decomposition, showing your work – do not simply state it. Also describe clearly and concisely the meaning of the bias term. (5) b) Show how the difference in means estimand can be decomposed into the average treatment effect on the controls (ATC) and a bias term. Again derive the decomposition and describe clearly the meaning of the bias term. (6) c) Now show how the difference in mean estimands can be decomposed into the ATE and a bias term. Again describe the meaning of the bias term. (Hint: you may want to start by showing how the ATE relates to the ATT and ATC, using the law of iterated expectations). (6) 1
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