Consider the potential outcomes framework from Section 2 7a, where y i (0) and y i (1) are the potential outcomes in each treatment state (i) Show that if we could observe y i (0) and y i (1) for all i then an unbiased estimator of ate would be This is sometimes called the sample average treatment effect (ii) Explain why the observed sample averages, y 0 and y 1 , are not the same as y(0) and y(1), respectively, by writing y 0 and y 1 in terms of y i (0) and y i (1), respectively nELy(1) y (0) y(1) y(0) i 1

Question: Consider the potential outcomes framework from Section 2.7a, where y i (0) and y i (1) are the potential outcomes in each treatment state. (i)

Consider the potential outcomes framework from Section 2.7a, where y_i(0) and y_i(1) are the potential outcomes in each treatment state.

(i) Show that if we could observe y_i(0) and y_i(1) for all i then an unbiased estimator of τ_ate would be

nELy(1) – y.(0)] = y(1) – y(0). i=1

This is sometimes called the sample average treatment effect.

(ii) Explain why the observed sample averages, y̅₀ and y̅₁, are not the same as y̅(0) and y̅(1), respectively, by writing y̅₀ and y̅₁ in terms of y_i(0) and y_i(1), respectively.

nELy(1) y.(0)] = y(1) y(0). i=1

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