The potential outcomes framework in Section 3-7e can be extended to more than two potential outcomes. In fact, we can think of the policy variable, w, as taking on many different values, and then y(w) denotes the outcome for policy level w. For concreteness, suppose w is the dollar amount of a grant that can be used for purchasing books and electronics in college, y(w) is a measure of college performance, such as grade point average. For example, y(0) is the resulting GPA if the student receives no grant and y(500) is the resulting GPA if the grant amount is $500.

For a random draw i, we observe the grant level, w_i ≥ 0 and y_i = y(w_i). As in the binary program

evaluation case, we observe the policy level, wi , and then only the outcome associated with that level.

(i) Suppose a linear relationship is assumed:

y(w) = α + βw = v(0)

where y(0) = α + v. Further, assume that for all i, wi is independent of v_i. Show that for each i

we can write

y_i = α + βw_i + v_i

E(v_i | w_i) = 0.

(ii) In the setting of part (i), how would you estimate b (and a) given a random sample? Justify

your answer.

(iii) Now suppose that wi is possibly correlated with vi, but for a set of observed variables xij,

The first equality holds if wi is independent of vi conditional on (x_i1, . . . , x_ik) and the second

equality assumes a linear relationship. Show that we can write

What is the intercept c?

(iv) How would you estimate b (along with c and the gj) in part (iii)? Explain.

E(v,w, X, . . , xk) = E(v,X1, .. . , X) = 7+ YX + + YxXit-