18 The potential outcomes framework in Section 3-7e can be extended to more than two potential out- comes. 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, CHAPTER 3 Multiple Regression Analysis: Estimation 109 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, 20 and y, = y(w). As in the binary program evaluation case, we observe the policy level, w;, and then only the outcome associated with that level. (i) Suppose a linear relationship is assumed: y(w) = a + Bw + v(0) where y(0) = a + v. Further, assume that for all i, w, is independent of v, Show that for each i we can write y; = a + Bw, + 7 E(v./w) = 0. (ii) In the setting of part (1), how would you estimate B (and a) given a random sample? Justify your answer. (iii) Now suppose that w; is possibly correlated with v, but for a set of observed variables xija E(v, w, xij..., Xix) = E(v;|x;}..., Xix) = n + y + : The first equality holds if w, is independent of v, conditional on (Xippe , Xx) and the second equality assumes a linear relationship. Show that we can write y = y + Bw, + y + .*. + Yi tu E(uw;, tij. Xx) = 0. What is the intercept? (iv) How would you estimate B (along with y and the y;) in part (iii)? Explain. + YRA ...