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A dummy variable takes on only the values 0 and 1.

= y: Exercise 3.12 A dummy variable takes on only the values 0 and 1. It is used for categorical data, such as an individualís gender. Let d1 and d2 be vectors of 1ís and 0ís, with the i th element of d1 equaling 1 and that of d2 equaling 0 if the person is a man, and the reverse if the person is a woman. Suppose that there are n1 men and n2 women in the sample. Consider Ötting the following three equations by OLS y =  + d1 1 + d2 2 + e (3.53) y = d1 1 + d2 2 + e (3.54) y =  + d1 + e (3.55) Can all three equations (3.53), (3.54), and (3.55) be estimated by OLS? Explain if not. (a) Compare regressions (3.54) and (3.55). Is one more general than the other? Explain the relationship between the parameters in (3.54) and (3.55). (b) Compute 1 0 nd1 and 1 0 nd2; where 1n is an n 1 vector of ones. (c) Letting = ( 1 2) 0 ; write equation (3.54) as y = X +e: Consider the assumption E(xiei) = 0. Is there any content to this assumption in this s CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 101 Exercise 3.13 Let d1 and d2 be deÖned as in the previous exercise. (a) In the OLS regression y = d1 b1 + d2 b2 + ub; show that b1 is the sample mean of the dependent variable among the men of the sample (y1 ), and that b2 is the sample mean among the women (y2 ). (b) Let X (n k) be an additional matrix of regressors. Describe in words the transformations y = y d1y1 d2y2 X = X d1x 0 1 d2x 0 2 where x1 and x2 are the k 1 means of the regressors for men and women, respectively. (c) Compare e from the OLS regression y = X e + ee with b from the OLS regression y = d1 b1 + d2 b2 + X b + be: Exercise 3.14 Let b n = (X0 nXn) 1 X0 nyn denote the OLS estimate when yn is n 1 and Xn is n k. A new observation (yn+1; xn+1) becomes available. Prove that the OLS estimate computed using this additional observation is b n+1 = b n + 1 1 + x 0 n+1 (X0 nXn) 1 xn+1 X0 nXn 1 xn+1  yn+1 x 0 n+1 b n  : Exercise 3.15 Prove that R2 is the square of the sample correlation between y and yb: Exercise 3.16 Consider two least-squares regressions y = X1 e 1 + ee and y = X1 b 1 + X2 b 2 + be: Let R2 1 and R2 2 be the R-squared from the two regressions. Show that R2 2  R2 1 : Is there a case (explain) when there is equality R2 2 = R2 1 ? Exercise 3.17 For e 2 deÖned in (3.47), show that e 2  b 2 : Is equality possible? Exercise 3.18 For which observations will b (i) = b? Exercise 3.19 For the intercept-only model yi = + ei , show that the leave-one-out prediction error is eei =  n n 1  (yi y): Exercise 3.20 DeÖne the leave-one-out estimator of  2 , b 2 (i) = 1 n 1 X j6=i  yj x 0 j b (i) 2 : This is the estimator obtained from the sample with observation i omitted. Show that b 2 (i) = n n 1 b 2 eb 2 i CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 102 Exercise 3.21 Consider the least-squares regression estimators yi = x1i b1 + x2i b2 + ebi and the ìone regressor at a timeî regression estimators yi = e1x1i + ee1i yi = e2x2i + ee2i Under what condition does e1 = b1 and e2 = b2?