5. (15 points) This question involves a simple example that illustrates why 0'2 can be estimated by \"12 when we compute the heteroskedasticity robust standard errors. Consider a simple regression model that relates the wage (y) to college education: y=+1$+ua where a: = 1 for college graduates and :1: = 0 for non-college graduates. Var(u|:r = 1) = or? > 03 = Var('u.|:r = 0), so the regression errors display heteroskedasiticity. In a random sample, the rst n1 individuals are college graduates, and the remaining no = n n1 are noncollege graduates. (a) Show that the variance of the OLS estimate 31 can be written as: A 1_l22+ 0_E122 VGTOBI): n1( ,1) 0'1 \"U( n) 0'0 - 2 A2_i n1A2 2 A2_; an A2 A- (b) Suppose we estnnate 01 by 01 m 21:15:15\" and 0'0 by do m 212:,\" +1 u\" where u is the residual from the regression. Show the estimate of Varl) can be written as: 2am- m) @(6'1)= 33:12 [Hint: (i) substitute fir? and 5% into Varh); and (ii) 57 = %l