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1. Suppose MLR.1 through MLR.5 are satisfied in the model wage = Pot Pieduce + Wies where the index i stands for firm i and the index e = 1,...,m, stands for different employees within firm i, so that there are n different firms with m, employees each, i = 1,...,n. Unfortunately, we only observe the average of the employees for each firm. The regression we end up running is wage, = Bo + Beduc, +un, i=1, ...,n, where wage, = m, 2. wage,,u, =me, up , and so on. Show that the error in the regression with the averaged data is heteroskedastic, that is, Var(u, | educt,1, ..., educ;,m) changes with i, unless all firms have the same number of employees. 2. * Verify that if a variable x* is measured with error, but the variable is measured twice, then the population correlation coefficient between the two measurements is the reliability ratio. Denote the first measurement by x, = x" + v, and the second measurement by x, = x* + 12, where the measurement errors v, and v, are independent from each other and x*, and satisfy Var(v, ) = Var(v2) = n' > 0. Recall that the reliability ratio and the population correlation coefficient are, respectively, Var(x*) Cov(X1, X2) and Var(x*) + 72 Var(y. )Vary