(a) Using the fact that Cov(Ay) = ACov(y)AT for any m n matrix A, derive the variance-covariance matrix of the OLS estimator := ( 0, 1) T . What is the variance of 1? (b) Now, assume that the predictor x is normally distributed with small variance. To collect the data of x, consider two possible ways: (i) collect the complete random sample of x; (ii) Split the possible range of x into 100 mutually exclusive windows and collect the equal amount of sample from each slot. For both ways, the total number of sample is the same. Which way would be more efficient in terms of estimating 1? Why?

y = XB+E. 1 where Un x1 = (71: 12, . . . . Un) ; Xax2 = In and Enx1 = (61, (2, . . . . En)" ~ N(0, o'I). (a) Using the fact that Cov(Ay) = ACov(y)A for any m x n matrix A, derive the variance-covariance matrix of the OLS estimator 8 := (80, B1)". What is the variance of B,? (b) Now, assume that the predictor r is normally distributed with small variance. To collect the data of r, consider two possible ways: (i) collect the complete random sample of c; (ii) Split the possible range of r into 100 mutually exclusive windows and collect the equal amount of sample from each slot. For both ways, the total number of sample is the same. Which way would be more efficient in terms of estimating 81? Why