1 Use the provided dataset "HW5 data.csv" for this question. Consider the linear model y = X + where y is a n 1 vector of observations, X is the n p design matrix, is the p 1 vector of coefficients, 2 is the variance and is a n 1 vector of independent standard normal random variables. For the data set, n = 200 and p = 5. (1) (8 pts) Using the notation in the slides, matrices Q, P and are such that QXXQ = 2 P XQ = D = ( 0 ) , = diag(1, 2, . . . , p) where the (j ) are the square roots of the eigenvalues of XX. Show that if = P then E (i) = 0 and Var(i) = 1 for all i = 1, . . . , n. (2) (10 pts) If = Q, show that P X can be written as D. Hence, show that the original linear model can be written as zi = ii + i i = 1, . . . , p i i = p + 1