1. Given 21, . .., In ER, and assume that not all of r; are the same. Recall the simple linear regression model with Normally distributed errors: yi = Bo+ I;Bite; for i = 1, ...,n where 61, 62, . .., En ~ i.i.d. N(0, 02). Or equivalently, we can write Define y1 y2 1 Y = X= B = . .. . .. . . . . .. then an expression of the normal simple linear regression model in matrix terms is Y =XB+c, where c ~ N (0, a2In) . Let || . || be the L2-norm of an n-dimensional vector, defined for a E Rnx1 as: llall = Var tan + ... + a? The sum of squares of errors can be expressed as Q(8) = ||Y - XB||2. (a) Derive the MSE minimizer B in matrix from, using only Y and X. Hint: in matrix calculus, we have aB EB =IT, aB = (E+ET) B. (b) Now fix a p 2 1. Suppose r, E RPXI and B, E RPX1 are both p-dim vector, i.e. we are using p predictor variables to predict the 1-dim response variable Y. We write (1) I2 X = E Rnx(P+1) . . . . . . . 1 and we assume that rank(X) = p + 1. Will the minimizer B of Q(B) still take the same form as in (a)? Derive it. (c) Write the fitted values Y in matrix form for fixed p 2 1, using only Y and X. (d) Recall the residual vector is defined as e = Y - Y. Show eTX = 0 and eTY = 0. (e) Let In denote the n-dimensional all-one column vector. Recall the hat matrix H = x(X X) -1X . Calculate the value of H . In