(2) Consider a linear regression model y = XB + , where y E R", X E Rnx(+1), and E ~ N(0, 2 In). We assume that the model contains an intercept. Recall the notations y(i) E R- obtained by removing yi from the response vector y, and X(i) ER(n-1)x(p+1) obtained by removing the row xi from X. Let be the least squares estimate of the model y(i) = X(i)B +e, and 8 71 ) = 1/y(1) - XB()12/(n-p-2), the corresponding estimator for oz. all a12 (a) Show that if A = a 21 a22 is an arbitrary 3 x 2 matrix, then a31 a32 A'A = A') A(1) + all (all, a12). a12 Argue similarly that in our regression setting, X'X = X', X(i) + mixi. (b) Show that X'y = X's y(i) + yixi (c) Use the above to show that xiB = xi (Ip+1 + ( X's X(1) ) 'xxi) B() + yixi (Ip+1 + ( X's X() ',xi) (X'ox(o) . (d) Deduce that o(i) and E = yi - xiS are independent