Assume that the sum of all Y and x, values satisfy yi = Tji = 0

Assume that the sum of all Y and x, values satisfy yi = Tji = 0. i=1 i=1 We now have a new predictor Spy which is orthogonal to all the columns of X. That is, T Ip+1 = 0 for j = 1, ...,p. Moreover, ) X(p+1) = 0. i=1 Prove that the least squares estimate Sp+1 in the new regression (involving all p + 1 predict tors) is same as the univariate regression coefficient of regressing Y on Cp+1. Hint: Consider any matrix X and a new one X' = [X Xp+1]. Then the orthogonal projection matrices onto the column spaces of X and X' satisfy 1 PX = Px + (I - Px)Ep+1Up+(I - Px). p+1(l - Px )Ip+1