this is a proof problem from hw.

Consider linear fit Y ~ X1B1 + X232, which is regressing n x 1 observed vector Y against (X1, X2), X1 is an n x K matrix, X2 is an n x L matrix. Let B1 and B2 be the OLS coefficient. Consider the following partial regression. First, from the OLS fit of Y on X1, we obtain the residual vector Y. Then, from the column-wise OLS fit of X2 on X1 we obtain the residual matrix X2. Finally, from the OLS fit of Y on X2 we obtain the coefficient B2. Prove B2 = B2 (7 points) Hint: You can use the lemma without proof. Let Sil = (XX1) '+(X;X1) XiX2 (X2X2) XX(X X1)-1, S21 = SF2 = - (X2X2) X2X1(X;X1) and S22 = (XT X2) -1. Then X'XI XiX2 - 1 (X' x ) = S11 S12 X5X1 X2X2 S21 S.22