Consider linear fit Y ~ X181 + 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 = 32 (7 points) Hint: You can use the lemma without proof. Let Sil = (X X1) "'+(XiXi) XiX2 (XX2) XX (XiX1)-1, S21 = S42 = - (X2X2) XX1(X X1) and $22 = (XT X2)-1. Then (X X ) 1 = XiX1 XiX2 -1 S11 S12 XX1 X2X2 S21 S22