(b) Because we have which is Equation (18.27). The residual vector is We know that MXX is orthogonal to the columns of X:      X X   MXX (In PX) X X PXX X (XX)1 XX X 0 so the residual vector can be further written as which is Equation (18.28).    18.7. (a) We write the regression model, Yi 1Xi 2Wi ui, in the matrix form as    Y X W U with The OLS estimator is By the law of large numbers (because X and W are independent with means of zero); (because X and u are independent with means of zero); Thus (b) From the answer to (a) if E(Wu) is nonzero. (c) Consider the population linear regression ui onto Wi:   ui Wi ai   where E(Wu)/E(W2). In this population regression, by construction, E(aW) 0. Using this equation for ui rewrite the equation to be estimated as where A calculation like that used in part (a) can be used to show that
(b) Because we have which is Equation (18.27). The residual vector is We know that MXX is orthogonal to the columns of X: MXX  (In  PX) X  X  PXX  X X (XX)1 XX X  X  0 so the residual vector can be further written as which is Equation (18.28). 18.7. (a) We write the regression model, Yi  1Xi  2Wi  ui, in the matrix form as Y  X  W  U with The OLS estimator is By the law of large numbers (because X and W are independent with means of zero); (because X and u are independent with means of zero); Thus (b) From the answer to (a) if E(Wu) is nonzero. (c) Consider the population linear regression ui onto Wi: ui  Wi  ai where  E(Wu)/E(W2). In this population regression, by construction, E(aW)  0. Using this equation for ui rewrite the equation to be estimated as where A calculation like that used in part (a) can be used to show that
sw3_exeol_oddonly.docx