1 ▁X T = Q ▁X T ▁is ▁called ▁the ▁Moore – P en rose ▁pseudo in verse ▁matrix ▁of ▁X . ▁This ▁form ulation ▁highlight s ▁the ▁point ▁that ▁estimation ▁can ▁be ▁carried ▁out ▁if , ▁and ▁only ▁if , ▁there ▁is ▁no ▁perfect ▁mult icol linear ity ▁between ▁the ▁explan atory ▁variables ▁( which ▁would ▁cause ▁the ▁normal ▁matrix ▁to ▁have ▁no ▁inverse ). ▁ ▁After ▁we ▁have ▁estimated ▁ β , ▁the ▁fitted ▁values ▁( or ▁predicted ▁values ) ▁from ▁the ▁regression ▁will ▁be ▁▁▁ ▁where ▁P ▁= ▁X ( XT X ) − 1 XT ▁is ▁the ▁projection ▁matrix ▁onto ▁the ▁space ▁V ▁sp anned ▁by ▁the ▁columns ▁of ▁X . ▁This ▁matrix ▁P ▁is ▁also ▁sometimes ▁called ▁the ▁hat ▁matrix ▁because ▁it ▁" put s ▁a ▁hat " ▁onto ▁the ▁variable ▁y . ▁Another ▁matrix , ▁closely ▁related ▁to ▁P ▁is ▁the ▁anni h il ator ▁matrix ▁; ▁this ▁is ▁a ▁projection ▁matrix ▁onto ▁the ▁space ▁orthogonal ▁to ▁V . ▁Both ▁matrices ▁P ▁and ▁M ▁are ▁symmetric ▁and ▁ide mp ot ent ▁( mean ing ▁that ▁ ▁and ▁), ▁and ▁relate ▁to ▁the ▁data ▁matrix ▁X ▁via ▁ident ities ▁ ▁and ▁. ▁Matrix ▁M ▁creates ▁the ▁resid uals ▁from ▁the ▁regression : ▁▁▁ ▁Using ▁these ▁resid uals ▁we ▁can ▁estimate ▁the ▁value ▁of ▁ σ ▁ 2 ▁using ▁the ▁reduced ▁chi - squ ared ▁stat istic : ▁▁ ▁The ▁numer ator , ▁n − p , ▁is ▁the ▁statistical ▁degrees ▁of ▁freedom . ▁The ▁first ▁quantity , ▁s