▁the ▁identity ▁matrix . ▁The ▁variance - cov ar iance ▁matrix ▁of ▁the ▁resid uals , ▁M ▁r ▁is ▁given ▁by ▁ ▁Thus ▁the ▁resid uals ▁are ▁correl ated , ▁even ▁if ▁the ▁observations ▁are ▁not . ▁ ▁When ▁, ▁ ▁The ▁sum ▁of ▁resid ual ▁values ▁is ▁equal ▁to ▁zero ▁whenever ▁the ▁model ▁function ▁contains ▁a ▁constant ▁term . ▁Left - multip ly ▁the ▁expression ▁for ▁the ▁resid uals ▁by ▁X T : ▁ ▁Say , ▁for ▁example , ▁that ▁the ▁first ▁term ▁of ▁the ▁model ▁is ▁a ▁constant , ▁so ▁that ▁ ▁for ▁all ▁i . ▁In ▁that ▁case ▁it ▁follows ▁that ▁ ▁Thus , ▁in ▁the ▁motiv ational ▁example , ▁above , ▁the ▁fact ▁that ▁the ▁sum ▁of ▁resid ual ▁values ▁is ▁equal ▁to ▁zero ▁is ▁not ▁accident al , ▁but ▁is ▁a ▁consequence ▁of ▁the ▁presence ▁of ▁the ▁constant ▁term , ▁α , ▁in ▁the ▁model . ▁ ▁If ▁experimental ▁error ▁follows ▁a ▁normal ▁distribution , ▁then , ▁because ▁of ▁the ▁linear ▁relationship ▁between ▁resid uals ▁and ▁observations , ▁so ▁should ▁resid uals , ▁but ▁since ▁the ▁observations ▁are ▁only ▁a ▁sample ▁of ▁the ▁population ▁of ▁all ▁possible ▁observations , ▁the ▁resid uals ▁should ▁belong ▁to ▁a ▁Student ' s ▁t - distribution . ▁Student ized ▁resid uals ▁are ▁useful ▁in ▁making ▁a ▁statistical ▁test ▁for ▁an ▁out lier ▁when ▁a ▁particular ▁resid ual ▁appears ▁to ▁be ▁excess ively ▁large . ▁ ▁See ▁also ▁Iter atively ▁re weight ed ▁least ▁squares ▁H eter os ced astic ity - cons