▁equal ▁to ▁zero ▁for ▁any ▁conform al ▁vector , ▁v . ▁This ▁means ▁that ▁ ▁is ▁the ▁short est ▁of ▁all ▁possible ▁vectors ▁, ▁that ▁is , ▁the ▁variance ▁of ▁the ▁resid uals ▁is ▁the ▁minimum ▁possible . ▁This ▁is ▁illustrated ▁at ▁the ▁right . ▁ ▁Int rodu cing ▁ ▁and ▁a ▁matrix ▁K ▁with ▁the ▁assumption ▁that ▁a ▁matrix ▁ ▁is ▁non - sing ular ▁and ▁K T ▁X ▁= ▁ 0 ▁( cf . ▁Orth ogonal ▁pro jections ), ▁the ▁resid ual ▁vector ▁should ▁satisfy ▁the ▁following ▁equation : ▁ ▁The ▁equation ▁and ▁solution ▁of ▁linear ▁least ▁squares ▁are ▁thus ▁described ▁as ▁follows : ▁ ▁Another ▁way ▁of ▁looking ▁at ▁it ▁is ▁to ▁consider ▁the ▁regression ▁line ▁to ▁be ▁a ▁weight ed ▁average ▁of ▁the ▁lines ▁passing ▁through ▁the ▁combination ▁of ▁any ▁two ▁points ▁in ▁the ▁dataset . ▁Although ▁this ▁way ▁of ▁calculation ▁is ▁more ▁computation ally ▁expensive , ▁it ▁provides ▁a ▁better ▁intu ition ▁on ▁O LS . ▁▁ ▁Max imum ▁lik elihood ▁ ▁The ▁O LS ▁estim ator ▁is ▁identical ▁to ▁the ▁maximum ▁lik elihood ▁estim ator ▁( M LE ) ▁under ▁the ▁normal ity ▁assumption ▁for ▁the ▁error ▁terms .[ proof ] ▁This ▁normal ity ▁assumption ▁has ▁historical ▁importance , ▁as ▁it ▁provided ▁the ▁basis ▁for ▁the ▁early ▁work ▁in ▁linear ▁regression ▁analysis ▁by ▁Y ule ▁and ▁Pear son . ▁From ▁the ▁properties ▁of ▁M LE , ▁we ▁can ▁infer ▁that ▁the ▁O LS ▁estim ator ▁is ▁asympt ot ically ▁efficient ▁( in ▁the ▁sense ▁of ▁att aining