▁by ▁a ▁simple ▁formula , ▁especially ▁in ▁the ▁case ▁of ▁a ▁simple ▁linear ▁regression , ▁in ▁which ▁there ▁is ▁a ▁single ▁re gress or ▁on ▁the ▁right ▁side ▁of ▁the ▁regression ▁equation . ▁ ▁The ▁O LS ▁estim ator ▁is ▁consistent ▁when ▁the ▁re gress ors ▁are ▁ex ogen ous , ▁and ▁optimal ▁in ▁the ▁class ▁of ▁linear ▁un bi ased ▁estim ators ▁when ▁the ▁errors ▁are ▁hom os ced astic ▁and ▁serial ly ▁un cor related ▁. ▁Under ▁these ▁conditions , ▁the ▁method ▁of ▁O LS ▁provides ▁minimum - var iance ▁mean - un bi ased ▁estimation ▁when ▁the ▁errors ▁have ▁finite ▁var ian ces . ▁Under ▁the ▁additional ▁assumption ▁that ▁the ▁errors ▁are ▁normally ▁distributed , ▁O LS ▁is ▁the ▁maximum ▁lik elihood ▁estim ator . ▁ ▁O LS ▁is ▁used ▁in ▁fields ▁as ▁diverse ▁as ▁physics , ▁chem istry , ▁bi ology , ▁econom ics ▁( e con omet rics ), ▁data ▁science , ▁political ▁science , ▁psych ology ▁and ▁engineering ▁( e . g ., ▁control ▁theory ▁and ▁signal ▁processing ). ▁ ▁Linear ▁model ▁▁ ▁Suppose ▁the ▁data ▁consists ▁of ▁n ▁observations ▁{   y ,   x   }. ▁Each ▁observation ▁i ▁includes ▁a ▁scalar ▁response ▁y i ▁and ▁a ▁column ▁vector ▁x i ▁of ▁values ▁of ▁p ▁predict ors ▁( re gress ors ) ▁x ij ▁for ▁j ▁= ▁ 1 , ▁... , ▁p . ▁In ▁a ▁linear ▁regression ▁model , ▁the ▁response ▁variable , ▁, ▁is ▁a ▁linear ▁function ▁of ▁the ▁re gress ors :