▁estim ator ▁ ▁is ▁efficient ▁in ▁the ▁class ▁of ▁linear ▁un bi ased ▁estim ators . ▁This ▁is ▁called ▁the ▁best ▁linear ▁un bi ased ▁estim ator ▁( BL UE ). ▁E ffic iency ▁should ▁be ▁understood ▁as ▁if ▁we ▁were ▁to ▁find ▁some ▁other ▁estim ator ▁ ▁which ▁would ▁be ▁linear ▁in ▁y ▁and ▁un bi ased , ▁then ▁▁▁ ▁in ▁the ▁sense ▁that ▁this ▁is ▁a ▁non negative - defin ite ▁matrix . ▁This ▁theorem ▁establish es ▁optim ality ▁only ▁in ▁the ▁class ▁of ▁linear ▁un bi ased ▁estim ators , ▁which ▁is ▁quite ▁restrict ive . ▁Depending ▁on ▁the ▁distribution ▁of ▁the ▁error ▁terms ▁ ε , ▁other , ▁non - linear ▁estim ators ▁may ▁provide ▁better ▁results ▁than ▁O LS . ▁▁ ▁Assuming ▁normal ity ▁ ▁The ▁properties ▁listed ▁so ▁far ▁are ▁all ▁valid ▁regardless ▁of ▁the ▁underlying ▁distribution ▁of ▁the ▁error ▁terms . ▁However , ▁if ▁you ▁are ▁willing ▁to ▁assume ▁that ▁the ▁normal ity ▁assumption ▁holds ▁( that ▁is , ▁that ▁), ▁then ▁additional ▁properties ▁of ▁the ▁O LS ▁estim ators ▁can ▁be ▁stated . ▁ ▁The ▁estim ator ▁ ▁is ▁normally ▁distributed , ▁with ▁mean ▁and ▁variance ▁as ▁given ▁before : ▁▁ ▁where ▁Q ▁is ▁the ▁c of actor ▁matrix . ▁This ▁estim ator ▁reaches ▁the ▁C ram ér – R ao ▁bound ▁for ▁the ▁model , ▁and ▁thus ▁is ▁optimal ▁in ▁the ▁class ▁of ▁all ▁un bi ased ▁estim ators . ▁Note ▁that ▁unlike ▁the ▁G auss – Mark ov ▁theorem , ▁this ▁result ▁establish