▁{ xi ε i } ▁is ▁a ▁mart ing ale ▁difference ▁sequence , ▁with ▁a ▁finite ▁matrix ▁of ▁second ▁moments ▁. ▁▁ ▁Fin ite ▁sample ▁properties ▁ ▁First ▁of ▁all , ▁under ▁the ▁strict ▁ex ogene ity ▁assumption ▁the ▁O LS ▁estim ators ▁ ▁and ▁s 2 ▁are ▁un bi ased , ▁meaning ▁that ▁their ▁expected ▁values ▁co inci de ▁with ▁the ▁true ▁values ▁of ▁the ▁parameters :[ proof ] ▁▁ ▁If ▁the ▁strict ▁ex ogene ity ▁does ▁not ▁hold ▁( as ▁is ▁the ▁case ▁with ▁many ▁time ▁series ▁models , ▁where ▁ex ogene ity ▁is ▁assumed ▁only ▁with ▁respect ▁to ▁the ▁past ▁shock s ▁but ▁not ▁the ▁future ▁ones ), ▁then ▁these ▁estim ators ▁will ▁be ▁bi ased ▁in ▁finite ▁samples . ▁ ▁The ▁variance - cov ar iance ▁matrix ▁( or ▁simply ▁cov ar iance ▁matrix ) ▁of ▁ ▁is ▁equal ▁to ▁▁ ▁In ▁particular , ▁the ▁standard ▁error ▁of ▁each ▁coefficient ▁ ▁is ▁equal ▁to ▁square ▁root ▁of ▁the ▁j - th ▁diagonal ▁element ▁of ▁this ▁matrix . ▁The ▁estimate ▁of ▁this ▁standard ▁error ▁is ▁obtained ▁by ▁replacing ▁the ▁unknown ▁quantity ▁ σ 2 ▁with ▁its ▁estimate ▁s 2 . ▁Thus , ▁▁▁ ▁It ▁can ▁also ▁be ▁easily ▁shown ▁that ▁the ▁estim ator ▁ ▁is ▁un cor related ▁with ▁the ▁resid uals ▁from ▁the ▁model : ▁▁▁ ▁The ▁G auss – Mark ov ▁theorem ▁states ▁that ▁under ▁the ▁spher ical ▁errors ▁assumption ▁( that ▁is , ▁the ▁errors ▁should ▁be ▁un cor related ▁and ▁hom os ced astic ) ▁the