▁assumption ▁about ▁the ▁distribution ▁of ▁error ▁term ▁ ε i , ▁it ▁is ▁impossible ▁to ▁infer ▁the ▁distribution ▁of ▁the ▁estim ators ▁ ▁and ▁. ▁Nevertheless , ▁we ▁can ▁apply ▁the ▁central ▁limit ▁theorem ▁to ▁derive ▁their ▁asympt otic ▁properties ▁as ▁sample ▁size ▁n ▁goes ▁to ▁infinity . ▁While ▁the ▁sample ▁size ▁is ▁necessarily ▁finite , ▁it ▁is ▁custom ary ▁to ▁assume ▁that ▁n ▁is ▁" large ▁enough " ▁so ▁that ▁the ▁true ▁distribution ▁of ▁the ▁O LS ▁estim ator ▁is ▁close ▁to ▁its ▁asympt otic ▁limit . ▁ ▁We ▁can ▁show ▁that ▁under ▁the ▁model ▁assumptions , ▁the ▁least ▁squares ▁estim ator ▁for ▁ β ▁is ▁consistent ▁( that ▁is ▁ ▁converges ▁in ▁probability ▁to ▁ β ) ▁and ▁asympt ot ically ▁normal :[ proof ] ▁▁ ▁where ▁▁▁ ▁Inter val s ▁▁ ▁Using ▁this ▁asympt otic ▁distribution , ▁approximate ▁two - s ided ▁confidence ▁intervals ▁for ▁the ▁j - th ▁component ▁of ▁the ▁vector ▁ ▁can ▁be ▁constructed ▁as ▁▁ ▁  ▁at ▁the ▁ ▁confidence ▁level , ▁where ▁q ▁denotes ▁the ▁quant ile ▁function ▁of ▁standard ▁normal ▁distribution , ▁and ▁[ · ] j j ▁is ▁the ▁j - th ▁diagonal ▁element ▁of ▁a ▁matrix . ▁ ▁Similarly , ▁the ▁least ▁squares ▁estim ator ▁for ▁ σ 2 ▁is ▁also ▁consistent ▁and ▁asympt ot ically ▁normal ▁( prov ided ▁that ▁the ▁fourth ▁moment ▁of ▁ ε i ▁exists ) ▁with ▁limit ing ▁distribution ▁▁▁ ▁These ▁asympt otic ▁distributions ▁can ▁be ▁used ▁for ▁prediction , ▁testing ▁hypoth eses , ▁construct ing ▁other