(1) Cov(, ) E[( E( ))( E( ))]

Transcription

1 Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε ) = 0 for There are observaos he sample, so =,,3. Assume ha he hrd assumpo s volaed ad here s frs order auo-correlao, whch s characerzed by he followg equao v Where ρ < ad we make he followg assumpos abou v ) E(v ) = 0 ) Var(v ) = v ) Cov(v, v ) = 0 for v) Cov(ε -,v ) = 0 Wh auocorrelao, cosder he value of Cov(ε, ε -). By defo () Cov(, ) E[( E( ))( E( ))] bu because E( ) E( ) 0, equao () reduces o () Cov(, ) E[ ] Usg he fac ha we have frs-order auocorrelao, v, we ca subsue hs equao o () for ε (3) Cov(, ) E[ ] E[( v ) ] E[ ] E[ v ] Recall ha by defo Var( ) E[( E( )) ], bu because E( ) 0, he Var( ) E[ ]. Noce also ha we assume v ad ε - are ucorrelaed so Ev [ ] =0, ad herefore (4) Cov(, ) E[ ] E[ v ] Ths mehod ca easly be geeralzed o cosder he degree of covarace bewee ay wo me perods he daa. For example, suppose we are eresed Cov(, ) whch by defo equals

2 (5) Cov(, ) E[ ]. Noe ha we ca wre ε as a fuco of ε - ad ε - as a fuco of ε -. (6) v (7) v Ad herefore, subsug (7) o (6) we ge (8) ( v ) v v v Subsug (8) o (5) for ε, we ge (9) Cov(, ) E[( v v ) ] E[ ] E[ v ] E[ v ] Ad usg he facs ha E[ v ] E[ v ] 0 ad E[ ] we fd ha (0) Cov(, ) E[ ] We ca easly geeralze hs resul o oe ha f here s a u dfferece bewee he errors, he () Cov(, ) E[ ] I wha follows, we wll exame he bas ad varace of he OLS esmae for β a) assumg auocorrelao s prese, bu b) gorg hs fac he model Is Ubased? Recall he defo for () ( y y)( x x) ( x x) Noe also ha we ca drop he y he umeraor ad we ca subsue he rue defo of y ( ) o he model 0 x () ( y y)( x x) y ( x x) ( x )( x x) 0 ( x x) ( x x) ( x x) Break apar he erms he umeraor

3 (3) ( x x) x ( x x) ( x x) 0 ( x x) We ca smplfy he erms (3) usg he properes of summaos: ( x x) (0) ( x x) x ( x x) so Ad herefore, equao (3) reduces o our old fred (4) ( x x) ( x x) Takg he expecao of (4), oce ha ( x x) ( x x) E ( x x) (5) E[ ] E E[ ] E ( x x) ( x x) ( x x) Noe ha he gve he presece of auocorrelao, he key assumpo ha x s fxed has o bee alered. Therefore, we ca sll maa assumpo v) E(ε x ) = 0 ad as a resul, (6) E[ β ] = β Eve he presece of auocorrelao, OLS esmaes are ubased. 3

4 The varace of β he presece of auocorrelao : By defo, (7) Var( ) E[( E[ ]) ] Prevously, we demosraed ha β s a ubased esmae eve he presece of auocorrelao ad herefore, E[ β ]=β (8) Var ( ) E[( ) ] Lookg a equao (4) o he prevous page, oe ha he dfferece β - β s smply (9) ( x x) ( x x) where ( ) x x ( x x) Usg he defo of he varace ad equao (9) ( x x) x (0) ( Var ) E[( ) ] E E x x Where x x x. Because x s a cosa (x s cosdered fxed) we ca brg ousde he expecao. Therefore () Var( ) E x x Le s work wh he umeraor he far rgh had erm equao (8). Complee he square o hs erm geeraes. x x x x x x x x x x 3x3 x x... [ ] Whch ca be smplfed o read. () x x x Noce he secod erm, he summaos go from = o - ad =+ o. So whe =, we cosder he cross erms xx o xx whle whe =-, we oly cosder he erm, x x. Subsug () o (), we oba ha 4

5 (3) Var( ) E x x x E x E x x x By defo, x s fxed ad herefore, he frs erm reduces o somehg we already kow: (4) E x [ ] x E x x x x x x ( ) x x x x x x Lookg back a our oes for he basc OLS model, recall ha he secod erm equao (3) E x x drops ou compleely because we assume ha Cov(ε, ε ) =0. I hs case, however, we are allowg he errors o be correlaed. Usg resuls from page () of hs hadou, he expecao of a erm sde he summao s he followg (5) E[ x x ] x x E[ ] x x ad herefore, addg (5) ad (4) back o (3) (6) Var( ) ( x )( ) x x x Ad for lack of a beer erm, call hs varace V ac(). Noe he secod erm equao () ha >0, ad assumg auocorrelao s posve, he >0 as well. Fally, f he x s are posvely correlaed over me, he ( x x)( x x) >0. >0, If here s o auocorrelao he model, he rue varace of he esmae would be Var( ) ad for lack of a beer erm, call hs Var( ) = V ols. Therefore, f he daa exhbs frs-order auocorrelao ad we gore, we wll use V ols as he varace. However, V ols s coaed V ac() ad he secod erm equao (6) s posve, he by cosruco V ac() > V ols ad as resul, f we have posve auocorrelao he daa ad we gre hs fac, he esmaed varace wll be oo small, leadg o more Type I errors. x 5