Auto-correlation of Error Terms

Transcription

1 Auo-correlaio of Error Terms Pogsa Porchaiwiseskul Faculy of Ecoomics Chulalogkor Uiversiy (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy Geeral Auo-correlaio () YXβ + ν E(ν)0 V(ν) σ Σ where 0 is a x colum vecor of zeroes Σ is a x osiive-defiie symmeric marix. (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 3 Covered Toics GLS Assumios Deecio, e.g., Durbi-Waso d-es Breusch-Godfrey es Esimaio Mehods, e.g., Cochrae-Orcu Geeral Auo-correlaio () σ σ σ σ σ σ σ σ σ Σ σσij νi ν j i j where V(, ) for,,..., (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 4

2 Geeral Auo-correlaio (3) Sources of Auo-correlaio Ieria (aure) Sill-over effec over geograhical regio, e.g., coagio, migraio (aure) Sec. errors, e.g., Exclusio of auo-correlaed ideede variables Icorrec fucioal form Lagged erms (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 5 Geeralized LS () ΩYΩXβ + Ων where Ω is a x symmeric marix such ha ΩΩ Σ Noe ha E(Ων)0 ad V(Ων)σ I (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 7 Geeral Auo-correlaio (4) Effec of Igorig Pure Auo-correlaio OLS is uibiased bu o he bes Need ew esimae of Σ, e.g., Newey-Wes formula (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 6 Geeralized LS () If Σ is kow, aly OLS o BLU esimae β,σ T T ( ) ( ) ( ) ( ) βˆ ΩX ΩX ΩX ΩY T T X Σ X T ( Ω Y Xβˆ ) ( Ω Y Xβˆ ) ˆ ( ) ( ) K σ X Σ Y ( Y Xβˆ) Σ ( Y Xβˆ) T K Wha if Σ is ukow? Need more assumios? (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 8

3 Assumios Y X β + X β + + X β + ν i,..., i i i... Ki K i I addiio o CLRM assumios ARMA error erm (ν) weakly saioary error erm. Oherwise, esimaio will be ivalid. Why? (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 9 AR() error erm --() Grahical Tes e e e - e - ρ>0 ρ<0 where auo-correlaio e is he OLS has residual bee igored whe (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy AR() error erm --() Y X β + X β X β + ρν + ε -< ρ <,,..., ε ~ Whie Noise H 0 : ρ 0 H: ρ 0 K K Acce > No auo-correlaio Rejec > AR() (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 0 AR() error erm --(3) Grahical Tes (co d) e ρ0 Noe ha lo of e vs e - cao reveal higher AR i error erms (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy e -

4 AR() error erm --() Y X β + X β X β + ρν K K ν ρν + ρν ρ ν + ε ~ saioary ε ~ Whie Noise H : ρ ρ... ρ 0 0 H : ρ ρ... ρ 0 Acce > No auo-correlaio Rejec > AR() or lower (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 3 Durbi-Waso s d-es () DW ( e e ) e (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 5 Noe ha DW saisic follows a secial disribuio. Need a secial able o erform AR() es. See Table D.5A-B (Gujarai) Auo-correlaio Tess Saisical Tess Durbi-Waso s AR(). d-es. Good oly for DW s h-es (obsolee) BG s Serial Geeral Correlaio Auo-correlaio es or Rus ormal es error (o-arameric). erm or ARMA No form. reuire Good for small samle. (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 4 Durbi-Waso s d-es () DW ( e e e + e ) e e e e e e e ρˆx + ( ρˆ) > 0 DW 4 + e (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 6

5 Durbi-Waso s d-es (3) Give sigifica level(α) ad samle size (), read d L ad d U from DW ables. Crierio: 0 DW dl > ρ > 0 os. auo-corr. dl DW du > ρ 0 idecisive du DW 4 dl > ρ 0 zero auo-corr. 4 dl DW 4 du > ρ 0 idecisive 4 d DW 4 > ρ < 0 eg. auo-corr. U (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 7 Durbi-Waso s d-es (4) Noes DW is ivalid if here are lagged deede variables as exlaaory variables (AR Model). Their exisece is euivale o higher AR of error erms Good oly for AR(). DW saisic is usually a iem i he OLS reor. DW ca be used o roughly esimae ρ bu o SE give. (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 9 Durbi-Waso s d-es (4) Pos. Auocorr.? Zero? Auo-correlaio 0 d L d U 4-d U 4-d L 4 Noe ha he coclusio is o simly Acce or Rejec. Neg. Auocorr. (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 8 Breusch-Godfrey s es () Se Ru OLS by igorig auo-correlaio > residual νˆ Se ru OLS for νˆ X β+ Xβ XKβK + ρνˆ + ρνˆ ρ νˆ + ε (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 0

6 Breusch-Godfrey s es () Se 3 Do F-Tes or χ es for H 0 : ρ ρ... ρ 0 H : ρ ρ... ρ 0 Acce H 0 > o AR() auo-corr. Rejec > AR() or lower Noe ha BG es also gives esimae for ρ s ad heir SE s bu sill o he bes esimaes. (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy Cochrae-Orcu Mehod () For AR() error erm Se auo-correlaio Ru OLS by igorig > residual νˆ Se ru OLS for νˆ ρνˆ ρνˆ ρ νˆ ε > ρˆ, ρˆ,..., ρˆ Se 3 Trasform Y ad X s (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 3 Breusch-Godfrey s es (3) F χ cal ( RSSR RSSU) RSS ( K ) U ( TSSU RSSU) RSS ( K ) U K R ( ) R ~ χ ( ) cal ~ F(, K ) (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy Cochrae-Orcu Mehod () Se 3 Trasform Y ad X s Y Y ρˆy ρˆ Y... ρˆ Y X X ρˆ X ρˆ X... ρˆ X k k k, k, k, Se 4 Ru OLS for Y X β + X β X β + ν K K ν ν ρν ρν... ρ ν ε k,.., K (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 4

7 Cochrae-Orcu Mehod (3) V( ν ) V( ε ) σ >OLS is almos BLUE Se 5 Re-calculae ν usig ew νˆ Y X βˆ X βˆ... X βˆ If soluio does o sigificaly chage, so. Oherwise, go back o Se ˆ K K (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 5 ˆβ MA Par Esimaio () γˆ ( θ θ + θ θ θ θ ) ( θ + θ θ ) 3 γˆ ( θ θ + θ θ θ θ ) ( θ + θ θ ) M 3 4 γˆ ( θ θ ) ( θ + θ θ ) ukow euaios > solve for Add as se 4. i Cochrae-Orcu (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 7 ˆθ MA Par Esimaio () Noe ha ν ε + θ ε + θ ε θ ε Cov( ν, ν ) σ ( θ + θ θ ) Cov( ν, ν ) σ ( θ θ + θ θ θ θ ) M Cov 3 ( ν, ν ) σ ( θθ) (c) Pogsa Porchaiwiseskul, Faculy of Ecoomics, Chulalogkor Uiversiy 6