Lecture 18. Serial correlation: testing and estimation. Testing for serial correlation

Transcription

1 Lecure 8. Serial correlaion: esing and esimaion Tesing for serial correlaion In lecure 6 we used graphical mehods o look for serial/auocorrelaion in he random error erm u. Because we canno observe he u we used he OLS residuals e. We looked a Time series graph of e, =,, n. If here is serial correlaion his graph shows gradual changes in he e. Scaerplo of e versus e. If he AR() model u = ρ u ε holds, hen we expec ha he scaerplo is concenraed along a sraigh line hrough 0. Tess for serial/auocorrelaion also use he OLS residuals e.

2 Consider he linear regression model Y = β β X L β K X K u, =, K, n As wih ess for heeroskedasiciy we assume a paricular model for he auocorrelaion. Iniially we consider firsorder serial correlaion () u ρ u ε = < ρ < wih ε whie noise (he ε s are independen and have all he same variance and mean 0). If ρ = 0, hen u = ε and in ha case he random errors u saisfy Assumpion 4, i.e. here is no serial correlaion. Hence a es for serial correlaion is a es of H : ρ 0. 0 =

3 Firs sep is o find esimaor for ρ. If we replace u in () by e and esimae ρ by OLS we obain ρ = n e = n = e e This is also he firs-order auocorrelaion coefficien of he ime series e, =, K, n (see lecure 6). The obvious hing o do is o use ρ o es wheher = 0 ρ. Insead of ρ, a relaed quaniy is used, he Durbin-Wason saisic d d = n = ( e n = e e )

4 I can be shown d = ( ρ) Hence if ρ close o 0 (no auocorrelaion) hen d is close o. If ρ is close o, hen d is close o 0 and if ρ is close o, hen d is close o 4 posiive no negaive auocor. auocor. auocor. 0 4 c c L c U d Because negaive auocorrelaion is rare, he usual es is H : ρ 0 agains : ρ 0 0 = H. > We rejec (see graph) if d is small, i.e. close o 0. The criical value is han some number c greaer han 0 bu less han.

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6 Wih a 5% significance level we wan o have ha if H 0 is rue, hen he probababiliy of rejecion of H 0 Pr( d < c) =.05 If he errors have a normal disribuion (assumpion 5), hen he disribuion of d can be derived. This disribuion depends on he independen variables X, K, X K. Compare wih he - or F-disribuion ha do no depend on his. Some programs compue c exacly for he independen variables in your daase. This is easy wih curren compuers. If no, hen here is a able wih bounds c L and c U. These bounds are for exreme daases and he c for any daase is beween hem.

7 Example: for independen variables (do no coun he consan) and 5 observaions c L =.06 and c U = Hence if e.g. d =. we rejec and if d =. 7 we do no rejec. If d =.3 we do know wha o do (es is inconclusive). This is compuaional problem, because c can be compued. Consider regression log housing sars per head on log GNP per head and log morgage ineres rae The DW saisics is.93 wih n=3, k =, so ha c L =. 68 and we rejec he hypohesis of no serial correlaion.

8 Dependen Variable: LNHOUSINGCAP Mehod: Leas Squares Dae: /3/0 Time: 00:06 Sample: Included observaions: 3 Variable Coefficien Sd. Error -Saisic Prob. C LNGNPCAP LNINTRATE R-squared Mean dependen var.9996 Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion 0.99 Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)

9 Alernaive o DW es is he Lagrange Muliplier (LM) es. Also uses he OLS residuals e. The firs sep of he es is a linear regression wih dependen variable e and independen variables X, K, X K, e. Compue he R of his regression. The es saisic is LM = ( n ) R Noe ha we use n observaions in he regression. If H 0 : ρ = 0 is rue han LM has a chi-square disribuion wih d.f. We rejec if LM > c and if we wan a es wih a 5% significance level we find he criical value c from Pr( LM > c) =.05

10 Applicaion o housing sar daa LM = *.3 = 6.85 and he criical value for 5% significance is Again we rejec H 0. Esimaion wih serial correlaion Consider he linear regression Y = β β X u and u ρ u ε = AR() How do we esimae he regression parameers and ρ? As wih heeroskedasiciy we ransform he variables such ha we have a random error erm ha saisfies he assumpions -4. Hence we can apply OLS o he ransformed regression.

11 Dependen Variable: RESID0 Mehod: Leas Squares Dae: /4/0 Time: 3:56 Sample(adjused): Included observaions: afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C LNGNPCAP LNINTRATE RESID0LAG R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa.7797 Prob(F-saisic)

12 Because ε saisfies all he usual assumpions we mus ge his as he random error erm. Noe ε = u ρu Now do he subracion ρ Y Y β X u = β = ρβ ρβ X ρu () Y ρ Y = ( ρ) β β( X ρx ) ε Conclusion: if we ransform he dependen variable o Y ρ Y and he independen variable o X ρ X we can use OLS o esimae β. Noe ha he OLS esimaor of he consan does no esimae β, bu if we divide he OLS esimaor of he consan by ρ we ge an esimaor of β.

13 Problem wih his mehod: We do no know ρ. Soluion: Choose range of values for ρ, e.g. -.99, -.98,.,.98,.99 and esimae () for each of hese values. For each ρ compue he residuals e = Y ρ Y ( ρ) β β( X ρx ) and he sum of squared residuals. Choose he value of ρ and he OLS esimaors of β, β ha has he smalles sum of squared residuals. This he Hildreh-Lu procedure. Applicaion o consumpion and wages (billion 99$) for US Tes for AR() errors (DW and LM) Compare esimaes and sandard errors

14 Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:0 Sample: Included observaions: 36 Variable Coefficien Sd. Error -Saisic Prob. C WAGES R-squared Mean dependen var 8.78 Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)

15 Dependen Variable: RESID0 Mehod: Leas Squares Dae: /5/0 Time: 0: Sample(adjused): Included observaions: 35 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C WAGES RESID0LAG R-squared Mean dependen var.503 Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic)

16 Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:3 Sample(adjused): Included observaions: 35 afer adjusing endpoins Convergence achieved afer 8 ieraions Variable Coefficien Sd. Error -Saisic Prob. C WAGES AR() R-squared Mean dependen var Adjused R-squared S.D. dependen var 97.3 S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa.3556 Prob(F-saisic) Invered AR Roos.94

17 Alernaive inerpreaion of AR() errors The linear regression in () can be rewrien as () Y = ( ρ ) β ρy β X ρβ X ε This is a linear regression model wih independen variables Y, X, X. In he model wih only X as independen variable Y, X are omied and relegaed o he error erm. Because boh variables are economic ime series and change gradually he error erm is auocorrelaed. Compare () o he linear regression model (3) Y = γ γ Y γ 3X γ 4 X ε Noe ha (3) has 4 regression coefficiens and 4 () has 3. (3) becomes () if γ γ γ 3 =.

18 If we esimae (3) we find γ 4 =. 69, γ 3 =.78, γ =. 933 and hence γ γ.69 =.78 4 = 3.96 Noe ha (3) is more general and an alernaive o ().

19 Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:5 Sample(adjused): Included observaions: 35 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C CONSLAG WAGES WAGESLAG R-squared Mean dependen var Adjused R-squared S.D. dependen var 97.3 S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa.059 Prob(F-saisic)

20 Dependen Variable: RESID0 Mehod: Leas Squares Dae: /5/0 Time: 0:6 Sample(adjused): Included observaions: 34 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C WAGES WAGESLAG RESID0LAG R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic.7748 Durbin-Wason sa Prob(F-saisic)

21 Predicion wih AR() The predicion of Y is e X X Y X X X Y Y ρ β β β ρ β β ρβ β ρ β ρ ) ( ) ( = = = = Compare his wih = X Y β β for he linear regression wihou serial correlaion. The error in period can be prediced using he residual in period.