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.
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 =
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 )
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.
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.
Example: for independen variables (do no coun he consan) and 5 observaions c L =.06 and c U =. 550. 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.
Dependen Variable: LNHOUSINGCAP Mehod: Leas Squares Dae: /3/0 Time: 00:06 Sample: 963 985 Included observaions: 3 Variable Coefficien Sd. Error -Saisic Prob. C.58899.8047.478 0.0447 LNGNPCAP -0.066000 0.540505-0.09 0.9040 LNINTRATE -0.84 0.0894 -.0435 0.30 R-squared 0.09447 Mean dependen var.9996 Adjused R-squared 0.00356 S.D. dependen var 0.6095 S.E. of regression 0.569 Akaike info crierion -0.0886 Sum squared resid.08735 Schwarz crierion 0.99 Log likelihood 3.0933 F-saisic.03935 Durbin-Wason sa 0.9305 Prob(F-saisic) 0.3707
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
Applicaion o housing sar daa LM = *.3 = 6.85 and he criical value for 5% significance is 3.84. 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.
Dependen Variable: RESID0 Mehod: Leas Squares Dae: /4/0 Time: 3:56 Sample(adjused): 964 985 Included observaions: afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C 0.330654.84974 0.5734 0.7998 LNGNPCAP -0.3373 0.59679-0.390759 0.7006 LNINTRATE 0.4769 0.03398 0.7355 0.4786 RESID0LAG 0.586673 0.0889.809346 0.06 R-squared 0.345 Mean dependen var -0.00634 Adjused R-squared 0.96694 S.D. dependen var 0.806 S.E. of regression 0.95443 Akaike info crierion -0.6434 Sum squared resid 0.68756 Schwarz crierion -0.065763 Log likelihood 6.905473 F-saisic.73985 Durbin-Wason sa.7797 Prob(F-saisic) 0.075366
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.
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 959-994. Tes for AR() errors (DW and LM) Compare esimaes and sandard errors
Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:0 Sample: 959 994 Included observaions: 36 Variable Coefficien Sd. Error -Saisic Prob. C 64.7 59.80 6.9979 0.0000 WAGES 0.76968 0.030647 5.440 0.0000 R-squared 0.94885 Mean dependen var 8.78 Adjused R-squared 0.947347 S.D. dependen var 945.5435 S.E. of regression 6.966 Akaike info crierion 3.653 Sum squared resid 60056. Schwarz crierion 3.7399 Log likelihood -43.736 F-saisic 630.7330 Durbin-Wason sa 0.074 Prob(F-saisic) 0.000000
Dependen Variable: RESID0 Mehod: Leas Squares Dae: /5/0 Time: 0: Sample(adjused): 960 994 Included observaions: 35 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C 48.97566 3.6937 3.690880 0.0008 WAGES -0.06748 0.00674-3.9785 0.0004 RESID0LAG 0.9308 0.03763 4.7658 0.0000 R-squared 0.950467 Mean dependen var.503 Adjused R-squared 0.94737 S.D. dependen var 03.83 S.E. of regression 46.694 Akaike info crierion 0.6034 Sum squared resid 6955.55 Schwarz crierion 0.7367 Log likelihood -8.5596 F-saisic 307.044 Durbin-Wason sa.09648 Prob(F-saisic) 0.000000
Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:3 Sample(adjused): 960 994 Included observaions: 35 afer adjusing endpoins Convergence achieved afer 8 ieraions Variable Coefficien Sd. Error -Saisic Prob. C 566.4 93.594.75300 0.0096 WAGES 0.59497 0.38746 3.7440 0.0007 AR() 0.9437 0.04785 9.7054 0.0000 R-squared 0.997574 Mean dependen var 85.680 Adjused R-squared 0.99743 S.D. dependen var 97.3 S.E. of regression 47.06673 Akaike info crierion 0.683 Sum squared resid 70888.86 Schwarz crierion 0.7564 Log likelihood -8.8995 F-saisic 6580.7 Durbin-Wason sa.3556 Prob(F-saisic) 0.000000 Invered AR Roos.94
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 =.
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 ().
Dependen Variable: CONS Mehod: Leas Squares Dae: /5/0 Time: 0:5 Sample(adjused): 960 994 Included observaions: 35 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C 58.357 75.3676.0989 0.044 CONSLAG 0.9394 0.049885 8.7040 0.0000 WAGES 0.77754 0.88670.48643 0.085 WAGESLAG -0.69637 0.7958 -.47389 0.090 R-squared 0.9976 Mean dependen var 85.680 Adjused R-squared 0.99739 S.D. dependen var 97.3 S.E. of regression 47.3597 Akaike info crierion 0.66030 Sum squared resid 69508.48 Schwarz crierion 0.83806 Log likelihood -8.5553 F-saisic 4334.35 Durbin-Wason sa.059 Prob(F-saisic) 0.000000
Dependen Variable: RESID0 Mehod: Leas Squares Dae: /5/0 Time: 0:6 Sample(adjused): 96 994 Included observaions: 34 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. C 9.39565 4.09 0.666779 0.500 WAGES -0.93875 0.05809-0.9405 0.3537 WAGESLAG 0.0677 0.607 0.93848 0.3556 RESID0LAG 0.47740 0.6659.8735 0.0074 R-squared 0.767 Mean dependen var.97489 Adjused R-squared 0.38884 S.D. dependen var 45.84 S.E. of regression 4.97033 Akaike info crierion 0.493 Sum squared resid 5845.5 Schwarz crierion 0.605 Log likelihood -73.79 F-saisic.7748 Durbin-Wason sa.845907 Prob(F-saisic) 0.058508
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.