9.1 Inroducion 9.2 Lags in he Error Term: Auocorrelaion 9.3 Esimaing an AR(1) Error Model 9.4 Tesing for Auocorrelaion 9.5 An Inroducion o Forecasing: Auoregressive Models 9.6 Finie Disribued Lags 9.7 Auoregressive Disribued Lag Models
Figure 9.1
y = f( x, x, x,...) 1 2 y = f( y, x ) 1 y = f( x) + e e = f ( e 1)
Figure 9.2(a) Time Series of a Saionary Variable
Figure 9.2(b) Time Series of a Nonsaionary Variable ha is Slow Turning or Wandering
Figure 9.2(c) Time Series of a Nonsaionary Variable ha Trends
9.2.1 Area Response Model for Sugar Cane ln ln ( A) =β +β ln ( P) 1 2 ( ) ln ( ) A = β+β P + e 1 2 y = β+β x + e 1 2 e = ρ e + v 1
y = β+β x + e 1 2 e = ρ e + v 1 Ev = v =σ v v = s 2 ( ) 0 var( ) v cov(, s) 0 for 1<ρ< 1
Ee ( ) = 0 var( ) 2 σv =σ = 1 ρ 2 e e 2 2 ( e e ) k cov, = σρ k > 0 k e
corr( e, e ) k cov( e, e ) cov( e, e ) σρ = = = =ρ var( e ) var 2 k k k e 2 ( e ) var( e) σ k e k corr( e, e 1) = ρ yˆ = 3.893 +.776 x (se) (.061) (.277)
Figure 9.3 Leas Squares Residuals Ploed Agains Time
r xy ( x x)( y y) cov( x, y) = 1 = = var( x)var( y) ( x x) ( y y) T T T 2 2 = 1 = 1 r 1 cov( e, e ) = = var( e ) T 1 = 2 T = 2 ee 垐 eˆ 1 2 1
The exisence of AR(1) errors implies: The leas squares esimaor is sill a linear and unbiased esimaor, bu i is no longer bes. There is anoher esimaor wih a smaller variance. The sandard errors usually compued for he leas squares esimaor are incorrec. Confidence inervals and hypohesis ess ha use hese sandard errors may be misleading.
Sugar cane example The wo ses of sandard errors, along wih he esimaed equaion are: yˆ = 3.893 +.776 x (.061) (.277) 'incorrec' se's (.062) (.378) 'correc' se's The 95% confidence inervals for β 2 are: (.211, 1.340) (incorrec) (.006,1.546) (correc)
y = β+β x + e 1 2 e = ρ e + v 1 y = β+β x +ρ e + v 1 2 1 e = y β β x 1 1 1 2 1
ρ e =ρy ρβ ρβ x 1 1 1 2 1 y =β (1 ρ ) +β x +ρy ρβ x + v 1 2 1 2 1 ln( A ) = 3.899 +.888ln( P) e =.422e + v 1 (se) (.092) (.259) (.166)
I can be shown ha nonlinear leas squares esimaion of (9.24) is equivalen o using an ieraive generalized leas squares esimaor called he Cochrane-Orcu procedure. Deails are provided in Appendix 9A.
y =β (1 ρ ) +β x ρβ x +ρ y + v 1 2 2 1 1 y = δ+δ x +δ x +θ y + v 0 1 1 1 1 δ =β1(1 ρ) δ 0 =β2 δ 1 = ρβ2 θ 1 =ρ yˆ = 2.366 +.777 x.611 x +.404y 1 1 (se) (.656) (.280) (.297) (.167)
9.4.1 Residual Correlogram H : ρ = 0 H : ρ 0 0 1 z = Tr1 N(0,1) z = 34.404 = 2.36 1.96
9.4.1 Residual Correlogram r 1.96 1.96 or r T T 1 1 r k 1.96 1.96 or rk T T cov( e, e ) E( ee ) ρ = = k k k 2 var( e) E( e )
Figure 9.4 Correlogram for Leas Squares Residuals from Sugar Cane Example
y = β+β x + e 1 2 y = β (1 ρ ) +β x +ρy ρβ x + v 1 2 1 2 1
Figure 9.5 Correlogram for Nonlinear Leas Squares Residuals from Sugar Cane Example
y = β+β x +ρ e + v 1 2 1 = 2.439 F = 5.949 p-value =.021 y = β+β x +ρ e垐 + v 1 2 1 b + b x + e垐 =β +β x +ρ e + v? 1 2 1 2 1
e垐 = ( β b) + ( β b ) x +ρ e + v? 1 1 2 2 1 =γ +γ x +ρ e垐 + v 1 2 1 LM T R 2 = = = 34.16101 5.474
y =δ+θ y +θ y + L+θ y + v 1 1 2 2 p p CPI CPI 1 y = ( ln( CPI) ln( CPI 1) ) 100 100 CPI 1 INFLN =.1883 +.3733 INFLN.2179 INFLN +.1013INFLN 1 2 3 (se) (.0253) (.0615) (.0645) (.0613)
Figure 9.6 Correlogram for Leas Squares Residuals from AR(3) Model for Inflaion
y =δ+θ y +θ y +θ y + v 1 1 2 2 3 3 y = δ+θ y +θ y +θ y + v T+ 1 1 T 2 T 1 3 T 2 T+ 1 yˆ =δ+θ 垐 y +θ 垐 y +θ y T+ 1 1 T 2 T 1 3 T 2 =.1883 +.3733.4468.2179.5988 +.1013.3510 =.2602
y垐 =δ+θ 垐 y +θ 垐 y +θ y T+ 2 1 T+ 1 2 T 3 T 1 =.1883 +.3733.2602.2179.4468 +.1013.5988 =.2487 u = y yˆ = ( δ δ 垐 ) + ( θ θ ) y + ( θ θ 垐 ) y + ( θ θ ) y + v 1 T+ 1 T+ 1 1 1 T 2 2 T 1 3 3 T 2 T+ 1
u = v + 1 T 1 u = θ ( y yˆ ) + v =θ u + v =θ v + v 2 1 T+ 1 T+ 1 T+ 2 1 1 T+ 2 1 T+ 1 T+ 2 u =θ u +θ u + v = ( θ +θ ) v +θ v + v 2 3 1 2 2 1 T+ 3 1 2 T+ 1 1 T+ 2 T+ 3
σ = var( u ) =σ 2 2 1 1 v σ = var( u ) =σ (1 +θ ) 2 2 2 2 2 v 1 σ = var( u ) =σ [( θ +θ ) +θ + 1] 2 2 2 2 2 3 3 v 1 2 1 ( y垐 1.96 σ, y垐 + 1.96 σ ) T + j j T + j j
y = α+β 0x +β 1x 1+β 2x 2 + L+β qx q + v, = q+ 1, K, T E( y ) x s = β s WAGE WAGE 1 x = ( ln( WAGE) ln( WAGE 1) ) 100 100 WAGE 1
y = δ+δ x +δ x + L+δ x +θ y + L+θ y + v 0 1 1 q q 1 1 p p y =α+β x +β x +β x +β x + L+ e 0 1 1 2 2 3 3 =α+ β x + e s= 0 s s
Figure 9.7 Correlogram for Leas Squares Residuals from Finie Disribued Lag Model
INFLN =.0989 +.1149 PCWAGE +.0377 PCWAGE +.0593PCWAGE 1 2 (se) (.0288) (.0761) (.0812) (.0812) +.2361 PCWAGE +.3536 INFLN.1976 INFLN 3 1 2 (.0829) (.0604) (.0604)
Figure 9.8 Correlogram for Leas Squares Residuals from Auoregressive Disribued Lag Model
y = δ+δ x +δ x +δ x +δ x +θ y +θ y + v 0 1 1 2 2 3 3 1 1 2 2 β 垐 =δ =.1149 0 0 β 垐 =θβ 垐 +δ =.3536.1149 +.0377 =.0784 1 1 0 1 β 垐 =θβ 垐 +θ β 垐 +δ =.0643 2 1 1 2 0 2 β 垐 =θβ 垐 +θ β 垐 +δ =.2434 3 1 2 2 1 3 β 垐 =θβ 垐 +θ β? =.0734 4 1 3 2 2
Figure 9.9 Disribued Lag Weighs for Auoregressive Disribued Lag Model
Principles of Economerics, 3rd Ediion Slide 9-44
Principles of Economerics, 3rd Ediion Slide 9-45
y = β+β x + e e =ρ e + v 1 2 1 y =β +β x +ρy ρβ ρβ x + v 1 2 1 1 2 1 (9A.1) ( 1 ) ( ) y ρ y =β ρ +β x ρ x + v 1 1 2 1 (9A.2) y = y ρ y x = x ρ x x = ρ 1 2 1 1 1 Principles of Economerics, 3rd Ediion Slide 9-46
y = x β+ x β+ v 1 1 2 2 (9A.3) y β β x =ρ( y β β x ) + v 1 2 1 1 2 1 (9A.4) Principles of Economerics, 3rd Ediion Slide 9-47
y1 = β+ 1 x1β+ 2 e1 1 ρ y = 1 ρ β + 1 ρ xβ + 1 ρ e 2 2 2 2 1 1 1 2 1 y = x β+ x β+ e 1 11 1 12 2 1 (9A.5) Principles of Economerics, 3rd Ediion y = 1 ρ y x = 1 ρ 2 2 1 1 11 x = 1 ρ x e = 1 ρ e 2 2 12 1 1 1 (9A.6) Slide 9-48
σv var( e ) (1 ) var( ) (1 ) 1 ρ 2 2 2 2 1 = ρ e1 = ρ =σ 2 v Principles of Economerics, 3rd Ediion Slide 9-49
H : ρ = 0 H : ρ> 0 0 1 d = T ( e垐 ) 2 e 1 = 2 T = 1 eˆ 2 (9B.1) Principles of Economerics, 3rd Ediion Slide 9-50
d = T T T 2 2 e垐垐 + e 1 2 ee 1 = 2 = 2 = 2 T 2 eˆ = 1 T T T 2 2 e垐 垐 e 1 ee 1 = 2 = 2 = 2 2 T T T 2 2 2 e垐? e e = 1 = 1 = 1 = + (9B.2) 1+ 1 2r 1 Principles of Economerics, 3rd Ediion Slide 9-51
d 21 r ( ) 1 (9B.3) d d c Principles of Economerics, 3rd Ediion Slide 9-52
Figure 9A.1: Principles of Economerics, 3rd Ediion Slide 9-53
Figure 9A.2: Principles of Economerics, 3rd Ediion Slide 9-54
The Durbin-Wason bounds es. if d < d, rejec H : ρ= 0 and accep H : ρ> 0; Lc 0 1 if d > d, do no rejec H : ρ= 0; Uc if d < d < d he es is inconclusive. Lc Uc, 0 Principles of Economerics, 3rd Ediion Slide 9-55
y =α+β x +β x +β x +β x + L+ e =α+ β x + e 0 1 1 2 2 3 3 s s s= 0 y = δ+δ x +δ x + L+δ x +θ y + L+θ y + v 0 1 1 q q 1 1 p p Principles of Economerics, 3rd Ediion Slide 9-56
y = δ+δ x +θ y + v 0 1 1 (9C.1) y = δ+δ x +θ y 1 0 1 1 2 (9C.2) y = δ+δ x +θ y =δ+δ x +θ ( δ+δ x +θ y ) 0 1 1 0 1 0 1 1 2 =δ+θδ+δ x +θδ x +θ y 2 1 0 1 0 1 1 2 Principles of Economerics, 3rd Ediion Slide 9-57
y =δ+θδ+δ x +θδ x +θ ( δ+δ x +θ y ) 2 1 0 1 0 1 1 0 2 1 3 =δ+θδ+θ δ+δ x +θδ x +θ δ x +θ y 2 2 3 1 1 0 1 0 1 1 0 2 1 3 y =δ+θδ+θ δ+ L+θ δ 2 j 1 1 1 +δ x +θδ x +θ δ x + L+θ δ x +θ y 2 j j+ 1 0 1 0 1 1 0 2 1 0 j 1 ( j+ 1) (9C.3) j 2 j s j+ 1 1 1 L 1 0 1x s 1 y ( j+ 1) s= 0 =δ (1 +θ +θ + +θ ) + δ θ +θ Principles of Economerics, 3rd Ediion Slide 9-58
y =α+ s= 0 δ θ s 0 1 x s (9C.4) α=δ 2 (1 +θ +θ + 1 1 L) = 1 θ 1 δ y = α+ β x + e s s s= 0 Principles of Economerics, 3rd Ediion Slide 9-59
β =δ θ s s 0 1 δ β =δ +θ +θ + = θ 2 0 s 0(1 1 1 L) s= 0 1 1 Principles of Economerics, 3rd Ediion Slide 9-60
y = δ+δ x +δ x +δ x +δ x +θ y +θ y +v 0 1 1 2 2 3 3 1 1 2 2 (9C.5) β =δ 0 0 β =θβ +δ 1 1 0 1 β =θβ +θ β +δ 2 1 1 2 0 2 β =θβ +θ β +δ 3 1 2 2 1 3 β =θβ +θ β M 4 1 3 2 2 β =θβ 1 1+θ2β 2 for 4 s s s s (9C.6) Principles of Economerics, 3rd Ediion Slide 9-61
ˆ y + y + y 3 T T 1 T 2 yt + 1 = yˆ =α y +α(1 α ) y +α(1 α ) y +L 1 2 T+ 1 T T 1 T 2 (9D.1) (1 α ) yˆ =α(1 α ) y +α(1 α ) y +α(1 α ) y +... 2 3 T T 1 T 2 T 3 (9D.2) y垐 = α y + + (1 α ) y T 1 T T Principles of Economerics, 3rd Ediion Slide 9-62
Figure 9A.3: Exponenial Smoohing Forecass for wo alernaive values of α Principles of Economerics, 3rd Ediion Slide 9-63