9.1 introduction - rice universityecon446/hgl/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −−...
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![Page 1: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/1.jpg)
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9.1 Introduction
9.2 Lags in the Error Term: Autocorrelation
9.3 Estimating an AR(1) Error Model
9.4 Testing for Autocorrelation
9.5 An Introduction to Forecasting: Autoregressive Models
9.6 Finite Distributed Lags
9.7 Autoregressive Distributed Lag Models
![Page 3: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/3.jpg)
Figure 9.1
![Page 4: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/4.jpg)
1 2( , , ,...)t t t ty f x x x− −=
1( , )t t ty f y x−=
1( ) ( )t t t t ty f x e e f e −= + =
![Page 5: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/5.jpg)
Figure 9.2(a) Time Series of a Stationary Variable
![Page 6: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/6.jpg)
Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’
![Page 7: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/7.jpg)
Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’
![Page 8: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/8.jpg)
9.2.1 Area Response Model for Sugar Cane
( ) ( )1 2ln lnA P= β +β
( ) ( )1 2ln lnt t tA P e= β +β +
1 2t t ty x e= β +β +
1t t te e v−= ρ +
![Page 9: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/9.jpg)
1 2t t ty x e= β +β +
1t t te e v−= ρ +
2( ) 0 var( ) cov( , ) 0 fort t v t sE v v v v t s= = σ = ≠
1 1− < ρ <
![Page 10: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/10.jpg)
( ) 0tE e =
22
2var( )1
vt ee σ= σ =
−ρ
( ) 2cov , 0kt t k ee e k− = σ ρ >
![Page 11: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/11.jpg)
( )
2
2
cov( , ) cov( , )corr( , )var( )var( ) var
kkt t k t t k e
t t kt et t k
e e e ee eee e
− −−
−
σ ρ= = = = ρ
σ
1corr( , )t te e − = ρ
ˆ 3.893 .776 (se) (.061) (.277)
t ty x= +
![Page 12: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/12.jpg)
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Figure 9.3 Least Squares Residuals Plotted Against Time
![Page 14: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/14.jpg)
1
2 2
1 1
( )( )cov( , )
var( )var( ) ( ) ( )
T
t tt t t
xy T Tt t t t
t t
x x y yx yrx y x x y y
=
= =
− −= =
− −
∑
∑ ∑
11 2
12
12
垐cov( , )
var( ) ˆ
T
t tt t t
T
t tt
e ee ere e
−− =
−=
= =∑
∑
![Page 15: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/15.jpg)
The existence of AR(1) errors implies:
The least squares estimator is still a linear and unbiased estimator, but
it is no longer best. There is another estimator with a smaller
variance.
The standard errors usually computed for the least squares estimator
are incorrect. Confidence intervals and hypothesis tests that use these
standard errors may be misleading.
![Page 16: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/16.jpg)
Sugar cane example
The two sets of standard errors, along with the estimated equation are:
The 95% confidence intervals for β2 are:
ˆ 3.893 .776(.061) (.277) 'incorrect' se's(.062) (.378) 'correct' se's
t ty x= +
(.211,1.340) (incorrect)
(.006,1.546) (correct)
![Page 17: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/17.jpg)
1 2t t ty x e= β +β +
1t t te e v−= ρ +
1 2 1t t t ty x e v−= β +β +ρ +
1 1 1 2 1t t te y x− − −= −β −β
![Page 18: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/18.jpg)
1 1 1 2 1t t te y x− − −ρ = ρ −ρβ −ρβ
1 2 1 2 1(1 )t t t t ty x y x v− −= β −ρ +β +ρ −ρβ +
1ln( ) 3.899 .888ln( ) .422 (se) (.092) (.259) (.166)
t t t t tA P e e v−= + = +
![Page 19: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/19.jpg)
It can be shown that nonlinear least squares estimation of (9.24) is
equivalent to using an iterative generalized least squares estimator
called the Cochrane-Orcutt procedure. Details are provided in
Appendix 9A.
![Page 20: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/20.jpg)
1 2 2 1 1(1 )t t t t ty x x y v− −= β −ρ +β −ρβ +ρ +
0 1 1 1 1t t t t ty x x y v− −= δ + δ + δ + θ +
1 0 2 1 2 1(1 )δ = β −ρ δ = β δ = −ρβ θ = ρ
1 1ˆ 2.366 .777 .611 .404
(se) (.656) (.280) (.297) (.167)t t t ty x x y− −= + − +
![Page 21: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/21.jpg)
9.4.1 Residual Correlogram
0 1: 0 : 0H Hρ = ρ ≠
1 (0,1)z T r N=
34 .404 2.36 1.96z = × = ≥
![Page 22: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/22.jpg)
9.4.1 Residual Correlogram
1 11.96 1.96 or r r
T T≥ ≤ −
1.96 1.96 or k kr rT T
≥ ≤ −
2cov( , ) ( )
var( ) ( )t t k t t k
kt t
e e E e ee E e
− −ρ = =
![Page 23: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/23.jpg)
Figure 9.4 Correlogram for Least Squares Residuals fromSugar Cane Example
![Page 24: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/24.jpg)
1 2t t ty x e= β +β +
1 2 1 2 1(1 )t t t t ty x y x v− −= β −ρ +β +ρ −ρβ +
![Page 25: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/25.jpg)
Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example
![Page 26: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/26.jpg)
1 2 1t t t ty x e v−= β +β +ρ +
= 2.439 = 5.949 -value = .021t F p
1 2 1垐t t t ty x e v−= β +β +ρ +
1 2 1 2 1垐 ?t t t t tb b x e x e v−+ + = β +β +ρ +
![Page 27: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/27.jpg)
1 1 2 2 1
1 2 1
垐 ?( ) ( )
垐
t t t t
t t t
e b b x e v
x e v
−
−
= β − + β − +ρ +
= γ + γ +ρ +
2 34 .16101 5.474LM T R= × = × =
![Page 28: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/28.jpg)
1 1 2 2t t t p t p ty y y y v− − −= δ + θ + θ + + θ +L
( ) 11
1
ln( ) ln( ) 100 100t tt t t
t
CPI CPIy CPI CPICPI
−−
−
⎛ ⎞−= − × ≈ ×⎜ ⎟
⎝ ⎠
1 2 3.1883 .3733 .2179 .1013 (se) (.0253) (.0615) (.0645) (.0613)
t t t tINFLN INFLN INFLN INFLN− − −= + − +
![Page 29: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/29.jpg)
Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation
![Page 30: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/30.jpg)
1 1 2 2 3 3t t t t ty y y y v− − −= δ + θ + θ + θ +
1 1 2 1 3 2 1T T T T Ty y y y v+ − − += δ + θ + θ + θ +
1 1 2 1 3 2垐 垐ˆ.1883 .3733 .4468 .2179 .5988 .1013 .3510.2602
T T T Ty y y y+ − −= δ + θ + θ + θ
= + × − × + ×=
![Page 31: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/31.jpg)
2 1 1 2 3 1垐 垐垐
.1883 .3733 .2602 .2179 .4468 .1013 .5988
.2487
T T T Ty y y y+ + −= δ + θ + θ + θ= + × − × + ×=
1 1 1 1 1 2 2 1 3 3 2 1垐 垐ˆ ( ) ( ) ( ) ( )T T T T T Tu y y y y y v+ + − − += − = δ − δ + θ −θ + θ −θ + θ −θ +
![Page 32: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/32.jpg)
![Page 33: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/33.jpg)
1 1Tu v +=
2 1 1 1 2 1 1 2 1 1 2ˆ( )T T T T T Tu y y v u v v v+ + + + + += θ − + = θ + = θ +
23 1 2 2 1 3 1 2 1 1 2 3( )T T T Tu u u v v v v+ + + += θ + θ + = θ + θ + θ +
![Page 34: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/34.jpg)
2 21 1
2 2 22 2 1
2 2 2 2 23 3 1 2 1
var( )
var( ) (1 )
var( ) [( ) 1]
v
v
v
u
u
u
σ = = σ
σ = = σ + θ
σ = = σ θ + θ + θ +
( )垐 垐1.96 , 1.96T j j T j jy y+ +− ×σ + ×σ
![Page 35: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/35.jpg)
0 1 1 2 2 , 1, ,t t t t q t q ty x x x x v t q T− − −= α +β +β +β + +β + = +L K
( )ts
t s
E yx −
∂= β
∂
( ) 11
1
ln( ) ln( ) 100 100t tt t t
t
WAGE WAGEx WAGE WAGEWAGE
−−
−
⎛ ⎞−= − × ≈ ×⎜ ⎟
⎝ ⎠
![Page 36: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/36.jpg)
![Page 37: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/37.jpg)
![Page 38: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/38.jpg)
0 1 1 1 1t t t q t q t p t p ty x x x y y v− − − −= δ + δ + δ + + δ + θ + + θ +L L
0 1 1 2 2 3 3
0
t t t t t t
s t s ts
y x x x x e
x e
− − −
∞
−=
= α +β +β +β +β + +
= α + β +∑
L
![Page 39: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/39.jpg)
Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model
![Page 40: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/40.jpg)
1 2
3 1 2
.0989 .1149 .0377 .0593 (se) (.0288) (.0761) (.0812) (.0812)
.2361 .3536 .1976(.0829) (.0604) (.0604)
t t t t
t t t
INFLN PCWAGE PCWAGE PCWAGE
PCWAGE INFLN INFLN
− −
− − −
= + + +
+ + −
![Page 41: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/41.jpg)
Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model
![Page 42: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/42.jpg)
0 1 1 2 2 3 3 1 1 2 2t t t t t t t ty x x x x y y v− − − − −= δ + δ + δ + δ + δ + θ + θ +
0 0
1 1 0 1
2 1 1 2 0 2
3 1 2 2 1 3
4 1 3 2 2
垐 .1149
垐 垐 .3536 .1149 .0377 .0784
垐 垐 垐 .0643
垐 垐 垐 .2434
垐 垐 ? .0734
β = δ =
β = θ β + δ = × + =
β = θ β + θ β + δ =
β = θ β + θ β + δ =
β = θ β + θ β =
![Page 43: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/43.jpg)
Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model
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Slide 9-44Principles of Econometrics, 3rd Edition
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Slide 9-45Principles of Econometrics, 3rd Edition
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Slide 9-46Principles of Econometrics, 3rd Edition
(9A.2)
1 2 1 t t t t t ty x e e e v−= β +β + = ρ +
(9A.1)1 2 1 1 2 1t t t t ty x y x v− −= β +β +ρ −ρβ −ρβ +
( ) ( )1 1 2 11t t t t ty y x x v− −−ρ = β −ρ +β −ρ +
1 2 1 1 1t t t t t t ty y y x x x x∗ ∗ ∗− −= −ρ = −ρ = −ρ
![Page 47: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/47.jpg)
Slide 9-47Principles of Econometrics, 3rd Edition
(9A.4)
(9A.3)1 1 2 2t t t ty x x v∗ ∗ ∗= β + β +
1 2 1 1 2 1( )t t t t ty x y x v− −−β −β = ρ −β −β +
![Page 48: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/48.jpg)
Slide 9-48Principles of Econometrics, 3rd Edition
(9A.5)
1 1 1 2 1y x e= β + β +
2 2 2 21 1 1 2 11 1 1 1y x e−ρ = −ρ β + −ρ β + −ρ
1 11 1 12 2 1y x x e∗ ∗ ∗ ∗= β + β +
(9A.6)
2 21 1 11
2 212 1 1 1
1 1
1 1
y y x
x x e e
∗ ∗
∗ ∗
= −ρ = −ρ
= −ρ = −ρ
![Page 49: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/49.jpg)
Slide 9-49Principles of Econometrics, 3rd Edition
22 2 2
1 1 2var( ) (1 ) var( ) (1 )1
vve e∗ σ
= −ρ = −ρ = σ−ρ
![Page 50: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/50.jpg)
Slide 9-50Principles of Econometrics, 3rd Edition
(9B.1)
0 1: 0 : 0H Hρ = ρ >
( )21
2
2
1
垐
ˆ
T
t tt
T
tt
e ed
e
−=
=
−=∑
∑
![Page 51: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/51.jpg)
Slide 9-51Principles of Econometrics, 3rd Edition
(9B.2)
2 21 1
2 2 2
2
1
2 21 1
2 2 2
2 2 2
1 1 1
1
垐 垐2
ˆ
垐 垐
2垐 ?
1 1 2
T T T
t t t tt t t
T
tt
T T T
t t t tt t tT T T
t t tt t t
e e e ed
e
e e e e
e e e
r
− −= = =
=
− −= = =
= = =
+ −=
= + −
≈ + −
∑ ∑ ∑
∑
∑ ∑ ∑
∑ ∑ ∑
![Page 52: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/52.jpg)
Slide 9-52Principles of Econometrics, 3rd Edition
(9B.3)( )12 1d r≈ −
cd d≤
![Page 53: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/53.jpg)
Figure 9A.1:
Principles of Econometrics, 3rd Edition Slide 9-53
![Page 54: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/54.jpg)
Figure 9A.2:
Principles of Econometrics, 3rd Edition Slide 9-54
![Page 55: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/55.jpg)
The Durbin-Watson bounds test.
if the test is inconclusive.
Principles of Econometrics, 3rd Edition
0 1if , re
Slide 9-55
ject : 0 and accept : 0;Lcd d H H< ρ = ρ >
0if , do not reject : 0;Ucd d H> ρ =
,Lc Ucd d d< <
![Page 56: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/56.jpg)
Slide 9-56Principles of Econometrics, 3rd Edition
0 1 1 2 2 3 30
t t t t t t s t s ts
y x x x x e x e∞
− − − −=
= α +β +β +β +β + + = α + β +∑L
0 1 1 1 1t t t q t q t p t p ty x x x y y v− − − −= δ + δ + δ + + δ + θ + + θ +L L
![Page 57: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/57.jpg)
Slide 9-57Principles of Econometrics, 3rd Edition
(9C.2)
(9C.1)0 1 1t t t ty x y v−= δ + δ + θ +
1 0 1 1 2t t ty x y− − −= δ + δ + θ
0 1 1 0 1 0 1 1 2
21 0 1 0 1 1 2
( )t t t t t t
t t t
y x y x x y
x x y
− − −
− −
= δ + δ + θ = δ + δ + θ δ + δ + θ
= δ + θ δ + δ + θ δ + θ
![Page 58: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/58.jpg)
Slide 9-58Principles of Econometrics, 3rd Edition
(9C.3)
21 0 1 0 1 1 0 2 1 3
2 2 31 1 0 1 0 1 1 0 2 1 3
( )t t t t t
t t t t
y x x x y
x x x y
− − −
− − −
= δ + θ δ + δ + θ δ + θ δ + δ + θ
= δ + θ δ + θ δ + δ + θ δ + θ δ + θ
21 1 1
2 10 1 0 1 1 0 2 1 0 1 ( 1)
2 11 1 1 0 1 1 ( 1)
0
(1 )
jt
j jt t t t j t j
jj s j
t s t js
y
x x x x y
x y
+− − − − +
+− − +
=
= δ + θ δ + θ δ + + θ δ
+ δ + θ δ + θ δ + + θ δ + θ
= δ + θ + θ + + θ + δ θ + θ∑
L
L
L
![Page 59: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/59.jpg)
Slide 9-59Principles of Econometrics, 3rd Edition
(9C.4)0 10
st t s
sy x
∞
−=
= α + δ θ∑
21 1
1
(1 )1δ
α = δ + θ + θ + =−θ
L
0t s t s t
sy x e
∞
−=
= α + β +∑
![Page 60: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/60.jpg)
Slide 9-60Principles of Econometrics, 3rd Edition
0 1s
sβ = δ θ
2 00 1 1
0 1
(1 )1s
s
∞
=
δβ = δ + θ + θ + =
−θ∑ L
![Page 61: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/61.jpg)
Slide 9-61Principles of Econometrics, 3rd Edition
(9C.6)
(9C.5)0 1 1 2 2 3 3 1 1 2 2t t t t t t ty x x x x y y− − − − − tv= δ + δ + δ + δ + δ + θ + θ +
0 0
1 1 0 1
2 1 1 2 0 2
3 1 2 2 1 3
4 1 3 2 2
1 1 2 2 for 4s s s s− −
β = δ
β = θ β + δβ = θ β + θ β + δ
β = θ β + θ β + δβ = θ β + θ β
β = θ β + θ β ≥M
![Page 62: 9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −− − − σρ ====ρ σ corr( , )ee tt−1 =ρ ˆ 3.893 .776 (se) (.061) (.277) yx](https://reader033.vdocuments.net/reader033/viewer/2022042613/5fad6769da0952197535b014/html5/thumbnails/62.jpg)
Slide 9-62Principles of Econometrics, 3rd Edition
(9D.2)
(9D.1)
1 21ˆ
3T T T
Ty y yy − −
++ +
=
1 21 1 2ˆ (1 ) (1 )T T T Ty y y y+ − −= α +α −α +α −α +L
2 31 2 3ˆ(1 ) (1 ) (1 ) (1 ) .....T T T Ty y y y− − −−α = α −α +α −α +α −α +
1垐 (1 )T T Ty y y+ = α + −α
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Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α
Principles of Econometrics, 3rd Edition Slide 9-63