9.1 Introduction - Rice Universityecon446/HGL/ch09.pdfk ttk ttk ek ttk ttkte ee ee ee ee e −−...

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

Figure 9.1

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 −= + =

Figure 9.2(a) Time Series of a Stationary Variable

Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’

Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’

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 +

1 2t t ty x e= β +β +


2( ) 0 var( ) cov( , ) 0 fort t v t sE v v v v t s= = σ = ≠

1 1− < ρ <

( ) 0tE e =

22

2var( )1

vt ee σ= σ =

−ρ

( ) 2cov , 0kt t k ee e k− = σ ρ >

( )

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= +

Figure 9.3 Least Squares Residuals Plotted Against Time

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

−− =

−=

= =∑

∑

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.

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)

1 2t t ty x e= β +β +


1 2 1t t t ty x e v−= β +β +ρ +

1 1 1 2 1t t te y x− − −= −β −β

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−= + = +

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.

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− −= + − +

9.4.1 Residual Correlogram

0 1: 0 : 0H Hρ = ρ ≠

1 (0,1)z T r N=

34 .404 2.36 1.96z = × = ≥

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 = =

Figure 9.4 Correlogram for Least Squares Residuals fromSugar Cane Example

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 +

Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example

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 +

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= × = × =

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− − −= + − +

Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation

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

= + × − × + ×=

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 +

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 +

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

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

−−

−

⎛ ⎞−= − × ≈ ×⎜ ⎟

⎝ ⎠

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

Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model

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

− −

− − −

= + + +

+ + −

Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model

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 =

Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model

Slide 9-44Principles 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


(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 +


(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


22 2 2

1 1 2var( ) (1 ) var( ) (1 )1

vve e∗ σ

= −ρ = −ρ = σ−ρ


(9B.1)

0 1: 0 : 0H Hρ = ρ >

( )21

2

2

1

垐

ˆ

T

t tt

T

tt

e ed

e

−=

=

−=∑

∑


(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

− −= = =

=

− −= = =

= = =

+ −=

= + −

≈ + −

∑ ∑ ∑

∑

∑ ∑ ∑

∑ ∑ ∑


(9B.3)( )12 1d r≈ −

cd d≤

Figure 9A.1:

Principles of Econometrics, 3rd Edition Slide 9-53

Figure 9A.2:


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< <


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


(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


(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


(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 +∑


0 1s

sβ = δ θ

2 00 1 1

0 1

(1 )1s

s

∞

=

δβ = δ + θ + θ + =

−θ∑ L


(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


(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

Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α


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