Chapter 9

Chapter 9

Dynamic Models, Autocorrelation and Forecasting

Prepared by Vera Tabakova, East Carolina University

Chapter 9: Dynamic Models, Autocorrelation and Forecasting

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

Slide 9-2Principles of Econometrics, 3rd Edition

9.1 Introduction

Figure 9.1


9.1 Introduction


(9.1)

(9.2)

(9.3)

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

9.1 Introduction

Figure 9.2(a) Time Series of a Stationary Variable


9.1 Introduction

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


9.1 Introduction

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


9.2 Lags in the Error Term: Autocorrelation9.2.1 Area Response Model for Sugar Cane


(9.5)

1 2ln lnA P

(9.4) 1 2ln lnt t tA P e

(9.6)

1 2t t ty x e

1t t te e v

9.2.2 First-Order Autoregressive Errors


(9.9)

(9.8)

(9.10)

(9.7)1 2t t ty x e

1t t te e v

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

1 1



(9.13)

(9.12)

(9.11)( ) 0tE e

22

2var( )

1v

t ee

2cov , 0kt t k ee e k



(9.16)

(9.15)

(9.14)

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 e

ee e

1corr( , )t te e

ˆ 3.893 .776

(se) (.061) (.277)t ty x


Figure 9.3 Least Squares Residuals Plotted Against Time




(9.18)

(9.17)

1

2 2

1 1

( )( )cov( , )

var( )var( ) ( ) ( )

T

t tt t t

xy T T

t tt t

t t

x x y yx y

rx y x x y y

11 2

12

12

ˆ ˆcov( , )

var( ) ˆ

T

t tt t t

T

t tt

e ee e

re 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)

9.3.2 Nonlinear Least Squares Estimation


(9.21)

(9.20)

(9.22)

(9.19)1 2t t ty x e

1t t te e v

1 2 1t t t ty x e v

1 1 1 2 1t t te y x

9.3.2 Nonlinear Least Squares Estimation


(9.25)

(9.24)

(9.23)1 1 1 2 1t t te y x

1 2 1 2 1(1 )t t t t ty x y x v

1ln( ) 3.899 .888ln( ) .422

(se) (.092) (.259) (.166)t t t t tA P e e v

9.3.2a Generalized Least Squares Estimation

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.


9.3.3 Estimating a More General Model


(9.27)

(9.26)1 2 2 1 1(1 )t t t t ty x x y v

0 1 1 1 1t t t t ty x x y v

1 0 2 1 2 1(1 )

(9.28)1 1ˆ 2.366 .777 .611 .404

(se) (.656) (.280) (.297) (.167)

t t t ty x x y


9.4.1 Residual Correlogram


(9.29)

0 1: 0 : 0H H

1 (0,1)z T r N

(9.30)34 .404 2.36 1.96z




(9.31)

(9.32)

1 1

1.96 1.96 or r r

T T

1.96 1.96 or k kr r

T T

2

cov( , ) ( )

var( ) ( )t t k t t k

kt t

e e E e e

e E e


Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example




1 2t t ty x e

1 2 1 2 1(1 )t t t t ty x y x v


Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example


9.4.2 A Lagrange Multiplier Test


(9.33)

(9.34)

1 2 1t t t ty x e v

= 2.439 = 5.949 -value = .021t F p

1 2 1ˆ ˆt t t ty x e v

1 2 1 2 1ˆ ˆ ˆt t t t tb b x e x e v

9.4.2 A Lagrange Multiplier Test


(9.35)1 1 2 2 1

1 2 1

ˆ ˆ ˆ( ) ( )

ˆ ˆ

t t t t

t t t

e b b x e v

x e v

2 34 .16101 5.474LM T R



(9.36)

(9.37)

1 1 2 2t t t p t p ty y y y v

11

1

ln( ) ln( ) 100 100t tt t t

t

CPI CPIy CPI CPI

CPI

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




(9.38)1 1 2 2 3 3t t t t ty y y y v

1 1 2 1 3 2 1T T T T Ty y y y v

1 1 2 1 3 2ˆ ˆ ˆ ˆˆ

.1883 .3733 .4468 .2179 .5988 .1013 .3510

.2602

T T T Ty y y y



(9.39)2 1 1 2 3 1

ˆ ˆ ˆ ˆˆ ˆ

.1883 .3733 .2602 .2179 .4468 .1013 .5988

.2487

T T T Ty y y y

1 1 1 1 1 2 2 1 3 3 2 1ˆ ˆ ˆ ˆˆ ( ) ( ) ( ) ( )T T T T T Tu y y y y y v



(9.42)

(9.41)

(9.40)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

23 1 2 2 1 3 1 2 1 1 2 3( )T T T Tu u u v v v v



(9.43)

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

ˆ ˆ ˆ ˆ1.96 , 1.96T j j T j jy y



(9.44)0 1 1 2 2 , 1, ,t t t t q t q ty x x x x v t q T

( )ts

t s

E y

x

11

1

ln( ) ln( ) 100 100t tt t t

t

WAGE WAGEx WAGE WAGE

WAGE



(9.45)

(9.46)

0 1 1 1 1t t t q t q t p t p ty x x x y y v

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


Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model




(9.47)

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

0 0

1 1 0 1

2 1 1 2 0 2

3 1 2 2 1 3

4 1 3 2 2

ˆ ˆ .1149

ˆ ˆ ˆ ˆ .3536 .1149 .0377 .0784

ˆ ˆ ˆ ˆ ˆ ˆ .0643

ˆ ˆ ˆ ˆ ˆ ˆ .2434

ˆ ˆ ˆ ˆ ˆ .0734


Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model


Keywords


autocorrelation autoregressive distributed lag

models autoregressive error autoregressive model correlogram delay multiplier distributed lag weight dynamic models finite distributed lag forecast error forecasting HAC standard errors impact multiplier infinite distributed lag

interim multiplier lag length lagged dependent variable LM test nonlinear least squares sample autocorrelation function standard error of forecast error total multiplier form of LM test

Chapter 9 Appendices


Appendix 9A Generalized Least Squares Estimation

Appendix 9B The Durbin Watson Test

Appendix 9C Deriving ARDL Lag Weights

Appendix 9D Forecasting: Exponential Smoothing



(9A.2)

1 2 1 t t t t t ty x e e e v

(9A.1)1 2 1 1 2 1t t t t ty x y x v

1 1 2 11t t t t ty y x x v

1 2 1 1 1t t t t t t ty y y x x x x



(9A.4)

(9A.3)1 1 2 2t t t ty x x v

1 2 1 1 2 1( )t t t t ty x y x v



(9A.5)

1 1 1 2 1y x e

2 2 2 21 1 1 2 11 1 1 1y x e

1 11 1 12 2 1y x x e

(9A.6)

2 21 1 11

2 212 1 1 1

1 1

1 1

y y x

x x e e



22 2 2

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

1v

ve e

Appendix 9B The Durbin-Watson Test


(9B.1)

0 1: 0 : 0H H

2

12

2

1

ˆ ˆ

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

ˆ ˆ ˆ ˆ2

ˆ

ˆ ˆ ˆ ˆ2

ˆ ˆ ˆ

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

Appendix 9B 9B.1 The Durbin-Watson Bounds Test

Figure 9A.2:


Appendix 9B 9B.1 The Durbin-Watson Bounds Test

The Durbin-Watson bounds test.

if the test is inconclusive.


0 1if , reject : 0 and accept : 0;Lcd d H H

0if , do not reject : 0;Ucd d H

,Lc Ucd d d

Appendix 9C Deriving ARDL Lag Weights


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

0 1 1 1 1t t t q t q t p t p ty x x x y y v

Appendix 9C 9C.1 The Geometric Lag


(9C.2)

(9C.1)0 1 1t t t ty x y v

1 0 1 1 2t t ty x y

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



(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

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



(9C.4)0 10

st t s

s

y x

21 1

1

(1 )1

0t s t s t

s

y x e



0 1s

s

2 00 1 1

0 1

(1 )1s

s

Appendix 9C 9C.2 Lag Weights for More General ARDL Models


(9C.6)

(9C.5)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

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



(9D.2)

(9D.1)

1 21ˆ

3T T T

T

y y yy

1 21 1 2ˆ (1 ) (1 )T T T Ty y y y

2 31 2 3ˆ(1 ) (1 ) (1 ) (1 ) .....T T T Ty y y y

1ˆ ˆ(1 )T T Ty y y


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


Chapter 9

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