Chapter 9 Dynamic Models, Autocorrelation and Forecasting Prepared by Vera Tabakova, East Carolina University
Jan 14, 2016
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
Slide 9-3Principles of Econometrics, 3rd Edition
9.1 Introduction
Slide 9-4Principles of Econometrics, 3rd Edition
(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
Slide 9-5Principles of Econometrics, 3rd Edition
9.1 Introduction
Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’
Slide 9-6Principles of Econometrics, 3rd Edition
9.1 Introduction
Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’
Slide 9-7Principles of Econometrics, 3rd Edition
9.2 Lags in the Error Term: Autocorrelation9.2.1 Area Response Model for Sugar Cane
Slide 9-8Principles of Econometrics, 3rd Edition
(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
Slide 9-9Principles of Econometrics, 3rd Edition
(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
9.2.2 First-Order Autoregressive Errors
Slide 9-10Principles of Econometrics, 3rd Edition
(9.13)
(9.12)
(9.11)( ) 0tE e
22
2var( )
1v
t ee
2cov , 0kt t k ee e k
9.2.2 First-Order Autoregressive Errors
Slide 9-11Principles of Econometrics, 3rd Edition
(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
9.2.2 First-Order Autoregressive Errors
Slide 9-12Principles of Econometrics, 3rd Edition
9.2.2 First-Order Autoregressive Errors
Figure 9.3 Least Squares Residuals Plotted Against Time
Slide 9-13Principles of Econometrics, 3rd Edition
9.2.2 First-Order Autoregressive Errors
Slide 9-14Principles of Econometrics, 3rd Edition
(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
9.3 Estimating an AR(1) Error Model
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.
Slide 9-15Principles of Econometrics, 3rd Edition
9.3 Estimating an AR(1) Error Model
Sugar cane example
The two sets of standard errors, along with the estimated equation are:
The 95% confidence intervals for β2 are:
Slide 9-16Principles of Econometrics, 3rd Edition
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
Slide 9-17Principles of Econometrics, 3rd Edition
(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
Slide 9-18Principles of Econometrics, 3rd Edition
(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.
Slide 9-19Principles of Econometrics, 3rd Edition
9.3.3 Estimating a More General Model
Slide 9-20Principles of Econometrics, 3rd Edition
(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 Testing for Autocorrelation
9.4.1 Residual Correlogram
Slide 9-21Principles of Econometrics, 3rd Edition
(9.29)
0 1: 0 : 0H H
1 (0,1)z T r N
(9.30)34 .404 2.36 1.96z
9.4 Testing for Autocorrelation
9.4.1 Residual Correlogram
Slide 9-22Principles of Econometrics, 3rd Edition
(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
9.4.1 Residual Correlogram
Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example
Slide 9-23Principles of Econometrics, 3rd Edition
9.4.1 Residual Correlogram
Slide 9-24Principles of Econometrics, 3rd Edition
1 2t t ty x e
1 2 1 2 1(1 )t t t t ty x y x v
9.4.1 Residual Correlogram
Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example
Slide 9-25Principles of Econometrics, 3rd Edition
9.4.2 A Lagrange Multiplier Test
Slide 9-26Principles of Econometrics, 3rd Edition
(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
Slide 9-27Principles of Econometrics, 3rd Edition
(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
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-28Principles of Econometrics, 3rd Edition
(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
9.5 An Introduction to Forecasting: Autoregressive Models
Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation
Slide 9-29Principles of Econometrics, 3rd Edition
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-30Principles of Econometrics, 3rd Edition
(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
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-31Principles of Econometrics, 3rd Edition
(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
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-32Principles of Econometrics, 3rd Edition
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-33Principles of Econometrics, 3rd Edition
(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
9.5 An Introduction to Forecasting: Autoregressive Models
Slide 9-34Principles of Econometrics, 3rd Edition
(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
9.6 Finite Distributed Lags
Slide 9-35Principles of Econometrics, 3rd Edition
(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
9.6 Finite Distributed Lags
Slide 9-36Principles of Econometrics, 3rd Edition
9.6 Finite Distributed Lags
Slide 9-37Principles of Econometrics, 3rd Edition
9.7 Autoregressive Distributed Lag Models
Slide 9-38Principles of Econometrics, 3rd Edition
(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
9.7 Autoregressive Distributed Lag Models
Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model
Slide 9-39Principles of Econometrics, 3rd Edition
9.7 Autoregressive Distributed Lag Models
Slide 9-40Principles of Econometrics, 3rd Edition
(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
9.7 Autoregressive Distributed Lag Models
Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model
Slide 9-41Principles of Econometrics, 3rd Edition
9.7 Autoregressive Distributed Lag Models
Slide 9-42Principles of Econometrics, 3rd Edition
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
9.7 Autoregressive Distributed Lag Models
Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model
Slide 9-43Principles of Econometrics, 3rd Edition
Keywords
Slide 9-44Principles of Econometrics, 3rd Edition
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
Slide 9-45Principles of Econometrics, 3rd Edition
Appendix 9A Generalized Least Squares Estimation
Appendix 9B The Durbin Watson Test
Appendix 9C Deriving ARDL Lag Weights
Appendix 9D Forecasting: Exponential Smoothing
Appendix 9A Generalized Least Squares Estimation
Slide 9-46Principles of Econometrics, 3rd Edition
(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
Appendix 9A Generalized Least Squares Estimation
Slide 9-47Principles of Econometrics, 3rd Edition
(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
Appendix 9A Generalized Least Squares Estimation
Slide 9-48Principles of Econometrics, 3rd Edition
(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
Appendix 9A Generalized Least Squares Estimation
Slide 9-49Principles of Econometrics, 3rd Edition
22 2 2
1 1 2var( ) (1 ) var( ) (1 )
1v
ve e
Appendix 9B The Durbin-Watson Test
Slide 9-50Principles of Econometrics, 3rd Edition
(9B.1)
0 1: 0 : 0H H
2
12
2
1
ˆ ˆ
ˆ
T
t tt
T
tt
e ed
e
Appendix 9B The Durbin-Watson Test
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
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
Appendix 9B The Durbin-Watson Test
Slide 9-52Principles of Econometrics, 3rd Edition
(9B.3) 12 1d r
cd d
Appendix 9B The Durbin-Watson Test
Figure 9A.1:
Principles of Econometrics, 3rd Edition Slide 9-53
Appendix 9B 9B.1 The Durbin-Watson Bounds Test
Figure 9A.2:
Principles of Econometrics, 3rd Edition Slide 9-54
Appendix 9B 9B.1 The Durbin-Watson Bounds Test
The Durbin-Watson bounds test.
if the test is inconclusive.
Principles of Econometrics, 3rd Edition Slide 9-55
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
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
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
Slide 9-57Principles of Econometrics, 3rd Edition
(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
Appendix 9C 9C.1 The Geometric Lag
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
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
Appendix 9C 9C.1 The Geometric Lag
Slide 9-59Principles of Econometrics, 3rd Edition
(9C.4)0 10
st t s
s
y x
21 1
1
(1 )1
0t s t s t
s
y x e
Appendix 9C 9C.1 The Geometric Lag
Slide 9-60Principles of Econometrics, 3rd Edition
0 1s
s
2 00 1 1
0 1
(1 )1s
s
Appendix 9C 9C.2 Lag Weights for More General ARDL Models
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 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
Appendix 9D Forecasting: Exponential Smoothing
Slide 9-62Principles of Econometrics, 3rd Edition
(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
Appendix 9D Forecasting: Exponential Smoothing
Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α
Principles of Econometrics, 3rd Edition Slide 9-63