Dynamic Models, Autocorrelation and
Forecasting
Prepared by Vera Tabakova, East Carolina University

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
Principles of Econometrics, 3rd Edition
Slide 9-2
Figure 9.1
Principles of Econometrics, 3rd Edition
Slide 9-3
yt  f ( xt , xt 1 , xt 2 ,...)
(9.1)
yt  f ( yt 1 , xt )
(9.2)
yt  f ( xt )  et
Principles of Econometrics, 3rd Edition
et  f (et 1 )
(9.3)
Slide 9-4
Figure 9.2(a) Time Series of a Stationary Variable
Principles of Econometrics, 3rd Edition
Slide 9-5
Figure 9.2(b) Time Series of a Nonstationary Variable that is
‘Slow Turning’ or ‘Wandering’
Principles of Econometrics, 3rd Edition
Slide 9-6
Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’
Principles of Econometrics, 3rd Edition
Slide 9-7
9.2.1 Area Response Model for Sugar Cane
ln  A  1  2 ln  P 
ln  At   1  2 ln  Pt   et
(9.4)
yt  1  2 xt  et
(9.5)
et  et 1  vt
Principles of Econometrics, 3rd Edition
(9.6)
Slide 9-8
yt  1  2 xt  et
(9.7)
et  et 1  vt
(9.8)
E(vt )  0 var(vt )  v2 cov(vt , vs )  0 for t  s
(9.9)
1    1
Principles of Econometrics, 3rd Edition
(9.10)
Slide 9-9
E (et )  0
(9.11)
2

var(et )  e2  v 2
1 
(9.12)
cov  et , et k   e2k
Principles of Econometrics, 3rd Edition
k 0
(9.13)
Slide 9-10
corr(et , et k ) 
cov(et , et k )
var(et ) var  et k 
cov(et , et k ) e2k

 2  k (9.14)
var(et )
e
corr(et , et 1 )  
yˆt  3.893  .776 xt
(se) (.061) (.277)
Principles of Econometrics, 3rd Edition
(9.15)
(9.16)
Slide 9-11
Principles of Econometrics, 3rd Edition
Slide 9-12
Figure 9.3 Least Squares Residuals Plotted Against Time
Principles of Econometrics, 3rd Edition
Slide 9-13
T
rxy 
cov( xt , yt )
 ( xt  x )(yt  y )

var( xt )var( yt )
t 1
T
T
t 1
t 1
2
2
(
x

x
)
(
y

y
)
 t
 t
(9.17)
T
r1 
cov(et , et 1 )
var(et )

 eˆt eˆt 1
t 2
T
2
ˆ
e
 t 1
(9.18)
t 2
Principles of Econometrics, 3rd Edition
Slide 9-14
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.
Principles of Econometrics, 3rd Edition
Slide 9-15
Sugar cane example
The two sets of standard errors, along with the estimated equation are:
yˆt  3.893  .776 xt
(.061) (.277)
'incorrect' se's
(.062) (.378)
'correct' se's
The 95% confidence intervals for β2 are:
(.211,1.340)
(incorrect)
(.006,1.546)
(correct)
Principles of Econometrics, 3rd Edition
Slide 9-16
yt  1  2 xt  et
(9.19)
et  et 1  vt
(9.20)
yt  1  2 xt  et 1  vt
(9.21)
et 1  yt 1  1  2 xt 1
(9.22)
Principles of Econometrics, 3rd Edition
Slide 9-17
et 1  yt 1  1  2 xt 1
(9.23)
yt  1 (1  )  2 xt  yt 1  2 xt 1  vt
(9.24)
ln( At )  3.899  .888ln( Pt )
(se)
(.092) (.259)
Principles of Econometrics, 3rd Edition
et  .422et 1  vt
(.166)
(9.25)
Slide 9-18
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.
Principles of Econometrics, 3rd Edition
Slide 9-19
yt  1 (1  )  2 xt  2 xt 1  yt 1  vt
(9.26)
yt    0 xt  1 xt 1  1 yt 1  vt
(9.27)
  1 (1  )
0  2
1  2
yˆt  2.366  .777 xt  .611 xt 1  .404 yt 1
(se) (.656) (.280) (.297)
Principles of Econometrics, 3rd Edition
1  
(9.28)
(.167)
Slide 9-20
9.4.1 Residual Correlogram
H0 :   0
z  T r1
H1 :   0
N (0,1)
z  34  .404  2.36  1.96
Principles of Econometrics, 3rd Edition
(9.29)
(9.30)
Slide 9-21
9.4.1 Residual Correlogram
1.96
r1 
T
1.96
rk 
T
or
1.96
r1  
T
1.96
rk  
T
(9.31)
cov(et , et k ) E (et et k )
k 

var(et )
E (et2 )
(9.32)
Principles of Econometrics, 3rd Edition
or
Slide 9-22
Figure 9.4 Correlogram for Least Squares Residuals from
Sugar Cane Example
Principles of Econometrics, 3rd Edition
Slide 9-23
yt  1  2 xt  et
yt  1 (1  )  2 xt  yt 1  2 xt 1  vt
Principles of Econometrics, 3rd Edition
Slide 9-24
Figure 9.5 Correlogram for Nonlinear Least Squares Residuals
from Sugar Cane Example
Principles of Econometrics, 3rd Edition
Slide 9-25
yt  1  2 xt  et 1  vt
t = 2.439
F = 5.949
(9.33)
p-value = .021
yt  1  2 xt  eˆt 1  vˆt
(9.34)
b1  b2 xt  eˆt  1  2 xt  eˆt 1  vˆt
Principles of Econometrics, 3rd Edition
Slide 9-26
eˆt  (1  b1 )  (2  b2 ) xt  eˆt 1  vˆt
(9.35)
 1   2 xt  eˆt 1  vˆt
LM  T  R2  34  .16101  5.474
Principles of Econometrics, 3rd Edition
Slide 9-27
yt    1 yt 1  2 yt 2 
  p yt  p  vt
(9.36)
 CPI t  CPI t 1 
yt   ln(CPI t )  ln(CPI t 1 )  100  
 100
CPI t 1


INFLN t  .1883  .3733 INFLNt 1  .2179 INFLNt 2  .1013 INFLNt 3
(se)
(.0253) (.0615)
Principles of Econometrics, 3rd Edition
(.0645)
(.0613)
(9.37)
Slide 9-28
Figure 9.6 Correlogram for Least Squares Residuals from
AR(3) Model for Inflation
Principles of Econometrics, 3rd Edition
Slide 9-29
yt    1 yt 1  2 yt 2  3 yt 3  vt
(9.38)
yT 1    1 yT  2 yT 1  3 yT 2  vT 1
yˆT 1  ˆ  ˆ 1 yT  ˆ 2 yT 1  ˆ 3 yT 2
 .1883  .3733 .4468  .2179  .5988  .1013 .3510
 .2602
Principles of Econometrics, 3rd Edition
Slide 9-30
yˆT 2  ˆ  ˆ 1 yˆT 1  ˆ 2 yT  ˆ 3 yT 1
 .1883  .3733 .2602  .2179  .4468  .1013 .5988 (9.39)
 .2487
u1  yT 1  yˆT 1  (  ˆ )  (1  ˆ 1 ) yT  (2  ˆ 2 ) yT 1  (3  ˆ 3 ) yT 2  vT 1
Principles of Econometrics, 3rd Edition
Slide 9-31
Principles of Econometrics, 3rd Edition
Slide 9-32
u1  vT 1
(9.40)
u2  1 ( yT 1  yˆT 1 )  vT 2  1u1  vT 2  1vT 1  vT 2 (9.41)
u3  1u2  2u1  vT 3  (12  2 )vT 1  1vT 2  vT 3
Principles of Econometrics, 3rd Edition
(9.42)
Slide 9-33
12  var(u1 )  v2
22  var(u2 )  v2 (1  12 )
32  var(u3 )  v2 [(12  2 ) 2  12  1]
 yˆ
T j
 1.96  ˆ j , yˆT  j  1.96  ˆ j 
Principles of Econometrics, 3rd Edition
(9.43)
Slide 9-34
yt    0 xt  1xt 1  2 xt 2 
 q xt q  vt , t  q  1,
,T (9.44)
E ( yt )
 s
xt s
 WAGEt  WAGEt 1 
xt   ln(WAGEt )  ln(WAGEt 1 )  100  
 100
WAGEt 1


Principles of Econometrics, 3rd Edition
Slide 9-35
Principles of Econometrics, 3rd Edition
Slide 9-36
Principles of Econometrics, 3rd Edition
Slide 9-37
yt    0 xt  1 xt 1 
 q xt q  1 yt 1 
yt    0 xt  1 xt 1  2 xt 2  3 xt 3 

    s xt s  et
  p yt  p  vt (9.45)
 et
(9.46)
s 0
Principles of Econometrics, 3rd Edition
Slide 9-38
Figure 9.7 Correlogram for Least Squares Residuals from
Finite Distributed Lag Model
Principles of Econometrics, 3rd Edition
Slide 9-39
INFLN t  .0989  .1149 PCWAGEt  .0377 PCWAGEt 1  .0593 PCWAGEt 2
(se)
(.0288) (.0761)
(.0812)
(.0812)
 .2361 PCWAGEt 3  .3536 INFLN t 1  .1976 INFLN t 2
(.0829)
Principles of Econometrics, 3rd Edition
(.0604)
(9.47)
(.0604)
Slide 9-40
Figure 9.8 Correlogram for Least Squares Residuals from
Autoregressive Distributed Lag Model
Principles of Econometrics, 3rd Edition
Slide 9-41
yt    0 xt  1 xt 1  2 xt 2  3 xt 3  1 yt 1  2 yt 2  vt
ˆ 0  ˆ 0  .1149
ˆ 1  ˆ 1ˆ 0  ˆ 1  .3536  .1149  .0377  .0784
ˆ 2  ˆ 1ˆ 1  ˆ 2ˆ 0  ˆ 2  .0643
ˆ 3  ˆ 1ˆ 2  ˆ 2ˆ 1  ˆ 3  .2434
ˆ 4  ˆ 1ˆ 3  ˆ 2ˆ 2  .0734
Principles of Econometrics, 3rd Edition
Slide 9-42
Figure 9.9 Distributed Lag Weights for Autoregressive
Distributed Lag Model
Principles of Econometrics, 3rd Edition
Slide 9-43














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
Principles of Econometrics, 3rd Edition









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
Slide 9-44
Principles of Econometrics, 3rd Edition
Slide 9-45
yt  1  2 xt  et
et  et 1  vt
yt  1  2 xt  yt 1  1  2 xt 1  vt
(9A.1)
yt  yt 1  1 1     2  xt  xt 1   vt
(9A.2)
yt  yt  yt 1
Principles of Econometrics, 3rd Edition
xt2  xt  xt 1
xt1  1  
Slide 9-46
yt  xt11  xt22  vt
yt  1  2 xt  ( yt 1  1  2 xt 1 )  vt
Principles of Econometrics, 3rd Edition
(9A.3)
(9A.4)
Slide 9-47
y1  1  x12  e1
1  2 y1  1  2 1  1  2 x12  1  2 e1
y1  x11 1  x12 2  e1
y1  1  2 y1
(9A.5)
x11  1  2
(9A.6)
x12  1  2 x1
Principles of Econometrics, 3rd Edition
e1  1  2 e1
Slide 9-48
2

var(e1 )  (1  2 ) var(e1 )  (1  2 ) v 2  v2
1 
Principles of Econometrics, 3rd Edition
Slide 9-49
H0 :   0
H1 :   0
T
d
  eˆt  eˆt 1 
t 2
2
(9B.1)
T
2
ˆ
e
t
t 1
Principles of Econometrics, 3rd Edition
Slide 9-50
T
d
T
 eˆ   eˆ
t 2
2
t
2
t 1
t 2
T
 2 eˆt eˆt 1
t 2
T
2
ˆ
e
t
t 1
T

2
ˆ
e
t
t 2
T
2
ˆ
e
t
t 1
T

2
ˆ
e
 t 1
t 2
T
2
ˆ
e
t
t 1
(9B.2)
T
2
 eˆt eˆt 1
t 2
T
2
ˆ
e
t
t 1
 1  1  2r1
Principles of Econometrics, 3rd Edition
Slide 9-51
d  2 1  r1 
(9B.3)
d  dc
Principles of Econometrics, 3rd Edition
Slide 9-52
Figure 9A.1:
Principles of Econometrics, 3rd Edition
Slide 9-53
Figure 9A.2:
Principles of Econometrics, 3rd Edition
Slide 9-54
The Durbin-Watson bounds test.
 if d  d Lc , reject H 0 :   0 and accept H1 :   0;
 if d  dUc , do not reject H 0 :   0;

if d Lc  d  dUc , the test is inconclusive.
Principles of Econometrics, 3rd Edition
Slide 9-55
yt    0 xt  1 xt 1  2 xt 2  3 xt 3 
yt    0 xt  1 xt 1 
Principles of Econometrics, 3rd Edition

 et     s xt s  et
 q xt q  1 yt 1 
s 0
  p yt  p  vt
Slide 9-56
yt    0 xt  1 yt 1  vt
(9C.1)
yt 1    0 xt 1  1 yt 2
(9C.2)
yt    0 xt  1 yt 1     0 xt  1 (   0 xt 1  1 yt 2 )
   1  0 xt  10 xt 1  12 yt 2
Principles of Econometrics, 3rd Edition
Slide 9-57
yt    1  0 xt  10 xt 1  12 (  0 xt 2  1 yt 3 )
   1  12  0 xt  10 xt 1  120 xt 2  13 yt 3
yt    1  12 
 1j 
 1j 0 xt  j  1j 1 yt ( j 1)
 0 xt  10 xt 1  120 xt 2 
 (1  1   
2
1
Principles of Econometrics, 3rd Edition
(9C.3)
j
  )   01s xt s  1j 1 yt ( j 1)
j
1
s 0
Slide 9-58

yt     01s xt s
(9C.4)
s 0
  (1  1   
2
1

)
1  1

yt     s xt s  et
s 0
Principles of Econometrics, 3rd Edition
Slide 9-59
s  01s

 s  0 (1  1  
s 0
Principles of Econometrics, 3rd Edition
2
1

0
)
1  1
Slide 9-60
yt    0 xt  1 xt 1  2 xt 2  3 xt 3  1 yt 1  2 yt 2  vt
(9C.5)
0   0
1  10  1
2  11  20   2
(9C.6)
3  12  21  3
4  13  22
 s  1 s 1  2 s 2
Principles of Econometrics, 3rd Edition
for s  4
Slide 9-61
yˆT 1 
yT  yT 1  yT 2
3
yˆT 1  yT  (1  )1 yT 1  (1  )2 yT 2 
(9D.1)
(1  ) yˆT  (1  ) yT 1  (1  )2 yT 2  (1  )3 yT 3  ..... (9D.2)
yˆT 1  yT  (1  ) yˆT
Principles of Econometrics, 3rd Edition
Slide 9-62
Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α
Principles of Econometrics, 3rd Edition
Slide 9-63