CHAPTER 7
FORECASTING WITH AUTOREGRESSIVE (AR) MODELS
Figure 7.1 A Variety of Time Series Cycles
500
4
450
3
(in thousands)
2
1
0
-1
-2
400
350
300
250
-3
200
-4
10
20
30
40
Adjusted signal
50
60
70
Observed signal
80
90 91 92 93 94 95 96 97 98 99 00 01
Home sales in California (16 counties)
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7.1 Cycles
Figure 7.2 Deterministic Cycle
4
3
2
1
0
-1
-2
-3
-4
5
10
15
Y=2cos(0.5 t +0.78) + e
20
25
30
35
40
Y_determ=2cos(0.5 t +0.78)
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Figure 7.3 Unemployed Persons, 1989-2002 (Seasonally Adjusted)
A
B
D
C
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7.2.1 The AR(1) Process
Figure 7.4 Autoregressive Processes AR(1)
5.0
3.5
3.0
  0.4
2.5
4.0
2.0
3.5
1.5
3.0
1.0
2.5
0.5
2.0
0.0
200
225
250
275
  0 .7
4.5
300
325
350
375
400
1.5
200
225
250
275
300
350
375
400
325
350
375
400
Y
Y
400
23
22
325
  0.95
 1
360
21
320
20
280
19
18
240
17
200
16
15
200
225
250
275
300
Y
325
350
375
400
160
200
225
250
275
300
Y
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Figure 7.5 Autocorrelation Functions of Covariance-Stationary AR(1) Processes
  0 .4
  0.7
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  0.95
5
Figure 7.6 Time Series Plot and Autocorrelation Functions of AR(1)
with Negative Parameter
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6
4
2
0
-2
-4
-6
25
50
75
100
125
150
Y
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Figure 7.7 Per Capita Income Growth (California, 1969-2002)
12
10
8
6
4
2
0
1970
1975
1980
1985
1990
1995
2000
Personal Income Growth in CA
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7.2.2 AR(2) Process
Figure 7.8 Autoregressive Processes AR(2)
Yt  1  0.5Yt 1  0.4Yt 2   t
Yt  1  Yt 1  0.5Yt 2   t
1   2  0.5
Yt  1  0.5Yt 1  0.3Yt 2   t
1   2  0.1
6
5
5
4
4
3
3
2
2
1
1
0
0
-1
-1
-2
1  2  0.8
8
7
6
5
-2
600
625
650
675
Series 1
700
725
750
-3
600
4
3
2
625
650
675
700
725
750
1
600
Series 2
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625
650
675
700
725
750
Series 3
8
Figure 7.9 Nonstationary AR(2)
520
500
Yt  1  0.5Yt 1  0.5Yt 2   t
480
460
440
420
400
380
600
625
650
675
700
725
750
Series 4
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Figure 7.10 Autocorrelation Functions of Covariance-Stationary AR(2) Processes
Yt  1  Yt 1  0.5Yt 2   t
Yt  1  0.5Yt 1  0.4Yt 2   t
Yt  1  0.5Yt 1  0.3Yt 2   t
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Figure 7.11 U.S. Inflation Rate
20
15
10
5
0
-5
-10
-15
20
30
40
50
60
70
80
90
00
Inflation (cpi growth)
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Table 7.1 Estimation Results, U.S. Inflation Rate, AR(2) Model
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Table 7.2 Multistep Forecast of U.S. Inflation Rate
h 1
f t ,1  cˆ  ˆ1 Yt  ˆ2 Yt 1 
2004
 1.49  0.79  2.25  0.25  1.56 
 t21|t  ˆ 2  3.74 2
f (Yt 1 | I t )  N (  t 1|t ,  t21|t )
 N (2.90,3.74 2 )
 2.90
h2
f t , 2  cˆ  ˆ1 f t ,1  ˆ2 Yt 
 t2 2|t  ˆ 2 (1  ˆ1 2 ) 
2005
 1.49  0.79  2.90  0.25  2.25 
 3.74 2 (1  0.79 2 ) 
 3.25
 4.812
h3
f t ,3  cˆ  ˆ1 f t , 2  ˆ2 f t ,1 
 t23|t  ˆ 2 (1  ˆ12 
2006
 1.49  0.79  3.25  0.25  2.90 
 (ˆ2  ˆ12 ) 2 ) 
 3.36
f (Yt  2 | I t )  N (3.25,4.812 )
f (Yt 3 | I t )  N (3.36,5.03 2 )
 5.03 2
h4
f t , 4  cˆ  ˆ1 f t ,3  ˆ2 f t , 2 
2007
 1.49  0.79  3.36  0.25  3.25 
 t24|t  5.04 2
f (Yt  4 | I t )  N (3.37,5.04 2 )
 t25|t  5.04 2
f (Yt 5 | I t )  N (3.34,5.04 2 )
 t26|t  5.04 2
f (Yt  6 | I t )  N (3.32,5.04 2 )
 3.37
h 5
f t ,5  cˆ  ˆ1 f t , 4  ˆ2 f t ,3 
2008
 1.49  0.79  3.37  0.25  3.36 
 3.34
h6
f t , 6  cˆ  ˆ1 f t ,5  ˆ2 f t , 4 
2009
 1.49  0.79  3.34  0.25  3.37 
 3.32
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Figure 7.12 U.S. Inflation Rate, Multistep Forecast
16
estimation sample
forecasting sample
12
8
4
multistep forecast
0
-4
-8
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09
Actual data (CPI inflation)
Forecast
Upper bound, 95% confidence interval
Lower bound, 95% confidence interval
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7.2.3 AR(p) Process
Figure 7.13 Number of Unemployed People Looking for Part-Time Work
1900
1800
in thousands
1700
1600
1500
1400
1300
1200
1100
1000
1990 1992 1994 1996 1998 2000 2002
Unemployed looking for part time work
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7.3.1 Deterministic and Stochastic Seasonal Cycles
Figure 7.14 Deterministic Seasonality
4
Q4
Q4
Q4
3
2
1
0
Q2
Q2
Q3
Q3
Q3
-1
00
01
02
03
04
05
06
07
08
09
10
Y=1*Q1+0.5*Q2+0.5*Q3+3*Q4+e
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Figure 7.15 Stochastic Seasonality
3
2
1
0
-1
-2
-3
20
22
24
26
28
30
32
34
36
38
40
Y=0.8*Y(-4)+e
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7.3.2 Seasonal ARMA Models
Figure 7.16 Seasonal AR(1) and AR(2), Time Series Plots and Autocorrelograms
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Figure 7.17 Monthly Clothing Sales in the United States. Time Series Plot and
Autocorrelation Functions
22000
20000
18000
16000
14000
12000
10000
8000
6000
2003 2004 2005 2006 2007 2008 2009 2010
Clothing Sales (millions $)
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Figure 7.18 Seasonal MA(1) and MA(2), Time Series Plots and Autocorrelograms
5
5
4
4
3
3
2
2
1
1
0
0
-1
-1
20
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30
32
34
Y=2+0.9*e(-4)+e
36
38
40
20
22
24
26
28
30
32
34
36
38
40
Y=2-e(-4)+0.25*e(-8)+e
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7.3.3 Combining ARMA and Seasonal ARMA Models
Figure 7.19 Monthly Changes of Private Residential Construction in U.S. (Millions
of Dollars), Time Series Plot and Autocorrelograms
12000
8000
4000
0
-4000
-8000
02
03
04
05
06
07
08
09
10
Monthly Change in Residential Construction (millions $)
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Table 7.3 Monthly Changes in Residential Construction. Estimation Results of AR(1)
and S-AR(1) Model
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Figure 7.20 Monthly Changes in Residential Construction, Multistep Forecast
6000
estimation sample
forecasting sample
4000
2000
0
-2000
-4000
-6000
09M01 09M07 10M01 10M07 11M01 11M07 12M01
Lower bound, 95% confidence interval
Forecast
Lower bound, 95% confidence interval
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