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90 91 92 93 94 95 96 97 98 99 00 01
Home sales in California (16 counties)
(in
tho
usa
nd
s)
1
CHAPTER 7FORECASTING WITH AUTOREGRESSIVE (AR) MODELS
Figure 7.1 A Variety of Time Series Cycles
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1
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4
10 20 30 40 50 60 70 80
Adjusted signal Observed signal
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.2 Deterministic Cycle
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
7.1 Cycles
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5 10 15 20 25 30 35 40
Y=2cos(0.5 t +0.78) + e Y_determ=2cos(0.5 t +0.78)
3González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
Figure 7.3 Unemployed Persons, 1989-2002 (Seasonally Adjusted)
AB
C
D
4González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
Figure 7.4 Autoregressive Processes AR(1)
7.2.1 The AR(1) Process
0.0
0.5
1.0
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Y
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Y
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Y
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200 225 250 275 300 325 350 375 400
Y
7.0
1
4.0
95.0
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Figure 7.5 Autocorrelation Functions of Covariance-Stationary AR(1) Processes
4.0 7.0 95.0
6
Figure 7.6 Time Series Plot and Autocorrelation Functions of AR(1) with Negative Parameter
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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25 50 75 100 125 150
Y
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.7 Per Capita Income Growth (California, 1969-2002)
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12
1970 1975 1980 1985 1990 1995 2000
Personal Income Growth in CA
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.8 Autoregressive Processes AR(2)
7.2.2 AR(2) Process
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Series 1
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tttt YYY 21 5.01 tttt YYY 21 4.05.01 tttt YYY 21 3.05.01
5.021 1.021 8.021
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Figure 7.9 Nonstationary AR(2)
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600 625 650 675 700 725 750
Series 4
tttt YYY 21 5.05.01
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.10 Autocorrelation Functions of Covariance-Stationary AR(2) Processes
ttYtYtY 25.011 ttYtYtY 24.015.01 ttYtYtY 23.015.01
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.11 U.S. Inflation Rate
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0
5
10
15
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20 30 40 50 60 70 80 90 00
Inflation (cpi growth)
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Table 7.1 Estimation Results, U.S. Inflation Rate, AR(2) Model
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Table 7.2 Multistep Forecast of U.S. Inflation Rate1h
2004
90.2
56.125.025.279.049.1
ˆˆˆ 1211,
ttt YYcf
222|1 74.3ˆ tt
)74.3,90.2(
),()|(2
2|1|11
N
NIYf tttttt
2h
2005
25.3
25.225.090.279.049.1
ˆˆˆ 21,12,
ttt Yfcf
2
22
21
22|2
81.4
)79.01(74.3
)ˆ1(ˆ
tt
)81.4,25.3()|( 22 NIYf tt
3h
2006
36.3
90.225.025.379.049.1
ˆˆˆ 1,22,13,
ttt ffcf
2
2212
21
22|3
03.5
))ˆˆ(
ˆ1(ˆ
tt
)03.5,36.3()|( 23 NIYf tt
4h
2007
37.3
25.325.036.379.049.1
ˆˆˆ 2,23,14,
ttt ffcf
22|4 04.5 tt )04.5,37.3()|( 2
4 NIYf tt
5h
2008
34.3
36.325.037.379.049.1
ˆˆˆ 3,24,15,
ttt ffcf
22|5 04.5 tt )04.5,34.3()|( 2
5 NIYf tt
6h
2009
32.3
37.325.034.379.049.1
ˆˆˆ 4,25,16,
ttt ffcf
22|6 04.5 tt )04.5,32.3()|( 2
6 NIYf tt
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.12 U.S. Inflation Rate, Multistep Forecast
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12
16
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09
Actual data (CPI inflation)ForecastUpper bound, 95% confidence intervalLower bound, 95% confidence interval
estimation sample forecasting sample
multistep forecast
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.13 Number of Unemployed People Looking for Part-Time Work
7.2.3 AR(p) Process
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1700
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1900
1990 1992 1994 1996 1998 2000 2002
Unemployed looking for part time work
in th
ousa
nds
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.14 Deterministic Seasonality
7.3.1 Deterministic and Stochastic Seasonal Cycles
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00 01 02 03 04 05 06 07 08 09 10
Y=1*Q1+0.5*Q2+0.5*Q3+3*Q4+e
Q4 Q4Q4
Q3Q3
Q3
Q2Q2
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.15 Stochastic Seasonality
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0
1
2
3
20 22 24 26 28 30 32 34 36 38 40
Y=0.8*Y(-4)+e
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.16 Seasonal AR(1) and AR(2), Time Series Plots and Autocorrelograms7.3.2 Seasonal ARMA Models
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.17 Monthly Clothing Sales in the United States. Time Series Plot and Autocorrelation Functions
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18000
20000
22000
2003 2004 2005 2006 2007 2008 2009 2010
Clothing Sales (millions $)
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.18 Seasonal MA(1) and MA(2), Time Series Plots and Autocorrelograms
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20 22 24 26 28 30 32 34 36 38 40
Y=2+0.9*e(-4)+e
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20 22 24 26 28 30 32 34 36 38 40
Y=2-e(-4)+0.25*e(-8)+e
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.19 Monthly Changes of Private Residential Construction in U.S. (Millions of Dollars), Time Series Plot and Autocorrelograms
7.3.3 Combining ARMA and Seasonal ARMA Models
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12000
02 03 04 05 06 07 08 09 10
Monthly Change in Residential Construction (millions $)
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Table 7.3 Monthly Changes in Residential Construction. Estimation Results of AR(1) and S-AR(1) Model
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 7.20 Monthly Changes in Residential Construction, Multistep Forecast
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6000
09M01 09M07 10M01 10M07 11M01 11M07 12M01
Lower bound, 95% confidence intervalForecastLower bound, 95% confidence interval
estimation sample forecasting sample
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