1 chapter 7 forecasting with autoregressive (ar) models figure 7.1 a variety of time series cycles...

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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

-4

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0

1

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10 20 30 40 50 60 70 80

Adjusted signal Observed signal

González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.

2

Figure 7.2 Deterministic Cycle


7.1 Cycles

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1

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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

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3.5

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Y

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5.0

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Y

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200 225 250 275 300 325 350 375 400

Y

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Y

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1

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95.0


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Figure 7.5 Autocorrelation Functions of Covariance-Stationary AR(1) Processes

4.0 7.0 95.0

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Figure 7.6 Time Series Plot and Autocorrelation Functions of AR(1) with Negative Parameter


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Y


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Figure 7.7 Per Capita Income Growth (California, 1969-2002)

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1970 1975 1980 1985 1990 1995 2000

Personal Income Growth in CA


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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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Series 3

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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Series 4

tttt YYY 21 5.05.01


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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


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Figure 7.11 U.S. Inflation Rate

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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 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


14

Figure 7.12 U.S. Inflation Rate, Multistep Forecast

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Actual data (CPI inflation)ForecastUpper bound, 95% confidence intervalLower bound, 95% confidence interval

estimation sample forecasting sample

multistep forecast


15

Figure 7.13 Number of Unemployed People Looking for Part-Time Work

7.2.3 AR(p) Process

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Unemployed looking for part time work

in th

ousa

nds


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Figure 7.14 Deterministic Seasonality

7.3.1 Deterministic and Stochastic Seasonal Cycles

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Y=1*Q1+0.5*Q2+0.5*Q3+3*Q4+e

Q4 Q4Q4

Q3Q3

Q3

Q2Q2


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Figure 7.15 Stochastic Seasonality

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Y=0.8*Y(-4)+e


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Figure 7.16 Seasonal AR(1) and AR(2), Time Series Plots and Autocorrelograms7.3.2 Seasonal ARMA Models


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Figure 7.17 Monthly Clothing Sales in the United States. Time Series Plot and Autocorrelation Functions

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Clothing Sales (millions $)


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Figure 7.18 Seasonal MA(1) and MA(2), Time Series Plots and Autocorrelograms

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Y=2+0.9*e(-4)+e

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Y=2-e(-4)+0.25*e(-8)+e


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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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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

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09M01 09M07 10M01 10M07 11M01 11M07 12M01

Lower bound, 95% confidence intervalForecastLower bound, 95% confidence interval

estimation sample forecasting sample

1 chapter 7 forecasting with autoregressive (ar) models figure 7.1 a variety of time series cycles...

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ar2 process slide

ar1 process slide

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arp process slide

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