1 CHAPTER 3 STATISTICS AND TIME SERIES González-Rivera: Forecasting for Economics and Business,...
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1 CHAPTER 3 STATISTICS AND TIME SERIES González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc. 0 2000 4000 6000 8000 10000 12000 14000 16000 88 90 92 94 96 98 00 02 04 06 Index level -20 -15 -10 -5 0 5 10 15 88 90 92 94 96 98 00 02 04 06 M onthly returns to the D J index igure 3.1 Examples of Time Series Dow Jones Index
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Transcript of 1 CHAPTER 3 STATISTICS AND TIME SERIES González-Rivera: Forecasting for Economics and Business,...
1
CHAPTER 3STATISTICS AND TIME SERIES
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
0
2000
4000
6000
8000
10000
12000
14000
16000
88 90 92 94 96 98 00 02 04 06
Index level
-20
-15
-10
-5
0
5
10
15
88 90 92 94 96 98 00 02 04 06
Monthly returns to the DJ index
Figure 3.1 Examples of Time Series
Dow Jones Index
2
10
12%
Y
%12)10( YP
Figure 3.2 Probability Density Function
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
3.1 Stochastic Process and Time Series
3
1 2t
T……………..
}{ tY
1
2
T
1Y 2Y TY
Figure 3.3 Graphical Representation of a Stochastic Process
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
3.1.1 Stochastic Process
4
1 2t
T……………..
}{ tY
1
2
T
1Y 2Y TY..
.……….
1y
2y
Ty}{ ty
Figure 3.4 Graphical Representation of a Stochastic Process and a Time Series (Thick Line)
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
3.1.1 Time Series
2000
4000
6000
8000
10000
12000
1990 1992 1994 1996 1998 2000 2002
CLOSE
-20
-15
-10
-5
0
5
10
1990 1992 1994 1996 1998 2000 2002
RETURN
1 2 tT……………..
}{ tY
1
2
T
1Y 2Y TY
2 t……………..
}{ tY
12 T
1Y 2Y TY
1 T
Nonstationary
Stationary
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 3.5 Nonstationary and Stationary Stochastic Process
3.2.1 Stationarity
6
1 T2t……………..
}{ tY
12 T
1Y 2Y TY
Figure 3.6 Strongly Stationary Stochastic Process
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Table 3.1 Dow Jones Index and Returns
Date Index Index(-1)First differenceIndex-index(-1)
Return = 100 X(index-index(-1))/Index(-1)
Return = 100 Xlog(index)- log(index(-1))
3.2.2 Useful transformations of Nonstationary Processes
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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0
2000
4000
6000
8000
10000
12000
14000
16000
88 90 92 94 96 98 00 02 04 06
Dow Jones Index level
-1500
-1000
-500
0
500
1000
1500
88 90 92 94 96 98 00 02 04 06
First difference of Index
7.2
7.6
8.0
8.4
8.8
9.2
9.6
88 90 92 94 96 98 00 02 04 06
Log Index
-20
-15
-10
-5
0
5
10
15
88 90 92 94 96 98 00 02 04 06
First difference of the log Index (returns)
Figure 3.7 Dow Jones Index and Its Transformation to Returns
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1980 1985 1990 1995 2000 2005
Percent annual change
1400
1500
1600
1700
1800
1980 1985 1990 1995 2000 2005
Annual Hours Worked in Germany
Autocorrelation function
k 1 2 3 4 5 6 7 8 9 10
.22 .29 -.10 .16 -.01 .19 -.06 -.04 .09 .20k
-1.0
-0.5
0.0
0.5
1.0
1 2 3 4 5 6 7 8 9 10
displacement k
Figure 3.8 Annual Hours Worked per Person Employed in Germany
3.3 A New Tool of Analysis: The Autocorrelation Functions
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Table 3.2 Percentage Change in Working Hours in Germany: Calculation of the Autocorrelation Coefficients
1978 -1.0604
1979 -0.6699 -1.0604
1980 -1.1018 -0.6699
1981 -1.2413 -1.1018 -1.0604
1982 -0.6497 -1.2413 -0.6699
1983 -0.7536 -0.6497 -1.1018
1984 -0.6826 -0.7536 -1.2413
1985 -1.3733 -0.6826 -0.6497
1986 -1.1438 -1.3733 -0.7536
1987 -1.3533 -1.1438 -0.6826
1988 -0.3196 -1.3533 -1.3733
1989 -1.4574 -0.3196 -1.1438
1990 -1.4536 -1.4574 -1.3533
1991 -2.0234 -1.4536 -0.3196
1992 -0.5904 -2.0234 -1.4574
1993 -1.4550 -0.5904 -1.4536
1994 0.0264 -1.4550 -2.0234
1995 -0.7087 0.0264 -0.5904
1996 -0.9752 -0.7087 -1.4550
1997 -0.5249 -0.9752 0.0264
1998 -0.2026 -0.5249 -0.7087
1999 -0.7057 -0.2026 -0.9752
2000 -1.2126 -0.7057 -0.5249
2001 -0.8375 -1.2126 -0.2026
2002 -0.7745 -0.8375 -0.7057
2003 -0.4141 -0.7745 -1.2126
2004 0.2950 -0.4141 -0.8375
2005 -0.2879 0.2950 -0.7745
2006 -0.0492 -0.2879 -0.4141
Mean: -0.8026
Variance: 0.2905
(k= 1,3) 0.0651 -0.0282
(k= 1,3) 0.2240 -0.0970
tY 1tY 3tY
0
k
k
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 3.9 Percentage Change in Working Hours in Germany: Autocorrelations of Order 1 and 3
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
-2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4
1tY 3tY
tY
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 3.10 Annual Working Hours per Employee in the United States
1800
1810
1820
1830
1840
1850
1860
1800 1810 1820 1830 1840 1850 1860
USA(-1)
US
A
1800
1810
1820
1830
1840
1850
1860
1980 1985 1990 1995 2000 2005
USA
Autocorrelation function
k 1 2 3 4 5 6 7 8 9
.74 .36 .06 -.09 -.16 -.29 -.35 -.25 -.06k
González-Rivera: Forecasting for Economics and Business, Copyright © 2013 Pearson Education, Inc.
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Figure 3.11 Time Series: Annual Working Hours per Employee in the United States. Autocorrelation Function
3.3.2 Statistical Tests for Autocorrelation Coefficients
