Matakuliah
Tahun
Versi
: A0064 / Statistik Ekonomi
: 2005
: 1/1
Pertemuan 24
Deret Berkala, Peramalan, dan
Angka Indeks-2
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menghubungkan beberapa deret berkala
bagi penyusunan aangka indeks,
peramalan dengan menggunakan metode
rata-rata bergerak, dan exponential
smoothing
2
Outline Materi
• Metode Rata-rata Bergerak
• Metode Exponential Smoothing
• Angka Indeks
3
COMPLETE
12-4
BUSINESS STATISTICS
5th edi tion
Forecasting a Multiplicative Series:
Example 12-3
The forecast of a multiplica tive series :
Zˆ = TSC
Forecast for Winter 2002 (t = 17) :
Trend : ẑ = 152.26 - (0.837)(17) = 138.03
S = 1.1015
C  1 (negligibl e)
Zˆ = TSC
= (1)(138.03)(1.1015) = 152.02
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-5
BUSINESS STATISTICS
5th edi tion
Multiplicative Series: Review
Z  ( Trend )( Seasonal ( Cyclical )( Irregular )
 TSCI
MA  ( Trend )( Cyclical )
 TC
Z
TSCI

 SI
MA
TC
S = Average of SI (Ratio - to - Moving Averages)
Z
TSCI

 CTI (Deseasonalized Data)
S
S
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-6
BUSINESS STATISTICS
5th edi tion
12-5 Exponential Smoothing Methods
Smoothing is used to forecast a series by first removing sharp
variation, as does the moving average.
Weights Decline as we go back in
Time
Weights Decline as We Go Back in Time and Sum to 1
W ei g ht
Weight
0.4
0.3
0.2
0.1
0.0
-15
-10
-5
0
Lag
-10
McGraw-Hill/Irwin
Lag
Exponential smoothing is a forecasting
method in which the forecast is based in
a weighted average of current and past
series values. The largest weight is
given to the present observations, less
weight to the immediately preceding
observation, even less weight to the
observation before that, and so on. The
weights decline geometrically as we go
back in time.
0
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-7
BUSINESS STATISTICS
5th edi tion
The Exponential Smoothing Model
Given a weighting factor: 0 < w < 1:
2
3
Z t 1  w ( Z t )  w (1  w )( Z t 1 )  w (1  w ) ( Z t  2 )  w (1  w ) ( Z t  3 ) 
Since
2
3
Z t  w ( Z t 1 )  w (1  w )( Z t  2 )  w (1  w ) ( Z t  3 )  w (1  w ) ( Z t  4 ) 
2
3
(1  w ) Z t  w (1  w )( Z t 1 )  w (1  w ) ( Z t  2 )  w (1  w ) ( Z t  3 ) 
So
Z t 1  w(Z t )  (1  w)(Z t )
Z t 1  Z t  (1  w)(Z t - Z t )
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-8
BUSINESS STATISTICS
5th edi tion
Example 12-4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Z
925
940
924
925
912
908
910
912
915
924
943
962
960
958
955
*
McGraw-Hill/Irwin
w=.4
925.000
925.000
931.000
928.200
926.920
920.952
915.771
913.463
912.878
913.727
917.836
927.902
941.541
948.925
952.555
953.533
w=.8
925.000
925.000
937.000
926.600
925.320
914.664
909.333
909.867
911.573
914.315
922.063
938.813
957.363
959.473
958.295
955.659
Exponential Smo othing: w=0 .4 and w=0.8
960
950
w= .4
Day
940
930
920
910
0
5
10
15
Day
Original data:
Smoothed, w=0.4: ......
Smoothed, w=0.8: -----
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-9
5th edi tion
Example 12-4 – Using the Template
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-10
5th edi tion
12-6 Index Numbers
An index number is a number that measures the relative
change in a set of measurements over time. For example: the
Dow Jones Industrial Average (DJIA), the Consumer Price
Index (CPI), the New York Stock Exchange (NYSE) Index.
Value in period i
Index number in period i: = 100
Value in base period
Changing the base period of an index:
Old index value
New index value: = 100
Index value of new base
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-11
BUSINESS STATISTICS
5th edi tion
Index Numbers: Example 12-5
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
121
121
133
146
162
164
172
187
197
224
255
247
238
222
100.0
100.0
109.9
120.7
133.9
135.5
142.1
154.5
162.8
185.1
210.7
204.1
196.7
183.5
McGraw-Hill/Irwin
64.7
64.7
71.1
78.1
86.6
87.7
92.0
100.0
105.3
119.8
136.4
132.1
127.3
118.7
Price and Index (1982=100) of Natural Gas Price
250
Original
Index (1984)
P ric e
Index
Index
Year Price 1984-Base 1991-Base
150
Index (1991)
50
Aczel/Sounderpandian
1985
1990
1995
Year
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-12
5th edi tion
Consumer Price Index – Example 12-6
Consumer Price index (CPI): 1967=100
Example 12-6:
450
Adjusted Salary =
CPI
350
250
150
50
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995
Ye a r
McGraw-Hill/Irwin
Aczel/Sounderpandian
Year
1980
1981
1982
1983
1984
1985
Salary
100
CPI
Adjusted
Salary Salary
29500 11953.0
31000 11380.3
33600 11610.2
35000 11729.2
36700 11796.8
38000 11793.9
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-13
5th edi tion
Example 12-6: Using the Template
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
Penutup
• Deret Berkala pada dasarnya bertujuan
untuk mengidentifikasi faktor-faktor atau
komponen deret berkala (trend, variasi
musim, perilaku siklus dan variasi lainnya)
yang selanjutnya digunakan sebagai
landasan untuk meramalkan nilai-nilai
tersebut di masa mendatang
14