Matakuliah
Tahun
Versi
: A0064 / Statistik Ekonomi
: 2005
: 1/1
Pertemuan 23
Deret Berkala, Peramalan, dan
Angka Indeks-1
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menghubungkan beberapa deret berkala
bagi penyusunan peramalan dengan
menggunakan analisis trend, variasi
musim, dan perilaku siklus
2
Outline Materi
• Anailis Trend
• Variasi Musim dan Perilaku Siklus
3
COMPLETE
BUSINESS STATISTICS
12
•
•
•
•
•
•
•
12-4
5th edi tion
Time Series, Forecasting, and Index Numbers
Using Statistics
Trend Analysis
Seasonality and Cyclical Behavior
The Ratio-to-Moving-Average Method
Exponential Smoothing Methods
Index Numbers
Summary and Review of Terms
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-5
5th edi tion
12-1 Using Statistics
A time series is a set of measurements of a variable that are ordered
through time. Time series analysis attempts to detect and
understand regularity in the fluctuation of data over time.
Regular movement of time series data may result from a tendency
to increase or decrease through time - trend- or from a tendency to
follow some cyclical pattern through time - seasonality or cyclical
variation.
Forecasting is the extrapolation of series values beyond the region
of the estimation data. Regular variation of a time series can be
forecast, while random variation cannot.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-6
BUSINESS STATISTICS
5th edi tion
The Random Walk
A random walk:
Zt- Zt-1=at
or equivalently:
Zt= Zt-1+at
The difference between Z in time t and time t-1 is a random error.
Random Walk
15
10
5
Z
0
a
0
10
20
30
40
There is no evident regularity in a
random walk, since the difference in the
series from period to period is a random
error. A random walk is not
forecastable.
50
t
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-7
BUSINESS STATISTICS
5th edi tion
A Time Series as a Superposition of
Cyclical Functions
A Time Series as a Superposition of Two Wave
Functions and a Random Error (not shown)
z
4
3
2
1
0
-1
0
10
20
30
40
50
t
In contrast with a random walk, this series exhibits obvious
cyclical fluctuation. The underlying cyclical series - the
regular elements of the overall fluctuation - may be analyzable
and forecastable.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-8
5th edi tion
12-2 Trend Analysis: Example 12-1
The following output was obtained using the template.
Zt = 696.89 + 109.19t
Note: The template contains forecasts for t = 9 to t = 20 which corresponds to years 2002 to 2013 .
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-9
5th edi tion
12-2 Trend Analysis: Example 12-1
Straight line trend.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-10
5th edi tion
Example 12-2
The
forecast
for t = 10
(year 2002)
is 345.27
Observe that the forecast model is Zt = 82.96 + 26.23t
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-11
BUSINESS STATISTICS
5th edi tion
12-3 Seasonality and Cyclical Behavior
Monthly Sales of Suntan Oil
G ross Earnings: A nnual
20
100
11000
Pas sengers
E a rni ng s
Sa le s
200
Monthly Numbers of Airline Passengers
15
10
10000
9000
8000
7000
0
6000
5
J FMAMJ J ASONDJ FMAMJ J ASONDJ FMAMJ J A
80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
Time
Year
t
86 87 88
89 90 91
92 93
94 95
When a cyclical pattern has a period of one year, it is
usually called seasonal variation. A pattern with a period
of other than one year is called cyclical variation.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-12
BUSINESS STATISTICS
5th edi tion
Time Series Decomposition
• Types of Variation
Trend (T)
Seasonal (S)
Cyclical (C)
Random or Irregular (I)
• Additive Model
Zt = Tt + St + Ct + It
• Multiplicative Model
Zt = (Tt )(St )(Ct )(It)
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-13
5th edi tion
Estimating an Additive Model with
Seasonality
An additive regression model with seasonality:
Zt=0+ 1 t+ 2 Q1+ 3 Q2 + 4 Q3 + at
where
Q1=1 if the observation is in the first quarter, and 0
otherwise
Q2=1 if the observation is in the second quarter, and 0
otherwise
Q3=1 if the observation is in the third quarter, and 0
otherwise
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-14
BUSINESS STATISTICS
5th edi tion
12-4 The Ratio-to-Moving- Average
Method
A moving average of a time series is an average of a fixed
number of observations that moves as we progress down the
series.
Time, t:
1
Series, Zt:
15
Five-period
moving average:
Time, t:
Series, Zt:
2
12
3
11
4
18
15.4 15.6
1
2
3
15 12 11
(15 + 12 + 11
(12 + 11
(11
4
18
+ 18
+ 18
+ 18
5
21
6
16
7
14
8
17
9
20
16.0
17.2 17.6 17.0 18.0 18.4 17.8 17.6
5
6
7
8
9
21
16
14
17 20
+ 21)/5=15.4
+ 21 + 16)/5=15.6
+ 21 + 16 + 14)/5=16.0
10
18
10
18
11
21
11
21
12
16
12
16
13
14
14
19
13
14
14
19
. . . . .
(18 + 21 + 16 + 14 + 19)/5=17.6
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-15
5th edi tion
Comparing Original Data and Smoothed
Moving Average
Original Series and Five-Period Moving Averages
•
Moving Average:
– “Smoother”
– Shorter
– Deseasonalized
Z
20
–
15
Removes seasonal and
irregular components
– Leaves trend and
cyclical components
10
0
5
10
15
t
Z
t  TSCI  SI
MA TC
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-16
5th edi tion
Ratio-to-Moving Average
• Ratio-to-Moving Average for Quarterly
Data
Compute a four-quarter moving-average series.
Center the moving averages by averaging every
consecutive pair and placing the average between
quarters.
Divide the original series by the corresponding moving
average. Then multiply by 100.
Derive quarterly indexes by averaging all data points
corresponding to each quarter. Multiply each by 400
and divide by sum.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-17
BUSINESS STATISTICS
5th edi tion
Ratio-to-Moving Average: Example 12-3
Sales
170
148
141
150
161
137
132
158
157
145
128
134
160
139
130
144
McGraw-Hill/Irwin
Centered
Moving
Average
*
*
*
152.25
150.00
147.25
145.00
147.00
146.00
148.00
147.00
141.00
141.75
140.25
140.75
143.25
Average
*
*
151.125
148.625
146.125
146.000
146.500
147.000
147.500
144.000
141.375
141.000
140.500
142.000
*
*
Four-Quarter Moving Averages
Ratio
to
Average
*
*
93.3
100.9
110.2
93.8
90.1
107.5
106.4
100.7
90.5
95.0
113.9
97.9
*
*
Actual
Smoothed
Actual
Smoothed
170
160
Sales
Moving
Quarter
1998W
1998S
1998S
1998F
1999W
1999S
1999S
1999F
2000W
2000S
2000S
2000F
2002W
2002S
2002S
2002F
Simple
Moving
Aczel/Sounderpandian
150
Moving Average
Length:
4
140
MAPE:
MAD:
MSD:
130
0
5
10
7.816
10.955
152.574
15
Time
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-18
BUSINESS STATISTICS
5th edi tion
Seasonal Indexes: Example 12-3
Year
Winter
Quarter
Spring
Summer
1998
1999
2000
2002
110.2
106.4
113.9
93.8
100.7
97.9
Sum
Average
330.5
110.17
292.4
97.47
Fall
93.3
90.1
90.5
100.9
107.5
95.0
273.9
91.3
303.4
101.13
Sum of Averages = 400.07
Seasonal Index = (Average)(400)/400.07
Seasonal
Index
McGraw-Hill/Irwin
110.15
97.45
Aczel/Sounderpandian
91.28
101.11
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-19
BUSINESS STATISTICS
5th edi tion
Deseasonalized Series: Example 12-3
Original and Seasonally Adjusted Series
Seasonal
McGraw-Hill/Irwin
Index (S)
110.15
97.45
91.28
101.11
110.15
97.45
91.28
101.11
110.15
97.45
91.28
101.11
110.15
97.45
91.28
101.11
170
154.34
151.87
154.47
148.35
146.16
140.58
144.51
156.27
142.53
148.79
140.23
132.53
145.26
142.64
142.42
142.42
160
Sales
Deseasonalized
Quarter
Sales
Series(Z/S)*100
1998W
170
1998S
148
1998S
141
1998F
150
1999W
161
1999S
137
1999S
132
1999F
158
2000W
157
2000S
145
2000S
128
2000F
134
2002W
160
2002S
139
2002S
130
2002F
144
Aczel/Sounderpandian
150
140
130
1992W
1992S
1992S
1992F
t
Original
Deseasonalized - - -
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
12-20
BUSINESS STATISTICS
5th edi tion
The Cyclical Component: Example 12-3
Trend Line and Moving Averages
180
Sales
170
160
150
140
130
120
1992W
1992S
1992S
t
McGraw-Hill/Irwin
1992F
The cyclical component is
the remainder after the
moving averages have been
detrended. In this example,
a comparison of the moving
averages and the estimated
regression line:
Z  155.275  11059
.
t
illustrates that the cyclical
component in this series is
negligible.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-21
5th edi tion
Example 12-3 using the Template
The centered moving average, ratio to moving average, seasonal index, and deseasonalized
values were determined using the Ratio-to-Moving-Average method.
This displays just
a partial output
for the quarterly
forecasts.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-22
5th edi tion
Example 12-3 using the Template
Graph of the quarterly Seasonal Index
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-23
5th edi tion
Example 12-3 using the Template
Graph of the Data and the quarterly Forecasted values
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-24
5th edi tion
Example 12-3 using the Template
The centered moving average, ratio to moving average, seasonal index, and deseasonalized
values were determined using the Ratio-to-Moving-Average method.
This displays just
a partial output
for the monthly
forecasts.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-25
5th edi tion
Example 12-3 using the Template
Graph of the monthly Seasonal Index
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
12-26
5th edi tion
Example 12-3 using the Template
Graph of the Data and the monthly Forecasted values
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
Penutup
• Pembahasan materi dilanjutkan dengan
Materi Pokok 24 (Deret Berkala,
Peramalan, dan Angka Indeks-2)
27