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Fundamental Of Forecast Modeling
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1.0 INTRODUCTION
This chapter discuss about background of study, statement of problem, purpose, objective and
significance of the study.
1.1 BACKGROUND OF THE STUDY
This section will present detailed explanation of National Inflation Rate for Indonesia.
Inflation is the rate to measuring increase of goods price. There are certain processes to calculate
the Inflation in economic like we calculate GDP. So Inflation rate is important for the
government, academician, consumer also businessman to know economy situation for the
country. The rational why I am choose this topic is to open our mind and to know about our
neighbour economy. Then, with reference that I have in QMT 463 I can forecast one step ahead
with suitable models.
1.2 STATEMENT OF PROBLEM
The main problem in this case, is to choose the best fitted models to generate the forecast for
National Inflation Rate in Indonesia. By this guide it easy for me and other forecaster or
researcher to do this task.
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There are 10 stages in forecasting procedure that I must follow to complete this task.
i. Determine the purpose and objective of the forecasting exercise.
ii. Selection of relevant theory
iii. Collection data
iv. Getting to know your data
v. Initial model estimation
vi. Model evaluation and revision
vii. Initial forecast presentation
viii. Final revision
ix. Forecast distribution
x. Establish monitoring system
1.3 PURPOSE OF STUDY
The purpose of this assignment is to identify, choose, calculate the best fitted model for the set
data that I have. This study also explains the related graph which can explain the National
Inflation Rate in Indonesia.
1.4 OBJECTIVES OF STUDY
The objectives of this study are:
1.4.1 To study about National Inflation Rate of Indonesia.
1.4.2 To measure the one step ahead forecast with suitable model.
1.4.3 To analyze the data set and discuss on the component of time series (graph) that related to
the data set.
1.4.4 To search best fit model for the set data that I have.
3
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1.5 SIGNIFICANCE OF THE STUDY
The study of will present detailed explanation of National Inflation Rate for Indonesia.
Inflation is the rate to measuring increase of goods price. There are certain processes to calculate
the Inflation in economic like we calculate GDP. So Inflation rate is important for the
government, academician, consumer also businessman to know economy situation for the
country. The Government, academician, consumer also businessman can use this information to
the industry and society to increase the level of awareness of economy.
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2.0 METHODOLOGY
This chapter describes the methodology used to carry out the study on the benefit of eggshell
technology in industries. Only secondary research was used to get the data for the study. One set
data of National Inflation rate in Indonesia obtain through internet research will used as source of
information. The data was synthesized and summarized for the report, no primary research was
done, no interviews were conducted, no questionnaire will distribute and no observations will
make. These are the limitations of the study.
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3.0 FINDINGS AND DISCUSSIONS
I have search through the internet to get the data set which has 36 month data. This data set I get
from Indonesia Statistic Ministry Web. This data set is called External Data because obtained
outside the normal operational activities of the firms and are beyond the management’s control.
When data obtained from secondary sources they are known as “Secondary Data”. The data set
can be seen in Figure1 and Table1.
From the data set that I have, in this task, I must use five models and then choose the best fitted
model.
i. Naïve Model
ii. Simple exponential smoothing Model
iii. Decomposition Method
iv. ARRES Method
v. Holt-Winters
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Figure1 (National Inflation of Indonesia Graph)
The graph in Figure1 shows that the National Inflation of Indonesia from January 2002 to
December 2004. The highest rate in January 2002 and the lowest data in March 2003. The trend
for the graph is decrease.
The table1 in the next page shows that all data set that I get obtain through internet. This data set
about the National Inflation Rate of Indonesia.
National Inflation of Indonesia
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
Month
Inflation
Inflation
8
Table1 (Data set of Inflation Rate of Indonesia from January 2002-Disember2004)
Year Month Inflation
2002 1 1.99
2 1.50
3 -0.02
4 -0.24
5 0.80
6 0.36
7 0.82
8 0.29
9 0.53
10 0.54
11 1.85
12 1.20
2003 1 0.80
2 0.20
3 -0.23
4 0.15
5 0.21
6 0.09
7 0.03
8 0.84
9 0.36
10 0.55
11 1.01
12 0.94
2004 1 0.57
2 -0.02
3 0.36
4 0.97
5 0.88
6 0.48
7 0.39
8 0.09
9 0.02
10 0.56
11 0.89
12 1.04
9
3.1 NAIVE MODEL
Naïve with Trend Model
The application of this model is fairly common among organizations. One reason for its
popularity is that can be used even with fairly short time series. Thus, overcoming the common
problem in most organizations where insufficient data is a common phenomenon. Insufficient
data would prohibit the application of sophisticated modeling technique.
The one step ahead forecast is represented as, Ft+1 = yt (yt/ yt-1) where yt is the actual value at time
t, and yt-1 is the actual value in preceding period. This model implies that all future forecast can
be set to the equal the actual observed value in the most recent time period plus the growth rate
that is the trend value as measured by yt/ yt-1. Hence, if yt is greater than yt-1 then the trend is
upward and conversely if yt is less than yt-1 then trend is downward.
This model is highly sensitive to the change in the actual value. As such a sudden drop or sharp
increase in the value will severely affect the forecast. Furthermore, fitting this model type will
result in the loss of the first two observations in the series. On the other hand,this model can also
be used for short time series.
Fitting The Naïve With Trend Model With Excel
Table2 in the next page shows that how I am fitting Naïve with Trend Model with using Excel.
Firstly set the data like Table1. Then, make the column name fitted and type (D3*D3)/D2 in the
3rd
row. Then drag the box until one step ahead. Then, calculate its MSE to compare with other
model. The forecast value that I get for January 2005 is 1.22 .The MSE show that 24.23 and
the value of MAPE is 8.92 .
10
Table2 (Fitting Naïve with Trend Model with Excel)
Year Month t Inflation Fitted
2002 1 1 1.99
2 2 1.50
3 3 -0.02 1.13
4 4 -0.24 0.00
5 5 0.80 -2.88
6 6 0.36 -2.67
7 7 0.82 0.16
8 8 0.29 1.87
9 9 0.53 0.10
10 10 0.54 0.97
11 11 1.85 0.55
12 12 1.20 6.34
2003 1 13 0.80 0.78
2 14 0.20 0.53
3 15 -0.23 0.05
4 16 0.15 0.26
5 17 0.21 -0.10
6 18 0.09 0.29
7 19 0.03 0.04
8 20 0.84 0.01
9 21 0.36 23.52
10 22 0.55 0.15
11 23 1.01 0.84
12 24 0.94 1.85
2004 1 25 0.57 0.87
2 26 -0.02 0.35
3 27 0.36 0.00
4 28 0.97 -6.48
5 29 0.88 2.61
6 30 0.48 0.80
7 31 0.39 0.26
8 32 0.09 0.32
9 33 0.02 0.02
10 34 0.56 0.00
11 35 0.89 15.68
12 36 1.04 1.41
2005 1 37 1.22
11
Table2 ( Continue )
et et² (et/yt)*100
0
0
-1.15 1.32 5753
-0.24 0.06 100
3.68 13.54 460
3.03 9.16 841
0.66 0.43 80
-1.58 2.49 544
0.43 0.18 81
-0.43 0.18 79
1.30 1.69 70
-5.14 26.40 428
0.02 0.00 3
-0.33 0.11 167
-0.28 0.08 122
-0.11 0.01 76
0.31 0.09 147
-0.20 0.04 227
-0.01 0.00 29
0.83 0.69 99
-23.16 536.39 6433
0.40 0.16 72
0.17 0.03 17
-0.91 0.84 97
-0.30 0.09 53
-0.37 0.13 1828
0.36 0.13 100
7.45 55.50 768
-1.73 3.01 197
-0.32 0.10 66
0.13 0.02 33
-0.23 0.05 252
0.00 0.00 4
0.56 0.31 99
-14.79 218.74 1662
-0.37 0.14 36
12
3.2 SIMPLE EXPONENTIAL SMOOTHING MODEL
Some people call this model Single Exponential Smoothing Technique. But one thing is sure, it
is the simplest form of model within the family of the exponential smoothing technique. The
model requires only one parameter, that is the smoothing constant α to generate the fitted values
and hence forecast.
The advantage of this procedure is that it takes into account the most recent forecast. In Simple
Exponential Smoothing Model, the forecast for the next and all subsequent periods are
determined by adjusting the current period forecast by apportion of the difference between the
current forecast and current actual value. This is described in term of minimum errors.
Hence, if the recent forecast proves to be accurate, then it seems reasonable to base the
subsequent forecast on these estimates. Likewise, if recent predictions have been subjected to
large errors, then new forecast will also take this into consideration.
Another advantage of this technique is that it is requires the retention of only a limited amount
the data. There is no need to store data for many periods, because the historical profile is
recorded in concise form in the current smoothed statistic.
Ft+m = α yt + (1-α)Ft
The main thing in simple exponential smoothing is to choose best value of α. The first procedure
relies heavily not only on ones personal knowledge about the problem being evaluated and but
also on the amount of past experience one has with regard to the variable involved. For instance,
if one’s experience leads one to believe that past values can still contribute significantly the
necessary information needed to generate the forecast values, the small value of α is assigned.
Conversely, large value of α is used when one believes that only the most recent information are
important to generate the forecast value.
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The second procedure that require the application of certain measurement criterion that can be
used to determined the best value of α. This is called “error measurement”. Some people called it
Mean Square Error (MSE), Root Mean Square Error (RMSE) and Mean Absolute Percent Error
(MAPE). The main purpose of this procedure is to generate a set fitted values associated with
each value. This is with objective of choosing the alpha value such that when it applied to the
model it minimizes the error. More specifically, it is to search far an alpha that result I the
smallest error measurement.
Fitting The Exponential Using Excel
Firstly key in the data like Table3. Then, in the fitted column write the equation =(E3*C2)+((1-
E3)*D2) to get the fitted value. Then ,drag the box to get fitted data. From the Table3, forecast of
the January 2005 one step ahead is 1.22 . After that, calculate error, MSE and MAPE to compare
with other model.
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Table3 (Fitting Simple Exponential Smoothing Model with α = 0.9)
Year Month t Inflation Fitted
2002 1 1 1.99
2 2 1.50
3 3 -0.02 1.13
4 4 -0.24 0.00
5 5 0.80 -2.88
6 6 0.36 -2.67
7 7 0.82 0.16
8 8 0.29 1.87
9 9 0.53 0.10
10 10 0.54 0.97
11 11 1.85 0.55
12 12 1.20 6.34
2003 1 13 0.80 0.78
2 14 0.20 0.53
3 15 -0.23 0.05
4 16 0.15 0.26
5 17 0.21 -0.10
6 18 0.09 0.29
7 19 0.03 0.04
8 20 0.84 0.01
9 21 0.36 23.52
10 22 0.55 0.15
11 23 1.01 0.84
12 24 0.94 1.85
2004 1 25 0.57 0.87
2 26 -0.02 0.35
3 27 0.36 0.00
4 28 0.97 -6.48
5 29 0.88 2.61
6 30 0.48 0.80
7 31 0.39 0.26
8 32 0.09 0.32
9 33 0.02 0.02
10 34 0.56 0.00
11 35 0.89 15.68
12 36 1.04 1.41
2005 1 37 1.22
15
Et et² (et/yt)*100
0
0
-1.15 1.32 5753
-0.24 0.06 100
3.68 13.54 460
3.03 9.16 841
0.66 0.43 80
-1.58 2.49 544
0.43 0.18 81
-0.43 0.18 79
1.30 1.69 70
-5.14 26.40 428
0.02 0.00 3
-0.33 0.11 167
-0.28 0.08 122
-0.11 0.01 76
0.31 0.09 147
-0.20 0.04 227
-0.01 0.00 29
0.83 0.69 99
-23.16 536.39 6433
0.40 0.16 72
0.17 0.03 17
-0.91 0.84 97
-0.30 0.09 53
-0.37 0.13 1828
0.36 0.13 100
7.45 55.50 768
-1.73 3.01 197
-0.32 0.10 66
0.13 0.02 33
-0.23 0.05 252
0.00 0.00 4
0.56 0.31 99
-14.79 218.74 1662
-0.37 0.14 36
Total 872.12 ∑|(et/yt)*100| 321
MSE 24.23 MAPE 8.92
16
3.3 DECOMPOSITION METHOD
The process of generating the forecast values using this methodology is basically the reverse of
the process of decomposing the components. What is done here is to integrate the individual
components that have been identified and isolated earlier using past data points in the forecast
periods. This is made on the basis of either one assumptions used when the data were initially
analyze. For instance, if these components are assumed to be related in multiplicative manner,
such that y = T.S.C.I , then the forecast is simply the product of these components. Similarly, if
the assumption takes the additive form, y = T+S+C+I.
It should be note that the application of the decomposition method is basically made on a very
important assumption. It is assumed that the patterns or characteristics of the data as exhibited in
the past will be repeated in the future. Even if there is any change, it is not expected to seriously
affect the future estimates.
To make the job more easier in decomposition method, I have use a simple linear trend for this
purpose which can easily be extrapolated by using excel.
Where
T = α + βt
17
Figure2 (Linear Trend for National Inflation in Indonesia)
Inflation and Trend
y = -0.0071x + 0.7099
R2 = 0.0202
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
Month
Infl
ati
on
Inflation Linear (Inflation)
From the graph in figure2, we can see downward trend over 36 month period from January 2002
to December 2004. Based on adjusted seasonal indices, it is determined that the highest rate of
inflation in Indonesia is November. The highest being the month of November, recording an
index of 219.32 percent. The lowest rate is in March as evident with lowest index with 9.59
percent.
Y = -0.0071x + 0.0799
From the estimated linear equation, it ca be conclude that over the period time the National
Inflation Rate of Indonesia have been increase at average monthly rate of 0.0799.
18
Table4 (Decomposition Method)
Year Month t Inflation Moving Total Centered MT C.M.Average
2002 1 1 1.99
2 2 1.50
3 3 -0.02
4 4 -0.24
5 5 0.80
6 6 0.36
7 7 0.82 9.62 18.05 0.75
8 8 0.29 8.43 15.56 0.65
9 9 0.53 7.13 14.05 0.59
10 10 0.54 6.92 14.23 0.59
11 11 1.85 7.31 14.03 0.58
12 12 1.20 6.72 13.17 0.55
2003 1 13 0.80 6.45 12.11 0.50
2 14 0.20 5.66 11.87 0.49
3 15 -0.23 6.21 12.25 0.51
4 16 0.15 6.04 12.09 0.50
5 17 0.21 6.05 11.26 0.47
6 18 0.09 5.21 10.16 0.42
7 19 0.03 4.95 9.67 0.40
8 20 0.84 4.72 9.22 0.38
9 21 0.36 4.50 9.59 0.40
10 22 0.55 5.09 11.00 0.46
11 23 1.01 5.91 12.49 0.52
12 24 0.94 6.58 13.55 0.56
2004 1 25 0.57 6.97 14.30 0.60
2 26 -0.02 7.33 13.91 0.58
3 27 0.36 6.58 12.82 0.53
4 28 0.97 6.24 12.49 0.52
5 29 0.88 6.25 12.38 0.52
6 30 0.48 6.13 12.36 0.52
7 31 0.39 6.23
8 32 0.09
9 33 0.02
10 34 0.56
11 35 0.89
12 36 1.04
2005 1 37 -2.09
19
Table4 (continue)
Unadjusted SI Adjusted SI Linear Trend Deseasonalised Data
109.20 0.64 0.0182
15.89 0.57 0.0944
9.59 0.50 -0.0021
92.86 0.43 -0.0026
92.51 0.35 0.0086
49.17 0.28 0.0073
109.03 50.04 0.21 0.0164
44.73 113.16 0.14 0.0026
90.53 77.59 0.07 0.0068
91.08 90.68 0.00 0.0060
316.46 219.32 -0.07 0.0084
218.68 165.46 -0.14 0.0073
158.55 109.20 -0.21 0.0073
40.44 15.89 -0.28 0.0126
-45.06 9.59 -0.36 -0.0240
29.78 92.86 -0.43 0.0016
44.76 92.51 -0.50 0.0023
21.26 49.17 -0.57 0.0018
7.45 50.04 -0.64 0.0006
218.66 113.16 -0.71 0.0074
90.09 77.59 -0.78 0.0046
120.00 90.68 -0.85 0.0061
194.08 219.32 -0.92 0.0046
166.49 165.46 -0.99 0.0057
95.66 109.20 -1.07 0.0052
-3.45 15.89 -1.14 -0.0013
67.39 9.59 -1.21 0.0375
186.39 92.86 -1.28 0.0104
170.60 92.51 -1.35 0.0095
93.20 49.17 -1.42 0.0098
50.04 -1.49 0.0078
113.16 -1.56 0.0008
77.59 -1.63 0.0003
90.68 -1.70 0.0062
219.32 -1.78 0.0041
165.46 -1.85 0.0063
109.20 -1.92
20
Table5 (Adjusted Seasonal Indices)
Year 1 2 3 4 5 6 7 8 9 10 11 12
2002 109.03 44.77 90.53 91.08 316.46 218.68
2003 158.55 40.44 -45.06 29.78 44.76 21.26 7.45 218.66 90.09 120.00 194.08 166.49
2004 95.66 -3.45 67.39 186.39 170.60 93.20
Total 254.21 36.99 22.33 216.17 215.36 114.46 116.48 263.43 180.62 211.08 510.54 385.17
Mean 127.11 18.50 11.17 108.09 107.68 57.23 58.24 131.72 90.31 105.54 255.27 192.59
Adj Mean 109.20 15.89 9.59 92.86 92.51 49.17 50.04 113.16 77.59 90.68 219.32 165.46
21
Figure3 (Inflation, Deseasonalized Data and Linear Trend)
Infla
tion,
Des
easo
nalis
ed d
ata
and
Line
ar T
rend
-2.5
0
-2.0
0
-1.5
0
-1.0
0
-0.5
0
0.00
0.50
1.00
1.50
2.00
2.50
13
57
911
1315
1719
2123
2527
2931
3335
Mon
th
Inflation
Infla
tion
Line
ar T
rend
Dese
ason
alise
d Da
ta
22
3.4 ADDAPTIVE RESPONSE RATE EXPONENTIAL SMOOTHING (ARRES)
ARRES is different from other exponential method. It is because, other exponential method
discussed that, the value of parameter alpha used assumed constant for all time periods.
However, over time events may take place that affect the subsequent data behaviour. Some of
these events have been described earlier. For example, people may change their desire to buy a
certain product or there is change in the level of output as a result of technology change.
In these situation, to maintain the same value for alpha for all time periods may not be realistic
decision. Thus, the development of ARRES is an attempt to overcome this problem by
incorporating the effect of the changing pattern of the data series.
Ft+1 = αt yt + (1-αt) Ft
This indicates that the value of alpha is only appropriate at a particular period t, and maybe
different at different value of t.
As in any exponential smoothing technique, the appropriate initial values are required to start the
algorithm. In this case, value are for F0, α0, E0 and AET0.
Fitting ARRES With Excel
Firstly set up the data in the Table6,the make assumption alpha and beta with certain number
between 1 and 0. then in the fitted column write the equation =$H$2*C2+(1-$H$2)*C2 then drag
the box to the down. After that, calculate the MSE and retest the alpha and beta which have
smallest MSE.
23
Table6 ( Fitting ARRES with Excel )
Year Month Inflation Fitted Et Et
2002 1 1.99 1.99 0.00 0.00
2 1.50 1.99 -0.49 -0.05
3 -0.02 1.50 -1.52 -0.20
4 -0.24 -0.02 -0.22 -0.20
5 0.80 -0.24 1.04 -0.07
6 0.36 0.80 -0.44 -0.11
7 0.82 0.36 0.46 -0.05
8 0.29 0.82 -0.53 -0.10
9 0.53 0.29 0.24 -0.07
10 0.54 0.53 0.01 -0.06
11 1.85 0.54 1.31 0.08
12 1.20 1.85 -0.65 0.00
2003 1 0.80 1.20 -0.40 -0.04
2 0.20 0.80 -0.60 -0.09
3 -0.23 0.20 -0.43 -0.13
4 0.15 -0.23 0.38 -0.08
5 0.21 0.15 0.06 -0.06
6 0.09 0.21 -0.12 -0.07
7 0.03 0.09 -0.06 -0.07
8 0.84 0.03 0.81 0.02
9 0.36 0.84 -0.48 -0.03
10 0.55 0.36 0.19 -0.01
11 1.01 0.55 0.46 0.04
12 0.94 1.01 -0.07 0.03
2004 1 0.57 0.94 -0.37 -0.01
2 -0.02 0.57 -0.59 -0.07
3 0.36 -0.02 0.38 -0.02
4 0.97 0.36 0.61 0.04
5 0.88 0.97 -0.09 0.03
6 0.48 0.88 -0.40 -0.02
7 0.39 0.48 -0.09 -0.02
8 0.09 0.39 -0.30 -0.05
9 0.02 0.09 -0.07 -0.05
10 0.56 0.02 0.54 0.01
11 0.89 0.56 0.33 0.04
12 1.04 0.89 0.15 0.05
2005 1 1.04
24
AEt α β e² (et/yt)*100
0.00 0.90 0.10 0.00 0
1.05 0.90 0.10 0.24 33
1.05 0.90 0.10 2.31 7600
0.92 0.22 0.10 0.05 92
1.01 0.07 0.10 1.08 130
0.94 0.12 0.10 0.19 122
0.95 0.06 0.10 0.21 56
0.96 0.11 0.10 0.28 183
0.93 0.07 0.10 0.06 45
0.91 0.07 0.10 0.00 2
1.04 0.07 0.10 1.72 71
0.96 0.00 0.10 0.42 54
0.94 0.04 0.10 0.16 50
0.97 0.10 0.10 0.36 300
0.95 0.13 0.10 0.18 187
0.94 0.08 0.10 0.14 253
0.91 0.07 0.10 0.00 29
0.92 0.07 0.10 0.01 133
0.91 0.07 0.10 0.00 200
0.99 0.02 0.10 0.66 96
0.95 0.03 0.10 0.23 133
0.92 0.01 0.10 0.04 35
0.95 0.04 0.10 0.21 46
0.91 0.03 0.10 0.00 7
0.95 0.01 0.10 0.14 65
0.96 0.07 0.10 0.35 2950
0.94 0.03 0.10 0.14 106
0.97 0.04 0.10 0.37 63
0.91 0.03 0.10 0.01 10
0.95 0.02 0.10 0.16 83
0.91 0.03 0.10 0.01 23
0.94 0.05 0.10 0.09 333
0.91 0.06 0.10 0.00 350
0.96 0.01 0.10 0.29 96
0.94 0.04 0.10 0.11 37
0.92 0.05 0.10 0.02 14
∑e² 10.27 ∑|(et/yt)*100| 9826.65
MSE 0.27 MAPE 272.96
25
3.5 HOLT-WINTER’S METHOD
All earlier exponential models are good as long as they deal with non seasonal data. When
seasonality exists, a more suitable model is needed. Holt-Winters is one such technique that takes
into account the trend and seasonality factors.
Fitting Holt-Winters Using Excel
Holt-Winters consist of three basic equation that define the level component, the trend
component and the seasonality component. Two assumption can be made with regard to the
relationship of these component.
Level Component
Lt = α ( yt / st-s ) + ( 1-α ) ( Lt-1 + bt-1 )
Trend Component
bt = β ( Lt Lt-1 ) + ( 1-β ) bt-1
Seasonal Component
St = γ (yt / Lt ) + (1-γ) St-s
The forecast
Ft+m = (Lt + bt * m) St-s+m
As usual, when fitting the model, some initial value are required. For ease of computation, some
simple technique will discuss here.
Determine the Initial Value
To determine the initial value, a simple procedure used to take the average of the first 12 quarters
(month).
26
b0 = 1/s ( (ys+1 – y1 ) / s) + (ys+2 – y1 ) / s2) + ......
where s = 12 (represent the number of month in year)
The initial value of the seasonal component of the first 12 month are calculated by using the ratio
of the actual values to the mean of the first 12 values as represent by Lo in which
St = Yt / Lt
27
Table7 ( Fitting Holt winter using excel )
Year Month Inflation Lt bt St
2002 1 1.99 1.18
2 1.50 0.89
3 -0.02 -0.01
4 -0.24 -0.14
5 0.80 0.48
6 0.36 0.21
7 0.82 0.49
8 0.29 0.17
9 0.53 0.32
10 0.54 0.32
11 1.85 1.10
12 1.20 1.68 -0.39 0.71
2003 1 0.80 0.80 -0.83 1.17
2 0.20 0.17 -0.65 0.92
3 -0.23 15.36 13.61 -0.01
4 0.15 4.95 -8.01 -0.13
5 0.21 -0.26 -5.49 0.35
6 0.09 -0.81 -1.05 0.18
7 0.03 -0.32 0.34 0.43
8 0.84 3.90 3.83 0.18
9 0.36 2.46 -0.91 0.30
10 0.55 1.68 -0.79 0.32
11 1.01 0.91 -0.77 1.10
12 0.94 1.08 0.08 0.73
2004 1 0.57 0.62 -0.41 1.14
2 -0.02 0.03 -0.58 0.75
3 0.36 -23.69 -21.41 -0.01
4 0.97 -15.20 5.50 -0.12
5 0.88 0.09 14.31 1.31
6 0.48 4.99 5.84 0.17
7 0.39 2.89 -1.30 0.40
8 0.09 0.72 -2.08 0.17
9 0.02 -0.22 -1.06 0.26
10 0.56 1.14 1.11 0.34
11 0.89 1.10 0.08 1.07
12 1.04 1.37 0.26 0.73
2005 1 1.86 1.63 0.26 1.14
28
Fitted α β γ et e² (et/yt)*100
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
0.8 0.9 0.10 0 0 0
1.53 0.8 0.9 0.10 -0.73 0.53 91
-0.03 0.8 0.9 0.10 0.23 0.05 115
0.01 0.8 0.9 0.10 -0.24 0.06 102
-4.14 0.8 0.9 0.10 4.29 18.39 2859
-1.45 0.8 0.9 0.10 1.66 2.77 792
-1.23 0.8 0.9 0.10 1.32 1.75 1469
-0.91 0.8 0.9 0.10 0.94 0.88 3131
0.00 0.8 0.9 0.10 0.84 0.70 100
2.44 0.8 0.9 0.10 -2.08 4.32 577
0.50 0.8 0.9 0.10 0.05 0.00 10
0.98 0.8 0.9 0.10 0.03 0.00 3
0.10 0.8 0.9 0.10 0.84 0.70 89
1.35 0.8 0.9 0.10 -0.78 0.61 137
0.20 0.8 0.9 0.10 -0.22 0.05 1099
0.01 0.8 0.9 0.10 0.35 0.12 98
5.66 0.8 0.9 0.10 -4.69 22.02 484
-3.37 0.8 0.9 0.10 4.25 18.03 483
2.62 0.8 0.9 0.10 -2.14 4.57 445
4.66 0.8 0.9 0.10 -4.27 18.23 1095
0.28 0.8 0.9 0.10 -0.19 0.04 212
-0.41 0.8 0.9 0.10 0.43 0.18 2126
-0.41 0.8 0.9 0.10 0.97 0.94 173
2.48 0.8 0.9 0.10 -1.59 2.52 178
0.85 0.8 0.9 0.10 0.19 0.03 18
1.86 0.8 0.9 0.10 ∑e² 97.50 ∑|(et/yt)*100| 9448
MSE 2.71 MAPE 262.44
29
4.0 CONCLUSSION
1. Where you need to find the initial value of the models that you have used. Why was
the method ?
Determine the Initial Value
To determine the initial value, a simple procedure used to take the average of the first 12 quarters
(month).
b0 = 1/s ( (ys+1 – y1 ) / s) + (ys+2 – y1 ) / s2) + ......
where s = 12 (represent the number of month in year)
The initial value of the seasonal component of the first 12 month are calculated by using the ratio
of the actual values to the mean of the first 12 values as represent by Lo in which
St = Yt / Lt
2. You are to find the best fitted model. In the other words to find the best parameter
value.
The smallest MSE among the model is ARRES which its MSE = 0.27 . So, the best parameter
value among the model that I have used is ARRES.
But the smallest MAPE that I have is NAÏVE model which is MAPE = 8.92 . But the main
disadvantage of this measure lies in its relevancy as it is valid only for ratio scale data ( data with
meaning full zero ) . For this reason , MAPE is potentially explosive for large forecast error
when the actual value observation close to the zero. In addition, percentage measure do not treat
errors of overestimate and underestimate.
30
3. Present the result of your analysis. Which of the model do you think would perfom
the best forecast?
MODEL Naïve with Trend Single Exponential ARRES Method Holt Winters
α=0.9 α=0.9 α=0.9
β=0.1 β=0.1
γ=0.0
MSE 24.23 0.29 0.27 2.71
Forecast 1.22 1.02 1.04 1.86
From the table above, we can see the lowest MSE is 0.27 which is ARRES Method with α=0.9,
β=0.1 . So I choose ARRES Method as the best model in this task.
ARRES is different from other exponential method. It is because, other exponential method
discussed that, the value of parameter alpha used assumed constant for all time periods.
However, over time events may take place that affect the subsequent data behaviour. Some of
these events have been described earlier. For example, people may change their desire to buy a
certain product or there is change in the level of output as a result of technology change.
In this case, Inflation rate maybe change in certain time period because Income factor, GDP, cost
of production, price elasticity and other related factor.
In these situation, to maintain the same value for alpha for all time periods may not be realistic
decision. Thus, the development of ARRES is an attempt to overcome this problem by
incorporating the effect of the changing pattern of the data series.
31
5.0 APPENDICES
Table1 ( Fitting Simple Exponential α=0.1 )
Year Month Inflation Fitted α Et et² (et/yt)*100
2002 1 1.99 1.99 0.10 0.00 0.00 0
2 1.50 1.99 0.10 -0.49 0.24 33
3 -0.02 1.94 0.10 -1.96 3.85 9805
4 -0.24 1.74 0.10 -1.98 3.94 827
5 0.80 1.55 0.10 -0.75 0.56 93
6 0.36 1.47 0.10 -1.11 1.24 309
7 0.82 1.36 0.10 -0.54 0.29 66
8 0.29 1.31 0.10 -1.02 1.03 351
9 0.53 1.20 0.10 -0.67 0.46 127
10 0.54 1.14 0.10 -0.60 0.36 111
11 1.85 1.08 0.10 0.77 0.60 42
12 1.20 1.15 0.10 0.05 0.00 4
2003 1 0.80 1.16 0.10 -0.36 0.13 45
2 0.20 1.12 0.10 -0.92 0.85 462
3 -0.23 1.03 0.10 -1.26 1.59 548
4 0.15 0.91 0.10 -0.76 0.57 503
5 0.21 0.83 0.10 -0.62 0.38 295
6 0.09 0.77 0.10 -0.68 0.46 753
7 0.03 0.70 0.10 -0.67 0.45 2233
8 0.84 0.63 0.10 0.21 0.04 25
9 0.36 0.65 0.10 -0.29 0.09 82
10 0.55 0.62 0.10 -0.07 0.01 13
11 1.01 0.62 0.10 0.39 0.15 39
12 0.94 0.66 0.10 0.28 0.08 30
2004 1 0.57 0.68 0.10 -0.11 0.01 20
2 -0.02 0.67 0.10 -0.69 0.48 3465
3 0.36 0.60 0.10 -0.24 0.06 68
4 0.97 0.58 0.10 0.39 0.15 40
5 0.88 0.62 0.10 0.26 0.07 30
6 0.48 0.64 0.10 -0.16 0.03 34
7 0.39 0.63 0.10 -0.24 0.06 61
8 0.09 0.60 0.10 -0.51 0.26 571
9 0.02 0.55 0.10 -0.53 0.28 2664
10 0.56 0.50 0.10 0.06 0.00 11
11 0.89 0.51 0.10 0.38 0.15 43
12 1.04 0.54 0.10 0.50 0.25 48
2005 1 0.59 0.10 Total 19.16 ∑|(et/yt)*100| 6063
MSE 0.53 MAPE 168.4109
32
Table2 ( Fitting Simple Exponential α=0.5 )
Year Month Inflation Fitted α et et² (et/yt)*100
2002 1 1.99 1.99 0.50 0.00 0.00 0
2 1.50 1.99 0.50 -0.49 0.24 33
3 -0.02 1.75 0.50 -1.77 3.12 8825
4 -0.24 0.86 0.50 -1.10 1.22 459
5 0.80 0.31 0.50 0.49 0.24 61
6 0.36 0.56 0.50 -0.20 0.04 54
7 0.82 0.46 0.50 0.36 0.13 44
8 0.29 0.64 0.50 -0.35 0.12 120
9 0.53 0.46 0.50 0.07 0.00 12
10 0.54 0.50 0.50 0.04 0.00 8
11 1.85 0.52 0.50 1.33 1.77 72
12 1.20 1.18 0.50 0.02 0.00 1
2003 1 0.80 1.19 0.50 -0.39 0.15 49
2 0.20 1.00 0.50 -0.80 0.63 398
3 -0.23 0.60 0.50 -0.83 0.69 360
4 0.15 0.18 0.50 -0.03 0.00 23
5 0.21 0.17 0.50 0.04 0.00 20
6 0.09 0.19 0.50 -0.10 0.01 109
7 0.03 0.14 0.50 -0.11 0.01 364
8 0.84 0.08 0.50 0.76 0.57 90
9 0.36 0.46 0.50 -0.10 0.01 28
10 0.55 0.41 0.50 0.14 0.02 25
11 1.01 0.48 0.50 0.53 0.28 52
12 0.94 0.75 0.50 0.19 0.04 21
2004 1 0.57 0.84 0.50 -0.27 0.07 48
2 -0.02 0.71 0.50 -0.73 0.53 3632
3 0.36 0.34 0.50 0.02 0.00 5
4 0.97 0.35 0.50 0.62 0.38 64
5 0.88 0.66 0.50 0.22 0.05 25
6 0.48 0.77 0.50 -0.29 0.08 60
7 0.39 0.63 0.50 -0.24 0.06 60
8 0.09 0.51 0.50 -0.42 0.17 464
9 0.02 0.30 0.50 -0.28 0.08 1394
10 0.56 0.16 0.50 0.40 0.16 72
11 0.89 0.36 0.50 0.53 0.28 60
12 1.04 0.62 0.50 0.42 0.17 40
2005 1 0.83 0.50 Total 11.33 ∑|(et/yt)*100| 10742
MSE 0.31 MAPE 298.3958
33
Table3 ( Fitting Simple Exponential α=0.8 )
Year Month Inflation Fitted α et et² (et/yt)*100
2002 1 1.99 1.99 0.80 0.00 0.00 0
2 1.50 1.99 0.80 -0.49 0.24 33
3 -0.02 1.60 0.80 -1.62 2.62 8090
4 -0.24 0.30 0.80 -0.54 0.30 227
5 0.80 -0.13 0.80 0.93 0.87 116
6 0.36 0.61 0.80 -0.25 0.06 70
7 0.82 0.41 0.80 0.41 0.17 50
8 0.29 0.74 0.80 -0.45 0.20 155
9 0.53 0.38 0.80 0.15 0.02 28
10 0.54 0.50 0.80 0.04 0.00 7
11 1.85 0.53 0.80 1.32 1.74 71
12 1.20 1.59 0.80 -0.39 0.15 32
2003 1 0.80 1.28 0.80 -0.48 0.23 60
2 0.20 0.90 0.80 -0.70 0.48 348
3 -0.23 0.34 0.80 -0.57 0.32 247
4 0.15 -0.12 0.80 0.27 0.07 177
5 0.21 0.10 0.80 0.11 0.01 54
6 0.09 0.19 0.80 -0.10 0.01 108
7 0.03 0.11 0.80 -0.08 0.01 265
8 0.84 0.05 0.80 0.79 0.63 95
9 0.36 0.68 0.80 -0.32 0.10 89
10 0.55 0.42 0.80 0.13 0.02 23
11 1.01 0.52 0.80 0.49 0.24 48
12 0.94 0.91 0.80 0.03 0.00 3
2004 1 0.57 0.93 0.80 -0.36 0.13 64
2 -0.02 0.64 0.80 -0.66 0.44 3315
3 0.36 0.11 0.80 0.25 0.06 69
4 0.97 0.31 0.80 0.66 0.43 68
5 0.88 0.84 0.80 0.04 0.00 5
6 0.48 0.87 0.80 -0.39 0.15 82
7 0.39 0.56 0.80 -0.17 0.03 43
8 0.09 0.42 0.80 -0.33 0.11 371
9 0.02 0.16 0.80 -0.14 0.02 684
10 0.56 0.05 0.80 0.51 0.26 92
11 0.89 0.46 0.80 0.43 0.19 49
12 1.04 0.80 0.80 0.24 0.06 23
2005 1 0.99 0.80 Total 10.37 ∑|(et/yt)*100| 10453
MSE 0.29 MAPE 290.3682
34
Table4 ( Fitting ARRES with excel α=0.1 )
Yea
r
Mont
h
Inflatio
n
Fitte
d et Et
AE
t α β e² (et/yt)*100
200
2 1 1.99 1.99 0.00 0.00
0.0
0
0.1
0 0.10 0.00 0
2 1.50 1.99
-
0.49
-
0.05
1.0
5
0.1
0 0.10 0.24 33
3 -0.02 1.50
-
1.52
-
0.20
1.0
5
0.1
0 0.10 2.31 7600
4 -0.24 -0.02
-
0.22
-
0.20
0.9
2
0.2
2 0.10 0.05 92
5 0.80 -0.24 1.04
-
0.07
1.0
1
0.0
7 0.10 1.08 130
6 0.36 0.80
-
0.44
-
0.11
0.9
4
0.1
2 0.10 0.19 122
7 0.82 0.36 0.46
-
0.05
0.9
5
0.0
6 0.10 0.21 56
8 0.29 0.82
-
0.53
-
0.10
0.9
6
0.1
1 0.10 0.28 183
9 0.53 0.29 0.24
-
0.07
0.9
3
0.0
7 0.10 0.06 45
10 0.54 0.53 0.01
-
0.06
0.9
1
0.0
7 0.10 0.00 2
11 1.85 0.54 1.31 0.08
1.0
4
0.0
7 0.10 1.72 71
12 1.20 1.85
-
0.65 0.00
0.9
6
0.0
0 0.10 0.42 54
200
3 1 0.80 1.20
-
0.40
-
0.04
0.9
4
0.0
4 0.10 0.16 50
2 0.20 0.80
-
0.60
-
0.09
0.9
7
0.1
0 0.10 0.36 300
3 -0.23 0.20
-
0.43
-
0.13
0.9
5
0.1
3 0.10 0.18 187
4 0.15 -0.23 0.38
-
0.08
0.9
4
0.0
8 0.10 0.14 253
5 0.21 0.15 0.06
-
0.06
0.9
1
0.0
7 0.10 0.00 29
6 0.09 0.21
-
0.12
-
0.07
0.9
2
0.0
7 0.10 0.01 133
7 0.03 0.09
-
0.06
-
0.07
0.9
1
0.0
7 0.10 0.00 200
8 0.84 0.03 0.81 0.02
0.9
9
0.0
2 0.10 0.66 96
9 0.36 0.84
-
0.48
-
0.03
0.9
5
0.0
3 0.10 0.23 133
10 0.55 0.36 0.19
-
0.01
0.9
2
0.0
1 0.10 0.04 35
11 1.01 0.55 0.46 0.04
0.9
5
0.0
4 0.10 0.21 46
12 0.94 1.01 - 0.03 0.9 0.0 0.10 0.00 7
35
0.07 1 3
200
4 1 0.57 0.94
-
0.37
-
0.01
0.9
5
0.0
1 0.10 0.14 65
2 -0.02 0.57
-
0.59
-
0.07
0.9
6
0.0
7 0.10 0.35 2950
3 0.36 -0.02 0.38
-
0.02
0.9
4
0.0
3 0.10 0.14 106
4 0.97 0.36 0.61 0.04
0.9
7
0.0
4 0.10 0.37 63
5 0.88 0.97
-
0.09 0.03
0.9
1
0.0
3 0.10 0.01 10
6 0.48 0.88
-
0.40
-
0.02
0.9
5
0.0
2 0.10 0.16 83
7 0.39 0.48
-
0.09
-
0.02
0.9
1
0.0
3 0.10 0.01 23
8 0.09 0.39
-
0.30
-
0.05
0.9
4
0.0
5 0.10 0.09 333
9 0.02 0.09
-
0.07
-
0.05
0.9
1
0.0
6 0.10 0.00 350
10 0.56 0.02 0.54 0.01
0.9
6
0.0
1 0.10 0.29 96
11 0.89 0.56 0.33 0.04
0.9
4
0.0
4 0.10 0.11 37
12 1.04 0.89 0.15 0.05
0.9
2
0.0
5 0.10 0.02 14
200
5 1 1.04
∑e²
10.2
7
∑|(et/yt)*10
0| 9826.6
5
MS
E 0.27 MAPE 272.96
36
Table5 ( Fitting ARRES with excel α=0.5 )
Yea
r
Mont
h
Inflatio
n
Fitte
d et Et
AE
t α β e² (et/yt)*100
200
2 1 1.99 1.99 0.00 0.00
0.0
0
0.1
0 0.10 0.00 0
2 1.50 1.99
-
0.49
-
0.05
1.0
5
0.1
0 0.10 0.24 33
3 -0.02 1.50
-
1.52
-
0.20
1.0
5
0.1
0 0.10 2.31 7600
4 -0.24 -0.02
-
0.22
-
0.20
0.9
2
0.2
2 0.10 0.05 92
5 0.80 -0.24 1.04
-
0.07
1.0
1
0.0
7 0.10 1.08 130
6 0.36 0.80
-
0.44
-
0.11
0.9
4
0.1
2 0.10 0.19 122
7 0.82 0.36 0.46
-
0.05
0.9
5
0.0
6 0.10 0.21 56
8 0.29 0.82
-
0.53
-
0.10
0.9
6
0.1
1 0.10 0.28 183
9 0.53 0.29 0.24
-
0.07
0.9
3
0.0
7 0.10 0.06 45
10 0.54 0.53 0.01
-
0.06
0.9
1
0.0
7 0.10 0.00 2
11 1.85 0.54 1.31 0.08
1.0
4
0.0
7 0.10 1.72 71
12 1.20 1.85
-
0.65 0.00
0.9
6
0.0
0 0.10 0.42 54
200
3 1 0.80 1.20
-
0.40
-
0.04
0.9
4
0.0
4 0.10 0.16 50
2 0.20 0.80
-
0.60
-
0.09
0.9
7
0.1
0 0.10 0.36 300
3 -0.23 0.20
-
0.43
-
0.13
0.9
5
0.1
3 0.10 0.18 187
4 0.15 -0.23 0.38
-
0.08
0.9
4
0.0
8 0.10 0.14 253
5 0.21 0.15 0.06
-
0.06
0.9
1
0.0
7 0.10 0.00 29
6 0.09 0.21
-
0.12
-
0.07
0.9
2
0.0
7 0.10 0.01 133
7 0.03 0.09
-
0.06
-
0.07
0.9
1
0.0
7 0.10 0.00 200
8 0.84 0.03 0.81 0.02
0.9
9
0.0
2 0.10 0.66 96
9 0.36 0.84
-
0.48
-
0.03
0.9
5
0.0
3 0.10 0.23 133
10 0.55 0.36 0.19
-
0.01
0.9
2
0.0
1 0.10 0.04 35
11 1.01 0.55 0.46 0.04
0.9
5
0.0
4 0.10 0.21 46
12 0.94 1.01 - 0.03 0.9 0.0 0.10 0.00 7
37
0.07 1 3
200
4 1 0.57 0.94
-
0.37
-
0.01
0.9
5
0.0
1 0.10 0.14 65
2 -0.02 0.57
-
0.59
-
0.07
0.9
6
0.0
7 0.10 0.35 2950
3 0.36 -0.02 0.38
-
0.02
0.9
4
0.0
3 0.10 0.14 106
4 0.97 0.36 0.61 0.04
0.9
7
0.0
4 0.10 0.37 63
5 0.88 0.97
-
0.09 0.03
0.9
1
0.0
3 0.10 0.01 10
6 0.48 0.88
-
0.40
-
0.02
0.9
5
0.0
2 0.10 0.16 83
7 0.39 0.48
-
0.09
-
0.02
0.9
1
0.0
3 0.10 0.01 23
8 0.09 0.39
-
0.30
-
0.05
0.9
4
0.0
5 0.10 0.09 333
9 0.02 0.09
-
0.07
-
0.05
0.9
1
0.0
6 0.10 0.00 350
10 0.56 0.02 0.54 0.01
0.9
6
0.0
1 0.10 0.29 96
11 0.89 0.56 0.33 0.04
0.9
4
0.0
4 0.10 0.11 37
12 1.04 0.89 0.15 0.05
0.9
2
0.0
5 0.10 0.02 14
200
5 1 1.04
∑e²
10.2
7
∑|(et/yt)*10
0| 9826.6
5
MS
E 0.27 MAPE 272.96
