Forecast Modelling (Single Variable)
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Fundamental Of Forecast Modeling

Fundamental Of Forecast Modeling

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  • 1. 1.0 INTRODUCTIONThis chapter discuss about background of study, statement of problem, purpose, objective andsignificance of the study.1.1 BACKGROUND OF THE STUDYThis 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 calculatethe Inflation in economic like we calculate GDP. So Inflation rate is important for thegovernment, academician, consumer also businessman to know economy situation for thecountry. The rational why I am choose this topic is to open our mind and to know about ourneighbour economy. Then, with reference that I have in QMT 463 I can forecast one step aheadwith suitable models.1.2 STATEMENT OF PROBLEMThe main problem in this case, is to choose the best fitted models to generate the forecast forNational Inflation Rate in Indonesia. By this guide it easy for me and other forecaster orresearcher to do this task. 1
  • 2. 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 revisionvii. Initial forecast presentationviii. Final revision ix. Forecast distribution x. Establish monitoring system1.3 PURPOSE OF STUDYThe purpose of this assignment is to identify, choose, calculate the best fitted model for the setdata that I have. This study also explains the related graph which can explain the NationalInflation Rate in Indonesia.1.4 OBJECTIVES OF STUDYThe 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. 2
  • 3. 3
  • 4. 1.5 SIGNIFICANCE OF THE STUDYThe 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 calculatethe Inflation in economic like we calculate GDP. So Inflation rate is important for thegovernment, academician, consumer also businessman to know economy situation for thecountry. The Government, academician, consumer also businessman can use this information tothe industry and society to increase the level of awareness of economy. 4
  • 5. 2.0 METHODOLOGYThis chapter describes the methodology used to carry out the study on the benefit of eggshelltechnology in industries. Only secondary research was used to get the data for the study. One setdata of National Inflation rate in Indonesia obtain through internet research will used as source ofinformation. The data was synthesized and summarized for the report, no primary research wasdone, no interviews were conducted, no questionnaire will distribute and no observations willmake. These are the limitations of the study. 5
  • 6. 3.0 FINDINGS AND DISCUSSIONSI have search through the internet to get the data set which has 36 month data. This data set I getfrom Indonesia Statistic Ministry Web. This data set is called External Data because obtainedoutside 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 setcan 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 fittedmodel. i. Naïve Model ii. Simple exponential smoothing Model iii. Decomposition Method iv. ARRES Method v. Holt-Winters 6
  • 7. Figure1 (National Inflation of Indonesia Graph) National Inflation of Indonesia 2.50 2.00 1.50 Inflation 1.00 0.50 0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 -0.50 Month InflationThe graph in Figure1 shows that the National Inflation of Indonesia from January 2002 toDecember 2004. The highest rate in January 2002 and the lowest data in March 2003. The trendfor the graph is decrease.The table1 in the next page shows that all data set that I get obtain through internet. This data setabout the National Inflation Rate of Indonesia. 7
  • 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 8
  • 9. 3.1 NAIVE MODELNaïve with Trend ModelThe application of this model is fairly common among organizations. One reason for itspopularity is that can be used even with fairly short time series. Thus, overcoming the commonproblem in most organizations where insufficient data is a common phenomenon. Insufficientdata 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 timet, and yt-1 is the actual value in preceding period. This model implies that all future forecast canbe set to the equal the actual observed value in the most recent time period plus the growth ratethat is the trend value as measured by yt/ yt-1. Hence, if yt is greater than yt-1 then the trend isupward 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 sharpincrease in the value will severely affect the forecast. Furthermore, fitting this model type willresult in the loss of the first two observations in the series. On the other hand,this model can alsobe used for short time series.Fitting The Naïve With Trend Model With ExcelTable2 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 the3rd row. Then drag the box until one step ahead. Then, calculate its MSE to compare with othermodel. The forecast value that I get for January 2005 is 1.22 .The MSE show that 24.23 andthe value of MAPE is 8.92 . 9
  • 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 10
  • 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 11
  • 12. 3.2 SIMPLE EXPONENTIAL SMOOTHING MODELSome people call this model Single Exponential Smoothing Technique. But one thing is sure, itis the simplest form of model within the family of the exponential smoothing technique. Themodel requires only one parameter, that is the smoothing constant α to generate the fitted valuesand hence forecast.The advantage of this procedure is that it takes into account the most recent forecast. In SimpleExponential Smoothing Model, the forecast for the next and all subsequent periods aredetermined by adjusting the current period forecast by apportion of the difference between thecurrent 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 thesubsequent forecast on these estimates. Likewise, if recent predictions have been subjected tolarge 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 amountthe data. There is no need to store data for many periods, because the historical profile isrecorded in concise form in the current smoothed statistic.Ft+m = α yt + (1-α)FtThe main thing in simple exponential smoothing is to choose best value of α. The first procedurerelies heavily not only on ones personal knowledge about the problem being evaluated and butalso 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 thenecessary 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 areimportant to generate the forecast value. 12
  • 13. The second procedure that require the application of certain measurement criterion that can beused to determined the best value of α. This is called “error measurement”. Some people called itMean 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 witheach value. This is with objective of choosing the alpha value such that when it applied to themodel it minimizes the error. More specifically, it is to search far an alpha that result I thesmallest error measurement.Fitting The Exponential Using ExcelFirstly 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 ofthe January 2005 one step ahead is 1.22 . After that, calculate error, MSE and MAPE to comparewith other model. 13
  • 14. 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 14
  • 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 36Total 872.12 ∑|(et/yt)*100| 321MSE 24.23 MAPE 8.92 15
  • 16. 3.3 DECOMPOSITION METHODThe process of generating the forecast values using this methodology is basically the reverse ofthe process of decomposing the components. What is done here is to integrate the individualcomponents that have been identified and isolated earlier using past data points in the forecastperiods. This is made on the basis of either one assumptions used when the data were initiallyanalyze. 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, ifthe 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 veryimportant assumption. It is assumed that the patterns or characteristics of the data as exhibited inthe past will be repeated in the future. Even if there is any change, it is not expected to seriouslyaffect the future estimates.To make the job more easier in decomposition method, I have use a simple linear trend for thispurpose which can easily be extrapolated by using excel.WhereT = α + βt 16
  • 17. Figure2 (Linear Trend for National Inflation in Indonesia) Inflation and Trend 2.50 2.00 1.50 Inflation 1.00 0.50 0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 -0.50 Month y = -0.0071x + 0.7099 R2 = 0.0202 Inflation Linear (Inflation)From the graph in figure2, we can see downward trend over 36 month period from January 2002to December 2004. Based on adjusted seasonal indices, it is determined that the highest rate ofinflation in Indonesia is November. The highest being the month of November, recording anindex of 219.32 percent. The lowest rate is in March as evident with lowest index with 9.59percent.Y = -0.0071x + 0.0799From the estimated linear equation, it ca be conclude that over the period time the NationalInflation Rate of Indonesia have been increase at average monthly rate of 0.0799. 17
  • 18. Table4 (Decomposition Method)Year Month t Inflation Moving Total Centered MT C.M.Average2002 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 0.552003 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 0.562004 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 1.042005 1 37 -2.09 18
  • 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 19
  • 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 90.31 105.54 255.27 192.59Adj 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 20
  • 21. Inflation,Deseasonalised data and Linear Trend 2.50 2.00 1.50 1.00 0.50 0.00 Inflation -0.50 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 -1.00 Figure3 (Inflation, Deseasonalized Data and Linear Trend) -1.50 Inflation -2.00 Linear Trend -2.50 Deseasonalised Data Month21
  • 22. 3.4 ADDAPTIVE RESPONSE RATE EXPONENTIAL SMOOTHING (ARRES)ARRES is different from other exponential method. It is because, other exponential methoddiscussed 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 ofthese events have been described earlier. For example, people may change their desire to buy acertain 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 realisticdecision. Thus, the development of ARRES is an attempt to overcome this problem byincorporating the effect of the changing pattern of the data series.Ft+1 = αt yt + (1-αt) FtThis indicates that the value of alpha is only appropriate at a particular period t, and maybedifferent at different value of t.As in any exponential smoothing technique, the appropriate initial values are required to start thealgorithm. In this case, value are for F0, α0, E0 and AET0.Fitting ARRES With ExcelFirstly set up the data in the Table6,the make assumption alpha and beta with certain numberbetween 1 and 0. then in the fitted column write the equation =$H$2*C2+(1-$H$2)*C2 then dragthe box to the down. After that, calculate the MSE and retest the alpha and beta which havesmallest MSE. 22
  • 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 23
  • 24. AEt α β e² (et/yt)*1000.00 0.90 0.10 0.00 01.05 0.90 0.10 0.24 331.05 0.90 0.10 2.31 76000.92 0.22 0.10 0.05 921.01 0.07 0.10 1.08 1300.94 0.12 0.10 0.19 1220.95 0.06 0.10 0.21 560.96 0.11 0.10 0.28 1830.93 0.07 0.10 0.06 450.91 0.07 0.10 0.00 21.04 0.07 0.10 1.72 710.96 0.00 0.10 0.42 540.94 0.04 0.10 0.16 500.97 0.10 0.10 0.36 3000.95 0.13 0.10 0.18 1870.94 0.08 0.10 0.14 2530.91 0.07 0.10 0.00 290.92 0.07 0.10 0.01 1330.91 0.07 0.10 0.00 2000.99 0.02 0.10 0.66 960.95 0.03 0.10 0.23 1330.92 0.01 0.10 0.04 350.95 0.04 0.10 0.21 460.91 0.03 0.10 0.00 70.95 0.01 0.10 0.14 650.96 0.07 0.10 0.35 29500.94 0.03 0.10 0.14 1060.97 0.04 0.10 0.37 630.91 0.03 0.10 0.01 100.95 0.02 0.10 0.16 830.91 0.03 0.10 0.01 230.94 0.05 0.10 0.09 3330.91 0.06 0.10 0.00 3500.96 0.01 0.10 0.29 960.94 0.04 0.10 0.11 370.92 0.05 0.10 0.02 14 ∑e² 10.27 ∑|(et/yt)*100| 9826.65 MSE 0.27 MAPE 272.96 24
  • 25. 3.5 HOLT-WINTER’S METHODAll earlier exponential models are good as long as they deal with non seasonal data. Whenseasonality exists, a more suitable model is needed. Holt-Winters is one such technique that takesinto account the trend and seasonality factors.Fitting Holt-Winters Using ExcelHolt-Winters consist of three basic equation that define the level component, the trendcomponent and the seasonality component. Two assumption can be made with regard to therelationship of these component.Level ComponentLt = α ( yt / st-s ) + ( 1-α ) ( Lt-1 + bt-1 )Trend Componentbt = β ( Lt Lt-1 ) + ( 1-β ) bt-1Seasonal ComponentSt = γ (yt / Lt ) + (1-γ) St-sThe forecastFt+m = (Lt + bt * m) St-s+mAs usual, when fitting the model, some initial value are required. For ease of computation, somesimple technique will discuss here.Determine the Initial ValueTo determine the initial value, a simple procedure used to take the average of the first 12 quarters(month). 25
  • 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 ratioof the actual values to the mean of the first 12 values as represent by Lo in whichSt = Yt / Lt 26
  • 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 27
  • 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 28
  • 29. 4.0 CONCLUSSION1. Where you need to find the initial value of the models that you have used. Why wasthe method ?Determine the Initial ValueTo 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 ratioof the actual values to the mean of the first 12 values as represent by Lo in whichSt = Yt / Lt2. You are to find the best fitted model. In the other words to find the best parametervalue.The smallest MSE among the model is ARRES which its MSE = 0.27 . So, the best parametervalue 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 maindisadvantage of this measure lies in its relevancy as it is valid only for ratio scale data ( data withmeaning full zero ) . For this reason , MAPE is potentially explosive for large forecast errorwhen the actual value observation close to the zero. In addition, percentage measure do not treaterrors of overestimate and underestimate. 29
  • 30. 3. Present the result of your analysis. Which of the model do you think would perfomthe 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 1.86From 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 methoddiscussed 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 ofthese events have been described earlier. For example, people may change their desire to buy acertain 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, costof production, price elasticity and other related factor.In these situation, to maintain the same value for alpha for all time periods may not be realisticdecision. Thus, the development of ARRES is an attempt to overcome this problem byincorporating the effect of the changing pattern of the data series. 30
  • 31. 5.0 APPENDICESTable1 ( 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 31
  • 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 32
  • 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 33
  • 34. Table4 ( Fitting ARRES with excel α=0.1 )Yea Mont Inflatio Fitte AE r h n d et Et t α β e² (et/yt)*100200 0.0 0.1 2 1 1.99 1.99 0.00 0.00 0 0 0.10 0.00 0 - - 1.0 0.1 2 1.50 1.99 0.49 0.05 5 0 0.10 0.24 33 - - 1.0 0.1 3 -0.02 1.50 1.52 0.20 5 0 0.10 2.31 7600 - - 0.9 0.2 4 -0.24 -0.02 0.22 0.20 2 2 0.10 0.05 92 - 1.0 0.0 5 0.80 -0.24 1.04 0.07 1 7 0.10 1.08 130 - - 0.9 0.1 6 0.36 0.80 0.44 0.11 4 2 0.10 0.19 122 - 0.9 0.0 7 0.82 0.36 0.46 0.05 5 6 0.10 0.21 56 - - 0.9 0.1 8 0.29 0.82 0.53 0.10 6 1 0.10 0.28 183 - 0.9 0.0 9 0.53 0.29 0.24 0.07 3 7 0.10 0.06 45 - 0.9 0.0 10 0.54 0.53 0.01 0.06 1 7 0.10 0.00 2 1.0 0.0 11 1.85 0.54 1.31 0.08 4 7 0.10 1.72 71 - 0.9 0.0 12 1.20 1.85 0.65 0.00 6 0 0.10 0.42 54200 - - 0.9 0.0 3 1 0.80 1.20 0.40 0.04 4 4 0.10 0.16 50 - - 0.9 0.1 2 0.20 0.80 0.60 0.09 7 0 0.10 0.36 300 - - 0.9 0.1 3 -0.23 0.20 0.43 0.13 5 3 0.10 0.18 187 - 0.9 0.0 4 0.15 -0.23 0.38 0.08 4 8 0.10 0.14 253 - 0.9 0.0 5 0.21 0.15 0.06 0.06 1 7 0.10 0.00 29 - - 0.9 0.0 6 0.09 0.21 0.12 0.07 2 7 0.10 0.01 133 - - 0.9 0.0 7 0.03 0.09 0.06 0.07 1 7 0.10 0.00 200 0.9 0.0 8 0.84 0.03 0.81 0.02 9 2 0.10 0.66 96 - - 0.9 0.0 9 0.36 0.84 0.48 0.03 5 3 0.10 0.23 133 - 0.9 0.0 10 0.55 0.36 0.19 0.01 2 1 0.10 0.04 35 0.9 0.0 11 1.01 0.55 0.46 0.04 5 4 0.10 0.21 46 12 0.94 1.01 - 0.03 0.9 0.0 0.10 0.00 7 34
  • 35. 0.07 1 3200 - - 0.9 0.0 4 1 0.57 0.94 0.37 0.01 5 1 0.10 0.14 65 - - 0.9 0.0 2 -0.02 0.57 0.59 0.07 6 7 0.10 0.35 2950 - 0.9 0.0 3 0.36 -0.02 0.38 0.02 4 3 0.10 0.14 106 0.9 0.0 4 0.97 0.36 0.61 0.04 7 4 0.10 0.37 63 - 0.9 0.0 5 0.88 0.97 0.09 0.03 1 3 0.10 0.01 10 - - 0.9 0.0 6 0.48 0.88 0.40 0.02 5 2 0.10 0.16 83 - - 0.9 0.0 7 0.39 0.48 0.09 0.02 1 3 0.10 0.01 23 - - 0.9 0.0 8 0.09 0.39 0.30 0.05 4 5 0.10 0.09 333 - - 0.9 0.0 9 0.02 0.09 0.07 0.05 1 6 0.10 0.00 350 0.9 0.0 10 0.56 0.02 0.54 0.01 6 1 0.10 0.29 96 0.9 0.0 11 0.89 0.56 0.33 0.04 4 4 0.10 0.11 37 0.9 0.0 12 1.04 0.89 0.15 0.05 2 5 0.10 0.02 14200 10.2 ∑|(et/yt)*10 9826.6 5 1 1.04 ∑e² 7 0| 5 MS E 0.27 MAPE 272.96 35
  • 36. Table5 ( Fitting ARRES with excel α=0.5 )Yea Mont Inflatio Fitte AE r h n d et Et t α β e² (et/yt)*100200 0.0 0.1 2 1 1.99 1.99 0.00 0.00 0 0 0.10 0.00 0 - - 1.0 0.1 2 1.50 1.99 0.49 0.05 5 0 0.10 0.24 33 - - 1.0 0.1 3 -0.02 1.50 1.52 0.20 5 0 0.10 2.31 7600 - - 0.9 0.2 4 -0.24 -0.02 0.22 0.20 2 2 0.10 0.05 92 - 1.0 0.0 5 0.80 -0.24 1.04 0.07 1 7 0.10 1.08 130 - - 0.9 0.1 6 0.36 0.80 0.44 0.11 4 2 0.10 0.19 122 - 0.9 0.0 7 0.82 0.36 0.46 0.05 5 6 0.10 0.21 56 - - 0.9 0.1 8 0.29 0.82 0.53 0.10 6 1 0.10 0.28 183 - 0.9 0.0 9 0.53 0.29 0.24 0.07 3 7 0.10 0.06 45 - 0.9 0.0 10 0.54 0.53 0.01 0.06 1 7 0.10 0.00 2 1.0 0.0 11 1.85 0.54 1.31 0.08 4 7 0.10 1.72 71 - 0.9 0.0 12 1.20 1.85 0.65 0.00 6 0 0.10 0.42 54200 - - 0.9 0.0 3 1 0.80 1.20 0.40 0.04 4 4 0.10 0.16 50 - - 0.9 0.1 2 0.20 0.80 0.60 0.09 7 0 0.10 0.36 300 - - 0.9 0.1 3 -0.23 0.20 0.43 0.13 5 3 0.10 0.18 187 - 0.9 0.0 4 0.15 -0.23 0.38 0.08 4 8 0.10 0.14 253 - 0.9 0.0 5 0.21 0.15 0.06 0.06 1 7 0.10 0.00 29 - - 0.9 0.0 6 0.09 0.21 0.12 0.07 2 7 0.10 0.01 133 - - 0.9 0.0 7 0.03 0.09 0.06 0.07 1 7 0.10 0.00 200 0.9 0.0 8 0.84 0.03 0.81 0.02 9 2 0.10 0.66 96 - - 0.9 0.0 9 0.36 0.84 0.48 0.03 5 3 0.10 0.23 133 - 0.9 0.0 10 0.55 0.36 0.19 0.01 2 1 0.10 0.04 35 0.9 0.0 11 1.01 0.55 0.46 0.04 5 4 0.10 0.21 46 12 0.94 1.01 - 0.03 0.9 0.0 0.10 0.00 7 36
  • 37. 0.07 1 3200 - - 0.9 0.0 4 1 0.57 0.94 0.37 0.01 5 1 0.10 0.14 65 - - 0.9 0.0 2 -0.02 0.57 0.59 0.07 6 7 0.10 0.35 2950 - 0.9 0.0 3 0.36 -0.02 0.38 0.02 4 3 0.10 0.14 106 0.9 0.0 4 0.97 0.36 0.61 0.04 7 4 0.10 0.37 63 - 0.9 0.0 5 0.88 0.97 0.09 0.03 1 3 0.10 0.01 10 - - 0.9 0.0 6 0.48 0.88 0.40 0.02 5 2 0.10 0.16 83 - - 0.9 0.0 7 0.39 0.48 0.09 0.02 1 3 0.10 0.01 23 - - 0.9 0.0 8 0.09 0.39 0.30 0.05 4 5 0.10 0.09 333 - - 0.9 0.0 9 0.02 0.09 0.07 0.05 1 6 0.10 0.00 350 0.9 0.0 10 0.56 0.02 0.54 0.01 6 1 0.10 0.29 96 0.9 0.0 11 0.89 0.56 0.33 0.04 4 4 0.10 0.11 37 0.9 0.0 12 1.04 0.89 0.15 0.05 2 5 0.10 0.02 14200 10.2 ∑|(et/yt)*10 9826.6 5 1 1.04 ∑e² 7 0| 5 MS E 0.27 MAPE 272.96 37