Open Journal of Saisics, 26, 6, 33-39 Published Online April 26 in SciRes. hp://www.scirp.org/journal/ojs hp://dx.doi.org/.4236/ojs.26.6226 Using Box-Jenkins Models o Forecas Mobile Cellular Subscripion Ian Siluyele, Sanley Jere 2 Zambia Informaion and Communicaions Technology Auhoriy (ZICTA), Lusaka, Zambia 2 Deparmen of Mahemaics and Saisics, Mulungushi Universiy, Kabwe, Zambia Received 29 November 25; acceped 23 April 26; published 26 April 26 Copyrigh 26 by auhors and Scienific Research Publishing Inc. This work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY). hp://creaivecommons.org/licenses/by/4./ Absrac In his paper, he Box-Jenkins modelling procedure is used o deermine an ARIMA model and go furher o forecasing. The mobile cellular subscripion daa for he sudy were aken from he adminisraive daa submied o he Zambia Informaion and Communicaions Technology Auhoriy (ZICTA) as quarerly reurns by all hree mobile nework operaors Airel Zambia, MTN Zambia and Zamel. The ime series of annual figures for mobile cellular subscripion for all mobile nework operaors is from 2 o 24 and has a oal of 5 observaions. Resuls show ha he ARIMA (, 2, ) is an adequae model which bes fis he mobile cellular subscripion ime series and is herefore suiable for forecasing subscripion. The model predics a gradual rise in mobile cellular subscripion in he nex 5 years, culminaing o abou 9.% cumulaive increase in 29. Keywords Mobile Cellular Subscripion, Box-Jenkins Mehodology, ARIMA Model, Auocorrelaion Funcion, Parial Auocorrelaion Funcion. Inroducion In Zambia, he peneraion of informaion and communicaion echnology (ICT) in general and mobile in paricularly, plays an imporan role in compilaion of he naional Gross Domesic Produc (GDP). There are hree (3) mobile cellular operaors in Zambia wih neworks spanning land area of almos 62,9 Km 2, represening 8% nework coverage. In 24, Zambia had 67.% of subscribers from he mobile cellular subsecor wih revenue conribuion of nearly K3.4 billion. A he end of December 24 he populaion of Zambia was esimaed a 5. million while mobile cellular subscripion (MCS) was. million. Sudies have shown ha diffusion of mobile elecommunicaion affecs he growh of GDP. Oher sudies have also shown ha a long run causal relaionship exiss beween growh in elecommunicaions and he growh How o cie his paper: Siluyele, I. and Jere, S. (26) Using Box-Jenkins Models o Forecas Mobile Cellular Subscripion. Open Journal of Saisics, 6, 33-39. hp://dx.doi.org/.4236/ojs.26.6226
of he economy boh a secoral and aggregae levels. Therefore, he imporance of invesmen in elecommunicaion subsecor is acknowledged world over. Globally, socio-economic effec and economic developmen due o improved elecommunicaion canno be repudiaed. Time series modelling is an imporan par of every field. I provides boh shor and long erm forecasing echniques. Effecive implemenaion of forecasing echniques maximises he prospec of adoping opimum sraegies. Lieraure shows ha researchers have used boh sochasic and deerminisic models o model and forecas elecommunicaion daa. However, sochasic models aribued o Box-Jenkins, he Auo Regressive Inegraed Moving Average (ARIMA) models have been found o be more efficien and reliable even for shor erm forecasing han he deerminisic models. Furher, sochasic models are disribuion-free as no assumpions are required abou he daa or parameer hence he adopion of he forecasing mehodology in his paper. 2. Mehod and Maerials The MCS daa for he sudy has been aken from he adminisraive daa submied o he Zambia Informaion and Communicaions Technology Auhoriy (ZICTA) as quarerly reurns by all hree mobile nework operaors (MNOs). The ime series of annual figures for MCS for all MNOs is from 2 o 24 and has a oal of 5 observaions. Each observaion (X ) in he ime series is sum oal of subscriber for Airel Zambia, MTN Zambia and Zamel i.e. X = α + β + γ = 2,, 24 where, α = number of subscribers for Airel Zambia, β = number of subscribers for MTN Zambia, and γ = number of subscribers for Zamel. Saisical Analysis Sysem (SAS) [] and Microsof Excel will be used o implemen he sochasic models and graphical represenaions respecively. 3. Sochasic Modelling The Box-Jenkins approach o forecasing was firs described by saisicians George Box and Gwilym Jenkins and was developed as a direc resul of heir experience wih forecas problems in he business, economic, and conrol engineering applicaions [2]. The Box-Jenkins mehodology is a sysemaic process which is implemened by using an ieraive process unil an adequae model is achieved. The procedure is achieved by a sep-by-sep process of model IDENTIFICATION, SPECIFICATION, ESTIMATION, DIAGNOSTIC and FORECAST. The ARIMA has hree parameers viz. auoregressive (p), differencing order (d) and he order of moving average (q). The generic Box-Jenkin models are denoed by ARIMA (p, d, q) given by ( )( µ ) θ( ) B X = B a where,, θ and a are auoregressive parameer, moving average parameer and residual respecively. The residuals are assumed o be iid Normal. Using he backshif operaor/ransformaion he equaion above, when d =, is expressed as Also as ( p) ( p) ( q = µ + θ θ ) B B X B B B B a. p p q θ X = Consan + ( B) a ( B) where, a is a whie noise process wih mean and variance σ 2 [3]. 4. Measures of Forecas Accuracy The saisics are used o compare how well models fi he ime series. Akaike Informaion Crierion (AIC) and Independen and idenically disribued. 34
Schwarz s Bayesian Informaion Crierion (SBC) are some of he measures of accuracy of forecas ha are widely used in SAS. Oher measures used include Mean Squared Error (MSE), Mean Absolue Percenage Error (MAPE) and Mean Absolue Deviaion (MAD). Forecas error is given by AIC and SBC are given by ( FE ) ( O ) ( F ) Forecas Error = Observed Value Forecas Value. = ( ( ) ) and SBC 2ln ( L) kln ( n) AIC 2 ln L k Oher measures of forecas accuracy are given by ( ) 2 n MSE = O F, n FE n O MAPE = n = +. FE and MAD =. n In hese formulas, L is he value of he likelihood funcion evaluaed a he parameer esimaes, n is he number of observaions, k is he number of esimaed parameers and =, 2,, n. 5. Idenificaion This is he foremos sep of he Box-Jenkins process of ime series modelling. A imeplo of he MCS is ploed in Figure (a) and checked for saionariy and inveribiliy using visual display of he ACF 2 and PACF 3 graphs. Figure (a) show ha he MCS ime series is no sable and herefore nonsaionary. The nonsaionary behaviour is confirmed by he ACF and PACF plos in Figure 2(a) and Figure 2(b) below. Therefore some sor of ransformaion of he series is necessary o make i mean and variance saionary 4. ARIMA models are designed o model saionary ime series. Convering a nonsaionary ime series o a saionary one hrough differencing (where needed) is an imporan par of he process of fiing an ARIMA model. Figure (b) and Figure (c) shows firs order and second order differenced MCS series, respecively. The ACF and PACF plos are shown in Figure 2(c), Figure 2(d), Figure 2(e) and Figure 2(f) below. ACF and PACF plos indicae ha he firs and second differenced MCS series are saionary hence require furher examinaion o esablish he mos suiable ransformaion for he MCS series. Table shows he deails of various ARIMA models along he forecas accuracy measures. An ARIMA model wih leas measures of accuracy paricularly he AIC and SBC is considered an efficien model for predicion. Therefore, for MCS ime series, he ARIMA (, 2, ) is an adequae (bes fi) model because i has he lowes values for AIC and SBC saisics. 6. Parameer Esimaion Table 2 shows he esimaed parameers and he associaed p-values a 5% level of significance. Only he auoregressive parameer is significanly differen from zero a 5% implying ha he consan and he parameer for he moving average coefficiens have lile or no effec on he model. The model variable and facors are given in Table 2. Hence, he mahemaical form of he ARIMA (, 2, ) is 2 a ( B) X =..54647B 7. Diagnisic Check ( ) Verificaion of goodness of fi of any model should include a es as o wheher he residuals form a whie noise process. Diagnisic check helps deermine if an esimaed model is saisically adequae. If he idenified model passes he diagnosic ess, he model is ready o be used for forecasing. If i does no, he diagnosic ess 2 Auocorrelaion funcion. 3 Parial auocorrelaion funcion. 4 A saionary ime series is one whose saisical properies such as mean, variance, auocorrelaion, ec. are all consan over ime. 35
MCS ' 2 8 6 4 2 Time Plo of MCS Series 2 2 4 6 8 2 4 Time (Year) MCS' Plo of Firs Differenced MCS Series 3 25 2 5 5-5 2 2 4 6 8 2 4 - Time (Year) (a) (b) Plo of Second Differenced MCS Series 3 2-2 2 4 6 8 2 4-2 -3 Time (Year) (c) Figure. Time plos for d =, d = and d = 2 of MCS series. Table. Measures of accuracy for seleced ARIMA models. ARIMA MODELS AIC SBC RMSE MAPE MAD ARIMA (,, ) 422.68 423.96 725,927.3 79.94 5,53.8 ARIMA (,, ) 423.6 424.98 682,365.52 89.63 444,45.63 ARIMA (,, 2) 425.7 427.63 679,685.36 87.49 436,79.8 ARIMA (2,, ) 425. 427.56 68,695.68 95.37 448,89.5 ARIMA (, 2, ) 44.26 45.39,86,694.5 93.33 772,56.46 ARIMA (, 2, ) 399.9 4.6 783,94.98 46.22 525,36.6 ARIMA (2, 2, ) 4.35 42.6 77,65.27 87.9 53,58.84 ARIMA (2, 2, 2) 46.7 48.89 937,548.54 6.59 684,86.4 Table 2. Esimaed parameer and significance ess. TYPE Coefficien SE coefficien -saisics p-value Lag Consan 5,796 967,576.52.648 MA ().99995.529..9928 AR ().54647.276.98.477 36
ACF When d = PACF When d =.8.8.6.6.4.4.2.2 -.2 2 3 4 5 6 7 8 9 2 3 -.2 2 3 4 5 6 7 8 9 2 -.4 -.4 -.6 -.6 -.8 -.8 - (a) - (b) ACF When d =.8 PACF When d =.8.6.4.2.6.4.2 -.2 2 3 4 5 6 7 8 9 2 3 -.2 2 3 4 5 6 7 8 9 2 -.4 -.6 -.8 -.4 -.6 - (c) -.8 (d) ACF When d = 2.8 PACF When d = 2.8.6.4.6.4.2.2 -.2 -.4 -.6 -.8 2 3 4 5 6 7 8 9 2 3 -.2 -.4 -.6 2 3 4 5 6 7 8 9 2 - -.8 (e) Figure 2. The ACF and PACF plos for d =, d = and d = 2 of MCS series. (f) 37
should indicae how he model ough o be modified, and a new cycle of idenificaion, esimaion and diagnosis is performed. The Auocorrelaion check for whie noise of an ARIMA (, 2, ) model in Table 4 p-values a 5% level of significance as shown above indicaes ha he model is good because he residuals are a whie noise. 8. Forecasing Box-Jenkins approach o forecasing saionary ime series is relaively simple. The forecas value of X + k given all observaions up unil n he k-sep ahead forecas is denoed by xˆ ( k ). Table 3 shows five year forecass for mobile cellular subscripion using ARIMA (, 2, ). The rajecory of he forecass from 25 o 29 is shown in Figure 3. Table 3. Auocorrelaion check for whie noise. To lag Chi-square DF Pr > ChiSq Auocorrelaions 6 5.72 6.4558.555.4.5.35.57.39 2 8.63 2.979.263.37.272.32.63.98 Table 4. Forecass for ARIMA (, 2, ) for mobile cellular subscripion. Forecass for mobile cellular subscripion Year Forecas Sd error 95% confidence limis 25,32,672 95,295 8,547,326 2,96,9 26,48,227,667,67 7,5,64 3,685,85 27,65,78 2,359,965 6,25,333 5,276,228 28,825,22 2,979,34 4,986,42 6,664,2 29,,36 3532,78 4,76,325 8,24,287 4 2 MCS (Thousands) 8 6 4 2 2 2 22 23 24 25 26 27 28 29 2 2 22 23 24 25 26 27 28 29 Acual (MNO Subscripion) Fied (Forecas) Figure 3. Forecasing wih ARIMA (, 2, ) for mobile cellular subscripion. 38
9. Discussion The ARIMA (, 2, ) is an adequae model which bes fis he mobile cellular subscripion ime series and is herefore suiable for forecasing subscripion. The poenial implicaion of his sudy is ha by developing forecasing models for predicing mobile cellular subscripion in advance on a regular basis is o suppor inernal decisions and planning as well as marke communicaion. The subscripion forecas baseline in his sudy uses hisorical daa from Airel Zambia, MTN Zambia and Zamel. The sudy also provides a model o foresee and allocae appropriae resources o mainain a seady increase in mobile cellular subscripion.. Conclusion In his paper, he Box-Jenkins modelling procedure is used o deermine an ARIMA model and go furher o forecasing. The mobile cellular subscripion daa for he sudy were aken from he adminisraive daa submied o he Zambia Informaion and Communicaions Technology Auhoriy (ZICTA) as quarerly reurns by all hree mobile nework operaors Airel Zambia, MTN Zambia and Zamel. The ime series of annual figures for mobile cellular subscripion for all mobile nework operaors is from 2 o 24 and has a oal of 5 observaions. Resuls show ha he ARIMA (, 2, ) is an adequae model which bes fis he mobile cellular subscripion ime series and is herefore suiable for forecasing subscripion. The model predics a gradual rise in mobile cellular subscripion in he nex 5 years, culminaing o abou 9.% cumulaive increase in 29. Acknowledgemens The auhors are hankful o Zambia Informaion and Communicaions Technology Auhoriy (ZICTA) for providing he daa, Deparmen of Mahemaics and Saisics, Mulungushi Universiy for using heir resources and all he people who helped in making commens on his paper. References [] SAS Insiue Inc. (24) SAS/ETS 3.2 User s Guide: The ARIMA Procedure. SAS Insiue Inc., Cary. [2] Box, G.E. and Jenkins, G.M. (994) Time Series Analysis: Forecasing and Conrol. 3rd Ediion, Prenice Hall, Englewood Cliffs. [3] Wei, W. (99) Time Series Analysis: Univariae and Mulivariae Mehods. Addison-Wesley Publishing Company, Inc., New York. 39