In this paper, the Box-Jenkins modelling procedure is used to determine an ARIMA model and go further to forecasting. The mobile cellular subscription data for the study were taken from the administrative data submitted to the Zambia Information and Communications Technology Authority (ZICTA) as quarterly returns by all three mobile network operators Airtel Zambia, MTN Zambia and Zamtel. The time series of annual figures for mobile cellular subscription for all mobile network operators is from 2000 to 2014 and has a total of 15 observations. Results show that the ARIMA (1, 2, 1) is an adequate model which best fits the mobile cellular subscription time series and is therefore suitable for forecasting subscription. The model predicts a gradual rise in mobile cellular subscription in the next 5 years, culminating to about 9.0% cumulative increase in 2019.
In Zambia, the penetration of information and communication technology (ICT) in general and mobile in particularly, plays an important role in compilation of the national Gross Domestic Product (GDP). There are three (3) mobile cellular operators in Zambia with networks spanning land area of almost 602,090 Km2, representing 80% network coverage. In 2014, Zambia had 67.1% of subscribers from the mobile cellular subsector with revenue contribution of nearly K3.4 billion. At the end of December 2014 the population of Zambia was estimated at 15.1 million while mobile cellular subscription (MCS) was 10.1 million.
Studies have shown that diffusion of mobile telecommunication affects the growth of GDP. Other studies have also shown that a long run causal relationship exists between growth in telecommunications and the growth of the economy both at sectoral and aggregate levels. Therefore, the importance of investment in telecommunication subsector is acknowledged world over. Globally, socio-economic effect and economic development due to improved telecommunication cannot be repudiated.
Time series modelling is an important part of every field. It provides both short and long term forecasting techniques. Effective implementation of forecasting techniques maximises the prospect of adopting optimum strategies. Literature shows that researchers have used both stochastic and deterministic models to model and forecast telecommunication data. However, stochastic models attributed to Box-Jenkins, the Auto Regressive Integrated Moving Average (ARIMA) models have been found to be more efficient and reliable even for short term forecasting than the deterministic models. Further, stochastic models are distribution-free as no assumptions are required about the data or parameter hence the adoption of the forecasting methodology in this paper.
The MCS data for the study has been taken from the administrative data submitted to the Zambia Information and Communications Technology Authority (ZICTA) as quarterly returns by all three mobile network operators (MNOs). The time series of annual figures for MCS for all MNOs is from 2000 to 2014 and has a total of 15 observations. Each observation (Xt) in the time series is sum total of subscriber for Airtel Zambia, MTN Zambia and Zamtel i.e.
where,
Statistical Analysis System (SAS) [
The Box-Jenkins approach to forecasting was first described by statisticians George Box and Gwilym Jenkins and was developed as a direct result of their experience with forecast problems in the business, economic, and control engineering applications [
where,
Also as
where, at is a white noise process with mean 0 and variance σ2 [
The statistics are used to compare how well models fit the time series. Akaike Information Criterion (AIC) and Schwartz’s Bayesian Information Criterion (SBC) are some of the measures of accuracy of forecast that are widely used in SAS. Other measures used include Mean Squared Error (MSE), Mean Absolute Percentage Error (MAPE) and Mean Absolute Deviation (MAD). Forecast error is given by
AIC and SBC are given by
Other measures of forecast accuracy are given by
In these formulas, L is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, k is the number of estimated parameters and
This is the foremost step of the Box-Jenkins process of time series modelling. A timeplot of the MCS is plotted in
Converting a nonstationary time series to a stationary one through differencing (where needed) is an important part of the process of fitting an ARIMA model.
The model variable and factors are given in
Verification of goodness of fit of any model should include a test as to whether the residuals form a white noise process. Diagnistic check helps determine if an estimated model is statistically adequate. If the identified model passes the diagnostic tests, the model is ready to be used for forecasting. If it does not, the diagnostic tests
ARIMA MODELS | AIC | SBC | RMSE | MAPE | MAD |
---|---|---|---|---|---|
ARIMA (1, 1, 0) | 422.68 | 423.96 | 725,927.03 | 79.94 | 500,531.81 |
ARIMA (1, 1, 1) | 423.06 | 424.98 | 682,365.52 | 89.63 | 444,145.63 |
ARIMA (1, 1, 2) | 425.07 | 427.63 | 679,685.36 | 87.49 | 436,179.08 |
ARIMA (2, 1, 1) | 425.00 | 427.56 | 681,695.68 | 95.37 | 448,809.15 |
ARIMA (1, 2, 0) | 404.26 | 405.39 | 1,086,694.05 | 93.33 | 772,506.46 |
ARIMA (1, 2, 1) | 399.91 | 401.61 | 783,904.98 | 46.22 | 525,306.06 |
ARIMA (2, 2, 1) | 400.35 | 402.61 | 770,165.27 | 87.90 | 530,058.84 |
ARIMA (2, 2, 2) | 406.07 | 408.89 | 937,548.54 | 116.59 | 684,186.40 |
TYPE | Coefficient | SE coefficient | t-statistics | p-value | Lag |
---|---|---|---|---|---|
Constant | 500,796 | 967,576 | 0.52 | 0.6048 | 0 |
MA (1) | −0.99995 | 111.0529 | −0.01 | 0.9928 | 1 |
AR (1) | 0.54647 | 0.2761 | 1.98 | 0.0477 | 1 |
should indicate how the model ought to be modified, and a new cycle of identification, estimation and diagnosis is performed.
The Autocorrelation check for white noise of an ARIMA (1, 2, 1) model in
Box-Jenkins approach to forecasting stationary time series is relatively simple. The forecast value of
To lag | Chi-square | DF | Pr > ChiSq | Autocorrelations | |||||
---|---|---|---|---|---|---|---|---|---|
6 | 5.72 | 6 | 0.4558 | 0.555 | 0.040 | −0.051 | −0.035 | −0.057 | −0.139 |
12 | 18.63 | 12 | 0.0979 | −0.263 | −0.307 | −0.272 | −0.132 | 0.063 | 0.098 |
Forecasts for mobile cellular subscription | ||||
---|---|---|---|---|
Year | Forecast | Std error | 95% confidence limits | |
2015 | 10,321,672 | 905,295 | 8,547,326 | 12,096,019 |
2016 | 10,418,227 | 1,667,167 | 7,150,640 | 13,685,815 |
2017 | 10,650,781 | 2,359,965 | 6,025,333 | 15,276,228 |
2018 | 10,825,202 | 2,979,034 | 4,986,402 | 16,664,002 |
2019 | 11,100,306 | 3532,708 | 4,176,325 | 18,024,287 |
The ARIMA (1, 2, 1) is an adequate model which best fits the mobile cellular subscription time series and is therefore suitable for forecasting subscription. The potential implication of this study is that by developing forecasting models for predicting mobile cellular subscription in advance on a regular basis is to support internal decisions and planning as well as market communication. The subscription forecast baseline in this study uses historical data from Airtel Zambia, MTN Zambia and Zamtel. The study also provides a model to foresee and allocate appropriate resources to maintain a steady increase in mobile cellular subscription.
In this paper, the Box-Jenkins modelling procedure is used to determine an ARIMA model and go further to forecasting. The mobile cellular subscription data for the study were taken from the administrative data submitted to the Zambia Information and Communications Technology Authority (ZICTA) as quarterly returns by all three mobile network operators Airtel Zambia, MTN Zambia and Zamtel. The time series of annual figures for mobile cellular subscription for all mobile network operators is from 2000 to 2014 and has a total of 15 observations. Results show that the ARIMA (1, 2, 1) is an adequate model which best fits the mobile cellular subscription time series and is therefore suitable for forecasting subscription. The model predicts a gradual rise in mobile cellular subscription in the next 5 years, culminating to about 9.0% cumulative increase in 2019.
The authors are thankful to Zambia Information and Communications Technology Authority (ZICTA) for providing the data, Department of Mathematics and Statistics, Mulungushi University for using their resources and all the people who helped in making comments on this paper.
Ian Siluyele,Stanley Jere, (2016) Using Box-Jenkins Models to Forecast Mobile Cellular Subscription. Open Journal of Statistics,06,303-309. doi: 10.4236/ojs.2016.62026