Forecasting Annual International Tourist Arrivals in Zambia Using Holt-Winters Exponential Smoothing

Tourism is one of the major contributors to foreign exchange earnings to Zambia and world economy. Annual International tourist arrivals in Zambia from 1995 to 2014 are considered in this paper. In this study we evaluated the model performance of Auto-Regressive Integrated Moving Average (ARIMA) and Holt Winters exponential smoothing (HWES). The error indicators: Mean percentage error (MPE), Mean absolute error (MAE), Mean absolute scaled error (MASE), Root-mean-square error (RMSE) and Mean absolute percentage error (MAPE) showed that HWES is an appropriate model with reasonable forecast accuracy. The HWES (α = 1, β = 0.1246865) showed smallest error than those of the ARIMA (0, 1, 2) models. Hence, the HWES (α = 1, β = 0.1246865) can be used to model annual international tourist arrivals in Zambia. Further, forecasting results give a gradual increase in annual international tourist arrivals of about 42% by 2024. Accurate forecasts are key to new investors and Policymakers. The Zambian government should use such forecasts in formulating policies and making strategies that will promote the tourism industry.


Introduction
Tourism is one of the major contributors to foreign exchange earnings for Zambia and other countries worldwide. According to [1] an international tourist (overnight visitor) is an individual who travels to a country other than that in which they reside for a period not exceeding 12 months and whose main purpose in visiting is other than an remunerated activity within the country visited.
Data collection methods for arrivals vary from one country to another. In some countries data are from border statistics. In other countries data are from tourism accommodation establishments. Tourist data refer to the number of arrivals, not to the number of people traveling. Thus a person who makes several travels to a country during a given period is counted each time as a new arrival. Tourism contributes highly to GDP, increasing the employment rate, source of revenue for local people, private sector, public sectors and government [2]. The significance of tourism has encouraged the authors to study the number of international tourist arrivals and attempt to make more accurate forecasting for future planning.
Zambia's tourist attraction includes 20 National Parks and 34 Game Management Areas (GMAs) with a total of 23 million hectares of land devoted to spectacular wildlife. Zambia has a rich array of traditional cultural festivities and events, including: Kuomboka Ceremony, Nc'wala Ceremony, Umutomboko Ceremony and LikumbiLya Mize Ceremony. One of the Seven Natural Wonders of the World is the Victoria Falls. The Falls plunge into the Zambezi River at about 550,000 cubic meters per second. The impact is so big such that falling water raises a cloud of vapor that can be seen more than 30 kilometers away.

Literature Review
Studies by [2]  and declines in the second period but experience seasonal fluctuations in the third period. Studies by [5] produced forecasts of international tourist arrival to Thailand during 2006-2010 using two methods: Structural and Trend Extrapolation Models. The Structural Model involved VAR model, GMM method, ARCH-GARCH method, ARCH-GARCH-M method, TARCH method, EGARCH method and PARCH method. The Trend Extrapolation Model involved Holt-winter method, ARIMA method, SARIMA method and Neural Network method. Structural Models gave SARIMA (0, 1, 1) (0, 1, 4) method as the best because of its low MAPE value. Trend Extrapolation Models gave VAR method because of its low MAPE value. SARIMA (0, 1, 1) (0, 1, 4) and VAR method predicted 15700656.00 million and 15985416.00 million in 2010 respectively. It was also concluded that the Thailand government tourism sector and private tourism industry sector should prepare adequately for a much more increase in number of international tourism arrival to Thailand during 2006-2010. They suggested an increase in the number of hotel, the number transportation, new tourism place, more unit of tourism polices, much more problem environment impact on tourism place, airport unit, budget for developing new tourism places and human training in tourism industry.
According to [6], tourism is a key sector and contributes significantly to foreign exchange earnings. Earnings from tourism in Kenya increased annually from Kenya Shillings 24. 3  Results in this paper also show that in order to improve tourism, a model that can give accurate forecast results is required so that hotel industry players can respond in good time to the anticipated changes in demand over time and also maximize returns on investments. The authors used the Box-Jenkins models to generate a forecasting model using quarterly data on bed occupancy by tourists visiting Kenya from 1974 to 2011. The SARIMA (1, 1, 2) (1, 1, 1) [5] was the best fit model for forecasting future quarterly demand on tourist accommodation in Kenya. They further concluded that this model should be used in forecasting future demands and maximize their returns on investment.
The study of [7] used a number of time series models of tourist arrivals and ARIMA (2, 2, 2) model was the best fit than logistic model. The models were ca- And therefore its application is limited to forecasting arrivals for businesses and government in the event that there are no substantial changes in the current environment. They also suggested for further research to include econometric forecasting techniques in order to address the critique above as well as the application of the current method to other sectors of tourism industry such as the accommodation industry.

Holt-Winters Exponential Smoothing Model (HWES)
where, α is the smoothing constant, β is the trend smoothing constants, t Y is raw data, t L is smoothed data and t T is the trend estimates.
The h-step-ahead forecast equation is ˆt h t t Y L hT + = + [9].
The main reason of choosing HWES model in this study is because Holt-Winters exponential smoothing technique can be used to forecast data containing trend.

Autoregressive Integrated Moving Average Model ARIMA
ARIMA models known as Box-Jenkins methodology have been found to be more popular, efficient and reliable even for short term forecasting. The ARIMA model consists of the following components called the order of autoregressive (AR) model (p), differencing order (d) and the order of moving average (MA) model (q). The Box-Jenkin models are denoted by ARIMA (p, d, q). "I" implies that the process needs to undergo differencing and when the modelling is done, the results undergo an integration process to produce forecasts and estimates. The MA, AR and ARMA are defined as follows: AR model: The combination of AR and MA gives ARMA model: where t φ is the autoregressive parameter at time t, t ε is the error term at time t and t θ is the moving-average parameter at time t [9].
The main reason of choosing ARIMA model in this study for the forecasting is because this model assumes and takes into account the non-zero autocorrelation between the successive values of the time series data.

The Error Measures for Model-Selection
There are several ways to evaluate forecasting models. The error indicators are the most used to compare how well models fit the time series. The best fit or forecasting model is one with minimal errors [9]. Forecast accuracy is measured Open Journal of Statistics by the difference between actual value and the forecasted value at time period t.
The error indicators considered in this paper are MPE, MAE, MASE, RMSE and MAPE defined as follows in Table 1.

Results and Discussion
Annual International tourist arrivals in Zambia from 1995 to 2014 are shown (see, Table 2). The time series plot (see Figure 1) shows that the Zambian annual international tourist arrivals is non-stationary for d = 0 and stationary for d = 1.
The Holt-Winters exponential smoothing (HWES) and autoregressive integrated moving average (ARIMA) models are compared to determine the fore-  Table 3).
For ARIMA model, the procedure is achieved by considering the following     Figure 3. Figure 3 shows that the model satisfies all required tests for a suitable model for Zambia's tourist data.
The results in Table 3 show that HWES (α = 1, β = 0.1246865) model performed better than the ARIMA (0, 1, 2) on tourist arrivals data for Zambia on account of smaller measures of accuracy. Hence, HWES (α = 1, β = 0.1246865) model was selected for forecasting Zambia tourist arrivals. Table 4 shows the forecast of Zambia tourist arrivals using the HWES (α = 1, β = 0.1246865). Ten step forecasts up to 2024 are reported with 80% and 95% confidence limits. Forecasting results show a gradual increase in annual international tourist arrivals of about 42% by 2024 (see, Figure 4).

Conclusion
Two models of univariate time-series analysis were considered in this study: HWES and ARIMA models. The best fit of the two models used in this study was picked based on the model indicating minimum errors. The HWES (α = 1, β = 0.1246865) showed smallest error than those of the ARIMA (0, 1, 2) models.      new investors and Policymakers. Therefore, the Zambian government should use such forecasts in formulating policies and making strategies that will promote tourism industry. Future research should go further and consider monthly and quarterly data so that seasonality models can be used. Also non-linear models such as ARCH and GARCH can be applied.