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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.

Tourism is one of the major contributors to foreign exchange earnings for Zambia and other countries worldwide. According to [

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. The fall has been known for centuries as Mosi-Oa-Tunya, meaning “The Smoke That Thunders” and it lie in the country’s tourist capital called Livingstone, south of Zambia. It was declared a World Heritage Site for its unique geological/geomorphologic significance. Other water includes the Kalambo Falls, Ntumbachushi Falls, Ngonye Falls and the Chishimba Fall [

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Holt-Winters exponential smoothing method is appropriate for forecasting non seasonal time series data. It is an extension of Simple Exponential Smoothing method and uses a linear combination of the previous values of a series for generating and modeling future values. It is applicable to time series data that has trend. Initial estimates and the slope of the trend are key to forecasting. The model for time series data Y t is defined as:

L t = α Y t + ( 1 − α ) ( L t − 1 + T t − 1 ) , 0 < α < 1

T t = β ( L t − L t − 1 ) + ( 1 − β ) T t − 1 , 0 < β < 1

where, α is the smoothing constant, β is the trend smoothing constants, Y t is raw data, L t is smoothed data and T t is the trend estimates.

The h-step-ahead forecast equation is Y ^ t + h = L t + h T t [

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.

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: Y t = ∑ i = 1 p ϕ i Y t − i + ε t ,

MA model: Y t = ∑ i = 1 q θ i ε t − i , and

The combination of AR and MA gives

ARMA model: Y t = ∑ i = 1 p ϕ i Y t − i + ε t + ∑ i = 1 q θ i ε t − i

where ϕ t is the autoregressive parameter at time t, ε t is the error term at time t and θ t is the moving-average parameter at time t [

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.

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 [

Annual International tourist arrivals in Zambia from 1995 to 2014 are shown (see,

The time series plot (see

The Holt-Winters exponential smoothing (HWES) and autoregressive integrated moving average (ARIMA) models are compared to determine the forecasting model for annual International tourist arrivals in Zambia from 1995 to 2014. HWES (α = 1, β = 0.1246865 ) model with a = 947,000.00, b = 39707.48, with error measures ME = −27309.61, RMSE = 92634.37, MAE = 71,582.83, MPE = −6.240377, MAPE = 12.24099 and MASE = 0.8586324 was considered as the best fit model (see,

For ARIMA model, the procedure is achieved by considering the following

Criteria | Formula | Criteria | Formula |
---|---|---|---|

MPE | 1 n ∑ i = 1 n 100 × ε i y i | RMSE | 1 n ∑ i = 1 n ε i 2 |

MAE | 1 n ∑ i = 1 n | ε i | | MAPE | 1 n ∑ i = 1 n | ε i x i | × 100 |

MASE | | ε i 1 n − 1 ∑ i = 1 n | Y t − Y i − 1 | | |

See [

Year | No. of Tourist Arrival | Year | No. of Tourist Arrival |
---|---|---|---|

1995 | 163,000 | 2005 | 669,000 |

1996 | 264,000 | 2006 | 757,000 |

1997 | 341,000 | 2007 | 897,000 |

1998 | 362,000 | 2008 | 812,000 |

1999 | 404,000 | 2009 | 710,000 |

2000 | 457,000 | 2010 | 815,000 |

2001 | 492,000 | 2011 | 920,000 |

2002 | 565,000 | 2012 | 859,000 |

2003 | 413,000 | 2013 | 915,000 |

2004 | 515,000 | 2014 | 947,000 |

Source: WTO, Yearbook of Tourism Statistics.

steps: identification, model selection, parameter estimation and diagnostic check. A code in R was used to obtain a best fit model ARIMA (0, 1, 2) model that fitted the Zambian tourist arrival data. A code in R was also used to obtain the estimated coefficients for the ARIMA (0, 1, 2) model with MA(2)= −0.2195. The ARIMA (0, 1, 2) model parameters are significant with ME = 37002.01, RMSE = 85705.65, MAE = 73770.52, MPE = 6.869655, MAPE = 13.38775 and MASE = −0.2792933. The ACF plot (original series d = 0) in

A code in R (version 0.99.903) was used to obtain results of residuals. The results Plots of ACF, Normal Q-Q and Histogram of Residuals for the ARIMA (0, 1, 2) analysis is shown in

The results in

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. Hence, the HWES (α = 1, β = 0.1246865 ) can be used to model annual international tourist arrivals in Zambia. Forecasting results give a gradual increase in annual international tourist arrivals of about 42% by 2024 resulting in an average growth rate of 7.6% at confidence interval 95%. Accurate forecasts are key to

TENTATIVE MODEL | ME | RMSE | MAE | MPE | MAPE | MASE |
---|---|---|---|---|---|---|

ARIMA (0, 1, 2) | 37002.01 | 85705.64 | 73770.52 | 6.869655 | 13.38775 | 0.8848736 |

HWES (α = 1, β = 0.1246865 ) | −27309.61 | 92634.37 | 71582.83 | −6.240376 | 12.24099 | 0.8586324 |

Year Point Forecast | Low 80 | High 80 | Low 95 | High 95 |
---|---|---|---|---|

2015 | 986,708 | 1,103,436 | 808,187 | 1,165,228 |

2016 | 1,026,415 | 1,202,087 | 757,748 | 1,295,082 |

2017 | 1,066,122 | 1,294,440 | 716,940 | 1,415,305 |

2018 | 1,105,830 | 1,384,854 | 679,099 | 1,532,561 |

2019 | 1,145,537 | 1,474,871 | 641,865 | 1,649,210 |

2020 | 1,185,245 | 1,565,207 | 604,143 | 1,766,347 |

2021 | 1,224,952 | 1,656,241 | 565,354 | 1,884,551 |

2022 | 1,264,660 | 1,748,188 | 525,167 | 2,004,152 |

2023 | 1,304,367 | 1,841,177 | 483,388 | 2,125,347 |

2024 | 1,344,075 | 1,935,285 | 439,897 | 2,248,252 |

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Jere, S., Banda, A., Kasense, B., Siluyele, I. and Moyo, E. (2019) Forecasting Annual International Tourist Arrivals in Zambia Using Holt-Winters Exponential Smoothing. Open Journal of Statistics, 9, 258-267. https://doi.org/10.4236/ojs.2019.92019