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The ARIMA model building methods were introduced by Box Jenkins in 1970, using differencing to make a series stationary. The method is also known as the Box-Jenkins autoregressive integrated moving average (ARIMA) method. Slutsky presented it in the form of Autoregressive (AR) and Moving Average (MA) components in 1937 (Makridakis and Hibon, 1997). The ARIMA approach is the most widely used univariate

forecasting model. It uses an interactive method of an empirically driven equation to systematically identify, estimate, diagnose, and forecast time series (Delurgio, 1998).

Since the introduction of the ARIMA methods, extensive studies on the building of ARIMA models for tourism forecasting have been carried out over the past decade. Makridakis and Hibon (1997) argue that the use of differencing in the ARIMA techniques to make data stationary, results in more accurate forecasts by the ARMA(1,1) model. Turner et al., (1997) discovered that the AR model with periodic data gave better forecasts than the ARIMA with non-periodic seasonal data. Chu’s study (1998b) compared the use of an ARIMA and sine wave nonlinear regression combined model with the ARIMA model, and found that the combined model had relatively lower forecast errors. Chu’s study (2004) went a step further to compare ARIMA forecasts with a cubic polynomial model and also found that ARIMA forecasts had lower errors. Importantly, these studies show that ARIMA models may not be the most accurate forecasting models even though they may be the best fitting model.

In recent years, modelling and forecasting of seasonality in tourism demand forecasting has gathered momentum. Several papers have been published (Bar On, 1975, Sutcliffe and Sinclair, 1980, Bulter, 1994, Lim and McAleer, 2000, Kim, 2001, Kulendran and Wong, 2005, Lim and McAleer, 1999, Koc and Altinay, 2007) and a review of seasonality by Koenig-Lewis and Bischoff (2005). Seasonality in tourism edited by Baum and Lundtorp (2001) contains 11 papers on this area. These studies have raised awareness of the positive and negative impacts caused by the fluctuation of seasonal change to tourism demand caused by changes in policy, ecology, society and culture as well as employment in tourist destination countries or regions.

One of the popular aspects of the ARIMA model proposed by Box and Jenkins (1976) is its capacity for generating seasonal tourism demand forecasts. Various studies have suggested that the seasonal ARIMA methods have been regarded as better forecasting models than either the econometric or other time-series models (Preez and Witt, 2003, Chu, 1998, Gonzales, 1996, Kulendran and King, 1997b).

Multiplicative seasonal ARIMA modelling contains many forecasting models, of which ARIMA1,4 and ARIMA1are the most common. ARIMA1,4 is used for modelling

stochastic non-stationary seasonality which requires first and fourth differences to achieve stationarity; whilst ARIMA1 uses only the first differences and seasonality is

modelled with a constant and three seasonal dummies. The best models are selected depending on the forecasting accuracy of the models. There are numerous studies on modelling seasonality and performance comparisons using ARIMA, in comparison with other forecasting models (Geurts and Ibrahim, 1975, Gonzales, 1995, Lim and McAleer, 1999, Chu, 1998, Goh and Law, 2002 , Kulendran and Shan, 2002a, Kulendran and Witt, 2003b, Turner and Witt, 2001, Kulendran and Wong, 2005). A study by Kulendran and Wong (2005) suggests that ARIMA₁ provides more accurate forecasts for a time series that has fewer seasonal variations, whereas ARIMA1,4 provides more accurate forecasts

for a time series that has strong seasonal variation. But this study is limited to one quarter ahead forecasting.

Compared to simple time series models, the Box-Jenkins model (Box and Jenkins, 1976) is more complex in function and form and has more stringent validity tests and data requirements than other non-causal techniques. ARIMA time series models have been criticized for their ambiguity and inability to address the determinants of tourism demand necessary for policy assessment (Lim and McAleer, 2000, Kulendran and King, 1997a, Kulendran, 1996). Nevertheless, ARIMA models have the advantage of not being limited by the need for accurate economic causal variable forecasts that in some cases are difficult to obtain.

The use of the ARIMA model for short-term forecasts has been widely accepted in tourism forecasting studies for its versatility and accuracy (Delurgio, 1998, Harvey and Todd, 1983, Flechtling, 1996). A study by Lim and McAleer (1999) compares ARIMA with the seasonal Autoregressive Integrated Moving Average model. Gonzalez and Moral (1996), Kulendran and King (1997b) and, Kulendran and Witt (2001) compare the

seasonal ARIMA and basic structural time-series models (BSM), structural causal models, no change models, and error-correction models. Chan et al. (2005) use ARMA and GARCH methods on tourism demand to Australia and related volatility. Studies on model comparisons (Martin and Witt, 1989a, Kulendran and King, 1997a, Kulendran and Witt, 2001) suggest that time-series models and the “no change” model are capable of generating more accurate tourism forecasts than econometric models. A study by Louvieris (2002) successfully used a multiplicative seasonal autoregressive integrated moving average (SRIMA) model to forecast Greece’s inbound tourism in the medium/long-term. The findings of his study support the assertion that the ARIMA methods are accurate, not only for the short term, but also for medium/long term forecasting. His study suggests that there are situations, where the normally accepted restriction of imposing artificial short-term forecasting horizons on ARIMA modelling methods can be relaxed (Louvieris, 2002).