This paper analyses stationary and non-stationary international tourism time series data by formally testing for the presence of unit roots and seasonal unit roots prior to estimation, model selection and forecasting. Various Box-Jenkins Autoregressive Integrated Moving Average (ARIMA) models are estimated over the period 1975(1)-1989(4) for tourist arrivals to Australia from Hong Kong, Malaysia and Singapore. The mean absolute percentage error and root mean squared error (RMSE) are used as measures of forecast accuracy. As the best fitting ARIMA model is found to have the lowest RMSE, this model is used to obtain post-sample forecasts. Tourist arrivals data for 1990(1)-1996(4) are compared with the forecast performance of the ARIMA model for each origin market. The fitted ARIMA model forecasts tourist arrivals from Singapore for the period 1990(1)-1996(4) very well. Although the ARIMA model outperforms the seasonal ARIMA models for Hong Kong and Malaysia, the forecasts of tourist arrivals are not as accurate as in the case of Singapore.