In the past decades, both causal and noncausal modeling techniques have been extensively applied in tourism demand forecasting (Song, Wong, and Chon, 2003). Noncausal modeling techniques which mainly feature time series modeling approaches typically forecast future tourism demand based on historic trends. Exponential smoothing and the Box–Jenkins procedure have been frequently used in these past studies. Causal modeling techniques often resort to econometric methods to forecast tourism demand taking into consideration the factors which purportedly influence demand changes in the model specification. A number of published empirical papers reveal that noncausal models are often superior to causal models in forecasting tourism demand (Chan et al., 1999; Dharmaratne, 1995; Kulendran and King, 1997; Kulendran and Witt, 2001; Turner et al., 1995; Witt et al., 1994; Witt et al., 2003; Witt and Witt, 1992). On the other hand, there exist in the literature many studies which provide evidence that econometric models are superior to time series models in forecasting tourism demand (Crouch et al., 1992; Hiemstra and Wong, 2002; Kim and Song, 1998; Song et al., 2000; Song and Witt, 2000; Song, Witt, and Jensen, 2003; Song, Wong, and Chon, 2003).
Depending on the forecasting horizon, data frequency, and origin-destination types, tourism demand forecasting accuracy may be different (Witt and Song, 2001). This raises the fundamental research question whether an alternative approach of using appropriately designed combination techniques instead of individual forecasts would generate better forecasts. Would a combination of the forecasts generated by individual techniques be more accurate? A large body of literature has been published in the general forecasting area on combination techniques with accompanying empirical results (Clemen, 1989), both empirical studies and simulation, which provide evidence that combining multiple forecasts often leads to better forecast accuracy than a single forecast.
At the most basic level, an intuitive combination method that can be used is that of a simple average of the single series forecasts that assigns each forecast equal weight without taking the historical performance of the individual forecasts into consideration. Extending this idea, other combination methods which rely on the weights based on the historical performance of the individual forecasts have also been developed. Dickinson (1973, 1975) developed the minimum-variance combining forecast model. Granger and Ramanathan (1984) suggested that the minimum-variance method can be extended to a regression-based method. They suggested an approach which viewed every single forecast as an independent variable and the actual value as the dependent variable, so that an ordinary least squares estimation can be carried out. Estimated coefficients can then be used as forecast weights. Furthermore, the restriction on having combined weights necessarily summing to one could be relaxed to obtain a better fit and forecasting performance. Subsequent studies in the related literature tended to apply regression-based combination forecasts, for example Mills and Stephenson (1987), Holden and Peel (1986), and Crane and Crotty (1967). More sophisticated and complex combination methods have been employed in empirical studies such as Diebold and Pauly (1987) and Granger and Ramanathan (1984), both of which featured an advanced time-varying parameter forecast combination method to estimate the coefficients in the combined regression. Figlewski (1983) and Chan et al. (1999) successfully used a principal component forecast combination method to forecast macroeconomic growth. A discussion of Bayesian issues relating to forecast combination is found in Min and Zellner (1993). These past studies point to the increasing importance and relevance of combining forecasts in mainstream forecasting methodological approaches.
An interesting issue raised in using forecast combination is what researchers call “forecast encompassing.” Granger and Newbold (1973) argued that, if forecast 2 does not contain more useful information than that presented in forecast 1, then the latter may be said to be conditionally efficient relative to forecast 2. This would mean that forecast 1 encompasses forecast 2 based on the argument presented in Chong and Hendry (1986). The null hypothesis that forecast 1 encompasses forecast 2 can be tested based on an OLS regression using the residual series of each individual forecast. Necessary assumptions which have to be imposed would be that the forecast errors have a zero mean and are normally distributed. Harvey et al. (1998) also suggested more robust encompassing test methods for the case of nonnormality in the forecast errors. It is noteworthy that encompassing tests are largely applied in the empirical studies (e.g. Ericsson and Marquez, 1993; Fair and Shiller, 1990; Fang, 2003; Holden and Thompson, 1997).
In the present study, one time series modeling approach (integrated seasonal autoregressive moving average, ARIMA) and three modern econometric approaches – autoregressive distributed lag model (ADLM), error correction model (ECM), and vector autoregressive (VAR) model – are applied to forecast international tourist flows to Hong Kong. Two combination methods are then used to combine the individual forecasts. Based on an evaluation of the forecasting accuracy among the single forecasts and the combination forecasts, an attempt is made to determine whether the forecast combination techniques could improve the forecast performance. Finally, a forecast encompassing test is conducted to gain insights into the circumstances in which combination forecasts are likely to outperform each single forecast and other cases where they are not.
This study focuses on the forecasts of Hong Kong tourism demand by residents from the top ten origin countries/regions, which are obtained by averaging the annual tourist arrivals to Hong Kong for the period from 2002 to 2004 using data from Visitor Arrival Statistics published by Hong Kong Tourism Board (2006). These are: mainland China (9.18 million), Taiwan (2.12 million), Japan (1.13 million), USA (0.91 million), Macau (0.49 million), South Korea (0.45 million), Singapore (0.39 million), UK (0.36 million), Australia (0.34 million), and Philippines (0.30 million).
Hong Kong tourism demand is measured by tourist arrivals from the ten considered origin countries/regions to Hong Kong. A study by Song, Wong, and Chon (2003) revealed that, based on strong economic considerations, the important factors that influence demand for travel to Hong Kong include own price, substitute prices and consumers’ income. This serves as a useful guide for selection of relevant economic factors in the specification of the current model.
The own price variable Pit can be defined by the relative CPI between Hong Kong and each origin country/region. The exchange rate between the origin and Hong Kong is used to adjust the relative price (see Song, Wong, and Chon, 2003). It is defined as follows:
where CPIHK and CPIi are the consumer price indexes of Hong Kong and origin country/ region i, respectively, and EXHK is the exchange rate.
Substitute price is another important factor influencing tourism demand. In this study, six countries/regions are considered to be the most important substitute destinations for Hong Kong: China, Taiwan, Singapore, Thailand, Korea and Malaysia. In instances where mainland China is used in the study as an origin country, Taiwan is excluded from the substitute travel destinations of Hong Kong as (up until recently) it is not possible for tourists from China to travel to Taiwan because of political reasons. The substitute destinations were selected on the basis that they share similar characteristics to Hong Kong in terms of location (eastern Asia) and likely reasons for visiting such as shopping, cuisine, nightlife and visiting temples and historical building sites. The substitute price variable Pst (see Song, Wong, and Chon, 2003) is given as:
where j = l, 2, ..., 6 denotes the six substitute destinations mentioned above, and wj is the share of the tourists traveling to the jth substitute country/region among all the tourists to countries/regions, which can be calculated from 
in which TAij is the tourist arrivals from country/region i to the substitute destination j. Since the number of tourists arriving at these destinations tends to change as time passes on, wij is not a certain value but variable at every point in time.
Consumers’ income is measured by the index of real GDP of these ten origin countries/ regions (see Song, Wong, and Chon, 2003).
Dummy variables are included in the demand models to capture the impacts of “one-off” events. Previous studies examining Hong Kong inbound tourism suggest that the handover of Hong Kong to China in 1997 and the “9/11” incident in 2001 are the most likely factors to have impacted Hong Kong tourism demand. For quarterly data, seasonal dummies are also considered:
Quarterly data 1984(1)–2004(2) are used in this investigation. The sample for 1984(1)–1999(2) is used for estimation and 1999(3)–2004(2) for ex post forecasting. Most of the data relating to tourist arrivals in Hong Kong is extracted from Visitor Arrivals Statistics monthly published by the Hong Kong Tourism Board. The tourist arrivals data from ten origin countries/regions to the six substitute destinations are from the Tourism Statistical Yearbook published by the UN World Tourism Organization (UNWTO, 2005). Exchange rate and GDP data are from the International Financial Statistics Online Service website of the International Monetary Fund (IMF, 2005). GDP data of Japan, USA and Australia has been seasonally adjusted by these countries respectively. All GDP data are transformed to the index form of 2000 = 100 based on constant prices. Taiwan’s exchange rate data is from Taiwan’s central bank website and its CPI and GDP data are from Taiwan Statistics Bureau’s (2005) website. Macau CPI and GDP data are from Macau Statistics Bureau. However, the latter’s quarterly GDP data are available only from 1998(1)–2004(2). The quarterly data for 1984(1)–1997(4) are estimated based on the annual GDP data for the period 1984–97. China’s CPI data for 1984(1)–1986(4) and its GDP data for 1984(1)–1998(4) are extracted from Zhang and Okawa (1997).
All variables are transformed to logarithmic form, as Witt and Witt (1995) have shown that most previous empirical studies suggest a log–log linear function can best explain the relationship between tourism demand and its determinants.
In this study, one time series modeling method (seasonal ARIMA) and three econometric methods (ADLM, ECM, and VAR) are considered in the empirical analysis of the data.
Seasonal ARIMA
Box and Jenkins (1976) suggested that seasonal autoregressive (SAR) and seasonal moving average (SMA) models may be very useful for modeling monthly or quarterly data. The seasonal ARIMA model in this study is specified based on the standard Box–Jenkins method. Since quarterly data are used, this method is appropriate for fitting the models for the ten origin countries/regions. Dummy variables are not included in these models; as mentioned earlier the noncausal forecasts are generated based on the historic trends of the time series themselves.
Autoregressive Distributed Lag Model (ADLM)
The ADLM specified below follows from the general-to-specific modeling approach. Firstly, a general ADLM for each country/region is specified:
where Qit denotes the tourist arrivals from origin country/region i, and Yit denotes the GDP index of country/region i. Pit and Pits are the own price and substitute prices respectively. Dummies include “one-off” events and seasonal dummies; n is the lag length of the independent variables. Eight lags (covering two years) are considered here as many past empirical studies indicate that the influence of these variables usually takes effect for no more than two years. Insignificant variables including dummy variables are removed from Equation (4) one by one starting from the least significant one, which has the lowest t-statistic. The filtering out of the insignificant variables is continued until all the independent variables are significant. All variables included in the final ADLM are statistically significant.
Error Correction Model (ECM)
The Engle and Granger (1987) two-stage approach is employed to establish the error correction model (ECM). Firstly, a seasonal integration test is performed on each variable to detect seasonal unit roots in the time series. Discussion of the seasonal unit roots test may be found in Dickey et al. (1984), Engle et al. (1989), Hylleberg et al. (1990) and Engle et al. (1993).
In this study we use the test for seasonal unit roots test developed by Hylleberg et al. (1990) (HEGY) to examine the integration properties (seasonal and non-seasonal) of the quarterly series used in this study. To illustrate this test, let γt represent the time series under consideration. Then the test can be written in the following forms:
where L is a lag operator. These three equations may be further written as
Equation (6) can be further modified to include (i) a constant, (ii) a constant and seasonal dummies, (iii) a constant and a time trend, (iv) a constant, a time trend and seasonal dummies. Therefore, for each series, five models are estimated.
Hylleberg et al. (1990) generated the critical values for quarterly seasonal unit root tests by Monte Carlo simulation and these critical values are the one-tailed t statistic test for π1 = 0 and π2 = 0 and F statistic test for π3=π4 = 0. If the null π1 = 0 cannot be rejected this suggests that the series γt has a non-seasonal unit root at frequency 0. If the null π2 = 0 cannot be rejected, this would mean that the series γt has an annual seasonal unit root at frequency ½. Accepting the null π3 =π4 = 0 suggests that the series γt has a biannual unit root at frequency ¼(¾). Since this method was developed by Hylleberg et al. (1990), it is often referred to as the HEGY test.
In this study the lagged values of δ4γt are added to Equation (6) to whiten the residuals without affecting the distribution of the t statistic under the null (Zhang and Okawa, 1997). The HEGY test shows that the price variables (both Pit and Pits) of each country/region only have non-seasonal unit roots (integrated at zero frequency). Tourist arrivals variables Qit for China, Taiwan, Japan, and Australia have non-seasonal unit roots while the same variable for Macau, USA, Singapore, and the Philippines have both non-seasonal and annual seasonal unit roots (integrated at the frequency 0 and ½). The series of tourist arrivals from the UK, however, has one non-seasonal unit root and two seasonal unit roots at frequency ½ and ¼ (¾). For the GDP variable Yit, a non-seasonal unit root is found in the Japan, USA, Singapore, and Australia series; a non-seasonal unit root and one biannual seasonal unit root are found in the Taiwan series; and a non-seasonal unit root and two seasonal unit roots are identified in the China, Macau, Korea, UK, and Philippines series.
The seasonal filter (1 + L) can be used to eliminate the seasonal unit root at annual frequency while (1+L2) removes the biannual seasonal roots. In addition, (1 + L + L2+L3) takes out both the annual and biannual seasonal unit roots. As a result of applying all these filters, the series are integrated at the same frequency 0. Following this, a general two step ECM may be specified.
VAR Model
The vector autoregressive (VAR) modeling method is a system estimation technique which treats all the variables as endogenous. Witt et al. (2003) and Witt et al. (2004) showed in their empirical studies that the VAR model generated relatively accurate medium- and long-term forecasts of tourism demand. In this study, all the explanatory variables are considered as endogenous. Only the items such as a constant, a time trend and dummies are treated as exogenous. An important question to address is how the lag length p of the VAR model is determined. If the lag length is too short, the model will not be able to capture the properties of the true data-generating process while, if the lag length is too long, the degrees of freedom would run out very quickly in the model estimation. In this study, the Aikake Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC) (explained in Song and Witt, 2000) are used to determine the lag lengths of the VAR model. Based on AIC and SBC, the lag lengths of the variables were found to be in the range between two and five for these ten countries/regions under consideration.
Forecasting Error
Different statistics are available to measure the errors of forecasts. Witt and Witt (1992) demonstrated that the MAPE (Mean Absolute Percentage Error) is a suitable measure for evaluating the forecasting performance of tourism demand models. This measure does not depend on the magnitudes of the demand variables being predicted, and hence is suitable for comparing the forecasting accuracy among different models as well as among different tourism origin countries/regions. MAPE is denoted as:
where et is the forecast errors series, γt is the actual value of the studied variable, and n is the length of the forecasting horizon. In our study 20 one-step-ahead forecasts are obtained. The MAPEs for the ten countries/regions are presented in Table 6.1.
From Table 6.1, we note that the ADL model exhibits the highest degree of modeling accuracy for three origins (China, Singapore, and UK); the ARIMA model performs best for another three origins (Taiwan, Macau, and Philippines); the ECM is ranked the best for two origin countries (Japan and Korea); and the VAR model is the best for another two origins (USA and Australia). There is no a single forecast method which can always perform well for all origin countries/regions. This result is consistent with the findings at Witt and Song (2001), who demonstrate that the forecast accuracy of each model varies with the origin—destination pair and no single forecasting method can generate the best forecast in all situations.
Table 6.1 Forecasting performance as measured by MAPE
Combination Methods and Empirical Results
Two forecasting combination methods are used in this study for the forecasting performance evaluation: the simple average method, which gives each forecast value the same weight; and a weighted average method in which the weights are decided based on the past errors associated with the individual forecasting techniques.
The simple average combination is straightforward in combining the forecasts. In practice this method has been frequently used in the literature. Makridakis and Winkler (1983) showed that the simple average of forecasts generated by different models tends to outperform the single forecasts on average. This method gives each forecast the same weight and it can be written as:
where wi is the weight assigned to model i and n is the number of forecasting models.
Clemen (1989) suggested that the variance—covariance methods of forecast combination tend to generate more accurate forecasts than those generated by individual models. In this method past errors of each original forecast are used to determine the weights in forming the combined forecasts. The basic idea of this method is illustrated here using a pair of forecasts. Let f1t and f2t be two unbiased forecasts of γt, the composite forecast fct can then be defined as
where 0 ≤ λ ≤ 1 Let
where i = 1, 2, c. Because of the assumption that the forecasts f1t and f2t are unbiased, eu has zero mean. From Equation (9) and (10) we can obtain:
with a variance of 
where 
and 
are the variances of the forecasting errors from the two models and 
is their covariance. 
is minimized by setting the weight λ equal to
The weight in Equation (12) is constrained to range between zero and one. In contrast to the simple average method, this method can assign higher weights to better forecasts and lower weights to poor ones. In practice, because 
and 
are unknown and the expected values of 
and 
would be equal to zero when the forecasts of the two models are unbiased, we could use the following alternative to calculate the weight:
A more straightforward method of calculating the weight was suggested by Bates and Granger (1969):
Equation (14) ignores the covariance between e1t and e2t. Some published empirical studies showed that in some cases the mean squared error obtained from Equation (14) is much smaller than that obtained by Equation (13) (see for example, Newbold and Granger, 1974; Winkler and Makridakis, 1983). According to Clements and Hendry (2002), this is perhaps due to the fact that these studies calculated the weights based on a relatively small number of observations. Extension to multiple forecasts cases is straightforward. If there are n individual forecasts to be combined, the weight of the nth single forecast can be expressed as:
Tables 6.2–3 show the accuracy of combining forecasts measured by MAPE in the period 1999(3) to 2004(2). Table 6.2 presents the results of the simple average method while Table 6.3 shows the results obtained from Equations (14) and (15). In the latter combination method the weights are generated from the past performance of the single forecasts, the weights generated from the previous 18 forecast observations are assigned to the 19th forecast, and this window continuously moves until the combination series from 1999(3) to 2004(2) is generated. For example, the forecast errors from 1999(4) to 2004(1) are used to form the weights for every single forecast for 2004(2); the forecast errors from 1999(3) to 2003(4) are used to form the weights for the single forecasts for 2004(1).
In Tables 6.2–3, asterisks are used to denote the fact that the combined forecast accuracy is better than the accuracy of the forecasts generated by each of the individual component models. The results show that all combined forecasts outperform the poorest individual forecast among all combined single forecasts while only just over 50 percent of combined forecasts outperform the corresponding best individual forecast. Furthermore, there is little difference in relative combination forecast versus individual forecast accuracy for the two forecast combination techniques.
That combined forecasts are not always more accurate than the component individual forecasts may be due to the fact that a particular model contains the information of the other models under consideration, and hence their combination may not necessarily improve forecasting accuracy. This can be tested by utilizing the encompassing test described in the subsequent section.
Table 6.2 MAPE of the single forecasts and the combined forecasts (simple average)
Note: The forecasts for 2003Q2 were excluded from the calculation of the MAPEs as this observation is likely to be an extreme outlier on account of SARS.
Table 6.3 MAPE of the single forecasts and the combined forecasts (weighted average)
Note: The forecasts for 2003Q2 were excluded from the calculation of the MAPEs as this observation is likely to be an extreme outliers on account of SARS.
Encompassing Test and Forecasting Performance
Consider the case where two forecasting models are involved; if one forecasting model includes some useful information that is not considered in the second model, we would say that the forecasts generated by the first model encompass the forecasts generated by the second model. In this case, the forecast accuracy of the combined forecasts usually is not as good as the better performing model. This can be demonstrated by replacing ect in Equation (11) with εt, the error of the combined forecast:
Granger and Newbold (1973, 1986) suggested that one could examine whether the forecasts generated by Model 2 contain more useful information than the forecasts obtained from Model 1 through estimating Equation (16). The encompassing test refers to the t statistic for λ = 0 with the alternative λ > 0. Granger and Newbold (1973) defined that forecasts generated by Model 1 are “conditionally efficient” to those obtained by Model 2 if the null hypothesis cannot be rejected. However, later literature (Chong and Hendry, 1986) defined this conditional forecasting efficiency as forecast 1 encompasses forecast 2.
Multiple forecasts combination cases are discussed in Harvey and Newbold (2000). Consider the multiple forecast combination:
Similar to the case where two sets of forecasts are combined, Equation (17) can be rewritten as:
The null hypothesis that forecast 1 encompasses forecast 2,.., K is: H0 :λ1 = λ2 = ... = λk-1 = 0 against the alternative that H0:λ1, λ2,..., λk-1 are not jointly equal to zero. Since the null λ1 = λ2 = ... = λk-1 = 0 is multi-dimensional, the F statistic should be used as the forecast encompassing test.
Three possible scenarios could occur:
Table 6.4 presents the encompassing test results. The t statistic is applied to two forecasts cases while the F statistic is used in multiple forecasts combinations. The asterisks denote that the null hypothesis λ= 0 (or λ1 = λ1 =... = λk-1 = 0) cannot be rejected at the 10 percent significance level. Comparing the encompassing test results with the forecasting combination results in Tables Tables 6.2–3, we can see that in more than 65 percent of cases among all combination forecasts, the encompassing tests confirm the forecasting combination results in that where a particular model encompasses the other model(s) included in the combination it would not be expected that combining forecasts would result in greater accuracy. But there are still some cases where the encompassing tests are not coincident with the combination results. In the China and Macau models the encompassing tests show that most combination forecasts should perform better than the best single forecast but the combining results in Tables Tables 6.2–3 do not generate similar results. One possible reason is that some regressed residuals series have one unit root which will make the regression unstable. Another possible reason is that if the forecasting errors are not normally distributed, the F test tends to be less robust (Harvey and Newbold, 2000). Moreover, the ordinary least squares (OLS) method is sensitive to outliers, which may lead to inconclusive results. In addition, the possibility of type I error is controlled in the OLS estimation but not type II error. As a result, type II error may occur and reduce the reliability of the regression results when the hypothesis cannot be rejected at the normal significance level.
Table 6.4 Encompassing test results
The objective was to examine whether tourism demand forecasting accuracy can be improved by combining the forecasts generated by various techniques, rather than just focus on the forecasts generated by individual techniques. In this study a seasonal ARIMA and three econometric methods (ADLM, ECM, and VAR) are used to forecast international tourist flows to Hong Kong. Forecasting accuracy is measured by MAPE.
Previous research in tourism forecasting has shown that no single forecasting technique is superior to others in all situations, as the relative performance of forecasting models tends to vary across origin–destination pairs. The empirical results from this study support previous findings as it is found that the performance of each forecasting model varies for different origins and there is no single technique which can perform best in all situations.
A forecasting combination exercise is conducted using two relatively simple forecast combination techniques to see whether the combined forecasts improve forecasting accuracy over the individual models. The empirical results show that forecast combination only improves forecasting performance in the tourism context in just over 50 percent of cases compared with the most accurate individual forecast. This result departs from many past studies in the general forecasting literature (Clemen, 1989), which conclude that forecasting combination generally improves forecasting performance over the individual component models. However, the theory underlying encompassing tests suggests that only if competing models do not encompass each other is it likely that forecast combination will improve forecasting performance over the individual models, and this is broadly supported by the empirical findings from this study and is also consistent with previous studies in the general forecasting literature. Therefore, it is recommended that forecasting combination should always be carried out in conjunction with the encompassing analysis. The two forecast combination techniques yield similar results in terms of their performance relative to individual forecasting techniques.
This study breaks new ground in tourism forecasting research by examining the impact of forecast combination on tourism forecasting accuracy using sophisticated econometric models, though it is only a preliminary attempt with two simple combination methods. The authors will extend this research in the future by employing more complex forecast combination methodologies and examining different forecasting horizons.
Bates, J. M. and Granger, C. W. J. (1969) ‘The combination of forecasts’. Operational Research Quarterly, 20(4), 451–68
Box, G. E. P. and Jenkins, G. M. (1976) Time series analysis: Forecasting and control (Rev. edn). San Francisco, CA: Holden-Day
Chan, Y. L., Stock, J. H. and Watson, M. W. (1999) ‘A dynamic factor model framework for forecast combination’. Spanish Economic Review, 1(2), 91–121
Chong, Y. Y. and Hendry, D. F. (1986) ‘Econometric evaluation of linear macro-economic models’. Review of Economic Studies, 53(175), 671–90
Clemen, R. T. (1989) ‘Combining forecasts: A review and annotated bibliography’. International Journal of Forecasting, 5(4), 559–83
Clements, M. P. and Hendry, D. F. (ed.) (2002) A comparison to economic forecasting. Malden, MA: Blackwell Publishers
Crane, D. B. and Crotty, J. R. (1967) ‘A two-stage forecasting model: Exponential smoothing and multiple regression’. Management Science, 13(8), B–501–B–507
Crouch, G. I., Schultz, L. and Valerio, P. (1992) ‘Marketing international tourism to Australia: A regression analysis’. Tourism Management, 13(2), 196–208
Dharmaratne, G. S. (1995) ‘Forecasting tourist arrivals in Barbados’. Annals of Tourism Research, 22(4), 804–18
Dickey, D. A., Hasza, D. P. and Fuller, W. A. (1984) ‘Testing for unit roots in seasonal time series’. Journal of the American Statistical Association, 79(386), 355–67
Dickinson, J. P. (1973) ‘Some statistical results on the combination of forecasts’. Operational Research Quarterly, 24(2), 253–60
Dickinson, J. P. (1975) ‘Some comments on the combination of forecasts’. Operational Research Quarterly, 26(1, Pt. 2), 205–10
Diebold, F. X. and Pauly, P. (1987) ‘Structural change and the combination of forecasts’. Journal of Forecasting, 6(1), 21–40
Engle, R. F. and Granger, C. W. J. (1987) ‘Co-integration and error correction: Representation, estimation and testing’. Econometrica, 55(2), 251–76
Engle, R. F., Granger, C. W. J. and Hallman, J. J. (1989) ‘Merging short- and long-run forecasts: An application of seasonal cointegration to monthly electricity sales forecasting’. Journal of Econometrics, 40(1), 45–62
Engle, R. F., Granger, C. W. J., Hylleberg, S. and Lee, H. S. (1993) ‘The Japanese consumption function’. Journal of Econometrics, 55(1/2), 275–98
Ericsson, N. R. and Marquez, J. (1993) ‘Encompassing the forecasts of US trade balance models’. Review of Economics and Statistics, 75(1), 19–31
Fair, R. C. and Shiller, R. J. (1990) ‘Comparing information in forecasts form econometric models’. American Economic Review, 80(3), 375–89
Fang, Y. (2003) ‘Forecasting combination and encompassing tests’. International Journal of Forecasting, 19(1), 87–94
Figlewski, S. (1983) ‘Optimal price forecasting using survey data’. Review of Economics and Statistics, 65(1), 13–21
Granger, C. W. J. and Newbold, P. (1973) ‘Some comments on the evaluation of economic forecasts’. Applied Economics, 5(1), 35–47
Granger, C. W. J. and Newbold, P. (1986) Forecasting economic time series (2nd edn). Orlando, FL: Academic Press
Granger, C. W. J. and Ramanathan, R. (1984) ‘Improved methods of combining forecasts’. Journal of Forecasting, 3(2), 197–204
Harvey, D. and Newbold P. (2000) ‘Tests for multiple forecast encompassing’. Journal of Applied Econometrics, 15(5), 471–82
Harvey, D. I., Leybourne, S. J. and Newbold, P. (1998) ‘Tests for forecast encompassing’. Journal of Business and Economic Statistics, 16(2), 254–59
Hiemstra, S. and Wong, K. K. F. (2002) ‘Factors affecting demand for tourism in Hong Kong’. Journal of Travel and Tourism Marketing, 13(1/2), 43–62
Holden, K. and Peel, D. A. (1986) ‘An empirical investigation of combinations of economic forecasts’. Journal of Forecasting, 5(4), 229–43
Holden, K. and Thompson, J. (1997) ‘Combining forecasts, encompassing and the properties of UK macroeconomic forecasts’. Applied Economics, 29(11), 1447–58
Hong Kong Tourism Board (2006) ‘Visitor Arrival Statistics’, Hong Kong, available at http://www.tourism.gov.hk/english/statistics/statistics_research.html
Hylleberg, S., Engle, R. F., Granger, C. W. J. and Yoo, B. S. (1990) ‘Seasonal integration and cointegration’. Journal of Econometrics, 44(1/2), 215–38
IMF (2005) International Financial Statistics, available at http://www.imf.org/external/data.htm (accessed June 2005)
Kim, S. and Song, H. (1998) ‘Analysis of tourism demand in South Korea: A cointegration and error correction approach’. Tourism Analysis, 3(1), 25–41
Kulendran, N. and King, M. L. (1997) ‘Forecasting international quarterly tourist flows using error-correction and time-series models’. International Journal of Forecasting, 13(3), 319–27
Kulendran, N. and Witt, S. F. (2001) ‘Cointegration versus least squares regression’. Annals of Tourism Research, 28(2), 291–311
Makridakis, S. and Winkler, R. L. (1983) ‘Averages of forecasts: Some empirical results’. Management Science, 29(9), 987–96
Mills, T. C. and Stephenson, M. J. (1987) ‘A time series forecasting system for the UK money supply’. Economic Modelling, 4(3), 355–69
Min, C. K. and Zellner, A. (1993) ‘Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates’. Journal of Econometrics, 56(1/2), 89–118
Newbold, P. and Granger, C. W. J. (1974) ‘Experience with forecasting univariate time series and the combination offorecasts’. Journal of the Royal Statistical Society. Series A (General), 137(2), 131–65
Song, H. and Witt, S. F. (2000) Tourism demand modelling and forecasting: Modern econometric approaches. Amsterdam: Pergamon Song, H., Romilly, P. and Liu, X. (2000) ‘An empirical study of outbound tourism demand in the UK’. Applied Economics, 32(5), 611–24
Song, H., Witt, S. F. and Jensen, T. C. (2003) ‘Tourism forecasting: Accuracy of alternative econometric models’. International Journal of Forecasting, 19(1), 123–41
Song, H., Wong, K. K. F. and Chon, K. K. S. (2003) ‘Modelling and forecasting the demand for Hong Kong tourism’. International Journal of Hospitality Management, 22(4), 435–51
Taiwan Statistics Bureau (2005) Statistical Yearbook, available at http://www.stat.gov.tw (accessed June 2005)
Turner, L. W., Kulendran, N. and Pergat, V. (1995) ‘Forecasting New Zealand tourism demand with disaggregated data’. Tourism Economics, 1(1), 51–69
UNWTO (2005) Compendium of Tourism Statistics, available at www.unwto.org (accessed June 2005) Winkler, R. L. and Makridakis, S. (1983) ‘The combination of forecasts’. Journal of the Royal Statistical Society. Series A (General), 146(2), 150–57
Witt, C. A., Witt, S. F. and Wilson, N. (1994) ‘Forecasting international tourist flows’. Annals of Tourism Research, 21(3), 612–28
Witt, S. and Song, H. (2001) ‘Forecasting future tourism flows’. In A. Lockwood and S. Medlik (eds) Tourism and hospitality in the 21st century. Oxford: Butterworth-Heinemann, pp. 106–18
Witt, S. F. and Witt, C. A. (1992) Modelling and forecasting demand in tourism. London: Academic Press
Witt, S. F. and Witt, C. A. (1995) ‘Forecasting tourism demand: A review of empirical research’. International Journal of Forecasting, 11(3), 447–75
Witt, S. F., Song, H. and Louvieris, P. (2003) ‘Statistical testing in forecasting model selection’. Journal of Travel Research, 42(2), 151–58
Witt, S. F., Song, H. and Wanhill, S. (2004) ‘Forecasting tourism-generated employment: The case of Denmark’. Tourism Economics, 10(2), 167–76
Zhang, X. T. and Okawa, T. (1997) ‘Cointegration and error correction: Theory and application with Mathematica. Osaka City: Seseragi Publishing Co.
