Discussion Paper No. 533

TIME SERIES FORECASTS
OF INTERNATIONAL TOURISM
DEMAND FOR AUSTRALIA
Christine Lim
andMichael McAleer

April 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Time Series Forecasts of International Tourism Demand for Australia

Christine Lim
School of Tourism and Hotel Management
Griffith University
PMB 50 Gold Coast Mail Centre
Queensland 9726
Australia
Tel:Fax:
E-mail:+617 5594 8824
+617 5594 8507
Christine.Lim@mailbox.gu.edu.auandMichael McAleer
Department of Economics
University of Western Australia
Western Australia 6009
Australia

Time Series Forecasts of International Tourism Demand for Australia

Abstract
This paper examines stationary and nonstationary time series by formally testing for the presenceof 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 theperiod 1975(1)-1989(4) for tourist arrivals to Australia from Hong Kong, Malaysia and Singapore.
The mean absolute percentage error (MAPE) and root mean squared error (RMSE) are used asmeasures of forecast accuracy. As the best fitting ARIMA model is found to have the lowest
RMSE, it is used to obtain post-sample forecasts. Tourist arrivals data for 1990(1) to 1996(4) arecompared with the forecast performance of the ARIMA model for each origin market. The fitted
ARIMA model forecasts tourist arrivals from Singapore between 1990(1)-1996(4) very well.
Although the ARIMA model outperforms the seasonal ARIMA models for Hong Kong and
Malaysia, the forecast of tourist arrivals is not as accurate as in the case of Singapore.
Keywords: Unit roots, seasonality, forecasting models, forecast accuracy, root mean square error,
ex post forecasts.

11. Introduction
Hong Kong, Malaysia and Singapore are Australia’s major tourist markets in Asia apart fromJapan. In terms of the international tourism market share of the three countries, Singapore isAustralia’s fifth major market, with Hong Kong and Malaysia occupying seventh and eighth
places, respectively. The average annual growth rates of tourist arrivals from Hong Kong,
Malaysia and Singapore during 1991-96 were 19.37%, 21.4% and 20.8%, respectively, which
increased from average growth rates of 17.7%, 9.7% and 16.1% over the period 1985-90. The
rise in inbound tourism from these markets could be attributed to the rapid economic growthexperienced by these countries in the first half of the 1990s. Inspite of the phenomenal growth of
inbound tourism from these source markets, which far exceeded the average growth rate ofinternational arrivals to Australia of 10.5% over the period 1990-96, little research on thesemarkets has been undertaken so as to understand their significant contributions to Australia’s
inbound tourism. Most of the research has been conducted on the four major markets to Australia,
namely Japan, New Zealand, United Kingdom and the United States.
Quantitative methods for generating forecasts of future outcomes using statistical procedures
involve the examination of current and historical seasonally unadjusted data. This knowledge canbe used to extrapolate the variable of interest. It is assumed that the process is stable over theforecast time horizon, but this assumption may only be valid for short-term forecasts. Two typesof quantitative forecasting models used are time series models and causal econometric models.
Time series models involve a statistical analysis which uses only the historical data of thevariable to be forecast. Causal models are based on the statistical analysis of data for otherrelated (explanatory) variables, and the use of these variables to forecast the dependent variable ofinterest.
At present there are numerous forecasting methods available and the empirical findings, which
are often in conflict, have given no clear guidelines as to the most appropriate methods forforecasting.

The literature on international tourism demand forecasting, based on different

univariate time series forecasting methods (see, for example, Geurts and Ibrahim, 1975; Choy,
1984; van Doorn, 1984; Martin and Witt, 1989; Chan, 1993; Witt et al., 1994; Turner et al., 1995,
1997; Frechtling, 1996; Kulendran and King, 1997; Chu, 1998; Kim, 1999), is numerous.
1Forecasting performances of the various models are affected by the type of data used (namely
monthly, quarterly or annual data), the forecasting horizon, and the country of origin. From somethese studies, it is clear that sophisticated procedures such as ARIMA models do not necessarily
forecast better than their simple counterparts.
There are a number of factors used to evaluate the effectiveness of a forecasting method, such as
forecasting accuracy, costs associated with the application of a forecasting procedure (forexample, installation and operating costs), and ease of application and interpretation of the output
from a forecasting method. Accuracy is often regarded as the dominant criterion for selecting aforecasting method.

The accuracy of a forecasting method is determined by analyzing theforecast error, which is defined as the actual value minus the forecast (or fitted) value of thevariable for time period t, namely:e t = A t − Ft
wheree t = forecast error at time t;

A t = actual tourist arrivals at time t;
Ft = forecast tourist arrivals at time t+1.
For instance, forecast optimization typically chooses a model that minimizes root mean squared
error (RMSE), which is calculated as:
RMSE =

1 n 2∑et .
n t =1In this paper, various Box-Jenkins (1970) autoregressive integrated moving average (ARIMA)
forecasting models are considered and their comparative performances analysed over a sample ofinternational tourism demand for Australia by each of three origin countries, namely Hong Kong,
Malaysia and Singapore. The ARIMA models provide a useful framework to understand how thetourism time series is generated. Unlike univariate smoothing models which are more commonly
used, the ARIMA approach requires a tourism time series to be tested for nonstationarity prior toundertaking estimation and forecasting exercise. If a series is nonstationary (that is, the series hasa mean and variance that are not constant over time), the series has to be differenced to transform
it to a stationary series, before generating forecasts. A stationary tourist arrival series typically2

provides better and more reliable forecasts. Very few of the recent published studies on tourism
forecasting have considered or presented tests for unit roots and seasonal unit roots before
estimating ARIMA models and using them for forecasting. Such tests and their implications will
be discussed in this paper for the historical data on individual tourists arrivals to Australia from
Hong Kong, Malaysia and Singapore.
The logarithms of quarterly tourist arrivals data (from the Australian Bureau of Statistics) for the
March quarter of 1975 to the December quarter of 1989 are used. One-quarter-ahead international
tourist arrivals forecasting accuracy, beyond the sample used for estimation, is evaluated for theperiod 1990(1)-1996(4) using various estimated ARIMA time series models. The motivation forthe choice of this period is twofold. First, between 1990 and 1996, Australia experienced thelargest average annual percentage growth in tourist arrivals from Asia of 23 percent. Comparisonof the out-of-sample forecast of the various ARIMA models would be considered useful for awide range of policy-making in the tourism and travel industry. Second, the significance of the
impact on international tourist arrivals to Australia due to the 1979 Oil Crisis, 1988 Bicentennial
Celebration and 1989 Air Pilots Dispute can be estimated.
2. ARIMA Forecasting Models
The logarithms of quarterly tourist arrivals from the three countries are used to capture themultiplicative effects in the levels of the variables. Using an autoregressive specification andordinary least squares estimation, current tourist arrivals can be forecast one quarter ahead, basedon a fourth-order process, as follows:A t = β 0 + β1 A t −1 + β 2 A t − 2 + β 3 A t −3 + β 4 A t − 4 + ε t .
Table 1 shows that the only significant lags in forecasting tourist arrivals are the second andfourth for Hong Kong, the first and fourth lags for Malaysia, and only the fourth lag for Singapore.
The influential work of Box and Jenkins (1970) shifted professional attention in time seriesmodelling away from stationary processes to a class of nonstationary processes and the related
ideas of the order of integration necessary to obtain stationary series. Furthermore, the BoxJenkins method is popular because of its generality since it can handle any stationary or
nonstationary time series, with and without seasonal elements. The frequent use in applied3

empirical work and availability in well-documented econometric computer programs have
perhaps contributed most to its popularity. The Box-Jenkins method for selecting an appropriate
autoregressive integrated moving average (ARIMA) model for estimating and forecasting a
univariate time series consists of identification, estimation and testing, and application. In theidentification phase, a general class of models applicable to a particular situation is examinedwith the aid of the sample correlograms, and autocorrelation and partial autocorrelation functions.
The original time series in logarithms are checked for stationarity using the augmented DickeyFuller (ADF) test for unit roots and, if necessary, the series are transformed by taking appropriate
differences to render the series stationary. A detailed explanation of the test procedure is given in
Lim and McAleer (2000a). The ADF tests which are performed sequentially show that the fourth
lag is significant, and the ADF test statistics with trend are –2.40, -1.83 and –2.52 for Hong Kong,
Malaysia and Singapore, respectively. Each of the calculated statistics exceeds the critical valueof –3.49 at the 5% significance value, so the null hypothesis of a unit root is not rejected, which
implies that each of the three tourist arrival series is nonstationary. Taking first differences
renders each series stationary, with the ADF statistics in all cases for lag length of three (that is, 3.77, -4.40 and –4.74 for Hong Kong, Malaysia and Singapore, respectively) being less than thecritical value of –2.91 at the 5% significance level.
Since tourism data exhibit varying seasonal patterns, it is imperative to test for the presence of
seasonal unit roots in univariate series. The test most often used is the HEGY test of Hylleberg,
Engle, Granger and Yoo (1990). A detailed explanation of the test procedure is given, forexample, in Lim and McAleer (2000b). Briefly, the relevant hypotheses to be tested are as
follows:
1) π 1 = 0 ; unit root at the zero frequency2) π 2 = 0 ; unit root at the semi-annual frequency3) π 3 = π 4 = 0 ; unit root at the annual frequency.
The results (see Table 2) indicate that quarterly tourist arrivals from Hong Kong, Malaysia andSingapore are each integrated at the zero and semi-annual frequencies, but not at the annual
frequency, I(1,1,0).

4

Subsequently, parameter estimation in phase two involves fitting various autoregressive (AR)models of order p and moving average (MA) models of order q. The best fitting parsimonious
model is selected, based on various criteria, such as statistically significant AR and MA estimatedcoefficients at the 5% level, absence of serial correlation, and optimisation of the Akaike
Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC).

Diagnostic checking

involves residual analysis to ensure that the estimated model has independent and identically
distributed errors. When a satisfactory model has been selected, it is used in the third phase toforecast future values of tourist arrivals from the three origin countries. Using least squares
estimation, the variance of the optimum i-period-ahead forecast will be less than the variance ofthe predicted series:
Var( A t + i ) = σ 2A = Var( Ft + i ) + σ 2ε .
When an ARIMA model has been fitted to a time series, the i-period-ahead forecast of tourist
arrivals is given by:
$ $
$ $
$
$
$
$
Ft + i = C$ + φ$ 1At + i −1 +...+ φ p + d A t + i − p − d + ε t + i − θ1ε t + i −1 −...− θ q ε t + i − q , i = 1, 2, ...

(8)In order to compare the ex post forecast accuracy of the various ARIMA models for 1990(1)1996(4), the best fitting ARIMA models are estimated separately for tourist arrivals series from1975(1) to 1989(4). The correlogram and unit root tests of the series before and, if necessary,
after differencing are examined for stationarity.

After empirical examination, the mostappropriate models for tourist arrivals from Hong Kong, Malaysia and Singapore are determined
as ARIMA(3,1,1), ARIMA(3,1,2) and ARIMA(4,1,0), respectively (with absolute t-ratios inparentheses):(1 + 1.32 L + 0.82 L2 + 0.52 L3 )(1 − L) log HK t = 0.04 + ε$ HKt − 0.97ε$ HK ( t −1)(10.7) (4.20)
AIC = -3.65,(4.15)(3.25)(38.4)

SBC = -3.47(Hong Kong)(1 + 0.69L + 0.98L2 + 0.65L3 )(1 − L )log M t = 0.03 + ε$ Mt − 0.95ε$ M( t −1)(6.68) (117)(6.52)AIC = -3.73,

SBC = -3.55(2.41)(27.9)(Malaysia)

5

(1 + 0.64 L + 0.58L2 + 0.6 L3 − 0.36 L4 )(1 − L)log S t = 0.04 + ε$ St(4.84) (4.26) (4.45) (2.67)
AIC = -3.95,(4.94)

SBC = -3.77.

(Singapore)

Since the specific ARIMA models that adequately describe tourist arrivals from Hong Kong,
Malaysia and Singapore are given above, the fitted models used for ex post forecasting tourist
arrivals from these origin countries are given as follows:
Flog HK ( t + i ) = 0.04 − 0.32 log HK t + i −1 + 0.50 log HK t +i − 2 + 0.30 log HK t +i −3+ 0.52 log HK t + i − 4 + εˆ HK ( t +i ) − 0.97εˆ HK ( t + i −1)
Flog M( t + i ) = 0.03 + 0.31 log M t + i −1 − 0.29 log M t + i −2 + 0.33 log M t + i −3
+ 0.65 log M t + i − 4 + ε$ M ( t + i ) − 0.95ε$ M( t + i −1)
Flog S(t + i ) = 0.04 + 0.36 log S t + i −1 + 0.06 log S t + i −2 − 0.02 log S t + i −3
+ 0.96 log S t + i − 4 − 0.36 log S t +i −5 + ε$ S(t +i )where t = 1989(4),(Hong Kong)(Malaysia)(Singapore)i = 1, 2, ...

The expected values of the future random errors are assumed to be zero.
As the individual quarterly international tourist arrivals to Australia from Hong Kong, Malaysia
and Singapore exhibit pronounced seasonality, the most appropriate multiplicative seasonal
ARIMA models selected for the various tourist arrivals series can be used to forecast future
observations, namely ARIMA(1,1,4)(0,1,0)4 for Hong Kong, ARIMA(2,1,1)(4,1,2)4 for Malaysia,
and ARIMA(0,1,4)(0,1,1)4 for Singapore, which are given as follows:(1 + 0.48L )(1 − L)(1 − L4 )log HK t = (1 + 0.93L4 )ε$ HKt(4.21)AIC = -3.98,(39.2)
SBC = -3.91
6(Hong Kong)(1 − 0.62 L2 )(1 + 0.42 L4 )(1 − L)(1 − L4 )log M t = (1 + 0.27L )(1 + 0.98L2 )ε$ Mt(6.81)AIC = -3.76,(3.06)(2.81)(4746)

SBC = -3.61(1 − L)(1 − L4 ) log S t = (1 + 0.62L4 )(1 + 0.56L)εˆ St(5.88)AIC = -4.17,(Malaysia)(Singapore)(5.06)

SBC = -4.10.

3. Magnitude of Forecasting Errors
With the final observation being tourist arrivals for the fourth quarter of 1989, Table 3 presents
the RMSE one-quarter-ahead forecast accuracy measure of the ARIMA and Multiplicative
Seasonal ARIMA models. For tourist arrivals from Hong Kong and Malaysia, the ARIMA model
forecasts better than the seasonal multiplicative ARIMA model, and the reverse holds for
Singapore. However, the mean absolute percentage error (MAPE) of the ARIMA model is lower
than that of the seasonal model for tourist arrivals from Singapore.
Using the best fitting model for each tourist arrivals series, the accuracy of post-sample forecasts
from 1990(1) to 1996(4) is obtained to examine the relationship between the fit of the model andforecast performance. Table 4 shows Theil’s (1966) U-coefficients for the ARIMA and seasonal
ARIMA forecasts, all of which are less than one. As the U-coefficients of the ARIMA model are
all less than that of the seasonal ARIMA for each series, these results suggest that the ARIMA
model performs better in forecasting tourist arrivals from the three origin countries for the period
1990(1) to 1996(4). Using the best fitting ARIMA model for post-sample forecasting, the sample
coefficient of correlation is computed as a goodness-of-fit measure to determine how well theactual values fit the future observations. This measure provides useful information as to how
well the model forecasts the data. Table 5 shows that the correlation coefficients of the ARIMA
model range from 0.67 to 0.95. Apparently, the fitted ARIMA model forecasts tourist arrivals
from Singapore very well, given that 95% of the variation in the tourist arrivals forecast is
associated with variations in actual tourist arrivals between 1990(1)-1996(4). Even though the
ARIMA model outperforms the other models in forecasting tourist arrivals from Malaysia, only
67% of the variation in the tourist arrivals forecast is associated with variations in the actual
tourist arrivals in the same period.
7

The fitted values, which are interpreted as the forecasts for the next quarter, are sufficiently close
to the actual values for tourist arrivals from Hong Kong, Malaysia and Singapore using the
ARIMA models, as shown in Figures 1-3. However, the forecasts from the seasonal ARIMA
models are not close to the actual values (these figures are available on request). There is no
systematic pattern in the residuals for the ARIMA models. Using 24 lags for each series, thecorrelograms of the residuals show that there are very few autocorrelations outside the bounds±0.3, or the 95% confidence interval. The autocorrelations which are not within two standard
errors of the mean include the sixth and tenth lags for Hong Kong, the eleventh lag for Malaysia,
and only marginally for the third, sixth and tenth lags for Singapore (these figures are alsoavailable on request). The Lagrange multiplier test for serial correlation, LM(SC), shows that thenull hypothesis of no serial correlation is not rejected, and hence serial correlation is deemed to
be absent in the residuals (see Table 6). The calculated F values, which lie between 0.83 and 3.03,
are all less than the critical F value at the 5% level.
Table 7 shows the sum of squared residuals (SSR) from the fitted ARIMA and seasonal ARIMA
nmodels for Hong Kong, Malaysia and Singapore: SSR = ∑ ê 2t , where ê t denotes the onet =1

quarter-ahead prediction error at time t. The standard deviation (SD) of residuals, calculated asσˆ e2 = SSE / n , is between 0.12 and 0.15, signifying the average deviations in one-quarterahead forecasts of tourist arrivals.
Table 8 shows a breakdown of RMSE for the fitted ARIMA and seasonal ARIMA models
according to the forecasting horizon for the three tourist arrivals series. It suggests that a shorter
forecasting horizon is more accurate than a longer forecasting horizon for tourist arrivals from
Malaysia and Singapore. However, the same argument does not hold according to the RMSE for
tourist arrivals from Hong Kong.
During the period 1975(1) to 1989(4), Australia experienced the Oil Price Crisis of 1979, theBicentennial Celebration in 1988 of European Settlement in Australia, and the Australian Air
Pilots Strike in1989-90. These one-off events could distort the estimation, testing and analysis of
the underlying process, which is critical in forecasting. In order to analyse the impact of these8

one-off events, intervention analysis with deterministic dummies allows the effects of these
exogenous shocks to be represented by dynamic ARIMA models, as follows (with absolute tratios in parentheses):(1 + 1.36L + 0.87 L2 + 0.55L3 )(1 − L) log HK t
(10.9) (5.23) (6.31)= 0.04 + 0.07D1 − 0.04D2 + 0.11D3 + εˆ HKt − 1.13εˆ HK ( t −1)(3.84) (1.48)(0.92)(1.19)(Hong Kong)(12.6)(1 + 0.7 L + 0.98L2 + 0.66L3 )(1 − L) log M t(6.55) (112) (6.40)= 0.03 + 0.02D1 − 0.04D2 − 0.03D3 + εˆ Mt − 0.96εˆ M ( t −1)(2.28) (0.49)(0.73)(0.51)(Malaysia)(26.6)(1 + 0.69L + 0.64L2 + 0.66L3 − 0.3L4 )(1 − L) log S t(4.89) (4.36) (4.56) (2.10)= 0.04 + 0.03D1 − 0.03D2 − 0.01D3 + εˆ St(4.70) (1.08)(0.96)(Singapore)(0.09)

where
D1 = dummy variable for the 1979 Oil Price Crisis;
D2 = dummy variable for the 1988 Bicentennial Celebration;
D3 = dummy variable for the 1989-90 Air Pilots Strike.
The impulse specification characterizes a temporary intervention, in which D1, D2 and D3 arezero for all periods except for the quarters in which the events occurred. These include impulse(or dummy) variables for the Oil Price Crisis for the period 1979(1)-1979(4), Bicentennial
Celebration for 1988(1)-1988(4) and Air Pilots Dispute for 1989(3)-1990(2). Witt et al. (1994)
obtained mixed results when their ARIMA models incorporated interventions. However, Table 9shows that none of the intervention variables used in this study is significant at the 5% level. In
addition, the constant term, and the autoregressive and moving average coefficients remain
significant at the 5% level.
94. Conclusion
This paper examines univariate time series ARIMA forecasting methods based on current andpast tourist arrivals from three Asian countries to Australia. A wide range of quantitative
forecasting techniques is available, from sophisticated regression and smoothing procedures tonaive models. Unlike many exponential smoothing procedures which attempt to fit the data to aparticular model, time series analysis of ARIMA models fits various models to historical data toobtain forecasts of tourist arrivals from Hong Kong, Malaysia and Singapore. Makridakis andHibon (1979) reported little or no improvement in empirical forecast accuracy from using
ARIMA models instead of simpler forecasting techniques, which seems to support Geurts andIbrahim’s (1975, p. 186) contention that: “... the model that fits best is not necessarily the one that
forecasts best”.
Unlike the numerous studies using the Box-Jenkins model, the following is undertaken in thispaper prior to computing forecast accuracy measures for both levels and logarithms:•

pre-testing for nonstationarity in the series by applying the augmented Dickey-Fuller tests forunit roots, and•model selection of the best fitting ARIMA model (using various criteria) to explain thepattern of tourist arrivals based on its preceding values, and current and previous random
errors for each origin or tourist-source country.

The use of such procedures, particularly tests for unit roots, improves the validity of using the
ARIMA models for forecasting and allows the forecaster to make informed judgments at eachstep as the results are presented by the statistical packages. Overall, this paper shows that bycomparing the root mean squared errors, lower post-sample forecast errors were obtained whentime series methods, such as the Box-Jenkins ARIMA and seasonal ARIMA models, were used.
The Box-Jenkins approach is often argued to be powerful but also complex to use in building
forecasting models. Besides forecasting accuracy, simplicity of technique and the cost aspects ofvarious techniques (which include labour skills, financial means, and time) are often stated as
being useful criteria considered by policy-makers and forecasters. Admittedly, in the Box-Jenkins
approach, the initial model selection stage encounters problems of practical implementation, and10

it can be time consuming if a large number of time series observations are to be analysed. Even if
van Doorn (1984, p. 25) is correct in the observation that: “the popularity and perceived
usefulness of a technique is directly related to the effort and sophistication required to implementthat technique”, the appropriateness of the use of the Box-Jenkins approach for tourism
practitioners should not be underestimated in the art of forecasting international tourist flows to aparticular destination. The Box-Jenkins approach is flexible and a broad class of general models
can be considered. Diagnostic tests are also used in this study to test the validity of the ARIMA
models selected, for example, the Lagrange multiplier test for serial correlation. The lack of suchdiagnostic checks in previous studies suggests that inferences from estimated models may behighly sensitive to the assumption of serial independence.

This paper has illustrated thechallenges that tourism planners, policy-makers and academics will encounter if ARIMA models
are to be used sensibly in building forecasting models.

Acknowledgements
The first author wishes to acknowledge the financial support of an Individual Research Grant,
Faculties of Economics & Commerce, Education and Law, University of Western Australia, and
the Australian Research Council. The second author wishes to acknowledge the financial supportof the Australian Research Council. This paper was revised while the second author was visiting
the Institute of Social and Economic Research at Osaka University.
References
Australian Bureau of Statistics. Various Issues of Short-Term Overseas Arrivals, 1975-1996,
Canberra.
Box, G.E.P. and G.M. Jenkins (1970). Time Series Analysis: Forecasting and Control, Holden
Day, San Francisco.
Chan, Y.-M. (1993). Forecasting Tourism: A Sine Wave Time Series Regression Approach.
Journal of Travel Research, 32, 58-60.
Choy, D. (1984). Forecasting Tourism Revisited. Tourism Management, 5, 171-176.
Chu, F.-L. (1998). Forecasting Tourist Arrivals. Journal of Travel Research, 36, 79-84.
Frechtling, D.C. (1996). Practical Tourism Forecasting, Butterworth-Heinemann, Oxford.
11

Geurts, M. and I. Ibrahim (1975). Comparing the Box-Jenkins Approach with the Exponentially
Smoothed Forecasting Model Application to Hawaii Tourists.

Journal of Marketing

Research, 12, 182-188.
Hylleberg, S., R.F. Engle, C.W.J. Granger and B.S. Yoo (1990). Seasonal Integration and
Cointegration. Journal of Econometrics, 99, 215-238.
Kim, J.H. (1999). Forecasting Monthly Tourist Departures from Australia. Tourism Economics,
5, 277-291.
Kulendran, N. and M.L. King (1997). Forecasting International Quarterly Tourist Flows using
Error-Correction and Time-Series Models. International Journal of Forecasting, 13, 319327.
Lim, C. and M. McAleer (2000a). Monthly Seasonal Variations: Asian Tourism to Australia.
Annals of Tourism Research, 28, 68-82.
Lim, C. and M. McAleer (2000b). A Seasonal Analysis of Asian Tourist Arrivals to Australia.
Applied Economics, 32, 499-509.
Makridakis, S. and M. Hibon (1979). Accuracy of Forecasting: An Empirical Investigation.
Journal of the Royal Statistical Society A, 142, 97-145.
Martin, C.A. and S.F. Witt (1989).

Forecasting Tourism Demand: A Comparison of the

Accuracy of Several Quantitative Methods. International Journal of Forecasting, 5, 7-19.
Turner L., N. Kulendran and V. Pergat (1995). Forecasting New Zealand Tourism Demand with
Disaggregated Data. Tourism Economics, 1, 51-69.
Turner, L.W., N. Kulendran and H. Fernando (1997). Univariate Modelling Using Periodic and
Non-Periodic Analysis: Inbound Tourism to Japan, Australia and New Zealand compared.
Tourism Economics, 3, 39-56.
Theil, H. (1966). Applied Economic Forecasting, North Holland, Amsterdam.
van Doorn, J.W.M. (1984). Tourism Forecasting and the Policymaker: Criteria of Usefulness.
Tourism Management, 5, 24-39.
Witt, C.A., S.F. Witt and N. Wilson (1994). Forecasting International Tourist Flows. Annals of
Tourism Research, 21, 612-628.

12

Table 1
Autoregressions of the Logarithm ofInternational Tourist Arrivals, 1975(1)-1989(4)VariableHong Kong

Malaysia

SingaporeConstant-0.01 (-0.04)

0.36 (1.04)

0.08 (0.36)

A t −10.20 (1.72)

0.19 (2.16)

0.04 (0.71)

A t −20.32 (2.76)-0.07 (-0.78)

0.05 (0.99)

A t −3-0.11(-0.91)

0.13 (1.46)-0.02 (-0.40)

A t −40.60 (5.09)

0.72 (8.86)

0.94 (17.6)

Note: t-statistics are given in parentheses.

13

Table 2
HEGY Tests for Seasonal Integration of Quarterly Tourist Arrivals

Origin Countryt( π 1 )t( π 2 )F( π 3 , π 4 )Hong Kong

-2.65

4.38

23.05

Malaysia

-1.46

4.06

10.73

Singapore

-2.35

2.24

15.45

Note: An intercept, three seasonal dummies and a time trend are included in the HEGY regressions.
n = 84 is the number of observations in each series. The critical values at the 5% level aretaken from Hylleberg et al. (1990) for 100 observations:t( π 1 ) = -3.53, t( π 2 ) = -2.94, and F( π 3 , π 4 ) = 6.60.

14

Table 3
RMSE for One-Quarter-Ahead Ex Post Forecasts of the Logarithm ofInternational Tourist Arrivals, 1975(1)-1989(4)OriginCountry

ARIMA

Seasonal
ARIMA

Hong Kong

0.016 (1)0.042 (2)

Malaysia

0.028 (1)0.061 (2)Singapore

0.035 (2)0.008 (1)Note: Figures in parentheses denote rankings.

15

Table 4
Theil’s U-Coefficients for Forecast Errors, 1990(1)-1996(4)Origin Country

ARIMA

Seasonal ARIMA

Hong Kong

0.468

0.717

Malaysia

0.439

0.728

Singapore

0.090

0.885

16

Table 5
Correlation Coefficients Between Actual and Predicted Values
Using Box-Jenkins ARIMA Models, 1990(1)-1996(4)Origin Country

ARIMA

Correlation
CoefficientHong Kong(3,1,1)

0.68

Malaysia(3,1,2)

0.67

Singapore(4,1,0)0.95

17

Table 6
Lagrange Multiplier Test for Serial Correlation
Origin Country

Model

F valuep-valueHong Kong

ARIMA(3,1,1)

3.03

0.06

Malaysia

ARIMA(3,1,2)

1.38

0.26

Singapore

ARIMA(4,1,0)

0.83

0.44

Note: The LM test statistics presented here are for second-order serial correlation. Test results forfourth-order serial correlation were qualitatively very similar.

18

Table 7
Sum of Squared Residuals (SSR) and Standard Deviation (SD) of Residuals
for ARIMA and Seasonal ARIMA Models
SSR
OriginCountrySD

ARIMA

Seasonal
ARIMA

ARIMA

Seasonal
ARIMA

Hong Kong

1.21

0.93

0.15
(56)

0.13
(54)

Malaysia

1.13

0.96

0.14
(56)

0.14
(49)

Singapore

0.88

0.79

0.13
(55)

0.12
(55)

Note: Sample sizes are given in parentheses.

19

Table 8
Forecast RMSE for Different Horizons
Forecast Horizon

Hong Kong

Malaysia

Singapore

1

0.016

0.03

0.03

2

0.03

0.07

0.03

3

0.59

0.20

0.01

4

0.44

0.08

0.02

5

0.13

0.009

0.05

6

0.02

0.10

0.03

7

0.05

0.17

0.007

8

0.07

0.002

0.01

RMSE: up to 4 quarters ahead

0.439

0.126

0.033

RMSE: 5 to 8 quarters ahead

0.073

0.151

0.038

RMSE: 1990(1)-1996(4)

0.362

0.169

0.096

(quarters ahead)20

Table 9
Impact Effects on International Tourist Arrivals to Australia ofInterventions in 1979, 1988 and 1989

OriginOil Crisis

Bicentennial

Air Pilots

Country

1979

Celebration

Dispute

1988

1989

0.07

-0.04

0.11

(1.48)(-0.92)(1.19)

0.02

-0.04

-0.03

(0.49)(-0.73)(-0.51)

0.03

-0.03

-0.006(1.08)(-0.96)(-0.09)Hong Kong

Malaysia

SingaporeNote: t-statistics are given in parentheses.

21Fig u re 1
Actu al, Fitted an d Residu als fro m ARIMA(3 ,1 ,1 ) Mo d el
o f Tou rist Arrivals fro m Hon g Ko n g , 19 7 5 (1 )-1 9 89 (4 )1.0Actual, Fitted and Residuals

0.5
0.0
0.4

-0.5

0.2

-1.0

0.0
-0.2
-0.4
-0.6
76 77 78 79 80 81 82 83 84 85 86 87 88 89
Year
Residual

Actual22

FittedFigure 2
Actual, Fitted and Residuals from ARIMA(3,1,2) Modelof Tourist Arrivals from Malaysia, 1975(1)-1989(4)1.0Actual, Fitted and Residuals

0.5
0.0
0.4

-0.5

0.2

-1.0

0.0
-0.2
-0.4
76 77 78 79 80 81 82 83 84 85 86 87 88 89
Year
Residual

Actual23

FittedFig u re 3
Actu al, Fitted an d Residu als fro m ARIMA(4 ,1 ,0 ) Mo d el
o f To u rist Arriv als fro m Sing ap o re, 1 9 7 5 (1 )-1 9 89 (4 )1.0Actual, Fitted and Residuals

0.5
0.0
0.4

-0.5

0.2

-1.0

0.0
-0.2
-0.4
77 78 79 80 81 82 83 84 85 86 87 88 89
Year
Residual

Actual24

Fitted