Combining volatility and smoothing forecasts of UK demand for international tourism

Tourism Management 30 (2009) 495–511

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Combining volatility and smoothing forecasts of UK demand for international tourism John T. Coshall* The Business School, London Metropolitan University, North Campus, Holloway Road, London N7 8HN, United Kingdom

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 April 2008 Accepted 5 October 2008

Univariate volatility models are applied to UK tourism demand to the country’s most popular international destinations. Volatility is a concept borrowed from Finance. The fact that significant volatility models are found for ten of the twelve destinations examined shows that the volatility concept has relevance to tourism demand. Volatility models are able to quantify the impacts of positive and negative shocks on tourism demand. The impacts of negative shocks vary in magnitude and duration according to the destination involved and the nature of the shock. The forecasting capability of these models has never been assessed in the tourism field. They are shown to generate highly accurate forecasts, but become optimal when combined with forecasts obtained from exponential smoothing models. Two methods of combining individual forecasts are considered. Bias in individual volatility and smoothing models and in combinations of them is examined. Ó 2008 Elsevier Ltd. All rights reserved.

Keywords: Volatility GARCH EGARCH Shock Forecast combination Bias

1. Introduction The last thirty years have seen many studies of international tourism demand forecasting by both tourism researchers and practitioners. Reliable forecasting underpins rational planning in tourism and related industries. It acts as a basis for the development of supply-side facilities including urban and rural transportation, heritage sites, hospitality and catering, promotion of attractions, retail, entertainment and other support services. It is also an aspect of pricing policies related to international transport, airport taxes, urban congestion charging and environmental quality. In that tourism makes a major contribution to nations’ trade performances, economic development and prosperity, reliable forecasting is needed to assist decision makers plan effectively and resourcefully. Forecasts of tourism volume are a prime requirement for destinations to foresee infrastructure and superstructure development needs (Sheldon, 1993). Quantitative approaches to tourism forecasting fall into two groups – causal econometric models and time series models. The former models select explanatory variables on the basis of economic theory. The most recent developments in this field include combinations of time varying parameter models (TVP), the linear almost ideal systems approach (LAIDS) and cointegration/ error correction models (ECM), particularly TVP–ECM (Li, Wong, Song, & Witt, 2006) and TVP–LAIDS (Li, Song, & Witt, 2006). * Tel.: þ44 (0) 20 7423 0000; fax: þ44 (0) 20 7133 3076. E-mail address: [email protected] 0261-5177/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tourman.2008.10.010

The focus in this paper is on the second group of models of tourism demand. Time series methods require only historical data related to the subject matter at hand. Foremost among this class of models is the well-documented univariate ARIMA approach, typically applied to long-haul travel movement (Chu, 2008; Kim, 1999; Kim & Moosa, 2001; Kulendran & Witt, 2003; Lim & McAleer, 2000a, 2000b; du Preez & Witt, 2003). The popularity of ARIMA models reflects their general ability to produce accurate forecasts (Chu, 1998a; Lim & McAleer, 2002). In a study of inbound tourism to Korea, recent research has extended the ARIMA approach by incorporating the concept of ‘volatility’ (Kim & Wong, 2006). This is referred to as ‘ARIMAvolatility’ modelling throughout this paper. The underlying premise is that international tourism demand is susceptible to the impact of shocks to the industry that lead to periods of relatively large upturns and downturns in activity, i.e. volatile behaviour. The concept of volatility is borrowed from Finance, motivated by the observation that large market returns (of either sign) tend to follow large returns, and small returns (of either sign) tend to follow small returns (Brooks, 2004). Clusters of volatile behaviour become evident over time. Negative shocks (or ‘bad news shocks’ in the parlance of Finance) that have the potential to generate volatile behaviour in tourism demand are well-documented in the literature. They include political instability (Gartner & Shen, 1992), terrorism (Bhattarai, Conway, & Shrestha, 2005; Coshall, 2003; Wahab, 1996), crime and violence (Tynon & Chavez, 2006), disease (Huan, Beaman, & Shelby, 2004), natural disasters (Milo & Yoder, 1991) and war (Ryan, 1991). Of these, terrorism has become the

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most significant threat to the tourism and hospitality industries in recent years (Israeli & Reichel, 2003). Examples of positive (or ‘good news’) shocks to tourism flows would be injections of capital investment at a destination or a marked increase in marketing activity. In Finance, there is evidence in several contexts that negative shocks are associated with greater volatility than are positive shocks of the same magnitude. Although this latter concept is most rarely studied in the tourism field, one study produced contrary findings to the effect that the impacts of positive and negative shocks on monthly inbound tourist arrivals to the Maldives are much the same (Shareef & McAleer, 2007). There have been very few applications of univariate volatility models in the tourism field, yet these models have the potential to assist policy formulation either before or at the moment of a shock. The industry needs to be able to quantify the likely impact of shocks to the demand system and have in place contingency plans to adapt to the impact of a volatile market. In this respect, Kim and Wong (2006) mention that incorporation of volatility into the modelling process may well lead to more accurate forecasts of international tourism demand. This paper is the first to formally test this notion and generally evaluates the volatility concept in the context of UK demand for international tourism to 12 destinations. The differential impacts of negative and positive shocks on volatility are also examined. Less frequently applied in the tourism demand literature are exponential smoothing time series models, despite evidence that they often provide adequate forecasts of directional and trend changes in tourism demand (Cho, 2003; Saunders, Sharp, & Witt, 1987; Witt, Newbould, & Watkins, 1992; Witt & Witt, 1991). The inherent rationale is that smoothing models incorporate parameters reflecting any trend and/or seasonality that is present. The parameters control how rapidly the model reacts to changes in the process that generates the time series (Gardner, 1985). Smoothing models have ready application for forecasting tourism demand, since they can react quickly to changes in economic conditions and recent observations tend to be assigned larger weights in the forecasting process. Naı¨ve models may also be included under the time series heading (Chan, 1993; Coshall, 2006). In particular, a Naı¨ve 2 process assumes that the growth rate in tourism demand at one particular time period equals the growth rate observed at the previous, equivalent time period. This model is often used as a standard for comparing the forecasting accuracy of competing models. However, it is worthy of note that Naı¨ve models can sometimes outperform more formal forecasting methods when applied to tourism demand (Turner & Witt, 2001). ARIMA-volatility models are here compared with exponential smoothing models and Naı¨ve 2 in terms of their forecasting accuracy for the most popular destinations for UK international tourism and over different forecasting horizons. Incorporation of the concept of volatility and the rapid reaction to changes in data patterns respectively justify the potential for the ARIMA-volatility and exponential smoothing approaches to forecasting international tourism demand. The question therefore arises as to whether the advantages of both methods can be pooled to generate combined forecasts that are significantly superior to those generated by the individual models. Application of combination forecasts in tourism is rare (Chu, 1998b; Li, 2007; Oh & Morzuch, 2005; Wong, Song, Witt, & Wu, 2007). While the study of Li (2007) concludes that combination forecasts are superior to individual forecasts in terms of accuracy, that of Wong et al. (2007) suggests that the relative performance of the methods varies according to the combination procedure used and the origin country or region studied. This paper adopts two commonly employed combination methods to pool ARIMA-volatility and exponential smoothing forecasts. A final consideration is that most empirical studies of international tourism demand concentrate on the identification of models with minimum forecasting error. Such models implicitly assume

that the obtained forecasts are unbiased, yet evidence suggests that this assumption is often invalid (Witt, Song, & Louvieris, 2003). It is argued that part of the model evaluation process should involve examination for bias in forecasts. Tests of forecasting bias are conducted for the individual and combined models used in this study. All evaluations are both model specific and destination specific. 2. Models and methodology This section explains the logic underlying the ARIMA-volatility models and describes application of exponential smoothing models. An important aspect of combining forecasts is to perform encompassing tests. Such tests examine whether competing forecasts may be combined in order to generate a forecast that is superior to the individual forecasts. Two methods for combining forecasts are described. Measures of forecasting accuracy used to compare competing models are presented. A test for forecasting bias is introduced. 2.1. Volatility models Although uncommon in tourism studies, volatility models have been very popular in empirical research in Finance and Econometrics since the early 1990s. The models are based on influential papers by Engle (1982) and Bollerslev (1986). All volatility models start off with a ‘mean equation’, which is commonly a standard ARIMA (as here) or regression model. Whichever is used, it contains error or residual term over time, et. At the root of volatility modelling is the distinction between conditional (stochastic) and unconditional (constant) errors. The conditional variance of the error terms is denoted by s2t and is time varying. Volatility modelling involves adding a ‘variance equation’ to the original mean equation and which in turn models the conditional variance. One of the most widely used volatility models goes under the name of a ‘generalised autoregressive conditional heteroscedasticity’ GARCH scheme and was developed by Bollerslev (1986). The conditional variance is modelled as:

s2t ¼ a0 þ

q X i¼1

ai e2ti þ

p X

bj s2tj ;

(1)

j¼1

where a0 > 0 and ai and bj  0 to eliminate the possibility of a negative variance. However, it has been argued that in practice, this constraint may be over-restrictive (Nelson & Cao, 1992; Tsai & Chan, 2008). The specification in (1) allows for the conditional variance to be dependent on past information. It is explained by past short-run shocks represented by the lag of the squared residuals (e2i ) obtained from the mean equation and by past longerrun conditional variances (s2j ). Eq. (1) is referred to as a GARCH(p,q) P P process. In GARCH models, ai þ bj should be less than unity to satisfy stationarity conditions. If the bj are all zero, Eq. (1) reduces to what is called an ARCH(q) process, which is the earliest form of volatility model developed by Engle (1982). It is rare for the order (p,q) of a GARCH model to be high; indeed the literature suggests that the parsimonious GARCH(1,1) is often adequate for capturing volatility in financial data (see, for example, Chen & Lian, 2005). A potential problem with applying the model of Eq. (1) to tourism demand data is that it presumes that the impacts of positive and negative shocks are the same or ‘symmetric’. This is because the conditional variance in these equations depends on the magnitude of the lagged residuals, not their sign. The possibility that a negative shock to tourism movement causes volatility to rise by more than would a positive shock of the same magnitude remains worthy of analysis. Such a consideration led to the development of ‘asymmetric’ volatility models, specifically the threshold GARCH (TGARCH) (Glosten, Jaganathan, & Runkle, 1993; Zakoı¨an, 1994) and the exponential GARCH (EGARCH) (Nelson, 1991).

J.T. Coshall / Tourism Management 30 (2009) 495–511

The threshold model is a simple extension of the GARCH scheme with extra term(s) added to account for possible asymmetries. TGARCH extends the GARCH(p,q) model of equation (1) via:

s2t ¼ a0 þ

q  X



ai e2ti þ fi e2ti Iti þ

i¼1

p X

bj s2tj ;

(2)

j¼1

where Iti are dummy variables equal to unity if eti < 0 i.e. a negative shock or ‘bad news’ and equal to zero if eti > 0 i.e. a positive shock or ‘good news’. If fi > 0 in Eq. (2), then a negative shock increases volatility. Again, the values of p and q tend to be low in empirical applications. The EGARCH(p,q) model of Nelson (1991) can also accommodate asymmetry and specifies the conditional variance in a different way:

loge s2t ¼ a0 þ

  X q   P X e  e ai  ti  þ fi ti þ bj loge s2tj : s s

i¼1

ti

ti

(3)

j¼1

One reason that EGARCH has been popular in Financial applications is that the conditional variance, s2t , is an exponential function, thereby removing the need for constraints on the parameters to ensure a positive conditional variance (Longmore & Robinson, 2004). The model also permits asymmetries via the fi term in (3) and if fi < 0, negative shocks lead to an increase in volatility. If fi ¼ 0, the model is symmetric. The values of p and q are very rarely high and the EGARCH model tends to be parsimonious. 2.2. Exponential smoothing models These models use a linear combination of the previous values of a series for generating forecasts of its future values. The main question is how much weight should be attached to each of the past observations, with the likelihood being that more recent readings have more influence on future forecasts of tourism demand than do observations a long way in the past. The simple, one parameter exponential smoothing model is applicable to series with no trend or seasonality and is defined as:

b b Y tþ1 ¼ aYt þ ð1  aÞ Y t ;

(4)

b where Y tþ1 is the forecasted value of the series at time (t þ 1), Yt is the observed value of that series at time t and a is the smoothing (or ‘weighting’) parameter with 0  a 1. The ‘optimal’ value of a is defined as that which minimises the sum of the squares of the errors (SSE) and is found by means of a grid search of the form a ¼ 0(0.1)1 or a ¼ 0(0.01)1. High values of a in Eq. (4) imply that the impact of historical observations dies out quickly and vice versa. Given the strong influence of seasonal factors on tourism demand, the simple smoothing model needs be adapted to incorporate this feature. Also tourism numbers to many international destinations have been steadily increasing over the last thirty years, so any such trends present need to be integrated into the model. Two other parameters, g, a smoothing parameter for the trend and d, a smoothing parameter for seasonal components, may be added to the simple model. Both g and d lie between 0 and 1 inclusive and their numerical values are again obtained by a grid search with the objective of minimising the SSE. Each of these three smoothing parameters is updated with its own exponential smoothing equation and the equations involved have been presented in a recent article in this journal (Cho, 2003). Suffice it to say that large values for g give more weight to recent estimates of the trend component, with small values giving more weight to historical estimates of the trend component. Similarly, large values of d give most weight to the most recent estimates of the seasonal component and small values give more weight to historical estimates of this component.

497

In particular, Winters’ additive and multiplicative exponential smoothing models incorporate these three parameters. The former model is appropriate for a series with a linear trend and a seasonal effect that does not depend on the level of the series. The latter model is appropriate for the same type of trend, but when the seasonal effect does depend on the level of the series.

2.3. Encompassing tests Encompassing tests are concerned with the situation of usually two alternative models and examine whether one model adds anything to the explanatory power of the other model (Holden & Thompson, 1997). If the forecasts obtained from model 1 do not contain information missing from model 2, then the forecasts obtained from model 1 are said to be ‘encompassed’ by those derived from model 2. If one forecast does not encompass another, then forecasting can be improved by combining the two individual forecasts; both models have something to contribute. To assess whether combining forecasts is worthwhile, it is here necessary to test null hypothesis that the forecasts obtained from an ARIMAb volatility model over a time period t ( Y VOLðtÞ ) encompass those b obtained from an exponential smoothing model ( Y EXPðtÞ ) over the same time period. (This is assuming that the Naı¨ve 2 model has no significant role to play.) Rejection of the null suggests that the two individual forecasts should be combined. Several tests have been developed over the years to test the above null (for a review, see Harris & Sollis, 2003, p. 250). It has been shown that many such encompassing tests are over-sized in the case of forecast errors that are not normally distributed. The HLN encompassing test (Harvey, Leybourne, & Newbold, 1998) overcomes this problem. Denoting forecast errors for t time periods b derived from an ARIMA-volatility model as eVOLðtÞ ¼ YðtÞ  Y VOLðtÞ , where Y(t)are the observed values over this time interval, and the equivalent forecast errors from an exponential smoothing model as b eEXPðtÞ ¼ YðtÞ  Y EXPðtÞ , then the test statistic is defined as:

c HLN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi; varðcÞ

(5)

where ct ¼ eVOL(t)$(eVOL(t)  eEXP(t)) and c is the mean of the ct. Under the above null, the HLN statistic is asymptotically standard normally distributed.

2.4. Combining forecasts b Let weighted, combined forecasts ( Y COMBINEDðtÞ ) for t time periods be defined as:

b b b Y COMBINEDðtÞ ¼ k Y VOLðtÞ þ ð1  kÞ Y EXPðtÞ ;

(6)

where k and (1  k) are weights assigned to the ARIMA-volatility and exponential smoothing models respectively and 0 < k < 1. A simple method of combining two forecasts is to take their arithmetic mean i.e. set k ¼ ½ in (6). This method is one of the two combination approaches applied in this study, since there is evidence that equal weights can be accurate for many types of forecasting (Armstrong, 2001). The forecast errors from the combined forecasts of Eq. (6) are:

b eCOMBINEDðtÞ ¼ YðtÞ  Y COMBINEDðtÞ or

b b eCOMBINEDðtÞ ¼ YðtÞ  k Y VOLðtÞ  ð1  kÞ Y EXPðtÞ which reduces to:

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eCOMBINEDðtÞ ¼ keVOLðtÞ þ ð1  kÞeEXPðtÞ :

(7)

The variance of the combined forecast errors in Eq. (7) is:

vareCOMBINEDðtÞ ¼ k2 vareVOLðtÞ þ ð1  kÞ2 vareEXPðtÞ   þ 2kð1  kÞcov eVOLðtÞ ; eEXPðtÞ ;

(8)

where ‘cov’ represents covariance of the forecasting errors from the two models. Differentiation with respect to k establishes that the weight (k*) which minimises the variance of the combined errors in Eq. (8) is:

h i  vareEXPðtÞ  cov eVOLðtÞ ; eEXPðtÞ :  k* ¼ vareVOLðtÞ þ vareEXPðtÞ  2cov eVOLðtÞ ; eEXPðtÞ

(9)

This method of establishing k* is called the ‘variance–covariance’ approach and has had application in generating combined forecasts for quarterly tourist arrivals in Hong Kong (Wong et al., 2007). However, there has been a suggestion that the ‘optimal’ weight of Eq. (9) is over-complex. Following the proposal of Bates and Granger (1969), Li (2007) neglected the sample covariance terms in (9) in a study of quarterly UK outbound tourism to the United States. Also, since var eEXP(t) and var eVOL(t) are unknown, P P they were replaced with tt ¼ 1 e2EXPðtÞ and tt ¼ 1 e2VOLðtÞ respectively, to derive the weight:

Pt

*

2 t ¼ 1 eEXPðtÞ : Pt 2 2 t ¼ 1 eEXPðtÞ þ t ¼ 1 eVOLðtÞ

k ¼ Pt

(10)

This article uses the weights k* and (1  k*) of Eq. (10) as a second method of combining forecasts from the two models.

2.5. Measures of forecasting accuracy and a test for bias The three individual and two combined models used in this study are assessed in terms of forecasting accuracy, as well as bias. Studies of international tourism demand have used a variety of measures to assess forecasting accuracy (Song, Witt, & Jensen, 2003). However, the mean absolute percentage error (MAPE) and root mean squared percentage error (RMSPE) are amongst the most commonly used measures of error magnitude. Also used in this analysis is the median absolute percentage error (MedAPE). These three accuracy criteria have the advantage of being measured in unit-free terms (Witt & Witt, 1991). One method of testing for bias in forecast errors that has been applied in the context of tourism demand is to regress the observed values against the forecasts via ordinary least squares (Witt et al., 2003):

bt þ h Yt ¼ a þ b Y t

(11)

Failure to reject the joint null that a ¼ 0 and b ¼ 1 in Eq. (11) by means of the Wald test indicates that the forecasts are unbiased. However, it has been argued that there are problems of interpretation with the above test, in that rejection of the null does not lead to clear alternatives (Holden & Peel, 1990; Sanders & Manfredo, 2001). Consequently, it has been suggested that focus should be on b t , and unbiasedness should be the forecast errors, et ¼ Yt  Y tested via the least squares regression:

et ¼ q þ ht

(12)

The null of unbiasedness, q ¼ 0, is tested with a t statistic (Pons, 2000; Witt et al., 2003).

3. Data Twelve time series are used for the modelling processes. They represent quarterly, outbound UK tourism numbers (thousands) by air to the twelve most frequently visited destinations in 2006. These data are collected by the UK International Passenger Survey (IPS) and are published by the UK Office for National Statistics (ONS) (HMSO, 1977–2008). The IPS is based on face-to-face interviews with a random sample of passengers as they enter or leave the UK by the major air, sea and channel tunnel routes. Recently, about 225,000 travellers have been interviewed each year, representing a 0.2% sample. In order of frequency of visits by UK tourists in 2006, the twelve destinations considered here are Spain, the United States, the Irish Republic, France, Germany, Italy, the Netherlands, the Canary Islands, Greece, Switzerland, Canada and Cyprus. The data span 1976 Q1 (quarter 1) to 2007 Q3 inclusive, save for the Canary Islands and Cyprus for which the first available observation is at 1979 Q4 and Germany, where data are considered only post-unification in 1991 Q1. Three years worth of data from 2004 Q4 to 2007 Q3 inclusive are held back for the purpose of forecast evaluation. It should be noted that the ONS reports Spain and the Canary Islands as separate destinations, due to the relatively high UK demand associated with each of them. The above twelve destinations represent 67.7% of global outbound tourism by UK residents by air in 2006, with Spain alone accounting for 15.0% of that global total. The ONS classifies the United States, Canada and the Canary Islands as ‘Rest of the World’ destinations and these three countries account for 54.8% of UK tourism to this sector. In two instances, the ONS details higher numbers of UK visitors in 2006 than are reported to Canada, but both are at a regional scale; they involve East Africa and the Middle East with no disaggregation by country, so these figures are not used here. Cyprus is classified as a ‘Mediterranean’ destination by the ONS and constitutes 40.1% of UK tourism to that area. The remaining eight European destinations above account for 74.6% of UK outbound tourism to that continent. Albania, Bulgaria, the Czech Republic, Hungary, Poland and Romania account for 7.8% of UK tourism to Europe, but they are grouped together as ‘Eastern Europe’ for reporting purposes by the ONS and are therefore not used. UK tourism numbers to these twelve destinations are plotted in Fig. 1A–L. Features common to a greater or lesser extent on these plots involve seasonality and upward trending behaviour. The tourism numbers to Spain (Fig. 1A) and Greece (Fig. 1I) are textbook examples of multiplicative (increasing amplitude) seasonality, whereas those to the Netherlands (Fig. 1G) reflect additive (constant amplitude) seasonality. The impact of the Gulf War of 1991 Q1 is particularly evident on the plots for the United States (Fig. 1B) and Cyprus (Fig. 1L) and represents a period of relative volatility. The Gulf War impact is manifest on many of the other plots, but on a more modest scale. The shock representing the event of September 11th 2001 is clearly evident on the plot for the United States. However, this shock generates less volatile behaviour on the plots for the Irish Republic, France, Germany and Italy (Figs. 1C–F) and appears to be absent for the remaining destinations. 4. Derivation of volatility and exponential smoothing models The first stage in the volatility modelling process is to specify the mean equation. The method of applying ARIMA models to tourism demand data has been well-described in the literature (Chen, Kang, & Yang, 2007; Kim & Moosa, 2001; Lim & McAleer, 2000a; Vu & Turner, 2006). ARIMA models are fitted up to and including 2004 Q3 for all twelve destinations. It is clear from the plots in Fig. 1 that seasonal differencing is required as part of the process of making each series stationary. A further consideration when applying

J.T. Coshall / Tourism Management 30 (2009) 495–511

12000

A

Spain

10000

000's of passengers

8000

6000

4000

2000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

6000

B

USA

5000

000's of passengers

4000

3000

2000

1000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. Numbers of UK air passengers to the twelve most popular destinations.

499

500

J.T. Coshall / Tourism Management 30 (2009) 495–511

4000

C

Irish Republic

000's of passengers

3000

2000

1000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

4000

D

France

000's of passengers

3000

2000

1000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. (continued).

ARIMA models is trend stationarity, for which formal tests are available. The Augmented Dickey Fuller (ADF) test is often applied to tourism data for this purpose (Lim & McAleer, 2000b). It has been argued that the ADF is not a robust test, so the Phillips–Perron (PP)

for trend stationarity is also conducted in this study. In both instances, the null is that a particular series possesses a unit root(s) i.e. is not trend stationary. Once stationarity has been established, examination of the autocorrelation function plot (ACF) and partial

J.T. Coshall / Tourism Management 30 (2009) 495–511

3500

E

501

Germany

000's of passengers

3000

2500

2000

1500

1000

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

4000

F

Italy

000's of passengers

3000

2000

1000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. (continued).

autocorrelation plot (PACF) over several quarterly lags suggests which autoregressive and/or moving average terms should be included in the ARIMA model. The choice of ARIMA model for the data is based on its being parsimonious, having significant

parameters, errors that are white noise and minimum Schwarz Bayesian criterion (SBC) (Schwarz, 1978). The second stage is to specify the conditional variance equation. Given that the literature emphasises the parsimonious nature of

502

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2500

G

Netherlands

000's of passengers

2000

1500

1000

500

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

2500

H

Canar Islands

000's of passengers

2000

1500

1000

500

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. (continued).

the GARCH and EGARCH models, all combinations of parameter values in the ranges p ¼ 0, 1, 2 and q ¼ 1, 2 are considered for each of the variance Eqs. (1)–(3), as per Angelidis, Benos, and Degiannakis (2004). Parameter estimates are obtained by means of maximum likelihood using the Berndt, Hall, Hall, and Hausman (1974) algorithm in the EViews 5 software package or the Marquardt algorithm if the BHHH fails to converge.

The form of the mean equations obtained is presented in the first column of Table 1. As indicated, data for some destinations required natural logarithms to be taken in order to reduce the variation. Adopting conventional notation, the parentheses in the (p,d,q)(P,D,Q)S notation enclose the respective orders of nonseasonal and seasonal parameters obtained and S ¼ 4 is the order of seasonality (quarterly). The orders of differencing required to

J.T. Coshall / Tourism Management 30 (2009) 495–511

4000

I

503

Greece

000's of passengers

3000

2000

1000

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

J

Switzerland

000's of passengers

1500

1000

500

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. (continued).

eliminate any trend and seasonality are represented by d and D respectively. Results for the ADF and PP tests are also presented in Table 1. Associated significance levels greater than 0.05 indicate a unit root(s) and that differencing is required. EViews permits

testing for unit roots in levels, first differences and second differences. Here, d ¼ 1 for all series requiring non-seasonal differencing. Table 2 presents the conditional variance equations associated with the mean equations of Table 1. (Note that EViews includes a constant in

504

J.T. Coshall / Tourism Management 30 (2009) 495–511

K

Canada

1400

1200

Canada

1000

800

600

400

200

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date

L

Cyprus

1200

1000

000's of passengers

800

600

400

200

0

Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 1976 1977 1978 1979 1981 1982 1983 1984 1986 1987 1988 1989 1991 1992 1993 1994 1996 1997 1998 1999 2001 2002 2003 2004 2006 2007

Date Fig. 1. (continued).

the variance equation by default.) Should more than one conditional variance equation have significant coefficients while also meeting any necessary constraints, then the model with the maximum log-likelihood (LL) statistic is reported in Table 2. The Ljung–Box Q(12) statistics test for remaining serial autocorrelation in the residuals for up to

twelve quarterly lags. All are non-significant indicating that the mean equations are not incorrectly specified. The QSQ(12) statistics test for remaining ARCH in the variance equation up to a lag of twelve quarters and are all non-significant, as is required in order to avoid model misspecification (Quantitative Micro Software, 2004).

J.T. Coshall / Tourism Management 30 (2009) 495–511

505

Table 1 Mean equations (ARIMA). Spain (logs)

Constant

AR(1)

ARIMA(1,0,0)(0,1,1)4 t statistic Significance

0.062 3.986 0.000 ADF t ¼ -3.768 p ¼ 0.004

0.761 13.689 0.000 PP t ¼ -3.793 p ¼ 0.004

SMA(1) -0.414 -4.372 0.000

United States

Constant

MA(1)

ARIMA(0,0,3)(0,1,0)4 t statistic Significance

144.169 3.912 0.000 ADF t ¼ -3.284 p ¼ 0.018

0.662 10.532 0.000 PP t ¼ -3.828 p ¼ 0.004

Republic of Ireland (logs)

SMA(1)

ARIMA(0,1,0)(0,1,1)4 t statistic Significance

-0.469 -6.476 0.000 ADF t ¼ -1.323 p ¼ 0.616

PP t ¼ -0.873 p ¼ 0.793

France

Constant

AR(1)

SAR(1)

SMA(1)

ARIMA(1,0,0)(1,1,1)4 t statistic Significance

85.722 3.974 0.000 ADF t ¼ -4.266 p ¼ 0.001

0.711 9.280 0.000 PP t ¼ -4.221 p ¼ 0.001

-0.920 -21.150 0.000

0.885 10.095 0.000

Germany

Constant

ARIMA(0,0,0)(0,1,0)4 t statistic Significance

61.756 8.625 0.000 ADF t ¼ -3.125 p ¼ 0.028

PP t ¼ -4.255 p ¼ 0.001

Italy

Constant

AR(1)

ARIMA(1,0,0)(0,1,0)4 t statistic Significance

73.391 3.292 0.001 ADF t ¼ -4.395 p ¼ 0.001

0.690 8.853 0.000 PP t ¼ -4.204 p ¼ 0.001

Netherlands

AR(1)

SMA(1)

ARIMA(1,1,0)(0,1,1)4 t statistic Significance

-0.300 -2.848 0.005 ADF t ¼ -0.230 p ¼ 0.930

-0.697 -8.586 0.000 PP t ¼ 0.864 p ¼ 0.995

Canary Islands (logs)

SMA(1)

ARIMA(0,1,0)(0,1,1)4 t statistic Significance

-0.653 -10.377 0.000 ADF t ¼ -1.674 p ¼ 0.441

PP t ¼ -2.530 p ¼ 0.112

Greece (logs)

Constant

AR(1)

ARIMA(1,0,0)(0,1,0)4 t statistic Significance

0.054 2.205 0.028 ADF t ¼ -5.594 p ¼ 0.000

0.552 7.707 0.000 PP t ¼ -5.691 p ¼ 0.000

Switzerland (logs)

MA(1)

ARIMA(0,1,1)(0,1,0)4 t statistic Significance

-0.805 -27.849 0.000 ADF t ¼ -2.708 p ¼ 0.076

PP t ¼ -2.582 p ¼ 0.100

Canada (logs)

Constant

AR(1)

ARIMA(1,0,0)(1,1,0)4 t statistic Significance

0.034 2.283 0.025 ADF t ¼ -2.953 p ¼ 0.043

0.662 8.749 0.000 PP t ¼ -5.709 p ¼ 0.000

Cyprus (logs)

Constant

AR(1)

ARIMA(1,0,0)(0,1,1)4 t statistic Significance

0.069 8.751 0.000 ADF t ¼ -4.167 p ¼ 0.001

0.705 41.402 0.000 PP t ¼ -4.480 p ¼ 0.000

Significant volatility models are obtained for ten of the twelve destinations, indicating that the volatility concept has distinct relevance in a tourism context. In Table 2, nine destinations involve EGARCH volatility models and only the United States involves a GARCH model. No volatility models are obtained for UK tourism

MA(2) 0.805 50.066 0.000

MA(3) 0.819 13.502 0.000

SAR(1) -0.350 -4.105 0.000 SMA(1) -0.172 -6.978 0.000

flows to the Netherlands or Canada, which is probably unsurprising given the relatively consistent behaviour over time evident in Fig. 1G and K. The presence of significant f coefficients in Table 2 indicates asymmetric responses to shocks to tourism demand. Negative shocks increase volatility in UK tourism numbers more

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Table 2 Conditional variance equations.

a0 Spain EGARCH(1,0) t Statistic Significance United States GARCH(1,0) t Statistic Significance

10.434 41.228 0.000 LL ¼ 139.468

a1

SBC ¼ 2.279

a2

b1

Q(12) ¼ 9.023 (0.530)

0.954 51.125 0.000 QSQ(12) ¼ 12.303 (0.265)

Q(12) ¼ 11.272 (0.258)

0.373 2.542 0.011 QSQ(12) ¼ 4.958 (0.838) 0.457 2.403 0.016 QSQ(12) ¼ 5.327 (0.914)

5042.467 3.290 0.000 LL ¼ 701.771

SBC ¼ 12.942

Republic of Ireland EGARCH(1,1) t Statistic Significance

8.923 8.029 0.000 LL ¼ 143.002

0.499 3.112 0.002 SBC ¼ 2.863

Q(12) ¼ 7.323 (0.772)

France EGARCH(0,1) t Statistic Significance

8.126 34.489 0.000 LL ¼ 610.755

0.740 3.153 0.002 SBC ¼ 11.788

Q(12) ¼ 4.471 (0.878)

QSQ(12) ¼ 10.627 (0.302)

Germany EGARCH(1,1) t Statistic Significance

0.213 0.249 0.803 LL ¼ 32,0849

1.305 4.853 0.000 SBC ¼ 12.891

Q(12) ¼ 17.070 (0.147)

0.879 9.353 0.000 QSQ(12) ¼ 7.648 (0.812)

Italy EGARCH(1,0) t Statistic Significance

0.965 0.866 0.386 LL ¼ 626.238

Q(12) ¼ 10.841 (0.457)

0.887 6.662 0.000 QSQ(12) ¼ 14.066 (0.229)

Q(12) ¼ 9.473 (0.578)

1.004 61.834 0.000 QSQ(12) ¼ 3.848 (0.974)

Q(12) ¼ 9.602 (0.567)

0.861 2.956 0.003 QSQ(12) ¼ 13.709 (0.250)

SBC ¼ 11.557

Netherlands

No model found

Canary Islands EGARCH(1,0) t Statistic Significance

11.069 44.461 0.000 LL ¼ 125.586

Greece EGARCH(1,0) t Statistic Significance

0.605 0.481 0.630 LL ¼ 81.711

SBC ¼ 1.315

1.110 3.017 0.003 LL ¼ 176.820

0.281 2.177 0.030 SBC ¼ 3.001

Q(12) ¼ 12.592 (0.321)

0.783 10.744 0.000 QSQ(12) ¼ 5.539 (0.902)

1.855 8.439 0.000 SBC ¼ 1.398

1.941 9.486 0.000 Q(12) ¼ 12.197 (0.202)

1.000 15.963 0.000 QSQ(12) ¼ 14.904 (0.136)

Switzerland EGARCH(1,1) t Statistic Significance Canada

No model found

Cyprus EGARCH(2,2) t Statistic Significance

12.925 24.544 0.000 LL ¼ 84.634

SBC ¼ 2.500

b2

f1 0.187 2.583 0.010 1.107 2.300 0.021

0.444 4.503 0.000

0.118 1.904 0.029

The significance levels associated with Q(12) and QSQ(12) are shown in brackets.

than would positive shocks of the same magnitude only in the cases of travel to Spain, Switzerland and the United States. Given that tourism is often perceived as a vulnerable, unstable industry (Sinclair & Tsegaye, 1990), one might have expected this finding on a greater scale. However, these results do support the finding of Shareef and McAleer (2007) that asymmetric impacts on tourism data are not particularly profound. While forecast evaluation is the major aim of this paper, it is worthwhile in passing to visualise how the GARCH/EGARCH models depict historical patterns of volatility in tourism demand and how destinations are differentially impacted by shocks. As an example, Fig. 2 plots the estimated conditional variances up to 2004 Q3 for three of the destinations examined – the United States, France and Cyprus. Eqs. (2) and (3) are used for this purpose, with f1 ¼ 0 indicating symmetric models for the latter two countries. The conditional variances are based on ‘static’ one-step-ahead forecasts of the demand variable. Static forecasts are generated using the historical values of the time series, since the latter are observable. Fig. 2 shows that UK demand for tourism to the United States became particularly volatile due to the negative shock of September

11th. Other volatile periods followed the first Gulf War and the US bombing of Libya in 1986. The same three volatile periods may be discerned for UK tourism to France, but the emphasis is different, with the most volatile period following the 1991 Gulf War. The shock of September 11th generated a relatively less volatile response, as did that of the Libyan bombing. The latter event heralded a decrease of more than 130,000 air journeys between the UK and France in 1986 Q2 and was contemporary with an increase is sea journeys between the two countries. UK tourism to Cyprus, on the other hand, reacted in a less volatile manner to these shocks. However, after the adverse effects of the Gulf War, there was a surge of tourist arrivals from the UK in 1992, with UK visitors accounting for around 50% of all tourists to the island that year (Clements & Georgiou, 1998). Table 3 reports the exponential smoothing models obtained for each destination. SPSS versions 15 and 16 offer the ‘Expert Modeler’ facility which establishes the optimal smoothing model. (This process compares the simple smoothing model, Holt’s and Brown’s linear trend models, the damped trend model, the simple seasonal models and Winters’ additive and multiplicative models).

J.T. Coshall / Tourism Management 30 (2009) 495–511

A United States

600,000

C Cyprus

.6

LOG(CYPRUS)

CONDITIONAL VARIANCE

CONDITIONAL VARIANCE

UNITED STATES 500,000 400,000 300,000 200,000 100,000 0

507

.5

.4

.3 .2 .1

.0 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104

80

82

84

86

88

90

92

94

96

98 100 102 104

B France

CONDITIONAL VARIANCE

35,000

FRANCE

30,000 25,000 20,000 15,000 10,000 5,000 0 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104

Fig. 2. Estimated conditional variances.

Normalised Bayesian Information Criteria (NBIC) are reported. NBIC is a score based on the mean squared error. It includes a penalty for the number of parameters in the model and the length of the series. This penalty removes the advantage of models with more parameters and simplifies model comparison (SPSS Inc., 2007). The values of the parameter d in Table 3 are typically high, reflecting the strong seasonality present in the series. Values for the trend smoothing parameter, g, are close to zero (save for Greece) indicating that the slopes of the series are relatively constant over time. 5. Forecasting accuracy 5.1. Model specific The ARIMA-volatility models of Tables 1 and 2 and the exponential smoothing models of Table 3 along with the Naı¨ve 2 model are used to produce quarterly ex post forecasts 1-, 2- and 3-years ahead for 2004 Table 3 Exponential smoothing models. Country

Model

Spain United States Republic of Ireland France Germany Italy Netherlands Canary Islands Greece Switzerland Canada Cyprus

Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’ Winters’

additive multiplicative additive multiplicative multiplicative multiplicative additive multiplicative multiplicative additive multiplicative multiplicative

a

g

d

NBIC

0.739 0.540 0.570 0.576 0.345 0.425 0.727 0.821 0.156 0.300 0.420 0.402

0.001 0.000 0.070 0.001 0.001 0.061 0.084 0.001 0.929 0.000 0.001 0.000

1.000 1.000 1.000 1.000 0.118 0.871 0.982 0.590 0.950 1.000 0.712 1.000

10.400 10.926 8.695 8.963 9.721 8.510 7.281 8.538 9.715 7.204 7.390 8.173

Q4 to 2007 Q3 inclusive for each destination. (Only ARIMA models are involved for the Netherlands and Canada, since no significant volatility components were found.) The forecasts generated are ‘dynamic’ as opposed to ‘static’. Since information for the hold back period 2004 Q4 and onwards - would be unavailable, dynamic forecasts use the previously forecasted values of the demand variable. Given quarterly data, the Naı¨ve 2 model is computed by assuming that the growth rate in tourism demand between times t and (t  4) is the same as that between times (t  4) and (t  8). All models are next recalibrated up to 2005 Q3 and 1- and 2years ahead forecasts are obtained for the period 2005 Q4 to 2007 Q3. The models are finally recalibrated up to 2006 Q3 and 1-year ahead forecasts obtained for 2006 Q4 to 2007 Q3. Thus, three 1year ahead, two 2-years ahead and one 3-years ahead forecasts are obtained for each model. Given twelve destinations and quarterly data, this produces a total of 144 forecasts 1-year ahead, 96 such forecasts 2-years ahead and 48 forecasts 3-years ahead. This process of recursively updating the sample permits forecasting evaluation that is model specific (Witt et al., 2003). In the present study, model recalibration never changed the significant terms included in the models. Critically, the Harvey–Leybourne–Newbold test rejected the hypothesis that the forecasts derived from the volatility models encompass those obtained from the exponential smoothing models at the 1-year horizon (HLN ¼ 3.048, p ¼ 0.000), at the 2-year horizon (HLN ¼ 2.934, p ¼ 0.002) and at the 3-year horizon (HLN ¼ 2.403, p ¼ 0.008). There are, therefore, clear benefits in taking advantage of the distinct characteristics of the volatility and exponential smoothing models. Combined forecasts are achieved by taking (i) the arithmetic mean of the volatility and smoothing forecasts for each quarter and (ii) via the variance–covariance method with weights obtained from Eq. (10).

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MAPE, RMSPE and MedAPE measures are used to compare the accuracy of the forecasts obtained from the volatility, exponential smoothing, Naı¨ve 2 and two combination approaches over the 1-, 2- and 3-year horizons. The results are presented in Table 4. One suggestion is that MAPE measures below 10% represent ‘‘highly accurate forecasting’’ (Lewis, 1982). In respect of tourism forecasting, Frechtling (2001) suggested adopting alternative criteria to the effect that models with a MAPE less than 50% of the MAPE of a relevant Naı¨ve model should be regarded as ‘‘highly accurate forecasts’’; models with a MAPE equal to between 50% and 100% of a Naı¨ve MAPE should be considered as ‘‘reasonably accurate forecasts’’ (Mentzer & Bienstock, 1998). The non-Naı¨ve models in Table 4 thus generate at worst ‘‘reasonably accurate forecasts’’, but for the most part ‘‘highly accurate’’ ones. Forecasts derived from the individual and combination models are ranked from most accurate (1) to least accurate (5). On all three accuracy measures and for all models, the forecasts become less precise as the forecasting horizon increases. The volatility and smoothing models outperform Naı¨ve 2 throughout. The exponential smoothing model surpasses the volatility model at the 3-year horizon on all measures. The results are more mixed for 1- and 2-year ahead forecasts, with often little difference in forecasting accuracy between the two models. Importantly, in all cases save for the 1-year ahead MedAPE measure, both of the combined forecasting methods are superior to those obtained from the three individual models over all time horizons. The variance–covariance method is the better of the two combination approaches two thirds of the time. Clearly, there are overwhelming benefits in combining volatility and smoothing forecasts from a model specific perspective. The test of unbiasedness of Eq. (12) was applied to the five models in Table 4 over the 1-, 2- and 3-year forecasting horizons (full details on request). The Newey–West covariance estimator was used in the least squares process, since it is consistent in the presence of both heteroscedasticity and autocorrelation of unknown form. In that w was found to be less than zero for each of the three time horizons, the ARIMA-volatility models tend to overestimate UK tourism demand. Conversely, positive w for all three horizons indicates that the exponential smoothing models generally underestimate demand. However, the t statistics failed to reject the null that w ¼ 0 for the three individual models and for the two combined models over all time horizons, save for the Naı¨ve 2 model at the 1-year horizon where the decision was marginal (t ¼ 2.260, p ¼ 0.025). The estimated biases are not statistically significant from zero at the 5% level. 5.2. Destination specific Turning to destination specific forecasting accuracy, Table 5 presents the optimal models obtained for each of the twelve

destinations according to the three accuracy criteria, whose numerical values are shown in brackets. The 3-year ahead time horizon is ignored, since it comprises observations for only four quarters per destination. All but one of these accuracy measures is below 10%. Table 5 shows that the efficacy of combining forecasts depends on the destination involved. This is analogous to findings that the relative forecasting accuracy of single models varies according to the origin-destination pair (Witt & Song, 2002). Forecast combination is clearly advantageous in the cases of UK tourism to Spain, Italy, Greece, and Cyprus and for 1-year ahead forecasts to Switzerland. Depending on the accuracy measure used, forecast combination also has some relevance for projecting 1-year ahead UK tourism numbers to the Republic of Ireland, France and Germany. Overall, combination forecasts are optimal for 36 (50%) of the 72 options in Table 5, with the variance–covariance method dominating the approach based on taking the mean of the two individual forecasts. Single exponential smoothing models tend to be superior for travel to the United States and the Netherlands, the ARIMA-volatility model is appropriate for the Canary Islands and a basic ARIMA model for Canada. Table 5 also confirms the consistency found between the MAPE and RMSPE measures of forecasting accuracy in studies of tourism demand (Li, Wong, Song, & Witt, 2006), but here within the novel contexts of volatility modelling and combination forecasts. MAPE and RMSPE suggest the same optimal scenario for 87.5% of the models in Table 5; MAPE and MedAPE agree for 62.5% of these models and RMSPE and MedAPE for 58.3%. The unbiasedness test of Eq. (12) using the Newey–West estimator was run for the 1- and 2-year forecasts obtained for each destination (full details on request). The measures in Table 5 indicate that the variance–covariance combination model is optimal for UK tourism to Greece over the 1-year horizon. The ARIMA-volatility model for Greece significantly overestimates demand (t ¼ 3.056, p ¼ 0.011), but this bias is compensated for when the model’s forecasts are combined with underestimates of demand generated by the smoothing model, in the sense that the variance–covariance forecasts for Greece are statistically unbiased (t ¼ 0.960, p ¼ 0.358). (The bias in the ARIMA-volatility model does, however, run through to the non-optimal, mean-combined forecasts for Greece (t ¼ 2.620, p ¼ 0.024).) All of the remaining forecasts derived from the models in Table 5 over the 1-year horizon are unbiased. Turning to the 2-year ahead forecasts, the ARIMA-volatility model for UK tourism to Greece is again biased (t ¼ 3.085, p ¼ 0.018), but this is (marginally) not translated into significant bias in the meancombined model (t ¼ 2.769, p ¼ 0.028), which the three measures in Table 5 indicate is optimal. The variance–covariance combination model for Cyprus is optimal over this time horizon. However, on this occasion, overestimates derived from the ARIMA-volatility model (t ¼ 3.362, p ¼ 0.012) do translate into significant bias in the

Table 4 Forecasting accuracy over different time horizons: all destinations. Horizon

Measure

Volatility

Exponential smoothing

Naı¨ve 2

Mean-combined

Var–cov method

1 year 2 years 3 years

MAPE MAPE MAPE

4.254 (4) 5.827 (3) 7.786 (4)

4.222 (3) 5.910 (4) 6.749 (3)

6.159 (5) 15.603 (5) 30.764 (5)

4.020 (2) 5.185 (2) 5.504 (1)

3.717 (1) 4.943 (1) 6.471 (2)

1 year 2 years 3 years

RMSPE RMSPE RMSPE

5.343 (3) 7.339 (3) 9.559 (4)

5.793 (4) 8.180 (4) 9.062 (3)

7.975 (5) 21.210 (5) 43.737 (5)

5.273 (2) 6.966 (2) 6.931 (1)

4.868 (1) 6.545 (1) 8.453 (2)

1 year 2 years 3 years

MedAPE MedAPE MedAPE

3.643 (4) 4.918 (4) 6.544 (4)

3.183 (2) 4.344 (3) 5.260 (3)

4.952 (5) 10.583 (5) 23.849 (5)

3.274 (3) 4.218 (2) 4.752 (1)

2.946 (1) 3.952 (1) 5.113 (2)

Combined forecasts

Mean-combined, combined forecasts from taking the arithmetic mean of the volatility and smoothing forecasts; Var–Cov, combined forecasts generated by the variance– covariance method.

J.T. Coshall / Tourism Management 30 (2009) 495–511

509

Table 5 The optimal forecasting model by destination. MAPE

RMSPE

MedAPE

Spain 1 year 2 years

Var–Cov (2.751) Var–Cov (3.665)

Var–Cov (4.206) Var–Cov (4.280)

Var–Cov (1.815) Mean-combined (2.423)

United States 1 year 2 years

Exp smooth (2.390) Exp smooth (4.556)

Exp smooth (2.782) Exp smooth (5.152)

ARIMA-Vol (2.319) Exp smooth (6.203)

Republic of Ireland 1 year 2 years

Var–Cov (3.696) Exp smooth (6.646)

Var–Cov (4.246) Exp smooth (7.063)

Naı¨ve 2 (3.295) Naı¨ve 2 (5.608)

France 1 year 2 years

Var–Cov (2.507) ARIMA-Vol (2.997)

Var–Cov (2.953) Exp smooth (3.509)

ARIMA-Vol (1.797) ARIMA-Vol (2.224)

Germany 1 year 2 years

Exp smooth (3.410) Exp smooth (3.751)

Var–Cov (4.237) Exp smooth (4.754)

Var–Cov (2.773) Var–Cov (2.762)

Italy 1 year 2 years

Var–Cov (4.606) Var–Cov (3.806)

Var–Cov (5.837) Var–Cov (5.052)

Mean-combined (3.611) Var–Cov (1.903)

MAPE

RMSPE

MedAPE

Netherlands 1 year 2 years

Exp smooth (2.677) Exp smooth (3.940)

Exp smooth (3.198) Exp smooth (4.663)

Exp smooth (2.525) Exp smooth (3.206)

Canary Islands 1 year 2 years

ARIMA-Vol (3.929) ARIMA-Vol (2.485)

ARIMA-Vol (4.574) ARIMA-Vol (3.145)

ARIMA-Vol (3.573) Mean-combined (1.473)

Greece 1 year 2 years

Var–Cov (4.152) Mean-combined (6.138)

Var–Cov (5.642) Mean-combined (6.750)

Var–Cov (2.400) Mean-combined (6.059)

Switzerland 1 year 2 years

Var–Cov (4.277) ARIMA-Vol (7.289)

Var–Cov (5.204) ARIMA-Vol (8.861)

Var–Cov (3.407) ARIMA-Vol (6.192)

Canada 1 year 2 years

ARIMA (3.570) ARIMA (2.952)

Var–Cov (4.251) ARIMA (4.071)

ARIMA (3.186) ARIMA (1.469)

Cyprus 1 year 2 years

Var–Cov (6.485) Var–Cov (8.039)

Var–Cov (8.256) Var–Cov (11.562)

Var–Cov (4.473) Var–Cov (4.219)

ARIMA-Vol, ARIMA-volatility model; Exp smooth, exponential smoothing model; Mean-combined, combined forecasts from taking the arithmetic mean of the volatility and smoothing forecasts; Var–Cov, combined forecasts generated by the variance–covariance method.

variance–covariance model (t ¼ 3.326, p ¼ 0.013). Overestimates in the single model generate significant overestimates in the combination model. The only remaining bias found is for the optimal exponential smoothing model for UK tourism to the United States. The model generates forecasts that are significant overestimates (t ¼ 3.676, p ¼ 0.008). Overall, there is no bias in the optimal 1-year ahead models of Table 5; there is bias in terms of 2-year ahead optimal forecasts for UK demand to the United States and Cyprus. 6. Conclusions This research is one of very few applications of univariate volatility modelling in the tourism field and importantly is the first to evaluate the forecasting potential of this class of models. In terms of time series modelling of UK demand for international tourism, both the ARIMA-volatility and exponential smoothing procedures have generated highly accurate forecasts over 1-, 2- and 3-year horizons. There is little to choose between the two models in respect of the three measures of forecasting accuracy used in this study. Generally, the ARIMA-volatility models tend to overestimate demand, while the smoothing models are inclined to underestimate it; however, in neither instance is the bias statistically significant. Both models are clearly superior to the benchmark Naı¨ve 2 process. The volatility models evidence that negative

shocks impact more on tourism demand than would positive shocks of equal magnitude for only three of the destinations studied. Also, periods of volatility tend to be short lived, save for the impact of September 11th on UK tourism to the United States. The tourism product is perhaps not as fragile as is often assumed. In this study context, the results of the encompassing tests clearly evidence the utility of combining the forecasts obtained from the ARIMA-volatility and smoothing models. In terms of purely model specific evaluations, combined forecasts are superior to the individual ones over all three time horizons examined, except when considering the MedAPE measure for 1-year ahead forecasts. The difference between the MAPE, RMSPE and MedAPE measures obtained for the ARIMA-volatility models and the optimal combined models is consistently below 1% when forecasting 1- and 2-years ahead. It is only at the 3-year horizon that this difference becomes relatively more marked. It is interesting to note that the variance–covariance combination method provides the optimal model when forecasting 1- and 2-years ahead, whichever evaluative measure is used. Conversely, the combination scheme based on taking the mean of the individual model forecasts is best for the 3-year ahead forecasts. This contradicts Li’s (2007) conclusion that more sophisticated methods of forecast combination perform better than simple mean-combination forecasts. However, these findings do confirm that the general superiority of

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combination forecasts over individual forecasts does not depend on the length of the forecasting horizons. Turning to destination specific evaluations, significant ARIMAvolatility models were established for tourism numbers to ten of the twelve countries examined here. Plots of the conditional variance over time such as Fig. 2 clearly portray the volatility associated with negative shocks like the bombing of Libya, the first Gulf War and September 11th. The models indicate that the magnitude and duration of volatile periods in tourism demand vary according to the nature of the shock and the destination involved. For example, a catastrophic event of the magnitude of September 11th had minimal impact on UK tourism to Cyprus, yet is associated with a distinctly volatile period in UK tourism to France and the United States. As a forecasting tool, the volatility model is marginally inferior to the combined and exponential smoothing models, except for 1- and 2-year forecasts of UK tourism to the Canary Islands and 2-year forecasts to Switzerland. Kim and Wong (2006) recommended further research to assess whether volatility modelling could improve the accuracy of forecasting tourism demand. The findings presented here suggest that in terms of forecasting tourism demand per se, the benefits of volatility modelling come to the fore when combined with an exponential smoothing time series model. The particular nature of volatility modelling suggests directions for further research. From the standpoint of policy formulation and planning in the tourism sector, the ability of volatility models singly or combined to forecast such as numbers of international tourist arrivals, their associated expenditures or growth in tourism is clearly relevant. This paper has illustrated how volatility models may be used to generate ‘conditional mean forecasts’. Importantly and uniquely, the models also have the ability to forecast volatility itself. It is straightforward to generate ‘conditional variance forecasts’ from the GARCH and EGARCH equations described in Section 2.1, once the parameter values have been established (see, for example, Harris & Sollis, 2003, p. 247). EViews provides this facility. This class of models, therefore, has the capacity to predict whether the next time period(s) will be risky (relatively high conditional variance) or not. The question of forecasting volatility has received serious attention in Finance, but there is the major question of how to evaluate forecasts of the conditional variance in a practical context. In a nutshell, the problems are that volatility as defined by the conditional variance is not directly observable and that assessing the accuracy of volatility forecasts requires a definition of future or ‘realised’ volatility. Advances in Finance in this area (Aguilar, 1999; Bollerslev & Wright, 2001; Gospodinov, Gavala, & Jiang, 2006) offer pointers as to how this problem could be resolved in studies of tourism demand. It should be noted that to address this problem, Financial research capitalises on the fact that data are often readily available at a high frequency level (by the minute, hour or day); in tourism studies, arrival numbers, expenditure and growth data are rarely available at higher than a monthly frequency. From a planning perspective, this should be a major research theme in the study of tourism demand, since incorporation of forecasts of market volatility into decisionmaking processes would assist development and investment strategies at the destination point. References Aguilar, J. (1999). GARCH, implied volatilities and implied distributions: An evaluation for forecasting purposes. Working Paper Series, 88. Monetary Policy Department, Central Bank of Sweden. Angelidis, T., Benos, A., & Degiannakis, S. (2004). The use of GARCH models in VaR estimation. Statistical Methodology, 1, 105–128. Armstrong, J. S. (2001). Combining forecasts. In J. S. Armstrong (Ed.), Principles of forecasting: A handbook for researchers and practitioners (pp. 1–19). Norwell, MA: Kluwer Academic Publishers. Bates, J. M., & Granger, C. W. J. (1969). The combination of forecasts. Operational Research Quarterly, 20, 451–468.

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