Forecasting tourist arrivals using time-varying parameter structural time series models

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1 Forecasing ouris arrivals using ime-varying parameer srucural ime series models HAIYAN SONG 1, GANG LI*, STEPHEN F. WITT* AND GEORGE ATHANASOPOULOS School of Hoel and Tourism Managemen The Hong Kong Polyechnic Universiy Hung Hom, Kowloon Hong Kong SAR *Faculy of Managemen and Law Universiy of Surrey Guildford GU2 7XH Unied Kingdom Deparmen of Economerics and Business Saisics Monash Universiy VIC 3800 Ausralia Absrac Empirical evidence has shown ha seasonal paerns of ourism demand and he effecs of various influencing facors on his demand end o change over ime. To forecas fuure ourism demand accuraely requires appropriae modelling of hese changes. Based on he srucural ime series model (STSM) and he ime-varying parameer (TVP) regression approach, his sudy develops he causal STSM furher by inroducing TVP esimaion of he explanaory variable coefficiens, and herefore combines he meris of he STSM and TVP model. This new model, he TVP-STSM, is employed for modelling and forecasing quarerly ouris arrivals o Hong Kong from four key source markes: China, Souh Korea, he UK and he USA. The empirical resuls show ha he TVP-STSM ouperforms all seven compeiors, including he basic and causal STSMs and he TVP model for oneo four-quarers-ahead ex pos forecass and one-quarer-ahead ex ane forecass. Keywords: Tourism demand forecasing, Seasonaliy, Sochasic, Sae space models 1 Corresponding auhor (hmsong@polyu.edu.hk). This aricle should be cied as: Song H, Li G, Wi SF, Ahanasopoulos G. (2011) 'Forecasing Touris Arrivals Using Time-Varying Parameer Srucural Time Series Models'. Inernaional Journal of Forecasing, 27 (3), pp

2 1. Inroducion Tourism conribues significanly o he economic growh of many counries and regions. Considering he rapid increase in inernaional ourism demand over he las few decades, accurae predicions of fuure rends of ourism demand are of paricular imporance o boh ourism policymakers and ourism business praciioners. Moreover, in mos desinaions ourism demand displays significan seasonal variaions. Seasonaliy affecs ourism in various differen ways and is responsible for difficulies in gaining access o capial, high risks of invesmen and business failures, he ineffecive uilisaion of resources and faciliies, and difficulies in mainaining a consisen service qualiy. However, seasonaliy is no always derimenal o he indusry, as he off-peak season has benefis such as ime for environmenal reclamaion and residen recovery (Buler, 1994). Regardless of he advanages and disadvanages of seasonaliy, a comprehensive knowledge of seasonal paerns of ourism demand and he accurae predicion of heir fuure values will conribue o effecive planning and operaions managemen, such as saffing, resource allocaion and capaciy managemen. Empirical evidence has shown ha seasonal paerns of ourism demand and he effecs of various influencing facors on demand end o change over ime. Forecasing fuure ourism demand accuraely herefore requires he appropriae modelling of boh seasonaliy and he effecs of he explanaory variables. Srucural ime series models (STSMs), which specify he rend, seasonal and cycle componens of a variable as sochasic, and he ime-varying parameer (TVP) regression approach, which relaxes he resricion on he consancy of he demand parameers over ime, have boh been inroduced ino ourism demand sudies and have demonsraed superior forecasing performance compared o deerminisic models. The aim of his sudy is o consruc a new economeric model ha develops he causal STSM furher by inroducing TVP esimaion of he explanaory variable coefficiens, and herefore combines he meris of he STSM and TVP model. This new TVP-STSM is expeced o forecas seasonal ourism demand more accuraely han previously used mehods. The empirical sudy evaluaes he forecasing accuracy of he proposed model for forecasing ouris arrivals o Hong Kong from four key source markes: China, Souh Korea, he UK and he USA. 2. Modelling and forecasing seasonal ourism demand A large body of lieraure on seasonal ourism demand analysis and forecasing has been published over he las wo decades, and has conribued significanly o our undersanding of he feaures of seasonal ourism demand. However, here are poenial problems wih hese sudies. Earlier sudies ended o regard he paerns of seasonaliy in ourism demand as consan. However, due o various changes, such as climae and weaher condiions, he populariy of ouris aciviies and desinaions, echnology and poliics, seasonaliy is no deerminisic. To overcome he assumpion of deerminisic seasonaliy, he STSM, iniially proposed by Harrison and Sevens (1976) and laer refined by Harvey (1989), was inroduced ino seasonal ourism demand sudies. 2

3 The basic STSM wihou he inclusion of explanaory variables (also known as he basic srucural model or BSM) decomposes a ime series ino is rend, seasonal, cycle and irregular componens and regards hese componens as sochasic. Hence, his model reflecs he seasonal variaion in ourism demand beer han he radiional consan seasonal ime series models. However, he BSM (and univariae ime series models in general) does no accoun for he effecs of economic deerminans on he variable of ineres. To overcome his limiaion, he BSM was developed furher o include causal variables in he model specificaion (known as he causal srucural model or CSM). Boh he BSM and he CSM have been applied in he ourism demand forecasing conex, and several sudies have demonsraed heir superior forecasing performance relaive o oher ime series alernaives (see for example González & Moral, 1995, 1996; Kulendran & King, 1997; Kulendran & Wi, 2001; Turner & Wi, 2001). However, in he sudies of boh Turner and Wi (2001) and Kulendran and Wi (2003), he CSM produced less accurae forecass han he BSM. One possible reason why he CSM could no generae more accurae forecass han he BSM is relaed o is reamen of he explanaory variables. Alhough he seasonaliy, rend and cycle in he CSM are all regarded as sochasic, he parameers of he explanaory variables are reaed as being consan over ime. This implies ha he economic srucure generaing he daa does no change. In a demand model specified in double-log form, consan parameers sugges ha he elasiciies of ourism demand are consan over he sample period, which is very resricive and ofen unrealisic. In realiy, he changing economic environmen may induce people o reac differenly o a given simulus a differen poins in ime. As he modificaions o he environmen are ransiory or ambiguous in some siuaions, he changes in he coefficiens are likely o follow a sochasic process (Lucas, 1976). However, he CSM does no ake his ino accoun, which may conribue o is unsaisfacory forecasing performance. To overcome he above limiaions of he radiional fixed-parameer esimaion of demand models, he ime-varying parameer (TVP) modelling approach was inroduced ino he ourism conex in he lae 1990s. This approach relaxes he resricion on he consancy of he coefficiens of he explanaory variables and allows for sochasic parameers so ha i can beer reflec he evoluion of demand elasiciies over ime. Previous empirical sudies employing he TVP echnique have shown ha models ha incorporae he TVP approach end o generae more accurae forecass han oher economeric models, especially in he shor erm. For example, Song and Wi (2000), Song, Wi, and Jensen (2003) and Wi, Song, and Louvieris (2003) examined he TVP model s performance in forecasing inernaional ourism demand relaive o oher fixedparameer economeric models and ime series models. The firs wo sudies assess he forecasing accuracy in erms of he error magniude using he mean absolue percenage error (MAPE) and roo mean square percenage error (RMSPE) measures. In boh sudies, he TVP model ouperforms all of he compeiors, including he auoregressive disribued lag model (ADLM), he vecor auoregressive (VAR) model, ECMs and he naïve no-change model in he one-year-ahead forecasing comparison. Wi e al. (2003) invesigae he forecasing performance of he TVP model in erms of boh error 3

4 magniude and direcional change. The comparison resuls show ha he TVP model is ranked second bes amongs seven candidaes in a one year ahead forecasing compeiion in boh assessmens. All hree empirical sudies show ha he TVP echnique is ranked eiher firs or second for shor-run (specifically one year ahead) forecasing, which implies ha he TVP model is highly suiable for shor-run ourism planning purposes. However, all of he above applicaions of he TVP model use annual ourism demand daa and he influence of seasonaliy on ourism demand is no examined in hese sudies. The above TVP models can readily incorporae seasonal dummies in order o forecas seasonal ourism demand (see, for example, Shen, Li, & Song, 2008), bu his implies deerminisic seasonaliy. The curren sudy represens he firs aemp o incorporae ime-varying parameers in srucural ime series models for seasonal ourism demand forecasing. This new modelling approach provides a comprehensive analysis of seasonal demand and is expeced o improve forecasing accuracy. The TVP-STSM is applied o forecasing he quarerly demand for Hong Kong ourism by ouriss from four key source markes. As wih mos ouris desinaions, Hong Kong ourism demand experiences significan seasonaliy. The forecasing accuracy of he newly developed TVP-STSM is compared wih ha of commonly used ime series and economeric forecasing models. The empirical resuls of his sudy will provide very useful informaion for he key ourism players and public agencies in formulaing heir ourism policies and evaluaing he effeciveness of hese. 3. Mehodology The TVP-STSM developed here is based on he TVP model and he STSM, which have common echnical feaures such as being wrien in sae space form and being esimaed using he Kalman filer algorihm. Based on he advanages of he wo models and heir common echnical foundaion, he TVP-STSM is expeced o show improved forecasing abiliy when dealing wih seasonal daa TVP-STSM specificaion The TVP-STSM can be represened in he following sae space form (SSF): y = µ + ψ + γ + X Γ + ε ε NID(0, H ) (1) + = µ + β ~ 2 µ 1 + v v ~ NID(0, σ ) (2) β = β δ δ ~ NID(0, σ δ ) ψ * ψ cosτ sinτ ψ + ω = ρ * * sinτ cosτ ψ ω v (3) * 2 ω, ω ~ NID(0, σ ω ) (4) 4

5 s = γ + 1 j κ κ ~ NID(0, σ κ ) j= 1 γ + (5) Γ + = T Γ + R η, Γ N( K, ), η NID(0, Q ), (6) 1 1 ~ 1 P1 ~ where y is a univariae ime series, decomposed ino is unobservable componens, including a rend componen ( µ ), a cycle componen ( ψ ), a seasonal componen ( γ ) and an irregular componen ( ε ). X is a vecor of causal variables and Γ is he corresponding vecor of coefficiens. As equaion (1) suggess, he TVP-STSM is linear in sae variables ( µ, ψ, γ and Γ ), which are governed in urn by firs-order linear ransiion equaions (equaions (2) o (6)). Equaions (2) and (3) specify he sochasic rend, where β is he slope of he rend. Equaion (4) refers o a sochasic cycle specificaion, where τ [ 0, π ] is he cyclical frequency, ρ [0,1] is he damping facor * of he cycle, and ψ appears by consrucion. The seasonal componen is defined in equaion (5) in such a sochasic way ha he seasonal paern is allowed o change over ime, where s is he number of seasons per year. I is ofen preferable o express he sochasic seasonaliy in rigonomeric form, similarly o he specificaion of he cyclical componen (see Harvey, 1989, for more deails). The whie noise disurbances of he rend, cycle and seasonal equaions (equaions (2) o (5)) are muually independen and σ v, σ δ, σ ω and σ κ are he corresponding variances. Unlike radiional regression models, he coefficiens of he causal variables in a TVP-STSM are specified as imevarying, as equaion (6) suggess. In mos economic applicaions, i is assumed ha T = I, where I is he ideniy marix. In his case, Γ follows a mulivariae random walk, and H, R, Q and T are sysem marices ha are iniially assumed o be known. Wih regard o he esimaion of he TVP-STSM wrien in SSF, he filering and smoohing algorihms can be employed, condiional on known sae and error sysem marices. As for he unknown parameers in hese marices, hey are esimaed using maximum likelihood esimaion mehods (Koopman e al., 2007). Deailed explanaions and furher references wih regard o he specificaion and esimaion of sae space models are given by Durbin and Koopman (2001), Hamilon (1994a,b), Harvey (1989) and Koopman, Shephard, and Doornik (1999) Specificaions of alernaive models The above TVP-STSM is a more general form of a srucural ime series model. The BSM, CSM and TVP models can all be regarded as special cases of he TVP-STSM specificaion. If no causal variables are incorporaed ino equaion (1), he TVP-STSM is reduced o a BSM: y = µ + ψ + γ + ε. If causal variables are included bu heir parameers are esimaed as ime-invarian, he TVP-STSM is reduced o a CSM: y = µ + ψ + γ + X Γ + ε, where Γ does no have a ime subscrip. If no unobservable 5

6 componens (apar from ε ) are included in he model, he TVP-STSM is reduced o he TVP model: y = X Γ + ε. Where seasonal daa are used, deerminisic seasonal dummies can readily be added o he TVP model. However, such a model is unable o analyse he sochasic seasonaliy or cycles of a ime series. Wih a more general specificaion, he TVP-STSM embodies he various advanages of he CSM and TVP model. This new model is beer able o explain he dynamics of an economic series, and is expeced o furher enhance forecasing performance. The meris of he TVP-STSM are empirically demonsraed in he following secion. 4. Empirical resuls and discussion The empirical applicaion focuses on modelling and forecasing ouris arrivals o Hong Kong from four represenaive leading source markes, namely China, Souh Korea, he UK and he USA. The dynamic effecs of he seasonaliy, rend, cycle and various influencing facors on he demand for ourism in Hong Kong are esimaed for hese four key source markes. In addiion, he forecasing accuracy of he TVP-STSM is compared wih ha of oher ime series and economeric models which are commonly used in ourism demand sudies The daa and variables The following demand funcion is proposed for modelling he demand for Hong Kong ourism by residens from a paricular origin counry: ln Di = f (lnyi, ln Phki, ln Psi, Dummies,Trend, Seasonal, Cycle), (7) where ln sands for he naural logarihm; D i is ourism demand, for which he variable of ouris arrivals from origin counry i is used as a proxy; Y i is he income level in origin counry i, measured by he gross domesic produc (GDP) a consan prices; and P hki is he ourism price in Hong Kong relaive o ha of he origin counry i, adjused by he relevan exchange raes, i.e., P hki ( CPI hk / EX hk ) ( CPI / EX ) =, (8) i i where CPI hk and CPI i are he CPI s for Hong Kong and origin counry i respecively; and EX hk and EX i are he exchange raes beween he Hong Kong dollar and he US dollar, and beween he currency of origin counry i and he US dollar, respecively. Ideally, he ourism price should include ouriss living coss, as well as he ravel cos o Hong Kong, bu due o he difficulies in obaining reliable ravel cos daa, he own price variable only conains he living cos componen. Furhermore, i has been repored ha he ravel cos variable is insignifican in many ourism demand models, such as ha of Smeral Wi, and Wi (1992). This is due mainly o he fac ha he average economy airfare is no considered o be a good proxy for he ravel cos variable. The subsiue 6

7 price variable P si in equaion (7) is defined as a weighed average index of seleced counries ourism prices. In he selecion of subsiue desinaions, no only he geographic characerisics bu also he culural characerisics are aken ino accoun. Singapore, Taiwan, Thailand, Souh Korea and Japan were seleced as subsiue desinaions in his sudy. In he case where Souh Korea is he origin counry under consideraion, i is excluded from he subsiue price calculaion. The subsiue price index is calculaed by weighing he exchange-rae adjused CPI of each subsiue desinaion according o is share of visior arrivals, and is given as, where 5 si = ( j / j ) j, (9) j= 1 P CPI EX w w j is he marke share of subsiue desinaion j, which is calculaed from, 5 wj = TTAj / TTAj, (10) j= 1 where TTA j is he oal number of visior arrivals o counry/region j. The above explanaory variables are commonly considered in ourism demand sudies and have generally been shown o have significan effecs on ourism demand (Wi & Wi, 1995). In addiion o he above explanaory variables, some dummies are included o capure he effecs on Hong Kong ourism demand of various one-off evens (such as he Gulf war in 1990, he Asian financial crisis in , he handover of Hong Kong o China in 1997, he 9/11 erroris aack in 2001 and SARS in 2003). The dummy variables ake he value of 1 for he period when he one-off even occurs and zero oherwise. In paricular, he handover of Hong Kong o China appeared o have had only a shor-erm effec, and hence he dummy variable is specified as an oulier insead of a sep inervenion. I should be noed ha since a one-off even may have differen influences (in eiher scale or ime period) on ourism demand from differen source markes, he specificaion of a given dummy variable may vary across he differen models. The rend, seasonal and cycle componens are specified as oulined in Secion 3, following Koopman, Harvey, Doornik, and Shephard (2007). In his sudy, quarerly daa are colleced over he period 1985Q1 2008Q4, wihin which he daa from 1985Q1 o 2004Q4 are employed for model esimaion and he res are used for forecasing comparisons. Visior arrivals are iniially colleced from he saisics of he Hong Kong Tourism Board (HKTB) on a monhly basis, and are hen aggregaed o give quarerly daa. The income, price and exchange rae daa are obained from he Inernaional Financial Saisics Yearbooks published by he Inernaional Moneary Fund (IMF). 7

8 4.2. Model esimaion The general-o-specific approach is adoped for he TVP-STSM esimaion. The iniial specificaion of he measuremen equaion of he TVP-STSM includes rend (level and slope), cycle, seasonal and irregular componens, hree explanaory variables, and some dummy variables as discussed above. The coefficiens of he explanaory variables are specified as ime-varying and are modelled as random walk processes. If he variances of he esimaed parameers are zero, he coefficiens of he explanaory variables are hen aken as fixed parameers in he re-esimaed model. Wih regard o he unobserved componens, including he rend, seasonal, cycle and irregular componens, i is useful o run BSMs o examine he properies (i.e. sochasic versus deerminisic processes) of hese componens in he various ouris arrivals series. In paricular, such an examinaion helps in idenifying wheher any cycles exis in he daa, so ha spurious cycles can be avoided in he TVP-STSM modelling sage. The cyclical propery check is paricularly useful when a small daa sample is used, as spurious cycles are more likely o occur in srucural ime series modelling. I should be noed ha some componens feaure (relaively weak) sochasic processes in a BSM, bu may become deerminisic in a TVP-STSM. In oher words, including explanaory variables and modelling heir coefficiens as random walk processes may conribue o explaining he variaion in he dependen variable. Therefore, i is worh checking hese properies again in a TVP-STSM. Since previous economeric sudies have suggesed ha ouriss incomes, relaive ourism prices and prices in compeing desinaions are he mos imporan deerminans of ourism demand for mos desinaions (Li, Song, & Wi, 2005), and are hus able o explain he level of ourism demand o a grea exen, i is boh appropriae and effecive o sar wih a smooh local linear rend model; i.e., he disurbance in he level equaion is se o zero bu he disurbance in he slope equaion is non-zero, as specified by Koopman e al. (2007, p. 172). Therefore, in he iniial TVP- STSM, he rend, seasonal and irregular componens are specified as sochasic, and he cycle componen is only included if srong evidence is shown in a BSM. Again, he variances of he esimaed parameers of hese unobserved componens will sugges wheher or no i is more appropriae o specify hem as deerminisic. The model is hen re-esimaed wih all componens se appropriaely. In addiion o he pre-seleced dummy variables, some counry-specific dummies are included, as suggesed by he daa plos. The compuer programme STAMP 8.10 is used o obain maximum likelihood esimaes of he model parameers. STAMP can conver a linear dynamic model such as he proposed TVP-STSM ino sae space form (a firs-order dynamic linear model), and applies he Kalman filer in is maximum likelihood procedure. Following he above procedure, he general TVP-STSM will be reduced o a simpler specificaion sep by sep. The final specificaion should pass all diagnosic ess, including he residual serial correlaion, heeroscedasiciy and normaliy ess. In addiion, he esimaion resuls need o mee cerain convergence crieria. Since he esimaion of a TVP-STSM is a maximisaion process ha is erminaed when hree convergence crieria hold (depending on differen seings of he criical values), a good TVP-STSM specificaion should mee high levels of he crieria. In STAMP s esimaion 8

9 repor, he message of srong convergence or very srong convergence is expeced o be obained. Once he model passes all of he diagnosic saisics and a leas srong convergence is achieved, i will be used for forecasing purposes. Table 1 summarises he resuls for he four esimaed models in he iniial esimaion sample 1985Q1 o 2004Q4. Figures 1 and 2 illusrae he evoluion over ime of various esimaed parameers in he four models. Table 1. TVP-STSM esimaes and diagnosics for ouris arrivals o Hong Kong. China Korea UK US Hyperparameers (Average percenage deviaions) Level Slope Seasonal Irregular lny i lnp hki lnp si Coefficiens Level 5.450*** [0.428] [1.501] [2.871] Slope 0.018*** [0.003] [0.008] [0.005] lny i 0.483*** 1.370** [0.197] [0.722] [1.401] lnp hki * [0.299] [0.584] [0.160] lnp si *** [0.274] [0.681] [0.309] Dum *** [0.027] [0.040] [0.026] Dum [0.027] Dum *** 0.576*** 0.626*** [0.029] [0.041] [0.027] Seasonal effecs Seasonal * [0.016] Seasonal [0.007] Seasonal [0.006] Seasonal * [0.018] Diagnosics 0.053* [0.027] [0.018] [0.017] [0.030] 0.037** [0.015] 0.033** [0.015] 0.069*** [0.013] 0.065*** [0.014] [3.377] [0.007] [1.655] [0.613] [0.331] [0.025] 0.090*** [0.026] 0.725*** [0.027] 0.029** [0.016] [0.015] 0.028* [0.014] 0.045*** [0.015] Normaliy H (23) Q (10, 7) r(1)

10 r(10) p.e.v e e e e 4 2 R R s Convergence very srong srong srong srong Noes: *, ** and *** denoe significance a 10%, 5% and 1% levels, respecively (one-ailed ess for he income and own-price variables); he above models are esimaed for he iniial esimaion sample 1985Q1 2004Q4; he values in brackes are sandard errors; Normaliy refers o he correced Bowman-Shenon error normaliy saisic, approximaely disribued as chi-square (2) under he null hypohesis; H(23) is a heeroscedasiciy saisic disribued as F(23, 23) based on a wo-ailed es (Commandeur & Koopman, 2007); Q(10, 7) is he Box-Ljung saisic disribued as chi-square (7); r(1) and r(10) are he serial correlaion coefficiens a he equivalen residual lags, approximaely normally disribued; p.e.v. refers o he predicion error variance; relaive goodness-of-fi agains a random walk plus drif and fixed seasonals. R measures he 2 s (a) China 0.10 (b) Korea (c) UK 0.10 (d) US Figure 1. Seasonal componens of he esimaed TVP-STSMs. 10

11 0.6 (a) LnY_China (b) LnP_China 1.5 (c) LnY_Korea (d) LnP_Korea (e) lny_uk (f) lnp_uk (g) lny_us (h) lnp_us (i) lnps_us Figure 2. Esimaion resuls of he TVP-STSMs a period 2004Q4. In general, he TVP-STSMs end o have fewer significan explanaory variables han more convenional regression models such as auoregressive disribued lag models (ADLMs). This is because he unobserved componens in a well-specified TVP-STSM are able o capure a large amoun of he variaion in he demand variable under sudy, and herefore he effecs of some explanaory variables may no longer be saisically significan. However, hey are reained in he final models if hey are correcly signed, for he following wo reasons. Firsly, hese explanaory variables have commonly been idenified as he mos imporan deerminans of he demand for Hong Kong ourism in pas ourism demand sudies (see, for example, Song, Wong, & Chon, 2003). Secondly, including hese variables enables he models o pass he diagnosic checks and o achieve srong convergence in model esimaion. These esimaed models are subsequenly examined wih regard o heir forecasing performances relaive o hose of oher commonly used economeric and ime series mehods. The summary saisics in Table 1 indicae ha he models are correcly specified, as all of he models repored pass all diagnosic ess a he 5% significance level. In paricular, no serial correlaion exiss in he residuals, which suggess ha he models have capured he dynamic naure of he dependen variable adequaely. Alhough he relaive goodnessof-fi measure R 2 s appears low in he Korea model, here is srong evidence ha he TVP- STSM specificaion is preferable o a random walk plus drif and fixed seasonals model, 11

12 2 which is he benchmark model in he R s calculaion. For example, he TVP-STSM passes all diagnosic ess, he random walk esimaes of he income and relaive-price variables are boh significan, and sochasic seasonaliy is eviden, as can be seen from he plo in Figure 1(b). Wih respec o he esimaes of various hyperparameers, i.e., he average percenage deviaions of he disurbances in he measuremen and sae equaions, i can be seen ha he ouris arrivals from differen source markes exhibi differen rends and seasonal paerns (see Table 1). All of he Hong Kong ouris arrivals series under sudy exhibi sochasic seasonaliy. For example, panels (b) (d) of Figure 1 show clear evidence of varying seasonal paerns in he Korea, UK and US markes, paricularly since he mid- 1990s, whereas panel (a) of Figure 1 shows ha he seasonal paern in he China marke changes in a much more gradual fashion. Cycles do no feaure in any markes over he observed sample period. This is consisen wih mos of he ourism demand lieraure using CSMs, such as he sudies of González and Moral (1995) and Kulendran and Wi (2003). I indicaes ha he key economic deerminans included have successfully explained he variaion in he ourism demand variable, and herefore i is no necessary o include an addiional cyclical componen (Moosa, 2000). The iniial rend specificaions hold in wo of he final models, hose for Korea and he UK. This indicaes ha a smooh local linear rend specificaion fis he ouris arrivals series from hese wo counries well. For China and he US, deerminisic slopes are sufficien o fi he arrivals series. Wih regard o he explanaory variables, he variances of he esimaed coefficiens make he ime-varying naure of he various demand elasiciies clear (see Table 1 and Figure 2). The income and own-price elasiciies vary more significanly over ime han he cross-price elasiciies. In paricular, he ime-varying income elasiciies are eviden in all four markes under sudy. By conras, a non-consan cross-price elasiciy appears in he US marke only. Wih regard o he regression effecs in he final sae, he origin income shows he mos significan effec (see Table 1). Alhough hey are no always saisically significan in he final sae, he esimaed coefficiens of he income and relaive price variables have he expeced signs in all cases, in line wih demand heory. As Table 1 shows, he TVP esimaes of he income variable are significan a he 5% significance level in boh he China and Korea models, bu insignifican in he UK and US markes. 2 I seems ha he effec of income on ourism demand is more eviden in developing counries han in developed counries. Wih respec o he magniudes of he esimaed income elasiciies, hey vary across markes. As far as he wo significan cases are concerned, he income elasiciy is lower in he China marke han in he Korea marke. This is probably because of he high proporion of day-rips among he Chinese 2 To provide robus findings, hose resuls which are boh economically and saisically significan are seleced for discussion. When referring o he quaniaive naure of he poin esimaes, saisically insignifican resuls are omied. 12

13 visiors who ravel o Hong Kong for business purposes. On he conrary, mos Korean visiors are overnigh leisure ouriss, and herefore heir income elasiciy is expeced o be higher. Panels (a) and (c) of Figure 2 illusrae he ime-varying effecs of income on ourism demand from he China and Korea markes, respecively. Boh graphs show a urning poin in he lae 1990s, bu opposie rends were followed aferwards. The decreasing income elasiciy in he Korea marke suggess he mauriy of his marke as far as he desinaion of Hong Kong is concerned. On he oher hand, as an emerging marke, China has experienced a boom in oubound ravel in recen years ha is refleced in he growh in ourism o Hong Kong. This is mainly aribuable o he srong, sable economic growh in China. Wih respec o he own-price elasiciies, he resuls generally sugges ha he demand for Hong Kong ourism has become less sensiive o he variaions in ourism prices in Hong Kong relaive o hose in he origin counries since he early 1990s. This is paricularly eviden in he Korea marke, as panel (d) of Figure 2 shows. Subsiue prices seem o be imporan for he UK marke, bu less imporan for he oher origins. In paricular, he significan posiive coefficien in he case of he UK suggess ha desinaions neighbouring Hong Kong are srongly regarded as subsiues by UK ouriss. As far as he inervenion effecs of he major one-off evens (which are esimaed as fixed parameers) are concerned, he SARS oubreak in 2003 had he mos significan impac on ouris arrivals from all of he source markes. I should be noed ha alhough he war in Iraq also occurred in 2003, i did no have a significan impac on ouris arrivals o Hong Kong. The saisics for UK and US oubound ourism boh show a posiive growh in 2003, indicaing ha he war in Iraq did no have a significan effec on he oubound ravel from hese wo counries. Therefore, he dummy variable for 2003 only capures he effec of he SARS oubreak. The handover of Hong Kong o China in 1997 is observed o have had a significan adverse effec on he UK marke, bu he effec on he mainland Chinese marke is insignifican. The Asian financial crisis in 1997 did no have a significan impac on Korean ouris arrivals o Hong Kong. The Sepember 2001 erroris aack did no have any significan impac on ourism demand in Hong Kong eiher, excep for he US marke. The effec of he Gulf war is no significan in any of he models. Hence, i is removed from he final model esimaion Forecas accuracy analysis To assess he forecas accuracy of he TVP-STSMs in comparison wih ha of oher economeric and ime series models, one-, wo-, hree-, four- and eigh-quarers-ahead forecas horizons are considered, and ex-pos dynamic forecass are generaed over he hold-ou period 2005Q1 2008Q4. A recursive forecasing echnique is used during he whole procedure, i.e., he models are firs esimaed over he period 1985Q1 o 2004Q4, hen his esimaed model is used o forecas Hong Kong inbound ouris arrivals from each of he four origin counries. The models are hen re-esimaed for he periods up o 2005Q1, 2005Q2,, 2008Q3, and forecass are generaed based on each re-esimaed model for 2005Q2 2008Q4, 2005Q3 2008Q4,, 2008Q4, respecively. As a resul, 16 13

14 one-sep-ahead forecass, 15 wo-seps-ahead forecass, 14 hree-seps-ahead forecass, 13 four-seps-ahead forecass and 9 eigh-seps-ahead forecass are generaed for each source marke under sudy. The forecas accuracy is evaluaed based on he mean absolue percenage error (MAPE) and he roo mean square percenage error (RMSPE). The seasonal naïve No Change 1 (consan value for each season) and No Change 2 (consan growh rae for each season) models, he SARIMA model, BSM, CSM, he TVP model, and ADLM are included in he comparison of he forecasing performances. Due o space consrains, he specificaions of hese models are omied here, bu echnical deails are given by Frechling (2001) for he seasonal naïve models and Kulendran and Wi (2003) and Song and Wi (2000) for he res Ex-pos forecas accuracy comparison rankings The resuls for he ex-pos forecas accuracy comparisons are repored in Table 2. Over all of he forecasing horizons being considered, he TVP-STSM is consisenly ranked firs among he eigh compeing models, as judged by boh MAPE and RMSPE wih only one excepion. For he shorer erm (one o hree quarers ahead) he TVP-STSM, CSM, TVP model and BSM, all of which belong o he family of sae space models, ake he op four posiions in he forecasing compeiion according o he MAPE. The very good shor-erm forecasing performances of he TVP model, CSM and BSM are in line wih he findings of pas ourism forecasing sudies, such as hose of Song and Wi (2000), González and Moral (1995) and Turner and Wi (2001). In paricular, he TVP-STSM, CSM and TVP model, all of which incorporae explanaory variables ino he model specificaions, generae he mos accurae one- and wo-quarers-ahead forecass. This indicaes ha inroducing explanaory variables may help improve he shor-erm forecas accuracy of sae space models. The ousanding forecasing performances of he TVP- STSM, CSM and BSM sugges ha i is mos appropriae o model seasonal ourism demand using he unobserved componens mehod, which incorporaes sochasic specificaions of rend, seasonal and cycle componens. Such specificaions help o capure he dynamics of seasonal ourism demand as accuraely as possible. The implicaion of he superior performance of he TVP-STSM compared o CSM is ha he demand feaures ha are no explained by he unobserved componens are likely o embody sochasic paerns, which are beer modelled as ime-varying parameers of he explanaory variables. Looking a he relaive forecas accuracies of he TVP-STSM and TVP model, which use he same esimaion mehod for he explanaory variables (i.e., he Kalman filer algorihm), he resuls show ha applying he TVP model o seasonal ourism demand forecasing simply by augmening he original TVP specificaion wih deerminisic seasonal dummies is no sufficien o capure he dynamic paerns of seasonal ouris arrivals. 14

15 Table 2. Comparison of ex pos forecasing accuracy rankings over differen forecasing horizons. Horizon Measure TVP-STSM CSM TVP BSM ADLM SARIMA Naïve 1 Naïve 2 1 quarer MAPE (1) (2) (3) (4) (6) (5) (7) (8) RMSPE (1) (2) (4) (5) (6) (3) (7) (8) 2 quarers MAPE (1) (2) (3) (4) (7) (6) (5) (8) RMSPE (1) (2) (3) (7) (6) (5) (4) (8) 3 quarers MAPE (1) (3) (4) (2) (8) (6) (5) (7) RMSPE (1) (3) (6) (4) (8) (5) (2) (7) 4 quarers MAPE (1) (5) (3) (2) (8) (6) (4) (7) RMSPE (1) (7) (6) (3) (8) (4) (2) (5) 8 quarers MAPE (1) (7) (5) (2) (8) (3) (6) (4) RMSPE (2) (8) (6) (1) (7) (3) (5) (4) Noe: The figures in parenheses denoe rankings. 15

16 As he forecasing horizon is exended, he TVP model and CSM generae relaively less accurae forecass. On he oher hand, wo ime series models, he BSM and he SARIMA model, show improved forecasing resuls. In paricular, he BSM ouperforms all of is compeiors excep he TVP-STSM over he eigh-quarersahead horizon according o MAPE. This indicaes ha for an economeric model o ouperform is ime series counerpars in he longer-erm (four and eigh quarers ahead) forecasing of seasonal ourism demand, boh sochasic seasonaliy and TVP esimaion of explanaory variable coefficiens should be considered. The ADLM ha incorporaes deerminisic seasonal dummies in he model specificaion always shows poor forecasing performance in he group. Similarly, he wo seasonal naïve models ha also assume deerminisic seasonaliy fail o generae saisfacory forecass over mos horizons. Once again, his comparison implies ha reaing seasonaliy as sochasic in a forecasing model s specificaion is more appropriae han regarding seasonaliy as deerminisic HLN saisical es of differences in forecasing accuracy The Harvey-Leybourne-Newbold (HLN) es proposed by Harvey, Leybourne, and Newbold (1997) is applied o examine saisically significan differences in forecasing accuracy beween he TVP-STSM and he compeing models. As Table 3 shows, he TVP-STSM clearly ouperforms all compeing models a he 5% significance level for one-quarer-ahead forecass, and a he 10% significance level for wo-quarers-ahead forecass. Wih respec o he hree-quarers-ahead forecass, saisical evidence can be found for he TVP-STSM s superior forecasing performance a he 10% significance level in all bu one case (BSM). As he forecasing horizon exends, he forecas accuracy declines for all models. Alhough he TVP-STSM s superioriy is less eviden saisically speaking, is four- and eighquarers-ahead forecass are sill saisically more accurae han hose of he CSM (a he 10% significance level) and ADLM (a he 1% significance level). Table 3. HLN saisical ess of he differences in forecasing accuracy levels beween TVP-STSM and oher models. Horizon CSM TVP BSM ADLM SARIMA Naïve 1 Naïve 2 1 quarer 3.244*** 3.400*** 1.923** 5.575*** 3.225*** 4.141*** 3.247*** 2 quarers 1.386* 1.761** 1.308* 3.717*** 2.858*** 2.818*** 2.766*** 3 quarers 1.438* 2.472*** *** 2.326** 1.589* 2.932*** 4 quarers 1.403* *** * 8 quarers 1.339* ** Noe: *, ** and *** denoe significance a 10%, 5% and 1% levels (one-ailed ess) Ex-ane forecas accuracy comparison rankings As has been previously discussed in Secion 2, models wih explanaory variables have he grea advanage over univariae ime series models of incorporaing he impac of imporan economic deerminans on ourism demand. However, he greaes disadvanage of hese models is ha when forecasing ino he fuure (where no variables have ye been observed) one mus produce forecass of he regressors before forecasing he regressand. In many cases one could argue ha i may acually 16

17 be more challenging o forecas he regressors han o forecas he dependen variable direcly. In his secion we examine he effec of no observing he explanaory variables over he hold-ou sample, bu forecasing hem. Hence, all forecasing is now performed ex-ane. In order o forecas he explanaory variables, exponenial smoohing mehods are employed by implemening he fully auomaed algorihm for forecasing univariae ime series using innovaions sae space models, as presened by Hyndman and Khandakar (2008) (see also Hyndman, Koehler, Ord, & Snyder, 2008). The ex-ane forecasing evaluaion resuls are presened in Table 4. As expeced, boh he MAPEs and RMSPEs have deerioraed (increased) for mos models wih explanaory variables when performing ex-ane forecasing compared o ex-pos. The only excepion is for he TVP models, where boh forecas error measures improve (i.e., become smaller) for all forecas horizons. This somewha counerinuiive and surprising resul has also appeared in oher sudies (see for example he discussion by Ahanasopoulos, Hyndman, Song, & Wu, 2010, and references herein, Secion 5.5). When comparing he rankings of he models beween ex-pos and ex-ane forecasing, i is found ha he TVP-STSM is sill ranked firs among all mehods for forecasing one quarer ahead. For longer horizons, he TVP model and he BSM mosly hold he number one and number wo rankings beween hem, wih he TVP-STSM being mosly ranked hird. Alhough he performance of he TVP-STSM has deerioraed compared o ex-pos forecasing, i sill produces saisfacory forecass. This is refleced in he resuls of he HLN ess, presened in Table 5, where he TVP-STSM shows significanly more accurae one-quarer-ahead forecass over all is compeiors excep he TVP model, and no mehods predic significanly beer han he TVP- STSM for longer horizons. 17

18 Table 4. Comparison of ex ane forecasing accuracy rankings over differen forecasing horizons. Horizon Measure TVP-STSM CSM TVP BSM ADLM SARIMA Naïve 1 Naïve 2 1 quarer MAPE (1) [1] (3) [2] (2) [3] (4) [4] (6) [6] (5) [5] (7) [7] (8) [8] RMSPE (1) [1] (4) [2] (2) [4] (5) [5] (7) [6] (3) [3] (6) [7] (8) [8] 2 quarers MAPE (3) [1] (4) [2] (1) [3] (2) [4] (7) [7] (6) [6] (5) [5] (8) [8] RMSPE (2) [1] (5) [2] (1) [3] (6) [7] (7) [6] (4) [5] (3) [4] (8) [8] 3 quarers MAPE (2) [1] (5) [3] (3) [4] (1) [2] (8) [8] (6) [6] (4) [5] (7) [7] RMSPE (4) [1] (7) [3] (5) [6] (2) [4] (8) [8] (3) [5] (1) [2] (6) [7] 4 quarers MAPE (3) [1] (6) [5] (1) [3] (2) [2] (8) [8] (5) [6] (4) [4] (7) [7] RMSPE (3) [1] (7) [7] (2) [6] (3) [3] (8) [8] (5) [4] (1) [2] (6) [5] 8 quarers MAPE (3) [1] (6) [7] (2) [5] (1) [2] (7) [8] (4) [3] (8) [6] (5) [4] RMSPE (5) [2] (8) [8] (3) [6] (1) [1] (7) [7] (2) [3] (6) [5] (4) [4] Noe: The figures in parenheses denoe rankings. The figures in he square brackes denoe rankings when ex-pos forecasing was performed; hese are exraced from Table 2. 18

19 Table 5. HLN saisical ess of differences in ex ane forecasing accuracy levels beween TVP-STSM and oher models. Horizon CSM TVP BSM ADLM SARIMA Naïve 1 Naïve 2 1 quarer 2.435*** *** 4.015*** 3.068*** 4.078*** 2.999*** 2 quarers *** 1.310* 1.687** 2.359** 3 quarers *** * 4 quarers *** quarers Noe: *, ** and *** denoe significance a 10%, 5% and 1% levels (one-ailed ess). 5. Concluding remarks This sudy has developed a complee sochasic parameer model, he TVP-STSM, in which all of he parameers of he rend, seasonal, cycle and explanaory variables are reaed as ime-varying. This TVP-STSM has been used o model ouris arrivals o Hong Kong from four key source markes. In addiion, he forecasing performance of he TVP-STSM has been examined and compared wih ha of seven commonly used ime series and economeric forecasing models. The empirical resuls show ha he newly developed TVP-STSM consisenly ouperforms ime series models, such as he Naïve 1 and Naïve 2 models, he SARIMA model and BSM, and economeric forecasing models, such as he TVP model, CSM and ADLM, for one- o four- and eigh-quarer-ahead forecasing. Is ousanding performance is paricularly eviden for one- o hree-quarers-ahead expos forecasing, and one-quarer-ahead ex-ane forecasing. Oher sae space models, including he CSM, BSM and he TVP model, all generae sound shor-erm (one o hree quarers ahead) forecass. These resuls sugges ha i is imporan o rea seasonaliy as sochasic (BSM and CSM), and also o esimae explanaory variables as ime-varying (TVP model) when modelling and forecasing seasonal ourism demand, and ha i is mos beneficial o consider boh joinly in one forecasing model (TVP-STSM). Alhough his sudy has demonsraed he advanages of he TVP-STSM over various ime series and economeric models, more research sill needs o be carried ou. Firsly, wih longer daases, he significance of he TVP esimaes of he explanaory variables in a TVP-STSM is likely o be improved. The esimaion of such a complex model wih large numbers of parameers and hyperparameers consumes a considerable number of degrees of freedom and is likely o affec he significance of he esimaes. Secondly, he TVP-STSM should be esed on a variey of origindesinaion pairs, in order o provide more robus empirical evidence on he meris of his new forecasing mehod. Thirdly, more economeric and ime series models could be inroduced ino he comparison in order o es he forecasing abiliy of he TVP- STSM furher. 19

20 Acknowledgemens The auhors would like o acknowledge he financial suppor of he Hong Kong UGC- GRF gran PolyU 5473/06H and daa suppor from Wei Guo. Ahanasopoulos wishes o acknowledge he financial suppor from he Ausralian Research Council gran DP Bios Haiyan Song is Chair Professor of Tourism in he School of Hoel and Tourism Managemen a The Hong Kong Polyechnic Universiy. His main research ineres is in ourism demand modeling and forecasing wih a paricular focus on he evaluaion of forecasing performance of various economeric models in he conex of ourism. Gang Li is Senior Lecurer in Economics in he Faculy of Managemen and Law a he Universiy of Surrey. Dr Li s research ineress include ourism economics and ourism demand forecasing. Sephen F. Wi is Emerius Professor of Tourism Forecasing in he Faculy of Managemen and Law a he Universiy of Surrey. His main research area is in ourism demand forecasing. George Ahanasopoulos is Senior Lecurer in he Deparmen of Economerics and Business Saisics, Monash Universiy, and his research ineress are in he areas of ime series analysis and forecasing. References Ahanasopoulos, G., Hyndman, R. J., Song, H., & Wu, D. (2010). The ourism forecasing compeiion. Inernaional Journal of Forecasing, forhcoming. Buler, R. (1994). Seasonaliy in ourism: Issues and problems. In A. Seaon (Ed.), Tourism: The sae of he ar (pp ). Chicheser: Wiley. Commandeur, J. J. F., & Koopman, S. J. (2007). An inroducion o sae space ime series analysis. New York: Oxford Universiy Press. Durbin, J., & Koopman, S. J. (2001). Time series analysis by sae space mehods. New York: Oxford Universiy Press. Frechling, D. C. (2001). Forecasing ourism demand: Mehods and sraegies. Oxford: Buerworh-Heinemann. González, P., & Moral, P. (1995). An analysis of he inernaional ourism demand in Spain. Inernaional Journal of Forecasing, 11, González, P., & Moral, P. (1996). Analysis of ourism rend in Spain. Annals of Tourism Research, 23, Hamilon, J. D. (1994a). Time series analysis (Chaper 13). Princeon, NJ: Princeon Universiy Press. Hamilon, J. D. (1994b). Sae space models. In R.F. Engle & D.L. McFadden (Eds.), Handbook of economerics (Vol. 4, pp ). Amserdam: Elsevier Science B.V. Harrison, P. J., & Sevens, C. F. (1976). Bayesian forecasing (wih discussion). Journal of Royal Saisical Sociey, Series B, 38,

21 Harvey, A. C. (1989). Forecasing srucural ime series models and he Kalman filer. New York: John Wiley. Harvey, D., Leybourne, S., & Newbold, P. (1997). Tesing he equaliy of predicion mean squared errors. Inernaional Journal of Forecasing, 13, Hyndman, R. J., & Khandakar, Y. (2008). Auomaic ime series forecasing: The forecas package for R, Journal of Saisical Sofware, 26, Hyndman, R. J., Koehler, A. B., Ord, K. & Snyder, R. D., (2008). Forecasing wih exponenial smoohing: he sae space appraoch. Springer-Verlag: Berlin- Heidelberg. Koopman, S. J., Harvey, A. C., Doornik, J. A., & Shephard, N. (2007). Srucural ime series analyser, modeller and predicor: STAMP 8. London: Timberlake Consulans. Koopman, S. J., Shephard, N., & Doornik, J. A. (1999). Saisical algorihms for models in sae space using SSF pack 2.2. Economerics Journal, 2, Kulendran, N., & King, M. (1997). Forecasing inernaional quarerly ourism flows using error correcion and ime series models. Inernaional Journal of Forecasing, 13, Kulendran, N., & Wi, S. F. (2001). Coinegraion versus leas squares regression. Annals of Tourism Research, 28, Kulendran, N., & Wi, S. F. (2003). Forecasing he demand for inernaional business ourism. Journal of Travel Research, 43, Li, G., Song, H., & Wi, S. F. (2005). Recen developmens in economeric modeling and forecasing. Journal of Travel Research, 44, Lucas, J. R. E. (1976). Economeric policy evaluaion: A criique. In K. Brumer & A. H. Melzer (Eds.), The Philip curve and labour markes (Vol. 1, pp ). Carnegie Rocheser Series on Public Policy, Journal of Moneary Economics (suppl.). Moosa, I. A. (2000). The cyclical behaviour of prices in he U.K.: Some srucural ime series evidence. Empirical Economics, 25, Shen, S., Li, G., & Song, H. (2008). An assessmen of combining ourism demand forecass over differen ime horizons. Journal of Travel Research, 47, Smeral, E., Wi, S. F., & Wi, C. A. (1992). Economeric forecass: Tourism rends o Annals of Tourism Research, 19, Song, H., & Wi, S. F. (2000). Tourism demand modelling and forecasing: Modern economeric approaches. Oxford: Pergamon. Song, H., Wi, S. F., & Jensen, T. C. (2003). Tourism forecasing: Accuracy of alernaive economeric models. Inernaional Journal of Forecasing, 19, Song, H., Wong, K. F., & Chon, K. S. (2003). Modelling and forecasing demand for Hong Kong ourism. Inernaional Journal of Hospialiy Managemen, 22, Turner, L. W., & Wi, S. F. (2001). Forecasing ourism using univariae and mulivariae srucural ime series models. Tourism Economics, 7,