Modeling Tourist Arrivals Using Time Series Analysis: Evidence From Australia

Transcription

1 Journal of Mahemaics and Saisics 8 (3): , 2012 ISSN Science Publicaions Modeling Touris Arrivals Using Time Series Analysis: Evidence From Ausralia 1 Gurudeo AnandTularam, 2 Vicor Siew Howe Wong and 3 Seyed AbdelhamidShobeiriNejad 1 Mahemaics and Saisics, Science Environmen Engineering and Technology (ENV), 2 Finance, School of Business, Finance and Economics 3 Science Environmen Engineering and Technology (ENV), Environmen Fuure Cenre, Griffih Universiy, Ausralia Absrac: Ausralian ourism has a logisic rend as Buler s model shows. The sagnaion has no been reached so opporuniies exis o increase ourism. The logisic model predics 7.2 million ouriss in 2015 bu ime series models of ARIMA and VAR improve he predicion and explain he daa. The ARIMA (2, 2, 2) fis well while he VAR lead o Granger causaliies beween he hree daa ses. A regression model (R 2 = 0.99) using Ausralian ouris arrival as a funcion of Europe and World arrivals allowed o furher undersand he Granger causaliy. The ARIMA model predics ouris numbers o be approximaely 6 million in The VAR echnique allowed impulse response analysis as well. A wo-way causaliy beween he ouris in Ausralia, Europe and World exiss, while impulse response indicaed differen effec paerns, where ouris arrivals increase in he firs period and declines in he second period bu experience seasonal flucuaions in he hird period. The sronges causaliies in were period 1 beween World and Europe; period 2-a one-way causaliy from Ausralia o World and period 3-a wo-way causaliies beween Ausralia, Europe and World. The impulse responses resuls were aligned wih he Buler heory. Keywords: ARIMA, auo-regression, mahemaical modeling, buler model, ourism modeling, granger causaliy, impulse response, variance decomposiion model, logisic regression, ime series INTRODUCTION The number of ouriss has significan impacs on global economy. These impacs can be couned as boh of negaive and posiive effecs. The mos imporan effec of ourism on economy can be known as number of changes on supply and demand chain in he desinaion which is he hos of ouriss. Touriss demand or simply ouris consumpion conribues o GDP, increasing he employmen rae, making new source of revenue for local people, privae and public secors and desinaion s governmen and so on. For insance, he consumpion inernaional and domesic ourism was approximaely AUD$ billion from July 2010 o year-ended June I shows ha he ourism share of GDP and oal employmen rae were 2.5 and 4.5%, respecively. Moreover, he share of ourism indusry in expor was 8% 0f oal Ausralian expor in (ABS, 2011). This significan impac is enough o Corresponding Auhor: Gurudeo AnandTularam, Senior Lecurer in Mahemaics and Saisics in School of Environmen, Environmen Fuure Cenre, Griffih Universiy, Ausralia 348 encourage researches o invesigae on number of ouris arrivals and aemp o make a more accurae predicion for fuure planning. The number of shor-erm inernaional arrivals o Ausralia has significanly grown over he las 6 decades from abou 60,000 o approximaely 6 million arrivals per year presenly. From 1956 o 2009 he Ausralian ourism indusry has been growing seadily following he rend of worldwide ourism and increasing over ime. However, in recen imes he number of ouris arrivals has saring slowing down and his may be jus eviden in Fig. 1. However when compared wih acual numbers Ausralian ourism is significanly less han eiher Europe or he World as shown in Fig. 2. This Fig. 1 shows he rend of changes in ouris arrivals o boh Ausralia and he world. The above model gives his chance o look a he rend of growh of boh Ausralia and he world inernaional arrivals a he same ime.

2 J. Mah. & Sa., 8 (3): , 2012 Fig. 1: Shor-Term Inernaional visiors Arrivals compared o he World (Based on ABS, 2011) Fig. 2: Low levels of ourism in Ausralia compared o Europe and he World 349

3 This Fig. 2 shows he rend of changes in inernaional arrivals o boh Europe and he world. For easier comparison in he nex graph he rend of inernaional arrival is shown. Ausralian invesmens on projecs faciliaing ourism are cosly and require a long-erm approach and planning in order o develop relevan infrasrucures (Song and Turner, 2006). The conribuion and impac of ourism o Ausralian economy is also well acceped and o caer for he arrivals much planning and developmen is indeed necessary for fuure needs in erms of infrasrucure and like. In order o prepare for he fuure, Ausralian governmens, companies and organizaions ofen rely on mahemaical models (Song and Turner, 2006). I is hen imporan for our models o accuraely predic arrival numbers a i is criical for boh public and privae organizaions budgeing and moneary posiions (Li e al., 2005). According o Song and Li (2008), ouris arrivals or ouris populaion a a paricular ime is he mos common variable o measure ourism demand and usually given by he oal number of ouris arrivals from an origin o a desinaion. The Touris Area Cycle Theory (TACT) is a concepual framework proposed by Buler (1980) for modeling. This framework is used o model ouris arrivals. In he firs par of his sudy a mahemaical a logisic model is developed based on he Touris Area Cycle Theory (Nejad and Tularam, 2010). However in his sudy an inegraed ime series ARIMA model is considered ogeher wih a linear mulivariae Vecor Auo-Regression (VAR) model is also for a more deailed sudy of drivers of Ausralian ourism in erms of Europe and he World. Similar analyses have been done in oher areas (Tularam, 2010; Tularam and Illahee, 2010). A Variance decomposiion model (VAR) can be used o sudy Granger Causaliy and impulse response for furhering undersanding of ourism numbers worldwide. The VAR model allows an examinaion of he naure of inegraion of he ourism marke when compared wih Europe and he World ourism numbers. In fac, by aggregaion of he concep of TACT and he sudy in his sudy, he number of ouris arrivals can be linked in boh Area Cycle as well as he world s ourism cycle. J. Mah. & Sa., 8 (3): , 2012 By he sar of he developmen sage, he desinaion s ourism faciliies in boh public and privae areas become well developed while in he fourh consolidaion sage, he ouriss number coninue o increase bu he desinaion now becomes well known and visied by many; hus no lised now as a prioriy for poenial ouriss in ha he rae of growh of he ouris numbers gradually declines unil finally, a sagnaion sage is reached when poenially all ouriss know he desinaion well including he faciliies on offer. In addiion, Buler (1980) argued wo possibiliies afer sagnaion sage is reached, namely; rejuvenaion or decline. The evoluion of ouris area s sages including: Exploraion, Involvemen, Developmen, Consolidaion and Sagnaion over ime. Mahemaical model:the iniial ouris model may be wrien as x = x 0 e m (- 0 ) where x is posiive and when = 0 given 0 = 0, he iniial value ouris number is greaer han zero (x 0 >0). When 0 = 0 he model may be wrien as x = x 0 m, represening an exponenial growh. A more comprehensive model is developed by combining Malhus law and he Verhuls assumpion (Nejad and Tularam, 2010); where he negaive effecs and leveling or sagnaion period are boh capured by he model. The model is given as: Where: X = β Lieraure review: Buler (1980) heoreically sudied he concepual cycle of ouris evoluion and idenified X x xɺ = m x (1) five sages; namely; exploraion, involvemen, X developmen, consolidaion and sagnaion (Fig. 3). The firs sage of exploraion ourism is no recognized The soluion o he differenial equaion developed as an economic aciviy in ha only a few people ravel above can be given by Eq. 2: o he desinaion. The involvemen sage is a ime period in which ouris numbers increase mainly due o X x = (2) m( a ) an increased awareness of he desinaion as a ouris base. 1 + e 350 x 1 β = 1 + e m( a) A represens ime a half he carrying capaciy and X being he maximum ouris capaciy. This model demonsraes ha he ouris numbers growh a ime +1 is proporional o consan growh rae m (as before); ouris number a las ime (as before); and discrepancy of ouris numbers a previous ime as a raio of maximum ouris numbers X. X The growh is proporional o x X Eq. 1:

4 J. Mah. & Sa., 8 (3): , 2012 Fig. 3: The evoluion of a ouris area (Adaped from Nejad and Tularam, 2010) Fig. 4: Fied growh rae and observed daa (Nejad and Tularam, 2010) The consan parameer a and he maximum capaciy X are boh assumed o be fixed for a paricularly desinaion. As shown, given 1 x β = he model may be wrien as m( a) 1 + e X = β ; where β is proporion (percenage) of ouris a ime o he capaciy (Nejad and Tularam, 2010). Based on he 2010 paper, he auhors have updaed he daa and recalculaed he predicions based on he model in Equaion 2 (Fig. 3). I was noed ha he Nejad and Tularam (2010) predicion was 5,594,500 arrivals using 351 he logisic model and he acual resul for 2010 a updaed in his sudy is 5,692,400 (wihin he 95% range of 5,594,500). The new model in his sudy includes he acual arrivals in 2010 and he new predicion is 5,878,709 (5,824,886-5,932,065) suggesing he heoreical model predicions are somewha overesimaes; however, reasonably good in general given ha cyclones, floods and financial crises have all impaced Queensland in his ime period. This Table 1 shows he predicion of ouris arrivals wihin 95% ran based on logisic model.

5 Table 1: Predicions of ouris arrivals o Ausralia by Nejad and Tularam (2010) Year Lower bound Poin esimae Upper bound ,757,412 5,817,344 5,871, ,827,883 5,884,440 5,935, ,890,755 5,943,801 5,991, ,946,695 5,996,177 6,040, ,996,348 6,042,281 6,082,646 Table 2: New logisic model predicions of ouris arrivals o Ausralia Year Lower bound Poin esimae Upper bound ,824,886 5,878,709 5,932, ,890,580 5,949,745 6,007, ,948,205 6,012,656 6,075, ,998,612 6,068,214 6,136, ,042,598 6,117,157 6,189, ,080,900 6,160,179 6,237,130 This Fig. 4 shows he model resuls of logisic regression inernaional arrivals o Ausralia. A 95% confidence inerval shows a good fi. The logisic model (Fig. 4) fis he daa well wih an R- squared value of The calculaed carrying capaciy for his model is 6,453,205 (6,325,129-6,581,159) wih he inernaional arrivals growh rae of ( ). Using he urning poin as he half of he carrying capaciy occurring around in he middle of 1993 ( ), he mahemaical model can now be wrien as: x = (3) ( ) 1 + e Nejad and Tularam (2010) analyzed heir resuls in erms of he Buler model and idenified he year for each sage. The updaed daa confirms he sages as follows: Exploraion (1973 o 1974) Involvemen ( o ) Developmen ( o ) Consolidaion ( o 2008) Sagnaion (2008-dependen o he world s rae) The heoreical model s 95% range predicions using Eq. 3 for inernaional Ausralian ouris arrivals are shown in Table 2. This Table 2 shows he predicion of ouris arrivals wihin 95% rang for-squared A more in deph analysis of he ourism daa is conduced in his sudy in erms of ime series using ARIMA and VAR mehods, similar o hose used in oher sudies (Roca and Tularam, 2012; Tularam e al., 2010; Tularam and Illahee, 2010). J. Mah. & Sa., 8 (3): , 2012 auoregressive and moving average parameers. ARIMA models explicily include differencing (Tularam, 2010). The hree ypes of parameers are: he auoregressive parameers (p), he number of differencing (d) and moving average parameers (q). For example ARIMA (2, 2, 2) conains 2 auoregressive (p) parameers and 2 moving average (q) parameers compued for he series differenced wice. In general form, ARIMA (p, d, q) model can be wrien using inercep form or wih lag noaion (L) as follows Eq. 4: 2 Model of ourism evoluion: wih εi ~ i.i.d(0, σ εi) and cov (ε y, ε z ) = 0 in marix form ARIMA-auoregressive inegraed moving average he above can be wrien as Eq. 6: model: The common ime series analysis may be conduced using auoregressive moving average models 1 b 12 y b 10 c11 c12 y 1 ε y = + + (6) known as ARIMA (p, d, q) (Box e al., 1994) including b 21 1 z b 20 c 21 c 22 z 1 ε z 352 p Y = ϕ Y + u + θ u i i j j i = 1 j= 1 Y (1 ϕ L ϕ L... ϕ L ) d 2 p 1 2 p = (1 θ L θ L... θ L )u 2 q 1 2 p q Where: Y = The ime series daa u = The usual error erm - u i ~i.id(0,σ 2 ) (4) Variance decomposiion-granger causaliy and impulse response analyses: The VAR modeling process allows more specific examinaions of he relaionships beween he hree daa series. The srucure of he VAR model is such ha i provides informaion abou a variable s forecasing abiliy for oher variables (Roca e al., 2010). Granger (1969) argued ha if a variable, or group of variables, y 1 is found o be helpful for predicing anoher variable, or group of variables, y 2 hen y 1 is said o Granger-cause y 2 ; oherwise i is said o fail o Grangercause y 2. The noion of Granger Causaliy does no imply rue Causaliy in ha i only implies forecasing abiliy (Roca and Tularam, 2012). In paricular, he srucure allows Granger Causaliy ess o be conduced ha may indicae wheher here is one or wo-way Granger Causaliy beween ouris arrivals in Ausralia, Europe and World. To sudy impacs over differen periods ime series daa was divided ino sub-periods, where a srucural break due could be idenified (financial crises); for example, Period 1 ( ) consised of he sock marke crash in Ocober 1987, Period 2 ( ) consised of he Sepember 11, 2001 aacks in he US and Period 3 ( ) was he Global Financial Crisis (GFC) in As an example, a general form of a wo-variable VAR (1, 1) wih k = 2 are shown in Eq. 5: y = b b z + c y + c z + ε y z = b b y + c y + c z + ε z (5)

6 The es of he join hypohesis ha none of he z s is a useful predicor, above and beyond lagged values of y, is called a Granger Causaliy es. I is noed ha Granger Causaliy simply refers o (marginal) predicive conen. The significance of he ouris modeling and predicion analysis can be srenghened furher using impulse response analysis o observe duraion and impac of ourism in one counry o anoher. The impulse response funcions races he effec of a shock o one counry on o he oher counries. Every srucural shock affecs every oher variable. Thus, we can consruc an impulse graph for each variable as he response o a cerain shock. For our VAR (1, 1) example, ineres may be in: The impulse response of y in response o a shock in he z-equaion, ε z The impulse response of z in response o a shock in he z-equaion, ε z The impulse response of y in response o a shock in he y-equaion, ε y The impulse response of z in response o a shock in he y-equaion, ε y The impulse response funcions will all have he same general shape and if he sysem is sable, he impulse responses will all approach zero. There will be a difference in he iming of he effecs. Finally, i is cusomary o se he size of he shock equal o is sandard deviaion. The impulse response hen shows he reacion o a shock of uni size. Impulse response analysis provides useful informaion for example, how, Ausralian ouris arrivals a a paricular ime is likely o respond o changes in Europe and World ouris numbers. Consider now he moving average represenaion of he muliple-equaion, VAR (m) model where he consan erms may be ignored: Y = ψ (L) x If E( x x ) = x such ha shocks are conemporaneously correlaed, hen he generalized impulse response funcion of Y i o a uni (one sandard deviaion) shock in X j is given by Eq. 7: 1/ 2 ( ) ( ) Ψ = σ e x e, (7) ij,h ii j i Where: σ ii = The i h diagonal elemen of x e i = A selecion vecor wih he i h elemen equal o one and all oher elemens equal o zero h = The horizon J. Mah. & Sa., 8 (3): , 2012 impac response of each variable o shocks o any oher variable may be deermined. RESULTS ARIMA: A code in R was used o obain a imer series ARIMA (2, 2, 2) model ha fied he Ausralian ouris arrival daa wih a high degree of i and low AIC, BIC (AIC = and BIC = -46.9) suggesing an excellen fied model: wih p = 2, q = 2 and d = 2 Eq. 8: Y (1 ϕ L ϕ L ) = (1 θ L θ L )u (8) The resuls of he Residuals, ACF and Lung-Box ess for he ARIMA analysis is shown in Fig. 5. This Fig. 5 shows ha he model saisfies all required ess for a suiable model for ouris daa. The esimaed coefficiens for he ARIMA mode are presen in Table 3 suggesing a close fi wih daa. This Table 3 shows he ARIMA (2,2,2) model parameers wih significances ME = , RMSE = and MAE = The forecas using he ARIMA (2,2,2) is given for 80 and 95% accuracy limis for he nex en years in Table 4 and he 2010 acual value fis raher well in he se. The 80 and 95% bounds are shown Figure for he predicions made up o year This Table 4 shows he predicion of ouris arrivals o Ausralia wihin 80 and 95% accuracy range. This Fig. 6 shows he forecas of Ausralian ouris arrivals based on he ARIMA model. Error bounds are highlighed in darker grey shows 80% and ligher grey shows 95% error bound. Figure 6 shows he ARIMA prediced number of Ausralian arrivals up o 2015; i is noed ha he 2010 predicion is a much a beer: 5,613,810 (5,353,809-5,873,811) compares well o he acual of 5,692,400 even when boh a flood and a second wave of he financial crises placed raher sudden shocks o ourism numbers. MR-mulivariae analysis: To furher cross check he naure of he predicions and examine he kind of relaionship beween Ausralian, Europe and he World, anoher analysis was underaken using he muliple regression mehod. The resuls of a Mulivariae linear Regression (MR) model based on Ausralian dependency on Europe and he World highlighed in Table 5 and Fig. 7. This Table 5 shows parameers for muliple regression analysis. *** indicaes 1% level of significance. The equaion is expressed as: Ausralia = - 2,445, (Europe) (World) + ε. This Fig. 7 shows he 95% error bound.as fied The impulse responses compued using he generalized mehod is invarian o re-ordering of he variables in he VAR. Since orhogonaliy is no imposed, meaningful inerpreaions of he iniial using he mulivariae analysis. 353

7 J. Mah. & Sa., 8 (3): , 2012 Fig. 5: ARIMA residuals and Ljung-Box saisic Fig. 6: Forecas for Ausralian ouris arrivals (80 and 95% error bound) 354

8 J. Mah. & Sa., 8 (3): , 2012 Fig. 7: 95% error bound on model fi using MR The mulivariae analysis shows a significan model ha he model highlighs he Ausralian dependency on he Europe and World numbers (Tularam and Keeler, 2006). The R-squared value was and he adjused R-squared Figure 8 furher shows he QQ plo demonsraing ha he Ausralian daa fi is in line wih he assumpions of a linear mulivariae regression mehod. These Fig. 9 and 10 show he closeness of fi of he mulivariae model-close o normaliy of he hisogram and linear QQ plo of MR model shows a good fi. Fig. 8: Hisogram and QQ plo of MR model Over years ( ) here has been a seady growh in he Ausralian ouris numbers bu during he number of ouriss fell down due o he fear of economic down urn, SARS, Iraq war among oher facors. The global financial crisis lowered numbers in bu he rae of reducion in Ausralia was less han he world and Europe and Europe numbers. The number of arrivals compared o Europe and he world inernaional movemen is ineresing. Europe has a long hisory of ourism and may have been he iniiaor of he indusry. I is no 355

9 surprising hen ha Europe is he world s leading ouris desinaion having abou 50% of he world s inernaional arrivals. This Table 5 shows he proporion of arrivals o Europe o he World. This Table 6 shows he proporion of arrivals o Ausralia o he World. This able shows average growh rae for World, Europe and Ausralia during Table 6 shows he world s inernaional arrivals average is around 6.3% while Europe has a rae around 5.9%. In he same period, Ausralia experienced an average growh rae of around 9%. This is around 1.5 imes he world s growh rae. This Fig. 11 shows he number of inernaional arrivals in Ausralia and World during The paern of Ausralian ouris numbers over ime is similar o he world growh as shown in Fig. 11. Furher analysis of Fig. 11 shows ha Ausralian arrivals was growing from 1956 o 1985 a an average rae of 0.04 million persons per year (Mp/yr), while he world was growing in he same period a an average rae of 8 Mp/yr; similarly, in he period 1985 o 2000, Ausralia was growing a 0.26 Mp/yr and he world was Mp/yr; and from 2000 o 2007, Ausralia was growing a Mp/yr while he world was growing a an average rae of 28.5 Mp/yr. Clearly, a significan growh has occurred in Ausralia in erms of acual numbers. Ausralian rae increased 6.5 imes from he firs o he second period, when he world increased by 3 imes respecively. Ausralia decreased by a quarer in he nex period, while he World was increasing a 1.1 imes ha of he earlier period rae. Ausralian increasing rae reached a maximum around 1990 and coninued o grow bu a a lower rae unil he 2007 financial crisis. While here is a dip afer 2007, he marke sared recovering. The effecs of boh he cyclone Yasi and Queensland floods do no appear in 2010 numbers because of a lag effec. J. Mah. & Sa., 8 (3): , 2012 significance. In he firs period ( ), he Causaliy was boh ways for Europe and he World and a 10% (low) level of significance from Europe o Ausralia. In his period, he economy is growing sronger for he European coninens and ouriss are ravelling o he European counries during 1950 o 1987 because here were 8 Olympics evens in he European counries (1952, 1956, 1960, 1964, 1968, 1972, 1976 and 1984). In he second period, he Ausralian economy is performing beer and here is one-way Causaliy from Ausralia o he World. This is possibly due o he Olympics in 2000 locaed in Sydney, when Ausralia araced millions of ouriss. The Causaliy is a 10% (weak) level of significance from Ausralia o he World. In he final period, economies are globalised wih more informaion coming from media and news; ouris can easily search for places o visi from he inerne wih increased aenion using markeing campaigns. The resuls of his is a wo-way Causaliy for all he counries a 1% (high) level of significance, excep for Ausralia a 10% (weak) level of significance o Europe and he World. Impulse response analysis: The impulse response analysis analyzed he duraion and impac of ourism in one counry o anoher by racing he effec of a shock o one counry on o he oher counries. Such dynamic relaionships are capured in he impulse response funcions found. Figure 13 shows he impulse responses for he full period, period 1, 2 and 3. In he full period, he responses are ploed over 100 periods/years. The responses are generally he same paern among he counries where he number of ouriss fall up o years, increase for years and declines aferwards. In he firs period ( ) may be characerized by Exploraion and Involvemen sages, where he responses show a sharp peak in ouris number afer 25 Granger causaliy: The VAR modeling process years for Ausralia, followed by he World and Europe. showed ineresing relaionships beween he hree daa The second period ( ) shows a decline in series. In paricular, he Granger Causaliy ess ourism for Europe and he World; however, Ausralia indicae if here is wo-way Granger Causaliy beween responses show an increase of ouriss coming from he ouris in Ausralia, Europe and World. As noed Europe and he World. The responses generally earlier, Period 1 ( ) consiss of he sock decrease/increase for he firs 5 years and gradually marke crash in Ocober 1987, Period 2 ( ) declining aferwards. This period is characerized by consiss of he Sepember 11, 2001 aacks in he US; he Developmen sage. In he las period (2002- and Period 3 ( ) refers o he Global Financial 2009), he responses show mixed signals. The Crisis (GFC, ). responses generally show signs of flucuaions every The flow and srengh of Causaliy or ineracion 5 years and i indicaed a season cycle and he beween he counries is summarized in Fig. 12. In he responses complees afer years. This is he whole sudy period, all counries are Granger causing period characerized by Consolidaion and he ouris arrivals beween one counry and he oher a Sagnaion sages of he Buler heory. 1% (high) level of significance; however, only This Fig. 12 shows he resuls of Granger causaliy Ausralia is causing he World a 5% (medium) level of esimaed based on he VAR model in a visual diagram. 356

10 J. Mah. & Sa., 8 (3): , 2012 Fig. 9: Percenage of share of Europe o he World ( ) Fig. 10: Percenage of share of Ausralia o he World ( ) Fig. 11: World and Ausralia inernaional arrivals

11 J. Mah. & Sa., 8 (3): , 2012 Fig. 12: Significan linkages beween ouris in Ausralia, Europe and World Table 3: Esimaed parameers of he ARIMA Coefficiens Coefficiens AR ** AR ** MA ** MA ** Table 4: Forecas for he nex 6 years based on ARIMA(2,2,2) Year Poin Low80% - high80% Low95% - high95% ,613,810 5,443,804-5,78,3815 5,353,809-5,873, ,679,011 5,403,521-5,954,501 5,257,686-6,100, ,764,731 5,417,662-6,111,800 5,233,934-6,295, ,854,525 5,448,805-6,260,244 5,234,030-6,475, ,938,902 5,468,510-6,409,294 5,219,499-6,658, ,016,012 5,462,093-6,569,930 5,168,86606,863,157 Table 5: MR model coefficiens for he ouris arrivals o Ausralia Model Coefficiens T Sig. Consan -2,445, *** Europe *** World *** Table 6: Average growh rae for World, Europe and Ausralia ( ) Growh rae (%) World 6.3 Europe 5.9 Ausralia 9.0 The Table 7 shows he Granger causaliy resuls esimaed based on he VAR model. The null hypohesis is ha he row variable does no Granger cause he column variable. All es saisics are Chisquare wih he p-values lised in brackes. ***, ** and * indicae significance a he 1, 5 and 10 percen levels respecively. The following regression esimaes was esimaed wih he following equaions. For he ineres of saving o 6,863,157 for space, we only show he full period equaions as follows (ε-n (0,1)): For example, he Granger Causaliy Equaions for he Full Period: Ausralia = Ausralia (-1) Europe (-1) + ε Ausralia = Ausralia (-1) World (-1) + ε Europe = Europe (-1) Ausralia (-1) + ε Europe = Europe (-1) World (-1) + ε World = World (-1) Ausralia (-1) + ε World = World (-1) Europe (-1) + ε This Fig. 13 shows he impulse responses analysis based on he VAR model. Concluding commens: In his sudy a number of models of ouris arrivals were developed and analyzed using ime series analyses. The Granger and impulse response analyses relaed o VAR modeling process was cross checked and jusified wih he use of a mulivariae linear model and ARIMA model was shown o be in line wih and beer han he heoreical model based on Buler. The models were calibraed using Ausralian ouris arrival daa ( ) and he modified Buler model predics growh predics around 7.2 million arrivals in 2015 bu a more realisic 6,160,179 and 95% range of arrivals may be given as 6,080,900-6,237,130 using he updaed heoreical logisic model presened in his sudy. The ARIMA (2, 2, 2) model predics using daa up o and including 2009 more accuraely predics for he year 2010 wih a value of 6,016,012 in The acual value is well wihin he 95% range of 5,168,866-6,863,157. Tha is, a 95% range of arrival numbers prediced lie wihin he range 5,618,866

12 J. Mah. & Sa., 8 (3): , 2012 Fig. 13: Impulse response for Ausralia, Europe and World - full period, period 1, 2 and 3 The resuls of he analysis show: There is a close Europe and indeed he World. The logisic model relaionship beween he ouris numbers recorded for derived from heory was compared wih he ime series Europe, Ausralia and he World even hough daa based ARIMA model and he ARIMA performed Ausralia s ouris arrives are very low in acual erms he beer in erms of he predicion for The when compared o he oher numbers of ouriss. heoreical model was no as accurae bu performed Ausralia ouris arrivals have been growing since 1974 well and can be used a benchmark for checking and have srong influences from ouris arrival o performance of oher models. 359

13 J. Mah. & Sa., 8 (3): , 2012 Table 7: Granger causaliy resuls Ausralia Europe World Full Period Ausralia (0.0022)*** (0.0120)** Europe (0.0012)*** (0.0000)*** World (0.0048)*** (0.0002)*** - Period 1 Ausralia (0.4396) (0.2638) Europe (0.0508)* (0.0698)* World (0.2085) (0.0334)** - Period 2 Ausralia (0.1591) (0.0800)* Europe (0.3042) (0.5190) World (0.4205) (0.7111) - Period 3 Ausralia (0.0659)* (0.0766)* Europe (0.0052)*** (0.0077)*** World (0.0000)*** (0.0000)*** - The ineracions sudied wih he VAR model allowed more in deph analyses based as he Granger Causaliy and impulse response calculaions were possible based in his srucure. The validiy of he models can be furher clarified by he examinaion of a muliple regression and furher analyses of he VAR in erm Granger Causaliy o provide he jusificaion and predicive abiliy of models. The naure of influence facors is beer exemplified by he impulse response analyses. Deailed analyses no only showed he naure of significan ineracions bu he impulse response furher showed ha he Buler s sages can be verified when he daa is sudied using imporan srucural breaks such as financial crises, SARS. Finally, he differenial equaion, ARIMA and VAR models all helped in he developmen of a beer undersanding he overall picure of he ourism sory suggesing ha in differen periods, ouris numbers from differen regions end o drive new shor erm arrivals around o Ausralia. Such informaion is paricularly useful for new invesors and developers as well as longer erm planners and governmen deparmens. REFERENCES ABS, Ausralian naional accouns: Tourism saellie accoun, Ausralian Bureau of Saisics. Box, G., G.M. Jenkins and G. Reinsel, Time Series Analysis: Forecasing and Conrol. 3rd Edn., Prenice Hall, ISBN-10: , pp: 592. Buler, R.W., The concep of a ouris area cycle of evoluion: Implicaions for managemen of resources. Canadian Geographer, 24: Granger, C.W.J., Invesigaing causal relaions by economeric models and cross-specral mehods. Economerica, 37: DOI: / Li, G., H. Song and S.F. Wi, Recen developmens in economeric modeling and forecasing. J. Travel Res., 44: DOI: / Nejad, S.A.H.S. and G.A. Tularam, Modeling ouris arrivals in desinaion counries: An applicaion o Ausralian. J. Mah. Sa., 6: DOI: /jmssp Roca, E. and G.A. Tularam, Which way does waer flow? An economeric analysis of he global price inegraion of waer socks. Applied Econom., 44: DOI: / Roca, E., V.S.H. Wong and G.A. Tularam, Are socially responsible invesmen markes worldwide inegraed. Accoun. Res. J., 23: DOI: / Song, H. and G. Li, Tourism demand modeling and forecasing. Tourism Manage., 29: Song, H. and L. Turner, Tourism Demand Forecasing. In: Inernaional Handbook on he Economics of Tourism, Dwyer, L. and P. Forsyh, (Ed.,) Edward Elgar, Chelenham. Tularam, G.A. and H.P. Keeler, The sudy of coasal groundwaer deph and saliniy variaion using ime-series analysis. Environ. Impac Assessmen Rev., 26: Tularam, G.A. and M. Illahee, Time series analysis of rainfall and emperaure ineracions in coasal cachmens. J. Mahemaics Saisics. 6: Tularam, G.A., Relaionship beween El Nino souhern oscillaion index and rainfall (Queensland, Ausralia). In. J. Susain. Dev. Plann., 5: Tularam, G.A., V.S.H. Wong and E.D. Roca, An analysis on Ausralian superannuaion funds volailiy using EGARCH approach. Sud. Regional Urban Plann.