Modelling international tourist arrivals and volatility fortaiwan

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1 8 World IMACS / MODSIM Congress Cairns Ausralia 3-7 Jul 009 p://mssanz.org.au/modsim09 Modelling inernaional ouris arrivals and volaili fortaiwan Cang C.-L. M. McAleer and D. Sloje 3 Deparmen of Applied Economics Naional Cung Hsing Universi Taiwan Economeric Insiue Erasmus Scool of Economics Erasmus Universi Roerdam Tinbergen Insiue Roerdam and Deparmen of Applied Economics Naional Cung Hsing Universi Taiwan 3 FTI Consuling and Deparmen of Economics Souern Meodis Universi USA cangcialin@ncu.edu.w Absrac: Inernaional ourism is a major source of expor receips for man counries worldwide. Aloug i is no e one of e mos imporan indusries in Taiwan (or e Republic of Cina) an island in Eas Asia off e coas of mainland Cina (or e People s Republic of Cina) e leading ourism source counries for Taiwan are Japan followed b USA Republic of Korea Malasia Singapore UK German and Ausralia. Tese counries reflec sor medium and long aul ouris desinaions. Aloug e People s Republic of Cina and Hong Kong are large sources of ourism o Taiwan e poliical siuaion is suc a ouriss from ese wo sources o Taiwan are repored as domesic ouriss. Dail daa from Januar 990 o 30 June 007 are obained from e Naional Immigraion Agenc of Taiwan. Te Heerogeneous Auoregressive (HAR) model is used o capure long memor properies in e daa. In comparison wi e HAR() model e esimaed asmmer coefficiens for GJR() are no saisicall significan for e HAR(7) and HAR(78) models so a eir respecive GARCH() counerpars are o be preferred. Tese empirical resuls sow a e condiional volaili esimaes are sensiive o e long memor naure of e condiional mean specificaions. Aloug asmmer is observed for e HAR() model ere is no evidence of leverage. Te QMLE for e GARCH() GJR() and EGARCH() models for inernaional ouris arrivals o Taiwan are saisicall adequae and ave sensible inerpreaions. However asmmer (oug no leverage) was found onl for e HAR() model and no for e HAR(7) and HAR(78) models. Kewords: Tourism demand inernaional ouris arrivals eerogeneous auoregressive model (HAR) condiional volaili asmmer leverage. 44

2 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan. Inroducion Taiwan (or e Republic of Cina) is an island in Eas Asia off e coas of mainland Cina (or e People s Republic of Cina). Inernaional ourism is a major source of expor receips for man counries worldwide and Taiwan is no excepion. Te mos well known ouris aracions in Taiwan include e Naional Palace Museum (Taipei) Nig Markes (especiall in Taipei) Taipei 0 formerl e world s alles building Sun Moon Lake (cenral iglands) and Taroko Naional Park (eas coas). Te mos imporan ourism source counries o Taiwan are Japan followed b USA Republic of Korea Malasia Singapore UK German and Ausralia wic reflec sor medium and long aul desinaions. Te ree mos imporan counries during e sample period ave been Japan USA and Republic of Korea. Aloug e People s Republic of Cina and Hong Kong are large sources of ourism o Taiwan e poliical siuaion is suc a ouriss from ese wo sources o Taiwan are repored as domesic ouriss. Tis paper models inernaional ouris arrivals and volaili (or squared deviaion from e mean) in inernaional ouris arrivals o Taiwan. Dail daa from Januar 990 o 30 June 007 are obained from e Naional Immigraion Agenc of Taiwan. B using dail daa we can approximae e modelling sraeg and analsis o ose applied o financial ime series daa. From a ime series perspecive ere are several reasons for using dail daa (see for example McAleer (009)). Jus o menion some dail daa allow invesigaing weer e ime series properies ave canged e ime series beaviour a oer frequencies can be obained b aggregaion of dail daa and e sample size is considerabl increased.. Daa Te daa se comprises dail inernaional ouris arrivals from Januar 990 o 30 June 007 giving 6390 observaions and are obained from e Naional Immigraion Agenc of Taiwan. On an annual basis e number of inernaional ouris arrivals o Taiwan as sown an average grow rae of around 4% per annum from 990 o 007. Te lowes grow rae was observed in 003 wi a decrease of 3.9% over e previous ear (due o e oubreak of SARS) wile e iges grow rae occurred in 004 wen ere was a significan increase of 36.58% over 003. In e sample period as a wole ere was an increase of around 75% in inernaional ouris arrivals o Taiwan wic would seem o indicae a reasonabl good performance in e ourism secor over e decade. Nevereless e annual average inernaional ouris arrivals grow rae reveals a ere is scope for a significan increase in inernaional ourism o Taiwan. In order o manage ourism grow and volaili i is necessar o model adequael inernaional ouris arrivals and eir associaed volaili. 3. Uni Roo Tess Sandard ADF uni roo ess are obained from e economeric sofware package EViews 6.0. Tere is no evidence of a uni roo in dail inernaional ouris arrivals o Taiwan in e model wi a consan and rend as e deerminisic erms or wi jus a consan. Tese empirical resuls allow e use of inernaional ouris arrivals daa o Taiwan o esimae alernaive univariae long memor condiional mean and condiional volaili models given in e nex secion. 4. Condiional Mean and Condiional Volaili Models Te alernaive ime series models o be esimaed for e condiional means of e dail inernaional ouris arrivals as well as eir condiional volailiies are discussed below. Dail inernaional ouris arrivals o Taiwan sow periods of ig volaili followed b oers of relaivel low volaili. One implicaion of is persisen volaili beaviour is a e assumpion of (condiionall) omoskedasic residuals is inappropriae. For a wide range of financial and ourism daa series ime-varing condiional variances can be explained empiricall roug e auoregressive condiional eeroskedasici (ARCH) model wic was proposed b Engle (98). Wen e ime-varing condiional variance as bo auoregressive and moving average componens is leads o e generalized ARCH(pq) or GARCH(pq) model of Bollerslev (986). Te lag srucure of e appropriae GARCH model can be cosen b informaion crieria suc as ose of Akaike and Scwarz aloug i is ver common o impose e widel esimaed GARCH() specificaion in advance. In e seleced condiional volaili model e residual series sould follow a wie noise process. Li e al. (00) provide an exensive review of recen eoreical resuls for univariae and mulivariae ime series models wi 45

3 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan condiional volaili errors and McAleer (005) reviews a wide range of univariae and mulivariae condiional and socasic models of financial volaili. Wen inernaional ouris arrivals daa displa persisence in volaili i is naural o esimae alernaive condiional volaili models. Te condiional volaili lieraure as been discussed exensivel in recen ears (see for example Li Ling and McAleer (00) McAleer (005) and McAleer Can and Marinova (007). Consider e saionar AR()- GARCH() model for dail inernaional ouris arrivals o Taiwan (or eir grow raes as appropriae) : = + + ε () < for =... n were e socks (or movemens in dail inernaional ouris arrivals) are given b: ε = η = ω + αε η ~ iid (0) + β () and ω > 0 α 0 β 0 are sufficien condiions o ensure a e condiional variance > 0. Te AR() model in equaion () can easil be exended o univariae or mulivariae ARMA(pq) processes (for furer deails see Ling and McAleer (003). In equaion () e ARCH (or α ) effec indicaes e sor run persisence of socks wile e GARCH (or β ) effec indicaes e conribuion of socks o long run persisence (namel α + β ). In equaions () and () e parameers are picall esimaed b e maximum likeliood meod o obain Quasi-Maximum Likeliood Esimaors (QMLE) in e absence of normali of η e condiional socks (or sandardized residuals). Since e GARCH process in equaion () is a funcion of e uncondiional socks e momens of ε need o be invesigaed. Ling and McAleer (003) sowed a e QMLE for GARCH(pq) is consisen if e second momen of ε is finie. For GARCH(pq) Ling and Li (997) demonsraed a e local QMLE is asmpoicall normal if e four momen of ε is finie wile Ling and McAleer (003) proved a e global QMLE is asmpoicall normal if e six momen of ε is finie. Using resuls from Ling and Li (997) and Ling and McAleer (00a 00b) e necessar and sufficien condiion for e exisence of e second momen of ε for GARCH() is α + β <. Te effecs of posiive socks (or upward movemens in dail inernaional ouris arrivals) on e condiional variance are assumed o be e same as e negaive socks (or downward movemens in dail inernaional ouris arrivals) in e smmeric GARCH model. In order o accommodae asmmeric beaviour Glosen Jagannaan and Runkle (99) proposed e GJR model for wic GJR() is defined as follows: = ( + ω + α + γi( η )) ε β (3) were ω > 0 α 0 α + γ 0 β 0 are sufficien condiions for > 0 I η ) is an indicaor variable defined b: ( I( η ) = 0 ε < 0 ε 0 as η as e same sign as ε. Te indicaor variable differeniaes beween posiive and negaive socks of equal magniude so a asmmeric effecs in e daa are capured b e coefficien γ. For financial daa i is expeced a γ 0 because negaive socks increase risk b increasing e deb o equi raio bu is inerpreaion need no old for inernaional ourism arrivals daa in e absence of a direc risk inerpreaion. Te asmmeric effec γ measures e conribuion of socks o bo sor run persisence γ α + and o long run persisence γ α + β +. I is no possible for leverage o be presen in e GJR model wereb negaive socks increase volaili and posiive socks of equal magniude decrease volaili. 46

4 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan An alernaive model o capure asmmeric beaviour in e condiional variance is e Exponenial GARCH (EGARCH()) model of Nelson (99) namel: log = + α η + γη + β log ω β < (4) were e parameers α β and γ ave differen inerpreaions from ose in e GARCH() and GJR() models. If γ = 0 ere is no asmmer wile γ < 0 and γ < α < γ are e condiions for leverage o exis wereb negaive socks increase volaili and posiive socks of equal magniude decrease volaili. 5. Esimaed Models and Discussion Te Heerogenous Auoregressive (HAR) model was proposed b Corsi (004) as an alernaive o model and forecas realized volailiies and is inspired b e Heerogenous Marke Hpoesis of Muller Dacorogna Dav Olsen Pice and Ward (993) and e asmmeric propagaion of volaili beween long and sor orizons. Corsi (004) sowed a e acions of differen pes of marke paricipans could lead o a simple resriced linear auoregressive model wi e feaure of considering volailiies realized over differen ime orizons. Te eerogenei of e model derives from e fac a differen auoregressive srucures are presen a eac ime scale (for furer deails see McAleer and Medeiros (008)). In is secion e HAR model is used o model oal inernaional ouris arrivals o Taiwan ogeer wi e ree condiional volaili models discussed in e previous secion. Te alernaive HAR() models o be esimaed o capure long memor are based on e following: = (5) were pical values of are one (dail daa) seven (weekl daa) and 8 (monl daa). In e empirical applicaion e ree models o be esimaed for inernaional ouris arrivals o Taiwan are as follows: = + + ε (6) = ε (7) = ε (8) wic will be referred o as e HAR() HAR(7) and HAR(78) models respecivel. Te condiional mean esimaes in Tables -3 sow a e HAR() HAR(7) and HAR(78) esimaes are all saisicall significan suc a e long memor properies of e daa are capured adequael. As e second momen condiion is less an uni in eac case and ence e weaker log-momen condiion (wic is no repored) is necessaril less an zero (see Tables -3) e regulari condiions are saisfied and ence e QMLE are consisen and asmpoicall normal and inferences are valid. Te EGARCH() model is based on e sandardized residuals so e regulari condiion is saisfied if β < and ence e QMLE are consisen and asmpoicall normal (see for example McAleer e al. (007)). Te GARCH() esimaes in Tables -3 for e HAR() HAR(7) and HAR(78) models of inernaional ouris arrivals o Taiwan sugges a e sor run persisence of socks lies beween 0.54 and 0.85 wile e long run persisence lies beween 0.36 and As e second momen condiion α + β < is saisfied e log-momen condiion is necessaril saisfied so a e QMLE are consisen and asmpoicall normal. Terefore saisical inference using e asmpoic normal disribuion is valid and e smmeric GARCH() esimaes are saisicall significan. If posiive and negaive socks of a similar magniude o inernaional ouris arrivals o Taiwan are reaed asmmericall is can be evaluaed in e GJR() model. Te asmmer coefficien is found o be posiive and significan for HAR() namel 0.37 wic indicaes a decreases in inernaional ouris arrivals increase volaili. Tis is a similar empirical oucome as is found in viruall all cases in finance were negaive socks (a is financial losses) increase risk (or volaili). Tus socks o inernaional ouris arrivals resemble financial socks and can be inerpreed as risk associaed wi inernaional ouris arrivals. Aloug asmmer is observed for e HAR() model 47

5 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan ere is no evidence of leverage. As e second momen condiion α + β + γ < is saisfied e log-momen condiion is necessaril saisfied so a e QMLE are consisen and asmpoicall normal. Terefore saisical inference using e asmpoic normal disribuion is valid and e asmmeric GJR() esimaes are saisicall significan. However in comparison wi e HAR() model e esimaed asmmer coefficiens for GJR() are no saisicall significan for e HAR(7) and HAR(78) models so a eir respecive GARCH() counerpars are o be preferred. Tese empirical resuls sow a e condiional volaili esimaes are sensiive o e long memor naure of e condiional mean specificaions. Te inerpreaion of e EGARCH model is in erms of e logarim of volaili. For inernaional ouris arrivals eac of e EGARCH() esimaes is saisicall significan for e HAR() model wi e size effec α being posiive and e sign effec γ being negaive. Te coefficien of e lagged dependen variable β is esimaed o be 0. wic suggess a e saisical properies of e QMLE for EGARCH() will be consisen and asmpoicall normal. As in e case of e GJR() model e esimaed asmmer coefficiens for EGARCH() are no saisicall significan for e HAR(7) and HAR(78) models. Tese empirical resuls sow a e volaili in e socks o inernaional ouris arrivals o Taiwan are sensiive o e long memor naure of e condiional mean specificaions. Acknowledgemens For financial suppor e firs auor wises o ank e Naional Science Council Taiwan and e second auor wises o ank e Ausralian Researc Council. References Bollerslev T. (986) Generalised auoregressive condiional eeroscedasici Journal of Economerics Corsi F. (004) A simple long memor model of realized volaili" Unpublised paper Universi of Souern Swizerland. Engle R.F. (98) Auoregressive condiional eeroscedasici wi esimaes of e variance of Unied Kingdom inflaion Economerica Glosen L. R. Jagannaan and D. Runkle (99) On e relaion beween e expeced value and volaili of nominal excess reurn on socks Journal of Finance Li W.K. S. Ling and M. McAleer (00) Recen eoreical resuls for ime series models wi GARCH errors Journal of Economic Surves Reprined in M. McAleer and L. Oxle (eds.) Conribuions o Financial Economerics: Teoreical and Pracical Issues Blackwell Oxford 00 pp Ling S. and W.K. Li (997) On fracionall inegraed auoregressive moving-average models wi condiional eeroskedasici Journal of e American Saisical Associaion Ling S. and M. McAleer (00a) Saionari and e exisence of momens of a famil of GARCH processes Journal of Economerics Ling S. and M. McAleer (00b) Necessar and sufficien momen condiions for e GARCH(rs) and asmmeric power GARCH(rs) models Economeric Teor Ling S. and M. McAleer (003) Asmpoic eor for a vecor ARMA-GARCH model Economeric Teor McAleer M. (005) Auomaed inference and learning in modeling financial volaili Economeric Teor 3-6. McAleer M. (009) Te Ten Commandmens for opimizing value-a-risk and dail capial carges o appear in Journal of Economic Surves. McAleer M. F. Can and D. Marinova (007) An economeric analsis of asmmeric volaili: eor and applicaion o paens Journal of Economerics McAleer M. and M. Medeiros (008) A muliple regime smoo ransiion eerogeneous auoregressive model for long memor and asmmeries Journal of Economerics 47() Muller U. M. Dacorogna R. Dav R. Olsen O. Pice and J. ward (993) Fracals and inrinsic ime - a callenge o economericians" in Proceedings of e XXXIX Inernaional AEA Conference on Real Time Economerics. Nelson D.B. (99) Condiional eeroscedasici in asse reurns: a new approac Economerica

6 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan Table : Esimaed Condiional Mean (HAR()) and Condiional Volaili Models Parameers GARCH GJR EGARCH 5** (48.85) 0.806** (0.007) ω ** (4864) 0.54** GARCH/GJR α GARCH/GJR β GJR γ 00** (47.) 0.86** (0.007) 8073** (560) 0.55** (0.00) 0.0 (0.08) 0.37** (0.043) EGARCH α EGARCH γ EGARCH β 004** (46.97) 0.87** (0.007).8** (0.54) 0.483** (0.0) -0.8** (0.06) 0.** (0.037) Second momen AIC BIC Jarque-Bera [p-value] Noes: Te dependen variable TA is inernaional ouris arrivals o Taiwan. Numbers in pareneses are sandard errors. Te log-momen condiion is necessaril saisfied as e second momen condiion is saisfied. AIC and BIC denoe e Akaike Informaion Crierion and Scwarz Baesian Informaion Crierion respecivel. ** denoes e esimaed coefficien is saisicall significan a %. 49

7 Cang e al. Modelling inernaional ouris arrivals and volaili fortaiwan Table : Condiional Mean (HAR(7)) and Condiional Volaili Models Parameers GARCH GJR EGARCH 3.34** (5.00) 0.99** (0.04) 0.64** ω 56553** 3 GARCH/GJR α GARCH/GJR β (068) 0.85** 0.47** (0.0) GJR γ 3.5** (5.9) 0.99** (0.04) 0.64** 5630** (06) 0.85** (0.07) 0.47** (0.0) (0.03) EGARCH α EGARCH γ EGARCH β 94.3** (49.58) 0.30** (0.03) 0.65** 9.563** (0.430) 0.50** (0.0) ** (0.03) Second momen AIC BIC Jarque-Bera [p-value] Table 3: Condiional Mean (HAR(78)) and Condiional Volaili Models Parameers GARCH GJR EGARCH 67.8** (54.6) 0.98** (0.04) 0.460** 3 (0.0) 0.08** 4 (0.09) ω 5379** (9854) 0.85** GARCH/GJR α GARCH/GJR 0.3** β (0.0) GJR γ 66.58** (54.59) 0.99** (0.04) 0.459** (0.0) 0.08** (0.09) ** (08) 0.83** (0.07) 0.30** (0.0) (0.03) EGARCH α EGARCH γ EGARCH β 44.40** (5.93) 0.37** (0.03) 0.445** (0.00) 0.08** (0.08) 0.03** (0.439) 0.50** (0.0) ** (0.03) Second momen AIC BIC Jarque-Bera [p-value]