Aggregation, Heterogeneous Autoregression and Volatility of Daily International Tourist Arrivals and Exchange Rates*

Transcription

1 Aggregaion, Heerogeneous Auoregression and Volailiy of Daily Inernaional Touris Arrivals and Exchange Raes* Chia-Lin Chang Deparmen of Applied Economics Naional Chung Hsing Universiy Taichung, Taiwan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands Revised: February

2 Absrac Tourism is a major source of service receips for many counries, including Taiwan. The wo leading ourism counries for Taiwan are Japan and USA, which are sources of shor and long haul ourism, respecively. As a srong domesic currency can have adverse effecs on inernaional ouris arrivals, daily daa from 1 January 1990 o 31 December 2008 are used o model world price, exchange raes, and ouris arrivals from he world, USA and Japan o Taiwan, and heir associaed volailiy. Inclusion of he exchange rae capures daily price effecs on world, US and Japanese ouris arrivals o Taiwan. The Heerogeneous Auoregressive (HAR) model is used o approximae he slowly decaying correlaions associaed wih he long memory properies in daily exchange raes and inernaional ouris arrivals, o es wheher alernaive shor and long run esimaes of condiional volailiy are sensiive o he long memory in he condiional mean, o examine asymmery and leverage in volailiy, and o examine he effecs of emporal and spaial aggregaion. For policy purposes, he empirical resuls sugges ha an arbirary choice of daa frequency or spaial aggregaion will no lead o robus findings as hey are generally no independen of he level of aggregaion used. Keywords: Inernaional ouris arrivals, exchange raes, GARCH, GJR, EGARCH, HAR, long memory, emporal and spaial aggregaion, daily and weekly effecs, asymmery, leverage. JEL Classificaions: C22, F31, G18, G32. 2

3 1. Inroducion Tourism is a major source of service receips for many counries, including Taiwan. The wo leading ourism counries for Taiwan, comprising a high proporion of world ouris arrivals o Taiwan, are Japan and USA, which are sources of shor and long haul ourism, respecively. Alhough more han hree million inernaional ouriss visied Taiwan in 2008, he major par of he Taiwan ouris indusry is suppored by domesic ourism. Taiwan s exensive nework of rains and highways makes i possible o raverse he counry (norh-souh) in less han wo hours by he new high speed rain, and in a few hours by car. The mos well known ouris aracions in Taiwan include he specacular Naional Palace Museum (Taipei), home o some of Chinese greaes aniquiies, he amazing Nigh Markes hroughou he counry, Taipei 101, formerly he world s alles building, relaxing Sun Moon Lake (near Puli in he cenral highlands), and sunning Taroko Naional Park in Hualian on he eas coas. A major purpose in ourism markeing is o increase oal ourism expendiure receips. If he daily expendiure per inernaional ouris were o be reasonably consan over he sample period, hen inernaional ouris arrivals and oal inernaional ourism expendiure would be highly correlaed. Moreover, he rae of growh in daily inernaional ourism expendiure and he rae of growh in daily inernaional ouris arrivals would hen be virually idenical. As i is well known ha a srong domesic currency can have adverse effecs on inernaional ouris arrivals, one of he primary purposes of he paper is o model daily ouris arrivals o Taiwan from he world, USA and Japan, and he world price and US$ / New Taiwan $ and Yen/ New Taiwan $ exchange raes, and heir respecive volailiies. Daily daa from 1 January 1990 o 31 December 2008 are obained from he Naional Immigraion Agency of Taiwan for daily world, US and Japanese ouris arrivals, he Bloomberg daabase for he wo foreign exchange raes, and he Reuers daabase for he world price. In order o manage inernaional ouris arrivals from he major ourism sources, as well as ourism growh and is corresponding volailiy, i is necessary o model adequaely inernaional ouris arrivals and heir associaed volailiy, especially in he presence of significan economic and financial shocks. In ligh of he global financial crisis, is significan economic impac on he ourism indusry in Taiwan and inernaionally, and he need for a speedy and informaive analysis of he level of inernaional ouris arrivals, heir growh raes and associaed volailiy, i is 3

4 essenial o use daily daa raher han he usual monhly, quarerly or annual daa ha have radiionally been used in previous empirical ourism sudies. Daily daa permi an appeal o he heoreical resuls available in financial economerics, and an approximaion of he modelling and forecasing sraegies widely used in financial ime series analysis. From a ime series perspecive, here are several reasons for using daily daa raher han lower frequencies such as monhly, quarerly or annual daa (see, for example, McAleer (2009)). In addiion o he use of much larger sample sizes han hose associaed wih monhly, quarerly or annual daa, he use of daily daa permis an examinaion of wheher he ime series properies have changed. The ime series behaviour a oher emporal frequencies, such as weekly daa, can be obained by aggregaion of daily daa, so ha emporal aggregaion effecs can be analysed. Moreover, approximae daily price elasiciies of he demand for inernaional ourism can be esimaed hrough he use of daily exchange raes, and he daily volailiy of inernaional ourism demand and exchange raes can be analysed. The use of daily daa enables more immediae responses o be acivaed in ligh of generaing daily esimaes and forecass of approximae price effecs hrough he exchange rae, and accurae daily forecass of ouris arrivals and heir growh raes, in response o significan economic and financial shocks. In addiion, he esimaion and forecasing of ime-varying condiional volailiies will enable more accurae confidence inervals for inernaional ouris arrivals and heir growh raes o be deermined on a daily basis. The remainder of he paper is organized as follows. Secion 2 presens he daily world, US and Japanese ouris arrivals, world price and exchange rae ime series daa. Secion 3 performs uni roo ess on he hree ouris arrivals series for daily and weekly daa. Secion 4 discusses approximae long memory condiional mean and condiional volailiy models for daily world, US and Japanese ouris arrivals, world price and wo exchange raes. The esimaed models and empirical resuls for he heerogeneous auoregressive (HAR) and hree univariae condiional volailiy models are discussed in Secion 5, as are he effecs of emporal aggregaion from daily o weekly daa. Finally, some concluding remarks are given in Secion Daa The daa se comprises daily ouris arrivals from he world, USA and Japan for he period 1 January 1990 o 31 December 2008, giving 6,940 observaions obained from he Naional Immigraion Agency of Taiwan, and an equivalen number of observaions for he US$ / New 4

5 Taiwan $ and Yen/ New Taiwan $ exchange raes, ha are obained from he Bloomberg daabase: Taipei Foreign Exchange Marke Developmen Foundaion (URL: hp:// The world price is obained from Reuers as he Inerconinenal Exchange calculaion of he US Dollar Index, which is he US $ relaive o a geomeric weighed mean of six currencies (namely, Euro, Canadian $, Japanese yen, Swedish krona, Pound serling and Swiss franc). Thus, if he US $ increases relaive o he world price, hen prices in he USA will be lower, hereby leading o reduced US ouriss o Taiwan. Moreover, he higher world price will have a posiive income effec for he res of he world, which will end o increase world ouris arrivals o Taiwan. Overall, he world price effec on world ouris arrivals o Taiwan would be expeced o be negaive. Figures 1-4 plo he daily and weekly ouris arrivals from he world, USA and Japan, and he world price and US$ / New Taiwan $ and Yen/ New Taiwan $ exchange raes, as well as heir respecive volailiies, where volailiy is defined as he squared deviaion from he sample mean. There is higher volailiy persisence a he end of he sample period, due primarily o he global financial crisis (for furher deails on he global financial crisis see, for example, McAleer (2009) and McAleer e al. (2009, 2009a, 2009b, 2010)). Daily and weekly ouris arrivals o Taiwan from he world, USA and Japan, and he corresponding daily and weekly exchange raes, have varied considerably over he sample period, which suggess ha he daily and weekly effecs of he approximae price movemens on inernaional ourism demand migh be capured using appropriae heerogeneous ime series and condiional volailiy models. The exchange rae effecs aside, here would seem o be considerable scope for a significan increase in ourism o Taiwan from he world, USA and Japan. 3. Uni Roo Tess Sandard uni roo ess based on he classic mehods of Dickey and Fuller (1979, 1981) and Phillips and Perron (1988) are available in he economeric sofware package EViews 6.0, and are repored in Table 1. There is no evidence of a uni roo in daily or weekly world, US and Japanese ouris arrivals o Taiwan in he model wih a consan and rend as he deerminisic erms, or wih jus a consan, so ha daily and weekly series o be modeled are saionary. These empirical resuls allow he use of world, US and Japanese ouris arrivals daa o Taiwan, and he hree exchange raes, o esimae alernaive univariae approximae long memory condiional 5

6 mean and condiional volailiy models given in he nex secion. Before doing so, i is useful o examine which daily exchange raes should be used for heir weekly counerpars. Alhough no repored here, we calculaed he correlaion coefficiens for he world price and Japanese and US exchange raes for he arihmeic and geomeric means of he seven daily prices and exchange raes, as well as for he seven days of he week, for purposes of selecing he appropriae world weekly price and weekly exchange raes for Japan and USA. The correlaion coefficiens are very close o one in all cases, and he arihmeic and geomeric means are idenical o hree decimal places. For his reason, he arihmeic means of he seven daily world prices and exchange raes are chosen as he respecive weekly prices and weekly exchange raes for he world, Japan and USA. 4. Condiional Mean and Condiional Volailiy Models The alernaive ime series models o be esimaed for he condiional means of daily and weekly world, US and Japanese ouris arrivals o Taiwan, as well as heir respecive condiional volailiies, are discussed below. As shown in Figures 1-4, boh daily and weekly world, US and Japanese ouris arrivals o Taiwan and he hree exchange raes show periods of high volailiy, followed by ohers of relaively low volailiy. One implicaion of his persisen volailiy behaviour is ha he assumpion of (condiionally) homoskedasic residuals is inappropriae. As discussed in Divino and McAleer (2009, 2010) and Chang and McAleer (2009), for example, for a wide range of daa series in finance, inernaional finance and ourism research, ime-varying condiional variances can be explained empirically hrough he auoregressive condiional heeroskedasiciy (ARCH) model (Engle (1982)). When he ime-varying condiional variance has boh auoregressive and moving average componens, his leads o he generalized ARCH(p,q), or GARCH(p,q) (Bollerslev (1986)), wih he lag srucure of he appropriae GARCH model ypically given as he widely esimaed GARCH(1,1) specificaion. Li e al. (2002) provide an exensive review of heoreical resuls for univariae and mulivariae ime series models wih condiional volailiy errors, and McAleer (2005) reviews a wide range of univariae and mulivariae, condiional and sochasic, models of financial volailiy. When he daily and weekly world, US and Japanese ouris arrivals daa, and he hree exchange rae series, display persisence in volailiy, as shown in Figures 1-4, i is naural o esimae alernaive condiional volailiy models. 6

7 The GARCH(1,1), GJR(1,1) and EGARCH(1,1) condiional volailiy models have been esimaed using monhly and daily ouris arrivals daa in several papers, including Chan, Lim and McAleer (2005), Hoi, McAleer and Shareef (2005, 2007), Shareef and McAleer (2005, 2007, 2008), Chang e al. (2009), Chang and McAleer (2009), and Divino and McAleer (2009, 2010). However, hese papers have no esimaed any spillover effecs beween ouris arrivals and exchange raes using daily and weekly daa, and have no examined world price effecs, and hence have no been able o capure any approximae price effecs affecing ourism demand, or he effecs of emporal and spaial aggregaion. The condiional volailiy lieraure has been discussed exensively in recen years (see, for example, Li, Ling and McAleer (2002), McAleer (2005), McAleer, Chan and Marinova (2007), and Caporin and McAleer (2009, 2010)). Consider he saionary AR(1)-GARCH(1,1) model for daily or weekly world, US and Japanese ouris arrivals o Taiwan, y : y 1 (1) 1 2 y 1, 2 for 1,..., n, where he shocks (ha is, movemens in inernaional ouris arrivals are given by: h, h ~ iid (0,1) h 2 1 1, (2) and 0, 0, 0 are sufficien condiions o ensure ha he condiional variance h 0. The AR(1) model in equaion (1) can easily be exended o univariae or mulivariae ARMA(p,q) processes (for furher deails, see Ling and McAleer (2003a)). In equaion (2), he ARCH (or ) effec indicaes he shor run persisence of shocks, while he GARCH (or ) effec indicaes he conribuion of shocks o long run persisence (namely, + ). The saionary AR(1)-GARCH(1,1) model can be modified o incorporae a non-saionary ARMA(p,q) condiional mean and a saionary GARCH(r,s) condiional variance, as in Ling and McAleer (2003b). In equaions (1) and (2), he parameers are ypically esimaed by he maximum likelihood mehod o obain Quasi-Maximum Likelihood Esimaors (QMLE) in he absence of normaliy of, he 7

8 condiional shocks (or sandardized residuals). The condiional log-likelihood funcion is given as follows: n l 1 1 n 2 log h. 2 1 h The QMLE is efficien only if is normal, in which case i is he MLE. When is no normal, adapive esimaion can be used o obain efficien esimaors, alhough his can be compuaionally inensive. Ling and McAleer (2003b) invesigaed he properies of adapive esimaors for univariae non-saionary ARMA models wih GARCH(r,s) errors. The exension o mulivariae processes is complicaed. As he GARCH process in equaion (2) is a funcion of he uncondiional shocks, i is necessary o examine he momens condiions of. Ling and McAleer (2003a) showed ha he QMLE for GARCH(p,q) is consisen if he second momen of is finie. Using resuls from Ling and Li (1997) and Ling and McAleer (2002a, 2002b), he necessary and sufficien condiion for he exisence of he second momen of for GARCH(1,1) is 1 and, under normaliy, he 2 2 necessary and sufficien condiion for he exisence of he fourh momen is ( ) 2 1. As discussed in McAleer e al. (2007), i was esablished by Elie and Jeanheau (1995) and Jeanheau (1998) ha he log-momen condiion was sufficien for consisency of he QMLE of a univariae GARCH(p,q) process (see Lee and Hansen (1994) for an analysis of he GARCH(1,1) process), while Boussama (2000) showed ha he log-momen condiion was sufficien for asympoic normaliy. Based on hese heoreical developmens, a sufficien condiion for he QMLE of GARCH(1,1) o be consisen and asympoically normal is given by he log-momen condiion, namely 2 E (log( )) 0. (3) However, his condiion is no easy o check in pracice, even for he GARCH(1,1) model, as i involves he expecaion of a funcion of a random variable and unknown parameers. Alhough he sufficien momen condiions for consisency and asympoic normaliy of he QMLE for he univariae GARCH(1,1) model are sronger han heir log-momen counerpars, he second momen 8

9 condiion is more sraighforward o check. In pracice, he log-momen condiion in equaion (3) would be esimaed by he sample mean, wih he parameers and, and he sandardized residual,, being replaced by heir QMLE counerpars. The effecs of posiive shocks (or upward movemens in daily or weekly inernaional ouris arrivals or exchange raes) on he condiional variance, h, are assumed o be he same as he negaive shocks (ha is, downward movemens in daily or weekly inernaional ouris arrivals or exchange raes) in he symmeric GARCH model. In order o accommodae asymmeric behaviour, Glosen, Jagannahan and Runkle (1992) proposed he GJR model, for which GJR(1,1) is defined as follows: h 2 I( )) h, (4) ( where 0, 0, 0, 0 are sufficien condiions for h 0, and I ) is an indicaor variable defined by: ( 1, I( ) 0, 0 0 as has he same sign as. The indicaor variable differeniaes beween posiive and negaive shocks of equal magniude, so ha asymmeric effecs in he daa are capured by he coefficien. For financial daa, i is expeced ha 0 because negaive shocks increase risk by increasing he deb o equiy raio, bu his inerpreaion need no hold for daily or weekly inernaional ouris arrivals or exchange raes in he absence of a direc risk inerpreaion. The asymmeric effec,, measures he conribuion of shocks o boh shor run persisence,, and o long run 2 persisence,. I is no possible for leverage o be presen in he GJR model, whereby 2 negaive shocks increase volailiy and posiive shocks of equal magniude decrease volailiy. 9

10 Ling and McAleer (2002a) showed ha he regulariy condiion for he exisence of he second momen for GJR(1,1) under symmery of is given by: 1 1, (5) 2 while McAleer e al. (2007) showed ha he weaker log-momen condiion for GJR(1,1) was given by: 2 E (ln[( I( )) ]) 0, (6) which involves he expecaion of a funcion of a random variable and unknown parameers. An alernaive model o capure asymmeric behaviour in he condiional variance is he Exponenial GARCH (EGARCH(1,1)) model of Nelson (1991), namely: log h h, 1 (7) 1 1 log 1 where he parameers, and have differen inerpreaions from hose in he GARCH(1,1) and GJR(1,1) models. If = 0, here is no asymmery, while < 0, and < < - are he condiions for leverage o exis, whereby negaive shocks increase volailiy and posiive shocks of equal magniude decrease volailiy. As noed in McAleer e al. (2007), here are some imporan differences beween EGARCH and he previous wo models, as follows: (i) EGARCH is a model of he logarihm of he condiional variance, which implies ha no resricions on he parameers are required o ensure h 0 ; (ii) momen condiions are required for he GARCH and GJR models as hey are dependen on lagged uncondiional shocks, whereas EGARCH does no require momen condiions o be esablished as i depends on lagged condiional shocks (or sandardized residuals); (iii) Shephard (1996) observed ha 1 is likely o be a sufficien condiion for consisency of QMLE for EGARCH(1,1); (iv) as he sandardized residuals appear in equaion (7), 1 would seem o be a sufficien condiion for he exisence of momens; and (v) in addiion o being a sufficien condiion for consisency, 1 is also likely o be sufficien for asympoic normaliy of he QMLE of EGARCH(1,1). 10

11 Furhermore, EGARCH capures asymmeries differenly from GJR. The parameers and in EGARCH(1,1) represen he magniude (or size) and sign effecs of he sandardized residuals, respecively, on he condiional variance, whereas and represen he effecs of posiive and negaive shocks of equal magniude, respecively, on he condiional variance in GJR(1,1). 5. Heerogeneous Models and Empirical Analysis The Heerogenous Auoregressive (HAR) model was proposed by Corsi (2009) as an alernaive o model and forecas realized volailiies, and is inspired by he Heerogenous Marke Hypohesis of Muller, Dacorogna, Dav, Olsen, Pice, and Ward (1993) and he asymmeric propagaion of volailiy beween long and shor horizons. Corsi (2009) showed ha he acions of differen ypes of marke paricipans could lead o a resriced auoregressive model wih he feaure of considering volailiies realized over differen ime horizons. The heerogeneiy of he model derives from he fac ha differen auoregressive srucures are presen a each ime scale (for furher deails, see McAleer and Medeiros (2008)). Alhough HAR models canno reproduce he heoreical hyperbolic decay raes associaed wih fracionally inegraed (or long memory) ime series models, hey can neverheless approximae quie accuraely and parsimoniously he slowly decaying correlaions associaed wih such long memory models. For his reason, HAR models may be inerpreed as simple resriced approximaions o long memory models. Alernaive HAR models will be used o model inernaional ouris arrivals o Taiwan from he world, USA and Japan, ogeher wih hree widely used univariae condiional volailiy models, namely GARCH, GJR and EGARCH, as discussed in he previous secion. The alernaive HAR(h) models o be esimaed o approximae long memory are based on he following: y, h y y 1 y2... yh 1 (8) h where ypical values of h are 1 (daily daa), 7 (weekly daa), and 28 (monhly daa). In he empirical applicaion, he hree HAR models for world, US and Japanese daily ouris arrivals o Taiwan are: 11

12 y y y y x (9) 1 21 y 1 22x 1 31y 1,7 32x 1, 7 (10) y x y x y x ,7 32 1,7 41 1, , 28 (11) and he wo HAR models for world, US and Japanese weekly ouris arrivals o Taiwan are: y y y x (12) y x y x ,4 32 1, 4. (13) The models in equaions (9)-(11) will be referred o as he HAR(1), HAR(1,7) and HAR(1,7,28) models, respecively, and hose in equaions (12)-(13) as he HAR(1) and HAR(1,4) models, respecively. The wo ses of models in (9)-(11) and (12)-(13) enable an assessmen of he effecs of emporal aggregaion from he daily o weekly daa frequency. Moreover, a comparison of he model of world ouris arrivals o Taiwan wih hose of US and Japanese ouris arrivals o Taiwan enable an examinaion of spaial aggregaion effecs on he HAR esimaes, shor and long run persisence of shocks on ouris arrivals, he exchange rae effecs, and he empirical regulariy condiions. The esimaed condiional mean and condiional volailiy models for he world, Japan and USA are given for he HAR(1) model in Tables 2-4 for daily daa and in Tables 5-7 for weekly daa (he resuls for he HAR(1,7) and HAR(1,7,28) models for daily daa, and HAR(1,4) model for weekly daa, are available on reques). The mehod used in esimaion was he Marquard algorihm. The condiional mean esimaes in Tables 2-7 show ha he HAR(1) esimaes for daily and weekly daa are all saisically significan. Thus, he approximae long memory properies of world, Japanese and US ouris arrivals o Taiwan are capured adequaely hrough he saisical significance of he approximae long memory variables. As he second momen condiions for he GARCH(1,1) and GJR(1,1) models are less han uniy in each case, he log-momen condiions are also necessarily saisfied. Thus, he regulariy condiions are saisfied, and hence he QMLE are consisen and asympoically normal, and inferences are valid. The EGARCH(1,1) model is based on he sandardized residuals, so he regulariy condiion 12

13 is saisfied if 1, and hence he QMLE are consisen and asympoically normal (see, for example, McAleer e al. (2007)). Alhough no repored here, he GARCH(1,1) esimaes in Tables 2-7 for he HAR(1) models of world, Japanese and US ouris arrivals o Taiwan sugges ha he shor and long run persisence of shocks for daily daa lie beween (0.220, 0.261) and (0.243, 0.429), respecively, for he world, beween (0.256, 0.326) and (0.418, 0.489), respecively, for Japan, and beween ((0.051, 0.054) and (0.980, 0,984), respecively, for USA. The corresponding shor and long run persisence of shocks for weekly daa lie beween (0.348, 0.411) and (0.441, 0.541), respecively, for he world, beween (0.104, 0.108) and (0.854, 0.861), respecively, for Japan, and beween ((0.343, 0.352) and (0.579, 0.650), respecively, for USA. Thus, he range of esimaes for he shor and long run persisence of shocks differs according o he world and he wo leading ourism sources o Taiwan, which reflecs he imporance of spaial aggregaion, as well as he daa frequency, which reflecs he imporance of emporal aggregaion. If posiive and negaive shocks o world, Japanese and US ouris arrivals o Taiwan of a similar magniude are reaed asymmerically, his can be evaluaed in he GJR(1,1) model. Alhough no repored here, asymmery (hough no leverage) was found in 7 of 9 cases for daily daa for he world, Japan and USA, and asymmery (hough no leverage) was found in 4 of 6 cases for weekly daa. Therefore, shocks o world, Japanese and US ouris arrivals o Taiwan can be inerpreed as risk associaed wih he corresponding ouris arrivals. Alhough asymmery is observed for he HAR(1) model for he world, Japan and USA for daily daa, and for he HAR(1) model for he world and USA for weekly daa, here is no evidence of leverage. Moreover, he hree HAR models sugges asymmery for Japan using daily daa, bu changes o symmery for Japan using wo models for weekly daa. Thus, hese empirical resuls show ha a deerminaion of symmery or asymmery arising from he condiional volailiy models is sensiive o he emporal aggregaion of daily o weekly daa. As he second momen condiion, 1 1, is ypically saisfied, he log-momen condiion is 2 necessarily saisfied, so ha he QMLE for he GJR(1,1) model are consisen and asympoically normal. Therefore, saisical inference using he asympoic normal disribuion is valid, and he asymmeric GJR(1,1) esimaes are saisically significan. 13

14 The inerpreaion of he EGARCH model is in erms of he logarihm of volailiy. For daily and weekly world, Japanese and US ouris arrivals o Taiwan, he EGARCH(1,1) esimaes were generally saisically significan for he various HAR models, wih he size effec,, and sign effec,, ypically being significan. The coefficien of he lagged dependen variable,, is esimaed o be less han uniy, which suggess ha he saisical properies of he QMLE for EGARCH(1,1) will be consisen and asympoically normal. Alhough no repored here, he world price and exchange rae effecs are always negaive for he HAR(1) model for daily and weekly daa, and are also generally negaive for he HAR(1,7) and HAR(1,7,28) models for daily daa and HAR(1,4) model for weekly daa. The expeced negaive price and exchange rae effecs generally do no change wih emporal aggregaion. In summary, he QMLE for he GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for daily and weekly world, Japanese and US ouris arrivals o Taiwan are saisically adequae and have sensible inerpreaions. The empirical resuls also show ha he volailiy in he shocks o daily and weekly world, Japanese and US ouris arrivals o Taiwan can be sensiive o he long memory naure of he condiional mean specificaions. 6. Concluding Remarks Alhough ourism is no ye one of he mos imporan service indusries in Taiwan, ouris arrivals from Japan and USA, he wo mos imporan source counries for Taiwan, reflec an increasing demand for shor and long haul ouris ravel. World ouris arrivals o Taiwan have been growing seadily, and reflec he spaial aggregaion of numerous ourism source counries. However, here is significan room for improvemen in ourism receips from he various ourism source counries. The poenial negaive impacs of mass ourism on he environmen, and hence on fuure world, Japanese and US ourism demand, mus be managed appropriaely. In order o manage such ourism growh, i is necessary o model adequaely world, Japanese and US ouris arrivals and heir associaed volailiy. As he exchange rae allows approximae daily price effecs on Japanese and US ourism arrivals o Taiwan o be capured, i is also necessary o analyse he Yen / New Taiwan $ and US$ / New Taiwan $ exchange raes, and he world price, as well as heir associaed volailiies. 14

15 The paper examined daily and weekly world, Japanese and US ouris arrivals o Taiwan from 1 January 1990 o 31 December 2008, and he world price and Yen / New Taiwan $ and US$ / New Taiwan $ exchange raes. The Heerogeneous Auoregressive (HAR) model was used o capure he approximae long memory properies in he ouris arrivals series. The empirical resuls showed ha he ime series of world, Japanese and US ouris arrivals o Taiwan, and he world price and wo exchange raes, were saionary. In addiion, he esimaed symmeric and asymmeric condiional volailiy models, specifically he widely used GARCH, GJR and EGARCH models all fi he daa exremely well. The esimaed models were able o accoun for he higher volailiy persisence ha was observed a he end of he sample period, due primarily o he global financial crisis. The empirical second momen condiion also generally suppored he saisical adequacy of he models of world, Japanese and US ouris arrivals o Taiwan, so ha saisical inferences were valid. Moreover, he esimaes resembled hose arising from financial ime series daa, wih boh shor and long run persisence of shocks, and asymmeric effecs of posiive and negaive shocks of equal magniude o volailiy. Alhough asymmery was observed for he HAR models using daily and weekly daa, here was no evidence of leverage. Overall, volailiy could be inerpreed as risk associaed wih shocks o world, Japanese and US ouris arrivals o Taiwan. Wih regard o he effecs of emporal and spaial aggregaion, i was found ha HAR effecs did no seem o be sensiive o emporal aggregaion, a deerminaion of symmery or asymmery arising from he condiional volailiy models was sensiive o he emporal aggregaion of daily o weekly daa, he expeced negaive price and exchange rae effecs generally did no change wih emporal aggregaion, and he range of esimaes for he shor and long run persisence of shocks were differen for he world, Japan and USA. Thus, boh spaial aggregaion and he daa frequency, or emporal aggregaion, were found o be imporan for esimaing he dynamic effecs of world prices and exchange raes, and heir respecive volailiies, on world, Japanese and US ouris arrivals o Taiwan. For policy purposes, hese empirical resuls sugges ha an arbirary choice of daa frequency or spaial aggregaion will no lead o robus findings as hey are generally no independen of he level of aggregaion used. Thus, a careful analysis of differen levels of emporal and spaial aggregaion needs o be underaken o obain sensible esimaes, regulariy condiions, shor and long run persisence of shocks o ouris arrivals, asymmery and leverage effecs, and he negaive effecs of he world price and exchange raes on inernaional ouris arrivals. 15

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17 Elie, L. and T. Jeanheau (1995), Consisency in heeroskedasic models, Compes Rendus de l Académie des Sciences, Série I, 320, (in French). Engle, R.F. (1982), Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, Glosen, L., R. Jagannahan and D. Runkle (1992), On he relaion beween he expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, Hoi, S., M. McAleer and R. Shareef (2005), Modelling counry risk and uncerainy in small island ourism economies, Tourism Economics, 11, Hoi, S., M. McAleer and R. Shareef (2007), Modelling inernaional ourism and counry risk spillovers for Cyprus and Mala, Tourism Managemen, 28, Jeanheau, T. (1998), Srong consisency of esimaors for mulivariae ARCH models, Economeric Theory, 14, Lee, S.W. and B.E. Hansen (1994), Asympoic heory for he GARCH(1,1) quasi-maximum likelihood esimaor, Economeric Theory, 10, Li, W.K., S. Ling and M. McAleer (2002), Recen heoreical resuls for ime series models wih GARCH errors, Journal of Economic Surveys, 16, Reprined in M. McAleer and L. Oxley (eds.), Conribuions o Financial Economerics: Theoreical and Pracical Issues, Blackwell, Oxford, 2002, pp Ling, S. and W.K. Li (1997), On fracionally inegraed auoregressive moving-average models wih condiional heeroskedasiciy, Journal of he American Saisical Associaion, 92, Ling, S. and M. McAleer (2002a), Saionariy and he exisence of momens of a family of GARCH processes, Journal of Economerics, 106, Ling, S. and M. McAleer (2002b), Necessary and sufficien momen condiions for he GARCH(r,s) and asymmeric power GARCH(r,s) models, Economeric Theory, 18, Ling, S. and M. McAleer, (2003a), Asympoic heory for a vecor ARMA-GARCH model, Economeric Theory, 19, Ling, S. and M. McAleer (2003b), On adapive esimaion in nonsaionary ARMA models wih GARCH errors, Annals of Saisics, 31,

18 McAleer, M. (2005), Auomaed inference and learning in modeling financial volailiy, Economeric Theory, 21, McAleer, M. (2009), The Ten Commandmens for opimizing value-a-risk and daily capial charges, Journal of Economic Surveys, 23, McAleer, M., F. Chan and D. Marinova (2007), An economeric analysis of asymmeric volailiy: heory and applicaion o paens, Journal of Economerics, 139, McAleer, M., J.-A. Jiménez-Marin and T. Perez Amaral (2009a), Has he Basel II Accord encouraged risk managemen during he financial crisis?, Available a SSRN: hp://ssrn.com/absrac= McAleer, M., J.-A. Jiménez-Marin and T. Perez Amaral (2009b), Opimal risk managemen before, during and afer he financial crisis, Available a SSRN: hp://ssrn.com/absrac= McAleer, M., J.-A. Jiménez-Marin and T. Perez Amaral (2010), Wha happened o risk managemen during he financial crisis?, o appear in R.W. Kolb (ed.), Lessons from he Financial Crisis: Causes, Consequences, and Our Economic Fuure, Wiley, New York, 2010, Available a SSRN: hp://ssrn.com/absrac= McAleer, M., T. Perez Amaral and J.-A. Jiménez-Marin (2009), A decision rule o minimize daily capial charges in forecasing value-a-risk, o appear in Journal of Forecasing, Available a SSRN: hp://ssrn.com/absrac= McAleer, M. and M. Medeiros (2008), A muliple regime smooh ransiion heerogeneous auoregressive model for long memory and asymmeries, Journal of Economerics, 147(1), 2008, Muller, U., M. Dacorogna, R. Dav, R. Olsen, O. Pice and J. ward (1993), Fracals and inrinsic ime - a challenge o economericians," in Proceedings of he XXXIXh Inernaional AEA Conference on Real Time Economerics. Nelson, D.B. (1991), Condiional heeroscedasiciy in asse reurns: a new approach, Economerica, 59, Phillips, P.C.B. and P. Perron (1988), Tesing for a uni roo in ime series regression, Biomerika, 75,

19 Shareef, R. and M. McAleer (2005), Modelling inernaional ourism demand and volailiy in small island ourism economies, Inernaional Journal of Tourism Research, 7, Shareef, R. and M. McAleer (2007), Modelling he uncerainy in inernaional ouris arrivals o he Maldives, Tourism Managemen, 28, Shareef, R. and M. McAleer (2008), Modelling inernaional ourism demand and uncerainy in Maldives and Seychelles: a porfolio approach, Mahemaics and Compuers in Simulaion, 78, Shephard, N. (1996), Saisical aspecs of ARCH and sochasic volailiy, in O.E. Barndorff- Nielsen, D.R. Cox and D.V. Hinkley (eds.), Saisical Models in Economerics, Finance and Oher Fields, Chapman & Hall, London, pp

20 Table 1. Uni Roo Tess Variables ADF Z={1} PP Z={1} ADF Z={1,} PP Z={1,} Daily World Touris Arrivals o Taiwan * ** ** ** Daily Japanese Touris Arrivals o Taiwan ** ** ** ** Daily US Touris Arrivals o Taiwan ** ** ** ** Variables ADF Z={1} PP Z={1} ADF Z={1,} PP Z={1,} Weekly World Touris Arrivals o Taiwan ** ** ** Weekly Japanese Touris Arrivals o Taiwan ** ** ** ** Weekly US Touris Arrivals o Taiwan ** ** ** ** Noes: The criical values for he ADF es are (-2.86) a he 1% (5%) level when Z = {1}, and (-3.41) a he 1% (5%) level when Z = {1, }. The criical values for he PP es are (-2.86) a he 1% (5%) level when Z = {1}, and (-3.41) a he 1% (5%) level when Z = {1, }. ** and * denoe he null hypohesis of a uni roo is rejeced a he 1% and 5% levels, respecively. 20

21 Table 2: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for World Daily Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH *** (122.1) *** (0.008) *** (1.047) *** (29091) GARCH/GJR 0.220*** (0.013) GARCH/GJR (0.022) 1300*** (118.6) 0.824*** (0.007) *** (1.007) *** (29755) 0.135*** (0.009) 0.044* (0.025) *** (118.4) 0.830*** (0.007) *** (0.999) *** (0.555) GJR 0.259*** (0.038) EGARCH 0.438*** (0.020) EGARCH *** (0.016) EGARCH 0.164*** (0.040) Diagnosics AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] [0.002] [0.002] [0.0003] Noes: The dependen variable is world daily ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** and * denoe he esimaed coefficiens are saisically significan a he 1% and 10% levels, respecively. 21

22 Table 3: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for Japanese Daily Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH GARCH/GJR GARCH/GJR *** (53.04) 0.672*** (0.009) *** (11.486) *** (13137) 0.256*** (0.015) 0.162*** (0.033) GJR *** (52.30) 0.682*** (0.009) *** (11.59) *** (12750) 0.363*** (0.024) 0.242*** (0.033) *** (0.026) EGARCH EGARCH EGARCH Diagnosics *** (53.33) 0.674*** (0.009) *** (11.83) 5.372*** (0.392) 0.326*** (0.018) 0.160*** (0.012) 0.563*** (0.031) AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] Noes: The dependen variable is Japanese daily ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** denoes he esimaed coefficiens are saisically significan a he 1% level. 22

23 Table 4: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for US Daily Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH GARCH/GJR GARCH/GJR *** (26.75) 0.592*** (0.010) *** (632.3) *** (78.25) 0.053*** (0.003) 0.931*** (0.004) GJR *** (27.15) 0.593*** (0.010) *** (639.4) *** (88.49) 0.060*** (0.004) 0.924*** (0.005) * (0.005) EGARCH EGARCH EGARCH Diagnosics *** (26.06) 0.588*** (0.010) *** (613.87) 0.098** (0.026) 0.117*** (0.006) 0.011*** (0.003) 0.982*** (0.003) AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] Noes: The dependen variable is US daily ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** and * denoe he esimaed coefficiens are saisically significan a he 1% and 10% levels, respecively. 23

24 Table 5: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for World Weekly Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH GARCH/GJR GARCH/GJR *** (1371.4) 0.900*** (0.013) (12.34) *** ( ) 0.411*** (0.048) 0.130** (0.053) GJR *** (1612) 0.894*** (0.015) (14.09) *** ( ) (0.038) 0.094*** (0.036) 0.637*** (0.110) EGARCH EGARCH EGARCH Diagnosics 5094*** (1428) 0.900*** (0.013) (12.74) 8.240*** (1.017) 0.539*** (0.059) *** (0.042) 0.492*** (0.060) AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] [0.244] [0.132] [0.278] Noes: The dependen variable is world weekly ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** and * denoe he esimaed coefficiens are saisically significan a he 1% and 5% levels, respecively. 24

25 Table 6: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for Japanese Weekly Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH GARCH/GJR GARCH/GJR *** (861.4) 0.633*** (0.027) *** (160.7) *** (538185) 0.104*** (0.026) 0.757*** (0.066) GJR 8416*** (861.1) 0.631*** (0.027) *** (166.6) *** (472402) 0.116*** (0.038) 0.782*** (0.059) (0.043) EGARCH EGARCH EGARCH Diagnosics *** (848.7) 0.631*** (0.026) *** (168.9) 1.694** (0.625) 0.185*** (0.041) 0.045* (0.026) 0.886*** (0.040) AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] [0.002] [0.004] Noes: The dependen variable is Japanese weekly ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** and * denoe he esimaed coefficiens are saisically significan a he 1% and 10% levels, respecively. 25

26 Table 7: Esimaed Condiional Mean (HAR(1)) and Condiional Volailiy Models for US Weekly Touris Arrivals o Taiwan Parameers GARCH GJR EGARCH GARCH/GJR GARCH/GJR *** (395.06) 0.751*** (0.022) *** (8521.3) *** (38322) 0.352*** (0.045) 0.298*** (0.647) GJR *** (324.62) 0.799*** (0.020) *** (7081.7) *** (35754) 0.650*** (0.101) 0.366*** (0.063) *** (0.104) EGARCH EGARCH EGARCH Diagnosics *** (289.74) 0.795*** (0.019) *** (6248.4) 3.614*** (0.687) 0.436*** (0.055) 0.257*** (0.039) 0.703*** (0.052) AIC BIC Jarque-Bera [p-value] Causaliy es [p-value] Noes: The dependen variable is US weekly ouris arrivals o Taiwan. Numbers in parenheses are sandard errors. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. AIC and BIC denoe he Akaike Informaion Crierion and Schwarz Bayesian Informaion Crierion, respecively. *** denoes he esimaed coefficiens are saisically significan a he 1% level. 26

27 Figure 1. Daily Touris Arrivals o Taiwan and Volailiy Daily World Touriss Volailiy Daily World Touriss 16,000 70,000,000 14,000 60,000,000 12,000 50,000,000 10,000 40,000,000 8,000 6,000 4,000 30,000,000 20,000,000 2,000 10,000, Daily Japanese Touriss Volailiy Daily Japanese Touriss 8,000 24,000,000 7,000 20,000,000 6,000 5,000 16,000,000 4,000 12,000,000 3,000 8,000,000 2,000 1,000 4,000, Daily US Touriss Volailiy Daily US Touriss 2,800 3,200,000 2,400 2,800,000 2,000 2,400,000 1,600 2,000,000 1,600,000 1,200 1,200, , ,

28 Figure 2. Weekly Touris Arrivals o Taiwan and Volailiy Weekly World Touriss Volailiy Weekly World Touriss 80,000 1,400,000,000 70,000 1,200,000,000 60,000 1,000,000,000 50, ,000,000 40, ,000,000 30,000 20,000 10, ,000, ,000,000 0 Weekly Japanese Touriss Volailiy Weekly Japanese Touriss 32, ,000,000 28, ,000,000 24,000 20, ,000,000 16, ,000,000 12, ,000,000 8,000 4,000 50,000, Weekly US Touriss Volailiy Weekly US Touriss 14,000 35,000,000 12,000 30,000,000 10,000 25,000,000 8,000 20,000,000 6,000 15,000,000 4,000 10,000,000 2,000 5,000,

29 Figure 3. Daily Exchange Raes and Volailiy Daily World Price Volailiy Daily World Price Daily Exchange Rae Yen/NT$ Volailiy Daily Exchange Rae Yen/NT$ Daily Exchange Rae US$/NT$ Volailiy Daily Exchange Rae US$/NT$

30 Figure 4. Weekly Exchange Raes and Volailiy Weekly World Price Volailiy Weekly World Price Weekly Exchange Rae Yen/NT$ Volailiy Weekly Exchange Rae Yen/NT$ Weekly Exchange Rae US$/NT$ Volailiy Weekly Exchange Rae US$/NT$