Modelling and Forecasting Daily International Mass Tourism to Peru

Transcription

1 CIRJE-F-651 Modelling and Forecasing Daily Inernaional Mass Tourism o Peru Jose Angelo Divino Caholic Universiy of Brasilia Michael McAleer Erasmus Universiy Roerdam and Tinbergen Insiue and CIRJE, Faculy of Economics, Universiy of Tokyo Augus 2009 CIRJE Discussion Papers can be downloaded wihou charge from: hp:// Discussion Papers are a series of manuscrips in heir draf form. They are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 Modelling and Forecasing Daily Inernaional Mass Tourism o Peru Jose Angelo Divino Deparmen of Economics Caholic Universiy of Brasilia Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Cener for Inernaional Research on he Japanese Economy (CIRJE) Faculy of Economics Universiy of Tokyo Augus 2009 The auhors are mos graeful o he Edior and hree referees for helpful commens and suggesions. The second auhor wishes o hank he Ausralian Research Council and Naional Science Council, Taiwan, for financial suppor. 1

3 Absrac Peru is a Souh American counry ha is divided ino wo pars by he Andes Mounains. The rich hisorical, culural and geographic diversiy has led o he inclusion of en Peruvian sies on UNESCO s World Heriage Lis. For he poenially negaive impacs of mass ourism on he environmen, and hence on fuure inernaional ourism demand, o be managed appropriaely require modelling growh raes and volailiy adequaely. The paper models he growh rae and volailiy (or he variabiliy in he growh rae) in daily inernaional ouris arrivals o Peru from 1997 o The empirical resuls show ha inernaional ouris arrivals and heir growh raes are saionary, and ha he esimaed symmeric and asymmeric condiional volailiy models all fi he daa exremely well. Moreover, he esimaes resemble hose arising from financial ime series daa, wih boh shor and long run persisence of shocks o he growh rae in inernaional ouris arrivals. Keywords: Daily Inernaional Tourim; Condiional Mean Models; Condiional Volailiy Models. JEL codes: C51; C53. 2

4 1. Inroducion Peru is a Souh American counry, bordering Ecuador and Colombia o he norh, Brazil o he eas, Bolivia o he souheas, Chile o he souh, and he Pacific Ocean o he wes (see Figure 1). I is he 20h larges counry in he world, wih a erriory of 1,285,220 km², and has a populaion of over 28 million (July 2007 esimae). The counry is divided ino wo pars by he Andes Mounains, which cross he erriory parallel o he Pacific Ocean. The eas of he Andes up o he border wih Brazil is covered by he Amazon rainfores, corresponding o abou 60% of he counry s erriory. I is in he Peruvian erriory, precisely a he mounain peak Nevado Mismi locaed in he Andes, ha he Amazon River has is glacial source. The larges rivers in he counry are inegral pars of he upper Amazon Basin. The counry s climae is influenced by he proximiy o he Equaor, he presence of he Andes, and he cold waers from some Pacific currens. As a resul of his combinaion, here is wide diversiy in he climae, ranging from he dryness of he coas, o he exreme cold of he mounain peaks, and o heavy rainfall in he Amazon region. Peru is one of he few areas in he world where here has been indigenous developmen of civilizaion, as he home of he Inca Empire, which emerged in he 15h Cenury as a powerful sae and he larges empire in pre-columbian America. The Inca Emperor was defeaed in 1532 by he Spanish, who imposed colonial dominaion of he counry. During his dominaion, silver mining wih Indian forced labour became he basic economic aciviy, rendering considerable revenues for he Spanish Crown. However, he Royal income was reduced considerably over he years due o widespread smuggling and ax evasion. The Spanish Crown ried o recover conrol over is colonies by a series of ax reforms, which yielded numerous revols across he coninen. Finally, afer successful miliary campaigns, Peru proclaimed independence from Spain in Culural diversiy is one of he major aracions of Peru, and arises from a combinaion of differen radiions over several cenuries. Imporan conribuions o is culural 3

5 diversiy include naive Indians, Spanish colonizers, and ehnic groups from Africa, Asia and Europe. There are also Pre-Inca and Inca culures, wih impressive achievemens in archiecure, such as he world famous holy ciy of Machu Picchu. Peruvian cuisine is also linked o he diversiy of he counry, and uses many differen ingrediens ha are combined hrough disincive echniques. Climaic differences conribue o he success of he Peruvian cuisine by allowing he producion and inegraion of a wide variey of flora and fauna in he counry. The music in Peru follows similar diversificaion, combining Andean, Spanish and African rhyhms, insrumens and expressions. In recen decades, a new ingredien given by he urbanizaion has influenced radiional Andean expressions and increased he musical variey. As a resul of such rich hisorical, culural and geographic environmens, Peru aracs shor, medium and long haul ouriss from all over he world o visi is erriory. The major desinaion of he counry is he region of Cuzco, accouning for abou 27% of inernaional ouris arrivals. In his region are locaed he ciy of Cuzco, which was he capial of he Inca Empire, he spiriual ciy of Machu Picchu, and he Sacred Valley of he Incas. Anoher imporan desinaion, which is visied by around 13% of inernaional ouriss, is he region of Lima, he capial of Peru, where he major aracion is he hisorical side of Lima. In hird place, which receives around 11% of oal inernaional ouriss, is he ciy of Arequipa, locaed in he valley of he volcanoes. Taking as a whole, hese hree regions receive around 51% of he inernaional ouriss o Peru. UNESCO s World Heriage Lis includes properies ha form par of he world s culural and naural heriage wih ousanding universal value. The Ciy of Cuzco and he Hisoric Sancuary of Machu Picchu were inscribed as World Heriage Sies in 1983, while he hisorical cenre of he Ciy of Arequipa was inscribed in There are presenly seven oher Peruvian sies in he World Heriage Lis. Owing o he desrucive effecs of unbridled mass ourism, he reenion of Machu Picchu on he World Heriage Lis is a 4

6 maer of grea imporance, even hough Machu Picchu, was recenly voed one of he New Seven Wonders of he World. In 2006, Peru received abou 908,000 ouriss from around he world. The majoriy of inernaional ouriss o Peru come from Norh America, wih around 37% of he oal, and from Europe, wih around 30%. In Norh America, he USA is he major source of ouriss o Peru, accouning for abou 31% of he oal. In Europe, he major sources of inernaional ouriss are Spain, Unied Kingdom, France, Germany and Ialy, wih average proporions ha range from 6% o 4% of he oal of Europeans who visi Peru annually. Souh America accouns for around 22% of inernaional ouris arrivals o Peru, wih Argenina, Colombia, Chile and Brazil being he major sources, wih each having shares of around 4% of he oal. Inernaional ourism has no ye achieved he saus of an imporan economical aciviy in Peru. According o he Minisry of Exernal Commerce and Tourism of Peru, he consumpion of inernaional ourism as a proporion of GDP increased from 0.5% in 1992 o 1.8% in However, afer a significan increase in he lae 1990 s, he paricipaion of inernaional ourism in GDP decreased o 1.4% in 2004, and has been hovering a around 1.8% since This represens inernaional ourism revenues of only $1.4 billion o he counry on an annual basis. Consequenly, here is clearly significan room for improvemen in inernaional ourism receips. However, he poenially negaive impacs of mass ourism on he environmen, and hence on fuure inernaional ourism demand, mus be managed appropriaely. In order o manage ourism growh and volailiy, i is necessary o model he growh and volailiy in inernaional ouris arrivals adequaely. The primary purpose of his paper is o model he growh and volailiy (ha is, he variabiliy in he growh rae) in inernaional ouris arrivals o Peru. Informaion from 1997 o 2007 is used on daily inernaional arrivals a he Jorge Chavez Inernaional Airpor in Lima, which is he only inernaional airpor in Peru. By using daily daa, we can approximae he modelling and managemen sraegy and risk analysis o hose 5

7 applied o financial ime series daa. Alhough he volailiy in inernaional ouris arrivals has been analyzed a he monhly ime series frequency in Chan, Lim and McAleer (2005), Divino and McAleer (2008), Hoi, McAleer and Shareef (2005, 2007), and Shareef and McAleer (2005, 2007, 2008), o he bes of our knowledge here is no oher work ha models daily inernaional ouris arrivals. The paper also conribues wih he recen lieraure applying economeric echniques on forecasing ourism demand, where imporan references are Ahanasopoulos e al. (2009), Bonham e al. (2009), and Gil-Alana e al. (2008). From a ime series perspecive, here are several reasons for using daily daa as compared wih lower frequency daa a he monhly or quarerly levels. Among oher reasons, McAleer (2008) discusses how daily daa can lead o a considerably higher sample size, provide useful informaion on risk in finance, lead o he deerminaion of opimal environmenal and ourism axes, enable aggregaion of high frequency daa o yield aggregaed daa wih volailiy, analyze ime series behavior a differen frequencies hrough aggregaed daa, invesigae wheher ime series properies have changed over ime, capure day-of-he-week effecs hrough differenial pricing sraegies in he ourism indusry, including airlines, ouris aracions and he accommodaion secor, and deermine opimal ourism markeing policies hrough exploiing day-of-he-week effecs o enable ourism operaors o formulae pricing sraegies and ourism packages o increase ouris arrivals in periods of low demand. The empirical resuls show ha he ime series of inernaional ouris arrivals and heir growh raes are 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. In paricular, he esimaed models are able o accoun for he higher volailiy persisence ha is observed a he beginning and end of he sample period. The empirical second momen condiion also suppors he saisical adequacy of he models, so ha saisical inference is valid. Moreover, he esimaes resemble hose arising from financial ime series daa, wih boh shor and long run persisence of shocks in he growh rae of inernaional ouris arrivals. Therefore, 6

8 volailiy can be inerpreed as risk associaed wih he growh rae in inernaional ouris arrivals 1. The remainder of he paper is organized as follows. Secion 2 presens he daily inernaional ouris arrivals ime series daa se and discusses he ime-varying volailiy. Secion 3 performs uni roo ess on boh he levels and logarihmic differences (or growh raes) of daily inernaional ouris arrivals for Peru. Secion 4 discusses alernaive condiional mean and condiional volailiy models for he daily inernaional ouris arrivals series. The esimaed models and empirical resuls are discussed in Secion 5. Finally, some concluding remarks are given in Secion Daa The daa se comprises daily inernaional ouris arrivals a he Jorge Chavez Inernaional Airpor, he only inernaional airpor in Peru, which is locaed in he ciy of Lima, he capial of Peru. The daa are daily, wih seven days each week, for he period 1 January 1997 o 28 February 2007, giving a oal of 3,711 observaions. The source of he daa was he Peruvian Minisry of Inernaional Trade and Tourism. Figure 2 plos he daily inernaional ouris arrivals, he logarihm of daily inernaional ouris arrivals, and he firs difference (ha is, he log-difference or growh raes) of daily inernaional ouris arrivals, as well as he volailiy of he hree variables, where volailiy is defined as he squared deviaion from he sample mean. There is higher volailiy persisence a he beginning and a he end of he period for he series in levels and logarihms, bu here is a single clear dominan observaion in he series in around This exreme observaion is 31 December 1999, which is higher han he ypical decrease in inernaional ouris arrivals in December each year. However, his observaion is no sufficienly influenial o affec he empirical resuls as here is no 1 See McAleer and da Veiga (2008a, 2008b) for some applicaions of risk modelling and managemen o forecas value-a-risk (VaR) hresholds and daily capial changes. 7

9 significan change when his observaion is deleed from esimaion and esing. An increasing deerminisic rend is presen for he whole period in boh series. The series in log differences is clearly rend saionary and does no show higher volailiy a he beginning or end of he sample, bu here is clear volailiy persisence. I is ineresing ha he single clear dominan observaion in he logarihmic series is mirrored in he log difference series. On an annual basis, he number of inernaional ouris arrivals o Peru has shown an average growh rae of 8.8%, as illusraed in Figure 3. The lowes growh rae was observed in 2000, wih an increase of jus 0.8% over he previous year, while he highes growh rae occurred in 2005, when here was a significan increase of 27.1% over In he sample period as a whole, here was an increase of around 110% in inernaional ouris arrivals o Peru, which would seem o indicae a reasonably good performance in he ourism secor over he decade. Neverheless, annual average inernaional ouris arrivals of 620,000 reveal ha here is scope for a significan increase in inernaional ourism o Peru. However, he poenially negaive impacs of mass ourism on he environmen, and hence on fuure inernaional ourism demand, mus be managed appropriaely. In order o manage ourism growh and volailiy, i is firs necessary o model growh and volailiy adequaely. In he nex secion we analyze he presence of a sochasic rend by applying uni roo ess before modeling he ime-varying volailiy ha is presen in he logarihmic and log-difference (or growh rae) series. 3. Uni Roo Tess I is well known ha radiional uni roo ess, primarily hose based on he classic mehods of Dickey and Fuller (1979, 1981) and Phillips and Perron (1988), suffer from low power and size disorions. However, hese shorcomings have been overcome by modificaions o he esing procedures, such as he mehods proposed by Perron and Ng 8

10 (1996), Ellio, Rohenberg and Sock (1996), and Ng and Perron (2001). I is worh menioning, however, ha he modified ess are also subjec o low power and size disorions under shor run persisence implied by GARCH componens, as shown in Cook (2006). Neverheless, size disorions migh be even greaer for he radiional Dickey-Fuller es, despie he sensiiviy of he modified ess o he degree of volailiy in he GARCH process. We applied he modified uni roo ess, given by MADF GLS and MPP GLS, o he ime series of daily inernaional ouris arrivals in Peru. In essence, hese ess use GLS derended daa and he modified Akaike informaion crierion (MAIC) o selec he opimal runcaion lag. The asympoic criical values for boh ess are given in Ng and Perron (2001). The resuls of he uni roo ess are obained from he economeric sofware package EViews 5.0, and are repored in Table 1. There is no evidence of a uni roo in he logarihm of daily inernaional ouris arrivals o Peru (LY) in he model wih a consan and rend as he deerminisic erms, so ha LY is rend saionary. For he model wih jus a consan, however, he null hypohesis of a uni roo is no rejeced a he 5% significance level. For he series in log differences (or growh raes), he null hypohesis of a uni roo is rejeced for boh specificaions under he MADF GLS es. These empirical resuls allow he use of boh levels and log differences in inernaional ouris arrivals o Peru o esimae he alernaive univariae condiional mean and condiional volailiy models given in he nex secion. 4. Condiional Mean and Condiional Volailiy Models The alernaive ime series models o be esimaed for he condiional means of he daily inernaional ouris arrivals, as well as heir condiional volailiies, are discussed below. As Figure 1 illusraes, daily inernaional ouris arrivals, logarihm of daily inernaional ouris arrivals, and he firs difference (ha is, he log difference or growh rae) of daily 9

11 inernaional ouris arrivals, o Peru show periods of high volailiy followed by ohers of relaively low volailiy. An obvious implicaion of his persisen volailiy behaviour is ha he assumpion of (condiionally) homoskedasic residuals is no appropriae empirically. I is well known ha, for a wide range of financial daa series, ime-varying condiional variances can be explained empirically hrough he auoregressive condiional heeroskedasiciy (ARCH) model, which was proposed by Engle (1982). When he imevarying condiional variance has boh auoregressive and moving average componens, his leads o he generalized ARCH(p,q), or GARCH(p,q), model of Bollerslev (1986). The lag srucure of he appropriae GARCH model can be chosen by informaion crieria, such as hose of Akaike and Schwarz, alhough i is very common o impose he GARCH(1,1) specificaion in advance. In he seleced condiional volailiy model, he residual series should follow a whie noise process. Li e al. (2002) provide an exensive review of recen 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 (logarihmic) inernaional ouris arrivals daa, as well as heir growh raes, display persisence in volailiy, as shown in Figure 1, i is naural o esimae alernaive condiional volailiy models. As menioned previously, he GARCH(1,1) and GJR(1,1) condiional volailiy models have been esimaed using monhly inernaional ourism arrivals daa in Chan, Lim and McAleer (2005), Hoi, McAleer and Shareef (2005, 2007), Shareef and McAleer (2005, 2007, 2008), and Divino and McAleer (2008). Consider he saionary AR(1)-GARCH(1,1) model for daily inernaional ouris arrivals o Peru (or heir growh raes, as appropriae), y : y = φ + φ + ε φ 1 (1) 1 2 y 1, 2 < 10

12 for Ζ, as in Mikosch and Sarica (2000), where he shocks (or movemens in daily inernaional ouris arrivals) are given by: ε = η h, h = ω + αε η ~ iid (0,1) βh 1, (2) and ω > 0, α 0, β 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 α ) effec indicaes he shor run persisence of shocks, while he GARCH (or β ) effec indicaes he conribuion of shocks o long run persisence (namely, α + β ). 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 η, he condiional shocks (or sandardized residuals). The condiional log-likelihood funcion is given as follows: n l = 1 = 1 n 2 log h + ε 2 = 1 h. The QMLE is efficien only if η is normal, in which case i is he MLE 2. When η 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. 2 See, for example, McAleer and da Veiga (2008a, b) for he use of alernaive univariae and mulivariae disribuions for financial daa. 11

13 As he GARCH process in equaion (2) is a funcion of he uncondiional shocks, he momens of ε need o be invesigaed. Ling and McAleer (2003a) showed ha he QMLE for GARCH(p,q) is consisen if he second momen of ε is finie. For GARCH(p,q), Ling and Li (1997) demonsraed ha he local QMLE is asympoically normal if he fourh momen of ε is finie, while Ling and McAleer (2003a) proved ha he global QMLE is asympoically normal if he sixh momen of ε 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 ε for GARCH(1,1) is α + β < 1 and, under normaliy, he necessary and sufficien condiion for he exisence of 2 2 he fourh momen is ( α + β ) + 2α < 1. 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( αη + β)) < 0. (3) As discussed in McAleer e al. (2007), Elie and Jeanheau (1995) and Jeanheau (1998) esablished 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 he proof in he case of GARCH(1,1)), while Boussama (2000) showed ha he log-momen condiion was sufficien for asympoic normaliy. 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 logmomen counerpars, he second momen condiion is far more sraighforward o check. In pracice, he log-momen condiion in equaion (3) would be esimaed by he sample 12

14 mean, wih he parameers α and β, and he sandardized residual, η, being replaced by heir QMLE counerpars. The effecs of posiive shocks (or upward movemens in daily inernaional ouris arrivals) on he condiional variance, h, are assumed o be he same as he negaive shocks (or downward movemens in daily inernaional ouris arrivals) 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 = ( ω + α + γi( η )) ε βh, (4) where ω > 0, α 0, α + γ 0, β 0 are sufficien condiions for h > 0, and I η ) is an indicaor variable defined by: ( 1, I( η ) = 0, ε < 0 ε 0 as η has he same sign as ε. The indicaor variable differeniaes beween posiive and negaive shocks of equal magniude, so ha asymmeric effecs in he daa are capured by he coefficien γ. For financial daa, i is expeced ha γ 0 because negaive shocks increase risk by increasing he deb o equiy raio, bu his inerpreaion need no hold for inernaional ourism arrivals daa in he absence of a direc risk inerpreaion. The asymmeric effec, γ, measures he conribuion of shocks o boh shor run γ γ persisence, α +, and o long run persisence, α + β Ling and McAleer (2002a) showed ha he regulariy condiion for he exisence of he second momen for GJR(1,1) under symmery of η is given by: 13

15 1 α + β + γ < 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[( α + γi( η )) η + β ]) < 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 ω, β < 1 (7) = + α η 1 + γη 1 + β log h 1 where he parameers α, β and γ have differen inerpreaions from hose in he GARCH(1,1) and GJR(1,1) models. Leverage, which is a special case of asymmery, is defined as γ < 0 and α < γ. 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 β < 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), β < 1 would seem o be a sufficien condiion for he exisence of momens; and (v) in addiion o 14

16 being a sufficien condiion for consisency, β < 1 is also likely o be sufficien for asympoic normaliy of he QMLE of EGARCH(1,1). Furhermore, EGARCH capures asymmeries differenly from GJR. The parameers α and γ in EGARCH(1,1) represen he magniude (or size) and sign effecs of he sandardized residuals, respecively, on he condiional variance, whereas α and α + γ represen he effecs of posiive and negaive shocks, respecively, on he condiional variance in GJR(1,1). 5. Esimaed Models The condiional mean model was esimaed as AR(1), ARMA(1,1), ARMA(1,2), ARMA(2,1) and ARMA(2,2) processes, wih AR(1) or ARMA(1,1) generally being empirically preferred on he basis of AIC and BIC (see Table 2). The esimaed condiional mean and condiional volailiy models for he logarihm of ouris arrivals and he log-difference (or growh rae) of ouris arrivals are given in Table 3. The mehod used in esimaion was he Marquard algorihm. As shown in he uni roo ess, he logarihmic and log difference (or growh rae) series are saionary. These empirical resuls are suppored by he esimaes of he lagged dependen variables in he esimaes of equaion (1), wih he coefficiens of he lagged dependen variable being significanly less han one in each of he esimaed six models. Significan ARCH effecs are deeced by he LM es for ARCH(1) for LY, hough no for DLY. The Jarque-Bera LM es of normaliy rejecs he null hypohesis in all six cases. As he second momen condiion holds in each case, and hence he weaker log-momen condiion (which is no repored) is necessarily less han zero (see Table 2), 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 is saisfied if β < 1, and hence he QMLE are consisen and asympoically normal (see, for example, McAleer a al. (2007)). 15

17 The GARCH(1,1) esimaes for he logarihm of inernaional ouris arrivals o Peru sugges ha he shor run persisence of shocks is while he long run persisence is As he second momen condiion, α + β < 1, is saisfied, he log-momen condiion is necessarily saisfied, so ha he QMLE are consisen and asympoically normal. Therefore, saisical inference using he asympoic normal disribuion is valid, and he symmeric GARCH(1,1) esimaes are saisically significan. If posiive and negaive shocks of a similar magniude o inernaional ouris arrivals o Peru are reaed asymmerically, his can be evaluaed in he GJR(1,1) model. The asymmery coefficien is found o be posiive, namely 0.309, which indicaes ha decreases in inernaional ouris arrivals increase volailiy. This is a similar empirical oucome as is found in virually all cases in finance, where negaive shocks (ha is, financial losses) increase risk (or volailiy). Thus, shocks o ouris arrivals and he growh rae of ouris arrivals resemble financial shocks. They can be inerpreed as risk associaed o ouris arrivals. Moreover, he long run persisence of shocks is esimaed o 1 be As he second momen condiion, α + β + γ < 1, is saisfied, he log-momen 2 condiion is necessarily saisfied, so ha he QMLE 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. The inerpreaion of he EGARCH model is in erms of he logarihm of volailiy. For he logarihm of inernaional ouris arrivals, each of he EGARCH(1,1) esimaes is saisically significan, wih he size effec, α, being posiive and he sign effec, γ, being negaive. The condiions for leverage are saisfied for LY, bu no for DLY. The coefficien of he lagged dependen variable, β, is esimaed o be 0.763, which suggess ha he saisical properies of he QMLE for EGARCH(1,1) will be consisen and asympoically normal. 16

18 The GARCH(1,1) esimaes for he log difference (or growh rae) of inernaional ouris arrivals o Peru sugges ha he shor run persisence of shocks is while he long run persisence is 0.891, which is very close o he corresponding esimaes for he logarihm of inernaional ouris arrivals. As he second momen condiion is saisfied, he log-momen condiion is necessarily saisfied, so ha he QMLE are consisen and asympoically normal, and hence he symmeric GARCH(1,1) esimaes are saisically significan. The GJR(1,1) esimaes for he log difference (or growh rae) of inernaional ouris arrivals o Peru sugges ha he asymmery coefficien is posiive a 0.187, which indicaes ha decreases in he growh rae in inernaional ouris arrivals increase volailiy. The shor run persisence of posiive shocks is 0.025, he shor run persisence of negaive shocks is (= ), and he long run persisence of shocks is As he second momen condiion is saisfied, he log-momen condiion is necessarily saisfied, so ha he QMLE are consisen and asympoically normal. Therefore, as in he case of asymmery in financial markes, saisical inference using he asympoic normal disribuion is valid, and he asymmeric GJR(1,1) esimaes are saisically significan. For he log difference (or growh rae) of inernaional ouris arrivals, each of he EGARCH(1,1) esimaes is saisically significan, wih he size effec, α, being posiive and he sign effec, γ, being negaive. The coefficien of he lagged dependen variable, β, is esimaed o be 0.913, which suggess ha he saisical properies of he QMLE for EGARCH(1,1) will be consisen and asympoically normal. Overall, he QMLE for he GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for boh he logarihm and log difference of inernaional ouris arrivals, are saisically adequae and have sensible inerpreaions. The esimaed condiional mean and condiional volailiy models for he logarihm of annualized ouris arrivals and he log-difference (or growh rae) of annualized ouris 17

19 arrivals are given in Table 4. The annualized series would appear o have a uni roo, whereas he growh rae does no. Significan ARCH effecs are deeced for LYMA, hough no for DLYMA. The Jarque-Bera LM es of normaliy rejecs he null hypohesis in only wo of six cases. The GARCH(1,1) model has shor run persisence of shocks of 0.15 and long run persisence of shocks of The GJR(1,1) model does no have significan asymmery, so ha GARCH(1,1) is preferred. The second momen condiion is saisfied, so he QMLE are consisen and asympoically normal, and he log-momen condiion is necessarily saisfied. The EGARCH(1,1) esimaes are significan, including he asymmery coefficien, albei marginally. However, he condiions for leverage are no saisfied for LYMA or DLYMA. Again, he QMLE are saisically adequae, so ha inferences are sensible and saisically valid. The correlaion marix of he forecass in logarihmic levels and logarihmic firs differences (or growh raes) are given in Tables 4 and 5. The forecass in Table 4 can be very high a 0.999, bu hey can also be much lower beween he annualized and original daa series, as depiced in Figures 4 and 6, respecively. However, all of he correlaions for he forecass in log-differences are very high in Table 5, which is capured in he annualized inernaional ouris forecass in Figures 5 and 7. These resuls sugges ha annualized figures are much easier o forecas and manage han are heir daily counerpars. The forecass presened in Figures 4 o 7 are ou-of-sample dynamic forecass derived from each esimaed model repored in Tables 3 and 4. The models were esimaed using daily inernaional ouris arrivals daa o Peru from 1/1/1997 o 2/28/2007. Then ou-ofsample daily forecass are calculaed for he period from 1/3/2007 unil 2/28/2008. Thus, Figures 4 o 7 plo he acual series and he daily forecass one year ahead. I is worh noing ha he high volailiy of he daily series makes i somewha difficul o predic he log-level and log-difference of inernaional ouris arrivals o Peru. In boh cases, as presened in Figures 4 and 6, respecively, he forecass are roughly able o idenify a rend in he daa. On he oher hand, for he annualized daily series ploed in 18

20 Figures 5 and 7, respecively, he models succeed in predicing he one-year ahead annualized series. Comparing he relaive performance of he alernaive models, here is no significan differences in he forecass arising from he GARCH, GJR, and EGARCH models. 6. Concluding Remarks The rich hisorical, culural and geographic diversiy ha arises from a combinaion of differen radiions over several cenuries has led o he inclusion of en Peruvian sies on UNESCO s World Heriage Lis of properies ha form par of he world s culural and naural heriage wih ousanding universal value. These sies, paricularly he Ciy of Cuzco, he Hisoric Sancuary of Machu Picchu, which was recenly voed one of he Seven New Wonders of he World, and he hisorical cenre of he Ciy of Arequipa, are he major aracions for shor, medium and long haul inernaional ouriss. As inernaional ourism has no ye achieved he saus of an imporan economic aciviy for Peru s finances, here is significan room for improvemen in inernaional ourism receips. However, he poenially negaive impacs of mass ourism on he environmen, and hence on fuure inernaional ourism demand, mus be managed appropriaely. In order o manage ourism growh and volailiy, i is necessary o model growh and volailiy adequaely. The paper modelled he growh and volailiy (or variabiliy in he growh rae) in daily inernaional ouris arrivals o Peru from 1997 o There are several benefis arising from using daily daa as compared wih lower frequency daa. Among oher reasons, daily daa capure day-of-he-week effecs as arrival paerns on he week-end, allowing for differenial pricing sraegies in he ourism indusry, as well as he deerminaion of opimal ourism markeing policies o increase ouris arrivals during periods of low demand. In addiion, he growh rae in daily ouris arrivals expendiure, which is of primary ineres in he ourism indusry, is virually idenical o he growh rae in daily 19

21 ouris arrivals because he growh rae in daily spending per ouris arrivals changes very lile over ime. The empirical resuls showed ha he ime series of inernaional ouris arrivals and heir growh raes are saionary. In addiion, he esimaed symmeric and asymmeric condiional volailiy models, namely he widely used GARCH, GJR and EGARCH models, all fi he daa exremely well. In paricular, he esimaed models were able o accoun for he higher volailiy persisence observed a he beginning and end of he sample period for boh he logarihm and log difference (or growh rae) of inernaional ouris arrivals. The empirical second momen condiion also suppored he saisical adequacy of he models, 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 o he growh raes of inernaional ouris arrivals. Therefore, volailiy can be inerpreed as risk associaed wih he growh rae in inernaional ouris arrivals. The forecass of daily inernaional ouris arrivals o Peru suggesed ha he ourism influx o he region is likely o be very small in he years ahead. This finding poins o he need for a much wider developmen sraegy of he susainable ouris indusry o he region. Given he hisorical, naural, and culural imporance of Peru, appropriae markeing sraegies should be direced oward aracing a greaer number of inernaional ouriss o he counry. The developmen of a susainable ourism indusry is essenial o income generaion, job creaion, and economic growh of Peru. Raional exploraion of ourism aciviy would help o bring economic progress o Peru wihou negaively affecing he naural environmen and he lives of local communiies. Exensions of he models and daa used in he paper o he mulivariae level using modern sysems mehods is a opic of curren research. For a heoreical comparison of alernaive dynamic models of condiional correlaions and condiional covariances, see McAleer e al (2008). The alernaive condiional volailiy models can also be used o forecas value-a-risk hresholds. A panel daa analysis of emporal and spaial 20

22 aggregaion of alernaive ouris desinaions, incorporaing condiional volailiy models, could also be a useful direcion of research. References Ahanasopoulos, G., Ahmed, R. A., & Hyndman, R. J. (2009). Hierarchical forecass for Ausralian domesic ourism. Inernaional Journal of Forecasing, 25, Bollerslev, T. (1986). Generalised auoregressive condiional heeroscedasiciy. Journal of Economerics, 31, Bonham, C., Gangnesa, B., & Zhoub, T. (2009). Modeling ourism: A fully idenified VECM approach. Inernaional Journal of Forecasing, 25, Boussama, F. (2000). Asympoic normaliy for he quasi-maximum likelihood esimaor of a GARCH model. Compes Rendus de l Academie des Sciences, Serie I, 331, (in French). Chan, F., Lim, C., & McAleer, M. (2005). Modelling mulivariae inernaional ourism demand and volailiy. Tourism Managemen, 26, Cook, S. (2006). The robusness of modified uni roo ess in he presence of GARCH. Quaniaive Finance, 6, Dickey, D. A., & Fuller, W. A. (1979). Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American Saisical Associaion, 74, Dickey, D. A., & Fuller, W. A. (1981). Likelihood raio saisics for auoregressive ime series wih a uni roo. Economerica, 49, Divino, J. A., Farias, A., Takasago, M., & Teles, V. K. (2007). Tourism and economic developmen in Brazil. Unpublished paper, Cenro de Excelencia em Turismo (CET- UnB). Universiy of Brasilia, Brazil. Divino, J. A., & McAleer, M. (2008). Modelling and forecasing susainable inernaional ourism demand for he Brazilian Amazon. Environmenal Modelling & Sofware. (Forhcoming). 21

23 Elie, L., & Jeanheau, T. (1995). Consisency in heeroskedasic models. Compes Rendus de l Académie des Sciences, Série I, 320, (in French). Ellio, G., Rohenberg, T. J., & Sock, J. H. (1996). Efficien ess for an auoregressive uni roo. Economerica, 64, Engle, R. F. (1982). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica, 50, Gil-Alana, L., A., Cunado, J., & Gracia, F. P. (2008). Tourism in he Canary Islands: forecasing using several seasonal ime series models. Journal of Forecasing, 27, Glosen, L., Jagannahan, R., & Runkle, D. (1992). On he relaion beween he expeced value and volailiy of nominal excess reurn on socks. Journal of Finance, 46, Hoi, S., McAleer, M. & Shareef, R. (2005). Modelling counry risk and uncerainy in small island ourism economies. Tourism Economics, 11, Hoi, S., McAleer, M., & Shareef, R. (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., & Hansen, B. E. (1994). Asympoic heory for he GARCH(1,1) quasimaximum likelihood esimaor. Economeric Theory, 10, Li, W. K., Ling, S., & McAleer, M. (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., & Li, W. K. (1997). On fracionally inegraed auoregressive moving-average models wih condiional heeroskedasiciy. Journal of he American Saisical Associaion, 92, Ling, S., & McAleer, M. (2002a). Saionariy and he exisence of momens of a family of GARCH processes. Journal of Economerics, 106,

24 Ling, S., & McAleer, M. (2002b). Necessary and sufficien momen condiions for he GARCH(r,s) and asymmeric power GARCH(r,s) models. Economeric Theory, 18, Ling, S., & McAleer, M. (2003a). Asympoic heory for a vecor ARMA-GARCH model. Economeric Theory, 19, Ling, S., & McAleer, M. (2003b). On adapive esimaion in nonsaionary ARMA models wih GARCH errors. Annals of Saisics, 31, McAleer, M. (2005). Auomaed inference and learning in modeling financial volailiy. Economeric Theory, 21, McAleer, M. (2008). The Ten Commandmens for opimizing value-a-risk and daily capial charges. Journal of Economic Surveys. (Forhcoming) McAleer, M., Chan, F., Hoi, S., & Lieberman, O. (2008). Generalized auoregressive condiional correlaion. Economeric Theory, 24, McAleer, M., Chan, F., & Marinova, D. (2007). An economeric analysis of asymmeric volailiy: heory and applicaion o paens. Journal of Economerics, 139, McAleer, M., & da Veiga, B. (2008a). Forecasing value-a-risk wih a parsimonious porfolio spillover GARCH (PS-GARCH) model. Journal of Forecasing, 27, McAleer, M., & da Veiga, B. (2008b). Single-index and porfolio models for forecasing value-a-risk hresholds. Journal of Forecasing, 27, Mikosch, T. & Sarica, C. (2000). Limi heory for he sample auocorrelaions and exremes of a GARCH (1; 1) process. Annals Saisics, 28, Nelson, D. B. (1991). Condiional heeroscedasiciy in asse reurns: a new approach. Economerica, 59, Ng, S., & Perron, P. (2001). Lag lengh selecion and he consrucion of uni roo ess wih good size and power. Economerica, 69, Perron, P., & Ng, S. (1996). Useful modificaions o some uni roo ess wih dependen errors and heir local asympoic properies. Review of Economic Sudies, 63, Phillips, P. C. B., & Perron, P. (1988). Tesing for a uni roo in ime series regression. Biomerika, 75,

25 Shareef, R., & McAleer, M. (2005). Modelling inernaional ourism demand and volailiy in small island ourism economies. Inernaional Journal of Tourism Research, 7, Shareef, R., & McAleer, M. (2007). Modelling he uncerainy in inernaional ouris arrivals o he Maldives. Tourism Managemen, 28, Shareef, R., & McAleer, M. (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

26 Figure 1 Map of Peru Source: Wikipedia 25

27 Figure 2 Inernaional Touris Arrivals and Volailiy Arrivals (Y) Volailiy of Y Volailiy of Y from GARCH Log of arrivals (LY) Volailiy of LY Volailiy of LY from GARCH Firs difference of arrivals (DLY) Volailiy of DLY Volailiy of DLY from GARCH 26

28 Figure 3 - Inernaional Touris Arrivals o Peru 27

29 Figure 4 Forecass of Inernaional Touris Arrivals o Peru in Log-Levels Figure 5 Forecass of Annualized Inernaional Touris Arrivals o Peru in Log-Levels 28

30 Figure 6 Forecass of Inernaional Touris Arrivals o Peru in Firs Differences Figure 7 Forecass of Annualized Inernaional Touris Arrivals o Peru in Firs Differences 29

31 Table 1 - Uni Roo Tess Variables MADF GLS MPP GLS Lags Z LY -3.84** ** 27 {1, } LY {1} ΔLY ** ** 0 {1, } ΔLY -5.27** {1} Noes: LY is he logarihm of inernaional ouris arrivals o Peru. The criical values for MADF GLS and MPP GLS a he 5% significance level are 2.93 and 17.3, respecively, when Z={1,}, and 1.94 and 8.1, respecively, when Z={1}. ** denoes he null hypohesis of a uni roo is rejeced a he 5% significance level. Table 2 - Informaion Crieria for Alernaive ARMA Models Variable IC AR\MA LY AIC 1-0,6553-0,9224-0,9301 BIC -0,6519-0,9174-0,9234 AIC 2-0,8290-0,9334-0,9376 BIC -0,8239-0,9267-0,9293 LYMA AIC 1-12,268-12,278-12,339 BIC -12,264-12,272-12,331 AIC 2-12,283-12,398-12,401 BIC -12,278-12,391-12,392 DLY AIC 1-0,7938-0,9304-0,9358 BIC -0,7905-0,9254-0,9291 AIC 2-0,8236-0,9413-0,9425 BIC -0,8185-0,9346-0,9341 DLYMA AIC 1-12,282-12,4-12,402 BIC -12,278-12,394-12,394 AIC 2-12,354-12,403-12,407 BIC -12,349-12,395-12,398 Noes: IC denoes informaion crieria, AIC is he Akaike informaion crierion, and BIC is he Schwarz informaion crierion. 30

32 Table 3 Esimaed Condiional Mean and Condiional Volailiy Models Dependen variable: LY Dependen variable: DLY Parameers GARCH GJR EGARCH GARCH GJR EGARCH φ 0.008* * 0.892* 0.854* 0.001* (0.002) 0.01* (0.07) (0.07) (0.06) (0.002) (0.002) φ * (0.01) ω 0.002* (0.000) GARCH/GJR α 0.118* (0.01) 0.879* (0.009) 0.005* (0.000) 0.027* (0.01) 0.883* (0.01) 0.976* (0.09) 0.452* 0.464* (0.016) 0.003* (0.000) * (0.01) (0.02) 0.003* (0.000) 0.452* (0.02) 0.473* (0.06) 0.025** 0.01) ( -- GARCH/GJR β 0.803* (0.02) GJR γ EGARCH α EGARCH γ EGARCH β 0.688* 0.025) * ( (0.02) 0.780* -- ( 0.02) * * -- ( 0.03) ( 0.02) * * ( 0.02) (0.02) * * (0.02) (0.01) * * ( 0.02) (0.01) Diagnosic Second momen ARCH(1) LM es [p-value] [0.000] [0.008] [0.001] [0.916] [0.766] Jarque-Bera [p-value] [0.000] [0.000] [0.000] [0.000] [0.000] [0.855] [0.000] Noes: LY is he logarihm of inernaional ouris arrivals o Peru, and DLY is he log difference (or growh rae). Numbers in parenheses are sandard errors. * The esimaed coefficien is saisically significan a he 1% significance level. ** The esimaed coefficien is saisically significan a he 5% significance level. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. 31

33 Table 4 Esimaed Condiional Mean and Condiional Volailiy Models Parameers φ * (0.000) φ * (0.000) ω Dependen variable: LYMA Dependen variable: DLYMA GARCH GJR EGARCH GARCH GJR EGARCH (0.000) GARCH/GJR α 0.150* (0.058) GARCH/GJR β 0.600* (0.16) GJR γ EGARCH α EGARCH γ EGARCH β * (0.000) 1.000* (0.000) (0.000) -- (0.09) 0.000* 0.001* (0.000) 1.000* (0.000) (0.000) 0.107* (0.019) * 0.000* (0.127) (0.000) 0.150** ( 0.066) * (0.008) 0.600** ( 0.15) * (0.011) 0.000* (0.000) 0.106* (0.019) 0.000* (0.000) 0.000* (0.000) 0.107* (0.018) 0.488* (0.111) 0.066* -- ( 0.009) 0.917* -- ( 0.011) (0.008) * * ( 0.017) (0.016) * * * * ( 0.007) (0.005) * (0.008) (0.007) Diagnosic Second momen ARCH(1) LM es [p-value] [0.000] [0.000] [0.029] [0.226] [0.257] Jarque-Bera [p-value] [0.001] [0.000] [0.633] [0.254] [0.441] [0.091] 0.92 [0.630] Noes: LYMA is he logarihm of annualized inernaional ouris arrivals o Peru, and DLYMA is he log difference (or growh rae). Numbers in parenheses are sandard errors. * The esimaed coefficien is saisically significan a he 1% significance level. ** The esimaed coefficien is saisically significan a he 5% significance level. The log-momen condiion is necessarily saisfied as he second momen condiion is saisfied. 32

34 Table 5 - Correlaion Marix: Forecass of he Series in Log-Levels GARCH- LY GARCH- LYMA GJR- LY GJR- LYMA EGARCH- LY EGARCH- LYMA Model GARCH-LY GARCH-LYMA GJR-LY GJR-LYMA EGARCH-LY EGARCH-LYMA Table 6 - Correlaion Marix: Forecass of he Series in Log-Differences GARCH- DLY GARCH- DLYMA GJR- DLY GJR- DLYMA EGARCH- DLY EGARCH- DLYMA Model GARCH-DLY GARCH-DLYMA GJR-DLY GJR-DLYMA EGARCH-DLY EGARCH-DLYMA