Tourism forecasting using conditional volatility models

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1 Tourism forecasing using condiional volailiy models ABSTRACT Condiional volailiy models are used in ourism demand sudies o model he effecs of shocks on demand volailiy, which arise from changes in poliical, social or economic condiions. Seasonal ARIMA models have been widely employed for forecasing purpose bu lile aenion has been given o examining he forecas accuracy of condiional volailiy models. This sudy invesigaes wheher he condiional volailiy models can ouperform seasonal ARIMA models in predicing ouris arrivals o Ausralia. One key resul is ha seasonaliy exiss in he volailiy of ouris arrivals o Ausralia. Hence, his paper adds a new dimension in he lieraure of modelling seasonaliy in ourism demand, by incorporaing seasonal effecs ino condiional volailiy models. Using daa on ouris arrivals from USA, UK, Japan and New Zealand o Ausralia, his sudy found ha condiional volailiy models ouperform seasonal ARIMA models in forecasing for all counries excep UK. Keywords: forecasing; ouris arrivals o Ausralia; seasonal ARIMA; condiional volailiy models INTRODUCTION Several empirical sudies have found ha ourism demand daa exhibi volailiy (Shareef and McAleer, 2005) and specifically, a negaive shock can have more impac on he volailiy in Japanese ouris arrivals o Ausralia han a posiive shock (Chan e al., 2005). Furhermore, according o Kim and Wong (2006), he volailiy in ourism demand daa can be influenced by he effecs of news shocks such as economic crises, oubreak of deadly diseases, naural disasers and war. Therefore, he ourism lieraure concludes ha modelling he volailiy in ourism demand is imporan because i can capure he occurrence of unexpeced evens. Convenionally, volailiy of ourism demand is modelled using condiional volailiy models. The models ha appeared in ourism lieraure are univariae generalised auoregressive condiional heeroscedasiciy (GARCH), univariae asymmeric GARCH (or GJR), vecor auoregressive moving average GARCH (VARMA-GARCH) and VARMA-asymmery GARCH (VARMA-AGARCH) models (Chan e al., 2005; Kim and Wong, 2006; Shareef and McAleer, 2005; Shareef and McAleer, 2007). Despie ha modelling demand volailiy has emerged in he lieraure, here are wo areas which sill require aenion. Firs, seasonaliy exiss in ourism demand volailiy daa. Figure 1 exhibis he volailiy of ouris arrivals o Ausralia and find ha seasonal paerns exis and vary across counries of origin. For insance, from 1991 o 2007, he highes spikes in ouris arrivals from New Zealand and Unied Kingdom occurred in he monhs of December and March, respecively. Hence, he figure suggess ha incorporaing seasonal dummies in he condiional volailiy models are necessary o capure hese effecs. Second, here is a dearh of research assessing he forecas accuracy of condiional volailiy models. As predicing ourism demand is very imporan in business planning, i is imperaive o invesigae wheher condiional volailiy models can ouperform oher compeing models in forecasing ourism demand daa.

2 Figure 1 Volailiy of he log ouris arrivals o Ausralia for oal and four counries (1991 o 2007) Japan New Zealand Toal Unied Kingdom USA The purposes of his paper are as follows. Firs, as seasonaliy exiss in ourism demand daa, his paper aemps o modify he radiional condiional volailiy models by incorporaing seasonal effecs ino he models. Thereafer, he models will be employed for he purpose of ou-sample forecasing. Second, as seasonal ARIMA models have been widely employed for forecasing ourism demand o Ausralia (Kim, 1999; Kim and Moosa, 2001; Kulendran and Wong, 2005; and Lim and McAleer, 2002), his sudy invesigaes wheher condiional volailiy models can ouperform seasonal ARIMA models in erms of forecasing ouris arrivals o Ausralia. The condiional volailiy models employed in his research are univariae GARCH and GJR models. The daa employed are he logarihm of he monhly shor-erm ouris arrivals from Japan, New Zealand, Unied Kingdom, USA and all source counries o Ausralia from January 1991 o May The logarihm daa are used because, based on Chan e al. (2005), daa on logarihms of

3 ouris arrivals o Ausralia are inegraed o zero afer aking is firs difference. For forecasing purpose, he daa beween June 2006 and May 2007 are used. METHODOLOGY In a ourism demand daa series, he condiional variance may no be consan. To ackle he problem of heeroscedasic condiional variances, Engle (1982) developed a volailiy model which incorporaed all pas errors. Bollerslev (1986) furher modified Engle s idea by including lagged condiional volailiy ino he volailiy funcion. Given he univariae condiional mean, y = E( y / F 1) + ε, (1) ε = h η, where = 1,,n and η ~ iid(0,1), y = series of reurns, F -1 = pas informaion available o ime and ε = error erm wih sochasic process, he univariae Generalised Auoregressive Condiional Heeroscedesiciy (GARCH) model is given as follows: h ω αε β (2) = h 1 where h = condiional variance, and h > 0 when ω > 0, α > 0 and β > 0. Noe ha α and α+β represen he shor erm and long erm persisence of shocks o reurns. ω = consan variance. The simpler model ha capures he asymmeric impacs of good and bad news is he GJR(1,1) model developed by Glosen e al. (1993) as follows: h 2 = + ( α + γi ( η )) 1 ε 1 + βh 1 ω, (3) where 1, ε < 0 I ( η ) =, h > 0 when ω > 0, α 0, α + γ 0 and β 0. 0, ε 0 In erms of he persisence of shocks, he impac of shor run posiive and negaive shocks is α and α + γ respecively. When η follows a symmeric disribuion, he shor-run persisence of shocks is α γ and long run persisence of shocks is α γ + β. To incorporae seasonal effecs ino he GARCH and GJR models, his paper aemps o include seasonal dummies in Equaion (2) and (3). Figure 2 illusraes he mehodology of specificaion of condiional volailiy models.

4 Figure 2 Flow char of mehodology Correlograms Model specificaion Esimaion Diagnosic ess: The null hypohesis is ha he specificaion of model is correc, i.e. here are no issues of serial correlaion and non-normaliy in he sandardized residual. Do no rejec he null hypoheses. Rejec he null hypoheses. Forecas EMPIRICAL RESULTS Esimaes of GARCH and GJR models In Tables 1 and 2, AR(1) coefficiens in he condiional mean of GARCH and GJR models are saisically significan for Japan, New Zealand and Toal, indicaing a high persisence of ouris visiaions o Ausralia. However, for moving average componens, only he daa on ouris arrival from New Zealand show highly significan. This implies ha unexpeced shocks in he previous period can influence he arrivals of New Zealand ouriss in he curren period. Furhermore, Tables 1 and 2 also reveal ha ouris arrivals o Ausralia are highly seasonal. In erms of he condiional volailiy esimaes for he GARCH model (Table 3), he ARCH effec (or he shor-run shock persisence) is posiive and only significan for New Zealand. This shows ha a posiive shock will increase he variaion of ouris arrivals from New Zealand and vice-versa for negaive shocks. Conversely, he β (or GARCH effecs) are significan for all counries, excep Japan. The signs of β are posiive, as anicipaed, for UK, USA and Toal, bu no for New Zealand. This sudy also discovers ha seasonal effecs exis in he condiional volailiy of Japan, New Zealand, UK and USA (Table 3), which indicaing ha he variaions in ouris arrivals from hese source counries o Ausralia can be caused by seasonaliy. The condiional volailiy esimaes for GJR model in Table 4 reveal ha α (or ARCH effecs) for Japan and Toal are significan bu he signs are no consisen wih expecaions. For he GARCH effecs, he resuls for USA and Toal are significan. However, he sign of β for USA is negaive, which does no saisfy he condiion of posiiviy of condiional variance. In

5 addiion, he hreshold effecs for New Zealand and Toal are significan a 5% and 1%, respecively, and he impacs of a negaive shock on he variaions of boh ses of daa are similar. This implies ha a negaive shock will resul in less flucuaion in Toal and New Zealand ouris arrivals o Ausralia. This sudy could no generae reliable esimaes for seasonal dummies in he condiional volailiy of GJR model, which requires furher invesigaion. In his research, he esimaion of GARCH and GJR models has undergone rigorous diagnosic ess o ensure here are no issues of serial correlaion and non-normaliy in he esimaed sandardized residuals (Table 5). Given his fac, he condiional mean and variance esimaes in Table 1 o 4 are considered o be robus. Forecas accuracy of condiional volailiy models By comparing forecas accuracy beween condiional volailiy and ARIMA models, Table 6 shows ha he forecas errors of GARCH and GJR models for Japan, UK and Toal are lower han for he ARIMA model. This implies ha condiional volailiy models perform beer in forecasing for hese daa series. For New Zealand daa, ARIMA model ouperforms GARCH models bu under-performs GJR models. For UK daa, boh GARCH and GJR models generae less accurae forecass han ARIMA models. Forecas errors of GARCH models for UK and Toal are lower han GJR models (Table 6). This oucome implies ha GARCH models can predic beer han GJR models for hese hree daa series. Conversely, for Japan, New Zealand and USA daa, GJR models provide more accurae forecas han GARCH models. CONCLUSION For he firs ime, his sudy demonsraes ha seasonaliy exiss in he volailiy of ouris arrivals o Ausralia. To capure he seasonal effecs, seasonal dummy variables were included in he condiional volailiy models. Furhermore, given ha seasonal ARIMA models have been widely employed in forecasing ouris arrivals o Ausralia, his paper inends o invesigae wheher condiional volailiy models provide beer forecass han seasonal ARIMA models. The resuls showed ha seasonal dummy variables in he condiional mean of GARCH and GJR models are saisically significan for all source counries. Furhermore, he empirical resuls of GARCH models revealed ha seasonal dummy variables were significan in he condiional variances for GARCH model for Japan, New Zealand, Unied Kingdom and USA daa, leading o he conclusion ha seasonaliy exiss in he volailiy of ouris arrivals o Ausralia. In erms of comparing forecas performance of condiional volailiy and ARIMA models, his sudy found ha he former model forecas beer for all source counries excep UK. Furhermore, by evaluaing forecas accuracy beween GARCH and GJR models, he empirical resuls showed ha GJR can generae relaively more accurae forecass for Japan, New Zealand and USA daa. Overall, his paper concludes ha condiional volailiy models can ouperform seasonal ARIMA models in predicing ouris arrivals o Ausralia. Neverheless, fuure research should compare he forecas accuracy of condiional volailiy models wih economeric models.

6 Table 1 Esimaion of condiional mean for GARCH model Daa Consan AR(1) MA(1) SMA(12) S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Japan NZ UK USA Toal Noe: The wo enries corresponding o each variable are heir esimaes (in bold) and heir Bollerslev and Wooldridge (1992) robus sandard errors, respecively. AR(p) denoes auoregressive a lag p. MA(q) denoes moving average a lag q. SMA(Q) denoes seasonal moving average a lag Q. S1 o S12 signify seasonal dummies from January o December. The AR, MA, SMA and seasonal dummies erms above are saisically significan a 5%. In he ineress of presenaion, hose insignifican regressors are no repored here.

7 Table 2 Esimaion of condiional mean for GJR model Daa Consan AR(1) AR(2) MA(1) SMA(12) S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Japan NZ UK USA Toal Noe: The wo enries corresponding o each variable are heir esimaes (in bold) and heir Bollerslev and Wooldridge (1992) robus sandard errors, respecively. AR(p) denoes auoregressive a lag p. MA(q) denoes moving average a lag q. SMA(Q) denoes seasonal moving average a lag Q. S1 o S12 signify seasonal dummies from January o December. The AR, MA, SMA and seasonal dummies erms above are saisically significan a 5%. In he ineress of presenaion, hose insignifican regressors are no repored here.

8 Table 3 Esimaion of condiional volailiy for GARCH model Daa ω α β S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 Japan * * * * * * * * * * * NZ * * * * * * * * * * * * * UK * * * USA * * * Toal Noe: ω = consan variance, α = ARCH effec, β = GARCH effec. The wo enries corresponding o each variable are heir esimaes (in bold) and heir Bollerslev and Wooldridge (1992) robus sandard errors, respecively. S1, S3 and S4 are seasonal dummies for January, March and April, respecively. *denoes significan a 1%. + denoes significan a 5%. In he ineress of presenaion, some of he insignifican seasonal dummy variables are no repored here.

9 Table 4 Esimaion of condiional volailiy for GJR model Daa ω α γ β Japan * NZ * UK USA * * Toal * * * Noe: ω = consan variance, α = ARCH effec, γ = hreshold effec, β = GARCH effec. The wo enries corresponding o each variable are heir esimaes (in bold) and heir Bollerslev and Wooldridge (1992) robus sandard errors, respecively. * denoes significan a 1%. + denoes significan a 5%. Table 5 Diagnosic ess of GARCH and GJR models Diagnosic ess on sandardised residuals Daa Models Ho: No serial correlaion a Ho: Normaliy b Japan GARCH [0.208] [0.5576] GJR [0.129] [0.5941] NZ GARCH [0.018] [0.268] GJR [0.03] [0.9126] UK GARCH [0.265] [0.1509] GJR [0.531] [0.2651] USA GARCH [0.178] [0.5642] GJR [0.145] [0.022] Toal GARCH [0.083] [0.8974] GJR [0.145] [0.6043] Noe: (a) The es saisics are obained from Q-saisics of collegram of sandardized residuals. (b) The es saisics are based on Jarque-Bera of normaliy ess. Figures in brackes are p-value.

10 Table 6 Summary of forecas errors for ARIMA and condiional volailiy models Models Daa Forecas error ARIMA GARCH GJR Japan RMSE MAE MAPE Theil coefficien NZ RMSE MAE MAPE Theil coefficien UK RMSE MAE MAPE Theil coefficien USA RMSE MAE MAPE Theil coefficien Toal RMSE MAE MAPE Theil coefficien

11 ACKNOWLEDGEMENTS The auhor is graeful o wo anonymous reviewers for heir commens and STCRC for heir financial suppor o aend CAUTHE 2008 conference in Gold Coas, Ausralia. REFERENCES Bollerslev, T., (1986) Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 31 (3), Bollerslev, T., & Wooldridge, J. M., (1992) Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariance Economeric Review, 11 (2), Chan, F., Lim, C., & McAleer, M., (2005) Modelling mulivariae inernaional ourism demand and volailiy Tourism Managemen, 26 (3), Glosen, L.R., Jagannahan, R., & Runkle, D., (1993) On he Relaion beween he Expeced Value and he Volailiy of he Normal Excess Reurn on Socks Journal of Finance, 48 (5), Kim, J. H., & Moosa, I., (2001) Seasonal behaviour of monhly inernaional ouris flows: specificaion and implicaions for forecasing models Tourism Economics, 7 (4), Kim, S. S., & Wong, K. K. F., (2006) Effecs of news shocks on inbound ouris demand volailiy in Korea Journal of Travel Research, 44 (4), Kulendran, N., & King, M. L., (1997) Forecasing inernaional quarerly ouris flows using error-correcion and ime-series models Inernaional Journal of Forecasing, 13 (3), Kulendran, N., & Wong, K. K. F., (2005) Modelling seasonaliy in ourism forecasing Journal of Travel Research, 44 (2), Lim, C., & McAleer, M., (2002) Time series forecass of inernaional ravel demand for Ausralia Tourism Managemen, 23 (4), Engle, R. F., (1982) Auoregressive condiional heeroscedasiciy wih esimaes of variance of Unied Kingdom inflaion Economerica, 50 (4), Shareef, R., & McAleer, M., (2005) Modelling inernaional ourism demand and volailiy in small island ourism economics Inernaional Journal of Tourism Research, 7 (6), Shareef, R., & McAleer, M., (2007) Modelling he uncerainy in monhly inernaional ouris arrivals o he Maldives Tourism Managemen, 28 (1),