SEASONAL ADJUSTMENT VERSUS SEASONALITY MODELING: Effect on Tourism Demand Forecasting

Transcription

1 Seasonal Asian-African Adjusmen Journal Versus of Economics Seasonaliy and Modeling: Economerics, Effec Vol. on Tourism 13, No. 1, Demand 2013: Forecasing 71 SEASONAL ADJUSTMENT VERSUS SEASONALITY MODELING: Effec on Tourism Demand Forecasing Amira GASMI * ABSTRACT In his sudy, we rea he seasonal variaion in monhly ime series in he conex of he Wesern- European ourism demand for Tunisia, by presening differen echniques of deecion of seasonaliy and he parameric and non-parameric approaches of seasonal adjusmen. Then, we compare he forecasing performance of hese mehods. The empirical resuls miliae in favour of he TRAMO-SEATS mehod. In fac, his approach provides he bes forecas. In erms of forecasing efficiency, we noe in addiion, ha he modelling of he seasonal variaion using seasonal ARIMA model (SARIMA) may lead o beer predicive resuls compared wih oher echniques of seasonal adjusmen used in his research, namely: he X-12-ARIMA, regression on seasonal dummies and he raio-o-moving average mehods. Keywords: seasonaliy, ourism demand, forecasing performance, seasonal adjusmen, seasonal modelling. JEL Classificaion: C22, C52, C53, L INTRODUCTION Seasonaliy is a major characerisic of he ourism aciviy. I reveals he influence of he seasons on he ourism demand. This phenomenon is relaed o weaher changes as well as insiuional facors (school holidays, professional vacaion, public (Chrismas or Easer), religious and commemoraive fesivals). The calendar can also generae a seasonal movemen in monhly ime series since he number of working days varies from one monh o anoher. I is also relaed o cerain socio-culural characerisics (spor pracices; social or religious habis). Taking ha ino accoun, he phenomenon of high and low seasons consiues a problem of size which worries he acors of he ourism field. Time series analysis aims o separae he shor-erm behaviour from ha of long-erm of an economic daa series and o give reliable forecass for hese separae componens and for he oaliy of he series. The seasonal variaions explain mos of he variaion in he growh raes of he majoriy of he economic ime series. In order o draw conclusions on he naure of he business cycles and * Laboraoire d Economie e Finance Appliquees IHEC Carhage, Tunisia, amira.gasmi@yahoo.fr

2 72 Amira Gasmi he long-erm growh, he radiional approach is o remove he seasonal componen of a series hrough he use of deerminisic seasonal dummies, seasonal differeniaion, or using he seasonal adjusmen echniques such as he X-12-ARIMA mehod. However, someimes we show ha i can be more appropriae o sudy he seasonal models hemselves (Lee and Siklos (1993), Reimers (1997)) since hey could give informaion on he behaviour of he economic agens which are exposed o changes of endencies a he momen of planning and he formaion of waiings. Thus, alhough seasonal variaions-correced daa can be useful, i is ypically recommended o use no adjused daa. Moreover, several recen empirical sudies showed ha many mehods of seasonal adjusmen lead o seriously denaured daa, in he sense ha he key properies such as he endencies, he business cycles, he nonlineariies are affeced by he seasonal adjusmen (Ghysels and Perron (1993), Ghysels, Granger and Siklos (1995), Hylleberg (1994), Miron (1996), Maravall (1995)). In conras, King and Kulendran (1997) evaluae several models, including he seasonal uni roos model, in he forecas of quarerly ouris arrivals in Ausralia coming from many counries. Their principal conclusion is ha compared o ime series models, he forecasing performance of he seasonal uni roos models is weak. This may be due o he lack of power of some uni roos ess. On he oher hand, Paap, Franses and Hoek (1997) use empirical and simulaion examples o demonsrae ha he negleced seasonal average changes can desroy considerably he forecasing performance of he univariae auoregressive processes. Thus, appropriae reamen of seasonaliy is imporan o make reliable forecass. In his aricle, we propose firsly, o sudy he seasonal aspec srongly characerizing he ourism ime series, by presening differen ess of seasonaliy deecion and various mehods of reamen of seasonaliy, in paricular seasonal adjusmen mehods versus seasonaliy modeling. Then, secondly, we will compare he forecasing performance of hese mehods. 2. ANALYSIS OF SEASONALITY 2.1. Deecion of Seasonaliy During he analysis of a ime series, i is necessary o idenify he seasonal variaion which can be probably observed. Various ypes of ess are se up o deec he presence of his componen Auocorrelaions Seasonaliy can be deeced graphically by examining auocorrelaion (ACF) and parial auocorrelaion funcions (PACF) necessary for he idenificaion of suiable ARIMA models. Indeed, he correlogram of a seasonal series ofen akes a sinusoidal form (see able 1) Tradiional Tess of Presence of Seasonaliy To es he presence of seasonal variaion, a mulipliciy of ess were suggesed, namely: he sable seasonaliy and moving seasonaliy ess which are Fisher ypes ess based on models of analysis of he variance o one (he monh or he quarer) and wo facors (he monh or he quarer and he year), respecively. Indeed, sable seasonaliy is a ype of seasonaliy which is repeaed a he same ime each year, and his sable aspec faciliaes he forecass. While he

3 Seasonal Adjusmen Versus Seasonaliy Modeling: Effec on Tourism Demand Forecasing 73 moving seasonaliy is represened by a movemen effec from one monh o anoher. This lack of sabiliy makes is forecas difficul. Lasly, we disinguish he idenifiable seasonaliy es compleing he ess evoked above. I is buil saring from he values of Fisher saisics of sable and moving seasonaliy ess (Lohian and Morry, 1978). The es saisic, noed T, is expressed as follows: 1/ T wih T1 and T2 2 FS T T 3F F If saisic T is lower han 1 so we concludes he presence of idenifiable seasonal componen and he seasonal adjusmen of he series is hen necessary Seasonal Uni Roos Tess The deecion of seasonal variaion can be done using seasonal uni roos ess. For his purpose, a cerain number of ess were implemened in he eighies and nineies, in paricular o es he seasonal variaion a order 4 and order 12. Table 1: Correlogram of Wesern-European Touris Arrivals Series M S.

4 74 Amira Gasmi Tes DHF (Dickey, Hasza and Fuller, 1984): i allows o es he null assumpion = 1 in he model x = x S +. Under H 0 rue, he series is seasonal and he filer S = (1 L S ) suggesed by Box & Jenkins (1970) is appropriae o adjus i (S being he period of seasonaliy). Tes HEGY (Hylleberg, Engle, Granger and Yoo, 1990): he lieraure on he concep of unis roos (e.g., Dickey, Bell and Miller (1986)) shows ha he assumpion of exisence of cerain filers of differeniaion amouns o emi he assumpion of presence of a cerain number of seasonal and non-seasonal unis roos in a ime series. This can be easily seen by wriing: S = (1 L S ), and by solving he equaion: (1 z S ) = 0. The general soluion o his equaion is: {1, cos(2k / S) + i sin(2k / S); wih he erm (2k / s) for k = 1, 2,, represens he corresponding seasonal frequency, giving S differen soluions which all of hem are on he circle uni. The HEGY mehod, mainly conceived for he quarerly series, was adaped o he monhly case hanks o Franses (1990) and of Beaulieu and Miron (1993) works. In fac, if S = 12, soluions of he equaion (1 z 12 ) = 0 are: 1 for he non-seasonal uni roo corresponding o frequency 0; and 11 seasonal uni roos , i, (1 3 i), (1 3 i), ( 3 i), ( 3 i) corresponding respecively o he following frequencies:,,,,, and o he operaors of differeniaion: (1+ L), (1+ L 2 ), (1+ L+ L 2 ), (1 L+ L 2 ), (1+ 3 L+ L 2 ) and (1 3 L+ L 2 ). S 1 Thus, a filer of differeniaion ( S ) can be wrien as: s (1 L)(1 L L ), and can hus be decomposed ino a par wih non-seasonal uni roo and a par wih (S 1) seasonal uni roos. In his es, we resor o he decomposiion of he polynomial (1-L 12 ), wih 12 roos unis and we consider he following form: ( L) z z z z z z z z z 8 1 1, 1 2 2, 1 3 3, 1 4 3, 2 5 4, 1 6 4, 2 7 5, 1 8 5, 2 z z z z 9 6, , , , 2 The variables z i are in such a way ha: z i = P i (L) y, where polynomials P i are defined as follows : P1 ( L) (1 L)(1 L )(1 L L ) P2 ( L) (1 L)(1 L )(1 L L ) P3 ( L) (1 L )(1 L L ) P L L L L L L P L L L L L L P6 ( L) (1 L )(1 L L )(1 L L ) P7 ( L) (1 L )(1 L L )(1 L L ) 12 P8 ( L) (1 L ) ( ) (1 )(1 3 )(1 ) ( ) (1 )(1 3 )(1 ) (1)

5 Seasonal Adjusmen Versus Seasonaliy Modeling: Effec on Tourism Demand Forecasing 75 Wih also: (L) is an auoregressive polynomial in L, and µ may conain a consan, 11 seasonal dummies and/or a rend. The variables z i are hen associaed o he differen roos of he polynomial. The equaion (1) is esimaed using leas squares ordinary mehod. We may carry ou ess for he parameers 1 and 2, and F ess associaed o he couples ( 3, 4 ), ( 5, 6 ), ( 7, 8 ), ( 9, 10 ) and ( 11, 12 ) : i is a quesion of esing he joined significance of he coefficiens. This amouns o es he assumpion of exisence of uni roos a he differen frequencies. For his purpose, we mus compare he es saisics relaed o he esimaed parameers wih he criical values provided by Franses (1990) and Beaulieu and Miron (1993). To check he exisence of he roos 1 and -1 corresponding o frequencies 0 and respecively, we carry ou wo individual ess on parameers 1 and 2. As for he oher seasonal uni roos, we can perform eiher joined ess whose null assumpion akes he form k = k 1 = 0 and his for he even values of k, from 4 o 12; or quie simply, individual ess, suggesed in Franses (1990), allowing o verify he non-saionariy of he ime series a all he seasonal frequencies and his by esing he null assumpion according o which here is a seasonal uni roo ( k = 0, k [3, 12]). However, i should be noed ha he applicaion of he OLS o he regression (1) is made where he order of (L) is given in such a way ha he errors are roughly whie noises, or a leas, non-auocorrelaed residuals. For his purpose, Hylleberg and al. (1990) and Engle and al. (1993) propose o inroduce addiional lags of he variable unil we obain non-auocorrelaed residuals Seasonal Adjusmen Mehods Seasonal adjusmen mehods can be classified in wo caegories, namely: parameric approach and nonparameric approach Non-parameric Approaches The X-12-ARIMA mehod: when he seasonal variaion is very apparen in he ime series, a firs approach consiss in removing such seasonal flucuaions by using a seasonal adjusmen programs. They are echniques allowing he idenificaion of he differen componens of he iniial series (rend-cycle, seasonaliy, irregular) by applying linear filers, which cancels or preserves a well defined componen (endency-cycle or seasonal variaion). The irregular one is represened hereafer by he residual of he decomposiion. These linear filers are moving averages which consiue he principal ool of Census X- 11 mehod buil from successive ieraions of moving averages of differen naures for beer esimaing he series componens. However, his echnique leads o a loss of informaion in he final end of he series. This gap is filled by he forecas of fuure values of he ime series before is seasonal adjusmen, and his using ARIMA models. I is wha made i possible o exend he X-11 echnique (Census Bureau, 1967) o X-11-ARIMA (Dagum, 1988) and hen o X-12-ARIMA (Findlay and Al, 1998). The laer conains he RegARIMA module which allows o deec and o remove any undesirable effec of he series (ouliers, calendar effecs ).

6 76 Amira Gasmi The raio o moving average mehod: Monhly values of he sudied series (X ) are divided by he moving average figure corresponding for each monh (MA ), and expressed in % o X generae he raio-o-moving average: Mraio 100. The moving average is calculaed MA as follows: 1 M X 2 X X 1 2X 2 X X 1 X, m 2 p m 2m p p p p 1 MA M ( X ) X 2X 2 X... 2X X A. Carpener (2003) reveals ha his moving average eliminaes seasonal variaions from monhly series, preserves linear rends and reduces of more han 90% he variance of a whie noise. These raios are weighed by he monh and hereafer will separae he seasonal and cyclic componens Parameric Approaches The regression mehod: his approach is based on he Buys-Ballo model (1847) which consiss in carrying ou he regression below, using seasonal dummies (S,i ) in such a way ha S,i akes value 1 if T corresponds o he seasonal period, and 0 if no. The model is wrien as follows: T 1 X S. 0 1 i, i i1 Wih: T being he period of seasonaliy (T = 4 for a quarerly series, T = 12 for monhly daa). The use of only (T 1) dummies makes i possible o avoid he problem of colineariy which could exis wih he vecor uni relaing o he consan. We esimae hus (T 1) seasonal coefficiens and we check he T h using he principle of conservaion of he surfaces : i 0. Seasonal adjusmen by mehod TRAMO-SEATS: TRAMO-SEATS program (Gomez & Maravall, 1996) belongs o he parameric seasonal adjusmen mehods based on he signal exracion. I is composed of wo independen subrouines bu which are complemenary since hey are generally used ogeher: TRAMO program (Time series Regression wih ARIMA noise, Missing observaions and Ouliers) falls under he same opic as ARIMA modelling, or more exacly, i is abou an exension o hese models. Is principle is in fac o model he iniial series using he univariae approach of Box & Jenkins via ARIMA or seasonal ARIMA (SARIMA) models, while deecing, esimaing and correcing as a preliminary he ouliers, he missing values, he calendar effecs (holidays, public holidays ) as well as srucural changes, likely o disurb he esimaion of he model coefficiens. T i1

7 Seasonal Adjusmen Versus Seasonaliy Modeling: Effec on Tourism Demand Forecasing 77 SEATS program (Signal Exracion ARIMA Time Series) comes o complee TRAMO procedure by decomposing he iniial series hus modelled in is componens (rend, cycle, irregular and seasonaliy) by signal exracion, using he specral analysis of he iniial series Seasonal Differeniaion and Seasonaliy Modelling The verificaion of he exisence of seasonal uni roos using specific ess such as DHF (1984) and HEGY (1990) requires special reamen of seasonaliy. The use of he filer (1 L s ), suggesed by Box and Jenkins (1970), o differeniae he seasonal series, depends on he fac ha he variable is non-saionary a frequency 0 and a all he seasonal frequencies (Pichery and Ouerfelli, 1998). The exisence of seasonal uni roos leads o model he seasonal variaion insead of correcing or removing i using seasonal adjusmen mehods. The mos largely used seasonal model is he muliplicaive seasonal ARIMA model or SARIMA(p,d,q)(P,D,Q)S proposed by Box & Jenkins (1970) as a generalizaion of ARIMA(p,d,q) models conaining a seasonal par and which is wrien in his form: ( L) ( L s ) d D y ( L) ( L s ) (2) p P s q Q Where: S is he period of seasonaliy (S = 12 for monhly daa, S = 4 for quarerly daa); = 1 L, S = 1 L S, p, P, q, Q are polynomials of degrees: p, P, q, Q and he roos are of module higher han 1; ( ) is a whie noise; d and D are respecively he orders of non seasonal and seasonal differeniaion. 3. THE DATA The Wesern-European marke being he principal marke ransmiing ouriss owards Tunisia, he empirical applicaion is carried ou based on he series of he Wesern- European ouris arrivals in Tunisia ransformed o logarihm and subsequenly noed LTOEU. The sample covers he period from January 1997 o December For he esimaion, we use he daa beween January 1997 and June 2009, he six remaining observaions are used for he ex-pos forecas and for he predicive performance evaluaion of he various mehods. Daa are provided by he Naional office of Tunisian Tourism relaing o he minisry of ourism and rade. Table 2 Resuls of Seasonaliy Tess F S F M T Value 304,959 9,797 0,2439 Decision Presence of sable Absence of moving Presence of idenifiable seasonaliy seasonaliy seasonaliy

8 78 Amira Gasmi Table 3 Resul of Tes DHF (1984) Wesern-European ouris arrivals in Tunisia Tes saisic Level 5% Decision Filer 1,166-5,84 Accepe H 0 (1- L 12 )] Table 4 Resul of Tes HEGY (1990) Wesern-European ouris arrivals in Tunisia Frequency Tes saisic Level 5% Decision Filer o -0,795-2,76 Accepe H 0 (1- L) / 6 3,187-1,85 Accepe H 0 (1 2 3 L L ) / 3 3,21-3,25 Accepe H 0 (1 L + L 2 ) / 2 6,080-3,25 Accepe H 0 (1 + L 2 ) 2/ 3-1,783-1,85 Accepe H 0 (1+ L + L 2 ) 5/ 6 1,61-1,85 Accepe H 0 (1 2 3 L L ) -4,027-2,76 Reje de H 0-4. EMPIRICAL RESULTS Deecion of he Seasonaliy The presence of seasonal variaion noed graphically in able 1 is confirmed hanks o he resuls of he combined es which indicaes he presence of an idenifiable seasonal variaion, since he es saisic provides a value lower han 1 (see able 2). This is marked hanks o he resuls of es HEGY presened in able 4. In fac, by using he comparison of he T-saisic calculaed in he able wih he criical values provided in Beaulieu and Miron (1993), his es reveals he presence of he non-seasonal uni roo 1 corresponding o he zero frequency. This allows us o conclude of he nonsaionariy of he variable. Hence, is differeniaion wih he filer (1 L) is required. Furhermore, he es leads o he accepance of he assumpion H 0 of presence of uni roos a all he seasonal frequencies, excep for he frequency. Consequenly, he produc of he filers indicaed in able 4 mus be applied o eliminae he seasonal and non-seasonal uni roos, ha is o say: (1 L) (1 + L 2 + L 4 + L 6 + L 8 + L 10 ). Taking ha ino accoun, we can conclude ha he suiabiliy of he applicaion of he filer (1 L 12 ) o a seasonal series, as i is recommended by Box & Jenkins (1970), depends on he fac ha he series is inegraed a he seasonal frequency zero and a all frequencies. This being, hese resuls imply ha he auomaic applicaion of he filer of seasonal differeniaion is likely o produce a specificaion error. The proof presened here indicaes ha he uni roos are someimes missing a cerain seasonal frequencies, hen heir presence have o be checked by using he es HEGY, raher han o impose hem a priori a all he frequencies.

9 Seasonal Adjusmen Versus Seasonaliy Modeling: Effec on Tourism Demand Forecasing 79 However, and by conrariey of simplificaion, and aking ino accoun he exisence of only one seasonal frequency where he assumpion H 0 is rejeced, we have preferred he applicaion of he filer of seasonal differeniaion (1 L 12 ) suggesed by Box & Jenkins (1970) and recommended by he es of Dikey, Hasza and Fuller (1984) whose resul arises in able 3. Comparison of he Seasonal Adjusmen Mehods Figures 1, 2 and 3 presen he series of he Wesern-European ouris arrivals adjused by he differen seasonal adjusmen mehods considered in his sudy. We propose o compare he forecasing performance. For his purpose, we followed he forecas process of Box & Jenkins (1970) and he seps of idenificaions, esimaion and validaion enabled us o reain he following forecasing models: ARIMA (2,1,2), SARMA (1,1) (1,1,1) 12, ARMED (1,1), ARIMA (2,1,2), ARIMA (2,1,1) and ARMA(1,1) for each one of hese mehods of reamen of he seasonal variaion, respecively: he filer of seasonal differeniaion (1-L 12 ) suggesed by es DHF (forecass 1), he X-12-ARIMA mehod (forecass 2), he raio-o-moving average echnique (forecass 3), he regression on seasonal dummies (forecass 4) and he TRAMO-SEATS program (forecass 5). To compare he forecasing efficiency of hese models, we reained various crieria of evaluaion of he predicive precision, namely: he MAPE, he RMSE, he RMSPE and he U- Theil inequaliy coefficien. The reading of able 5 makes i possible o conclude ha overall (six-monhs-ahead horizon), he TRAMO-SEATS seasonal adjusmen mehod allows o obain he mos precise forecass since hey admi he weakes evaluaion crieria, followed by he seasonal model SARIMA (second rank) and he X-12-ARIMA mehod (hird rank). Therefore, modelling seasonaliy by he recourse o he SARIMA model (applicaion of he filer of seasonal differeniaion (1-L 12 )) is more advised in erms of forecasing efficiency han he seasonal adjusmen by he X-12-ARIMA, he raio-o-moving average and he regression on seasonal dummies mehods. Figure 1: Non-parameric Seasonal Adjusmen Approaches

10 80 Amira Gasmi Figure 2: Seasonal Adjusmen wih Seasonal Dummies Figure 3: Seasonal Adjusmen wih he TRAMO-SEATS Mehod

11 Seasonal Adjusmen Versus Seasonaliy Modeling: Effec on Tourism Demand Forecasing 81 Table 5 Forecasing Performance of Seasonal Adjusmen Mehods Horizons * MAPE RMSE RMSPE U-Theil one monh (2) 0, ,4837 0,7226 0,3626% Forecass 1 2 monhs (2) 2, ,6082 2,7568 1,5085% (filer (1 L 12 )) 3 monhs (3) 2, ,5156 2,7411 1,2751% 6 monhs (2) 2, ,7616 3,0800 1,4653% one monh(3) 0, ,5948 0,7527 0,3749% 2 monhs (3) 2, ,8214 2,7680 1,5184% Forecass 2 3 monhs (2) 1, ,5787 2,0260 0,8725% (X-12-ARIMA) 6 monhs (3) 4, ,9351 4,5367 2,3034% one monh(4) 2, ,4143 1,1928% Forecass 3 2 monhs (4) 3, ,102 3,7609 1,7802% (raio-o-moving 3 monhs (4) 8, ,649 8,2483 3,7769% average) 6 monhs (5) 12, ,35 14,8750 4,9032% Forecass 4 one monh(5) 4, ,842 4,6037 2,2500% (regression on seasonal) 2 monhs (5) 4, ,418 4,5867 2,0788% dummies mehods) 3 monhs (5) 9, ,210 9,3456 4,3768% 6 monhs (4) 11, ,436 13,418 4,3701% one monh(1) 0, ,3498 0, ,0284% Forecass 5 2 monhs (1) 1, ,9875 1,6724 0,8833% (Tramo-Seas) 3 monhs (1) 1, ,527 1,3353 0,4448% 6 monhs (1) 1, ,821 1,4008 0,7291% MAPE: Mean Absolue Percenage Error; RMSE: Roo Mean Square Error; RMSPE: Roo Mean Square Percenage Error. (*): Figures in blue represen he forecass order by horizon.

12 82 Amira Gasmi This order is mainained for one-monh and wo-monh-ahead horizons. On he conrary, for he hree-monh-ahead horizon, he forecass resuling from he X-12-ARIMA mehod become beer han hose obained using he SARIMA model. In consequence, he forecasing performance of he various mehods can vary according o he horizon of forecas, which corroboraes wih he resuls found in preceding sudies (Wong and al., 2007; Shen and al., 2009; Shen and al., 2011). By elsewhere, he empirical evidence suggess ha he echniques of reamen of he seasonal variaion affec he forecasing performance of he models, and ha differs according o sochasic or deerminisic naure of he seasonal variaion. In effec, he resuls obained in his empirical exercise reveal ha he bes forecass resul from he TRAMO-SEATS and he X-12-ARIMA mehods and also from he seasonal ARIMA model which consider he sochasic seasonal variaion (Bourbonnais and Terraza, 2008). 5. CONCLUSION In his paper, we applied four seasonal adjusmen mehods: wo parameric mehods (TRAMO- SEATS and regression on seasonal dummies) and wo non-parameric ones (he X-12-ARIMA and he raio-o-moving average), o a monhly series represening he Wesern-European ouris arrivals in Tunisia. We compared he forecasing performance of hese mehods in paricular, seasonal adjusmen versus seasonaliy modelling. The obained resuls miliae in favour of he TRAMO-SEATS mehod. In fac, his approach provides he bes forecas a all he forecas horizons. Always in erms of forecasing performance, we have been able o noe ha he seasonaliy modelling using seasonal ARIMA (SARIMA) models may lead o beer predicive resuls compared wih he oher echniques of seasonal adjusmen, namely: he X-12-ARIMA, he regression on seasonal dummies and he raio-o-moving average. Consequenly, i could be someimes more appropriae o model he seasonal variaion raher han o resor o is correcion or suppression by he means of seasonal adjusmen mehods. Anoher conclusion ha we could draw from he resuls is ha he forecasing performance is influenced by he manner wih which he seasonal variaion is reaed in he series, i.e. i differs according o sochasic or deerminisic naure of seasonaliy. Indeed, he empirical resuls reveal ha he bes predicive performance rises from he TRAMO-SEATS program, he X-12-ARIMA mehod and of he SARIMA model which consider he sochasic seasonaliy (Bourbonnais and Terraza, 2008). This is on line wih oher researches which sugges he sochasic reamen of he seasonal variaion (for example, Shen and Al, 2009). Acknowledgemens I would like o hank Mr. Regis Bourbonnais (Paris-Dauphine Universiy) for his consrucive commens and helpful suggesions hroughou his work.

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