Seasonal Adjustment versus Seasonality Modelling: Effect on Tourism Demand Forecasting

Transcription

1 Advances in Management & Applied Economics, vol. 3, no.4, 203, 9-32 ISSN: (print version), (online) Scienpress Ltd, 203 Seasonal Adjustment versus Seasonality Modelling: Effect on Tourism Demand Forecasting Amira Gasmi Abstract In tis study, we treat te seasonal variation in montly time series in te context of te Western-European tourism demand for Tunisia, by presenting different tecniques of detection of seasonality and te parametric and non-parametric approaces of seasonal adjustment. Ten, we compare te forecasting performance of tese metods. Te empirical results militate in favour of te TRAMO-SEATS metod. In fact, tis approac provides te best forecast. In terms of forecasting efficiency, we note in addition, tat te modelling of te seasonal variation using seasonal ARIMA model (SARIMA) may lead to better predictive results compared wit oter tecniques of seasonal adjustment used in tis researc, namely: te X-2-ARIMA, regression on seasonal dummies and te ratioto-moving average metods. JEL Classification numbers: C22, C52, C53, L83. Keywords: seasonality, tourism demand, forecasting performance, seasonal adjustment, seasonal modelling. Introduction Seasonality is a major caracteristic of te tourism activity. It reveals te influence of te seasons on te tourism demand. Tis penomenon is related to weater canges as well as institutional factors (scool olidays, professional vacation, public (Cristmas or Easter), religious and commemorative festivals). Te calendar can also generate a seasonal movement in montly time series since te number of working days varies from one mont to anoter. It is also related to certain socio-cultural caracteristics (sport practices; social or religious abits). Laboratoire d Economie et Finance Appliquées IHEC Cartage, Tunisia. Article Info: Received : February 23, 203. Revised : Marc 28, 203. Publised online : July, 203

2 20 Amira Gasmi Taking tat into account, te penomenon of ig and low seasons constitutes a problem of size wic worries te actors of te tourism field. Time series analysis aims to separate te sort-term beaviour from tat of long-term of an economic data series and to give reliable forecasts for tese separate components and for te totality of te series. Te seasonal variations explain most of te variation in te growt rates of te majority of te economic time series. In order to draw conclusions on te nature of te business cycles and te long-term growt, te traditional approac is to remove te seasonal component of a series troug te use of deterministic seasonal dummies, seasonal differentiation, or using te seasonal adjustment tecniques suc as te X-2-ARIMA metod. However, sometimes we sow tat it can be more appropriate to study te seasonal models temselves (Lee and Siklos (993), Reimers (997)) since tey could give information on te beaviour of te economic agents wic are exposed to canges of tendencies at te moment of planning and te formation of waitings. Tus, altoug seasonal variations-corrected data can be useful, it is typically recommended to use not adjusted data. Moreover, several recent empirical studies sowed tat many metods of seasonal adjustment lead to seriously denatured data, in te sense tat te key properties suc as te tendencies, te business cycles, te non-linearities are affected by te seasonal adjustment (Gysels and Perron (993), Gysels, Granger and Siklos (995), Hylleberg (994), Miron (996), Maravall (995)). In contrast, King and Kulendran (997) evaluate several models, including te seasonal unit roots model, in te forecast of quarterly tourist arrivals in Australia coming from many countries. Teir principal conclusion is tat compared to time series models, te forecasting performance of te seasonal unit roots models is weak. Tis may be due to te lack of power of some unit roots tests. On te oter and, Paap, Franses and Hoek (997) use empirical and simulation examples to demonstrate tat te neglected seasonal average canges can destroy considerably te forecasting performance of te univariate autoregressive processes. Tus, appropriate treatment of seasonality is important to make reliable forecasts. In tis article, we propose firstly, to study te seasonal aspect strongly caracterizing te tourism time series, by presenting different tests of seasonality detection and various metods of treatment of seasonality, in particular seasonal adjustment metods versus seasonality modelling. Ten, secondly, we will compare te forecasting performance of tese metods. 2 Analysis of Seasonality 2. Detection of Seasonality During te analysis of a time series, it is necessary to identify te seasonal variation wic can be probably observed. Various types of tests are set up to detect te presence of tis component.

3 Seasonal Adjustment versus Seasonality Modelling Autocorrelations Seasonality can be detected grapically by examining autocorrelation (ACF) and partial autocorrelation functions (PACF) necessary for te identification of suitable ARIMA models. Indeed, te correlogram of a seasonal series often takes a sinusoidal form (see table ) Traditional tests of presence of seasonality To test te presence of seasonal variation, a multiplicity of tests were suggested, namely: te stable seasonality and moving seasonality tests wic are Fiser types tests based on models of analysis of te variance to one (te mont or te quarter) and two factors (te mont or te quarter and te year), respectively. Indeed, stable seasonality is a type of seasonality wic is repeated at te same time eac year, and tis stable aspect facilitates te forecasts. Wile te moving seasonality is represented by a movement effect from one mont to anoter. Tis lack of stability makes its forecast difficult. Lastly, we distinguis te identifiable seasonality test completing te tests evoked above. It is built starting from te values of Fiser statistics of stable and moving seasonality tests (Lotian and Morry, 978). Te test statistic, noted T, is expressed as follows: T T2 / 2 7 3FM T ( ) wit T and T2. 2 FS FS If statistic T is lower tan so we concludes te presence of identifiable seasonal component and te seasonal adjustment of te series is ten necessary. Table : Correlogram of Western-European tourist arrivals series

4 22 Amira Gasmi 2..3 Seasonal unit roots tests Te detection of seasonal variation can be done using seasonal unit roots tests. For tis purpose, a certain number of tests were implemented in te eigties and nineties, in particular to test te seasonal variation at order 4 and order 2. Test DHF (Dickey, Hasza and Fuller, 984): it allows to test te null assumption in te model xt xt s t. Under H 0 true, te series is seasonal and te filter ( s s L ) suggested by Box & Jenkins (970) is appropriate to adjust it (S being te period of seasonality). Test HEGY (Hylleberg, Engle, Granger and Yoo, 990): te literature on te concept of units roots (e.g., Dickey, Bell and Miller (986)) sows tat te assumption of existence of certain filters of differentiation amounts to emit te assumption of presence of a certain number of seasonal and non-seasonal units roots in a time series. Tis can be easily seen by writing: ( s s L ), and by solving te equation: ( z ) 0. s Te general solution to tis equation is:,cos(2 k / S) isin(2 k / S) ; wit te term (2 k/ s) for k =,2,, represents te corresponding seasonal frequency, giving S different solutions wic all of tem are on te circle unit. Te HEGY metod, mainly conceived for te quarterly series, was adapted to te montly case tanks to Franses (990) and of Beaulieu and Miron (993) works. In fact, if S = 2, solutions of te 2 equation ( z ) 0 are: for te non-seasonal unit root corresponding to frequency 0; and seasonal unit roots, i, ( 3 i), ( 3 i), ( 3 i), ( 3 i) corresponding respectively to te following frequencies:,,,,, and to te operators of differentiation: (+ L), (+ L 2 ), (+ L+ L 2 ), ( L+ L 2 ), (+ 3 L+ L 2 ) and ( - 3 L+ L 2 ). Tus, a filter of differentiation ( s ) can be written as: ( )( s L L L ), and can tus be decomposed into a part wit non-seasonal unit root and a part wit (S - ) seasonal unit roots. In tis test, we resort to te decomposition of te polynomial (-L 2 ), wit 2 roots units and we consider te following form: ( L) z8t t z, t 2z2, t 3z3, t 4z3, t 2 5z4, t 6z4, t 2 7z5, t 8z5, t 2 z z z z () 9 6, t 0 6, t 2 7, t 2 7, t 2 t Te variables z it are in suc a way tat: zit Pi ( L) yt, were polynomials P i are defined as follows 2 : 2 Artur Carpenter (2003).

5 Seasonal Adjustment versus Seasonality Modelling 23 P L L L L L ( ) ( )( )( ) P L L L L L ( ) ( )( )( ) P L L L L ( ) ( )( ) P L L L L L L ( ) ( )( 3 )( ) P L L L L L L ( ) ( )( 3 )( ) P L L L L L L ( ) ( )( )( ) P L L L L L L ( ) ( )( )( ) 2 P ( L) ( L ) 8 Wit also: ( L) is an autoregressive polynomial in L, and t may contain a constant, seasonal dummies and/or a trend. Te variables z it are ten associated to te different roots of te polynomial. Te equation () is estimated using least squares ordinary metod. We may carry out t tests for te parameters and 2, and F tests associated to te couples ( 3, 4),( 5, 6),( 7, 8),( 9, 0) and (, 2) : it is a question of testing te joined significance of te coefficients. Tis amounts to test te assumption of existence of unit roots at te different frequencies. For tis purpose, we must compare te test statistics related to te estimated parameters wit te critical values provided by Franses (990) and Beaulieu and Miron (993). To ceck te existence of te roots and - corresponding to frequencies 0 and respectively, we carry out two individual tests on parameters and 2. As for te oter seasonal unit roots, we can perform eiter joined tests wose null assumption takes te form k k 0 and tis for te even values of k, from 4 to 2; or quite simply, individual tests, suggested in Franses (990), allowing to verify te non-stationarity of te time series at all te seasonal frequencies and tis by testing te null assumption according to wic tere is a seasonal unit root ( k 0, k 3,2 ). However, it sould be noted tat te application of te OLS to te regression () is made were te order of ( L) is given in suc a way tat te errors are rougly wite noises, or at least, nonautocorrelated residuals. For tis purpose, Hylleberg and al. (990) and Engle and al. (993) propose to introduce additional lags of te variable until we obtain nonautocorrelated residuals. 2.2 Seasonal Adjustment Metods Seasonal adjustment metods can be classified in two categories, namely: parametric approac and nonparametric approac 3. 3 Bourbonnais and Terraza (2008) propose anoter classification according to te nature of te seasonal variation wic can be is flexible (stocastic: random in amplitude and/or period), tat is to say rigid (determinist: marked well and repetitive).

6 24 Amira Gasmi 2.2. Non-parametric approaces Te X-2-ARIMA metod: wen te seasonal variation is very apparent in te time series, a first approac consists in removing suc seasonal fluctuations by using a seasonal adjustment programs. Tey are tecniques allowing te identification of te different components of te initial series (trend-cycle, seasonality, irregular) by applying linear filters, wic cancels or preserves a well defined component (tendency-cycle or seasonal variation). Te irregular one is represented tereafter by te residual of te decomposition. Tese linear filters are moving averages wic constitute te principal tool of Census X- metod built from successive iterations of moving averages of different natures for better estimating te series components. However, tis tecnique leads to a loss of information in te final end of te series. Tis gap is filled by te forecast of future values of te time series before its seasonal adjustment, and tis using ARIMA models. It is wat made it possible to extend te X- tecnique (Census Bureau, 967) to X--ARIMA (Dagum, 988) and ten to X-2- ARIMA (Findlay and Al, 998). Te latter contains te RegARIMA module wic allows to detect and to remove any undesirable effect of te series (outliers, calendar effects ). Te ratio to moving average metod: Montly values of te studied series (X t ) are divided by te moving average figure corresponding for eac mont (MA t ), and expressed in % to generate te ratio-to-moving average: X t M ratio 00 MAt Te moving average is calculated as follows: M X 2 X... 2X 2X 2 X... 2 X X, m 2 p m 2m t p t p t t t t p t p MAt M2( Xt ) X t 6 2X t 5 2 X t X t 5 X t 6 24 A. Carpenter (2003) reveals tat tis moving average eliminates seasonal variations from montly series, preserves linear trends and reduces of more tan 90% te variance of a wite noise. Tese ratios are weigted by te mont and tereafter will separate te seasonal and cyclic components Parametric approaces Te regression metod: tis approac is based on te Buys-Ballot model (847) wic consists in carrying out te regression below, using seasonal dummies ( S ti, ) in suc a way tat Sti, takes value if T corresponds to te seasonal period, and 0 if not. Te model is written as follows: T X t S. t 0 i t, i t i Wit: T being te period of seasonality (T = 4 for a quarterly series, T = 2 for montly data). Te use of only (T - ) dummies makes it possible to avoid te problem of

7 Seasonal Adjustment versus Seasonality Modelling 25 colinearity wic could exist wit te vector unit relating to te constant 4. We estimate tus (T - ) seasonal coefficients and we ceck te T t using te principle of conservation of te surfaces 5 : i 0. T i Seasonal adjustment by metod TRAMO-SEATS: TRAMO-SEATS program (Gomez & Maravall, 996) belongs to te parametric seasonal adjustment metods based on te signal extraction. It is composed of two independent subroutines but wic are complementary since tey are generally used togeter: - TRAMO program (Time series Regression wit ARIMA noise, Missing observations and Outliers) falls under te same optic as ARIMA modelling, or more exactly, it is about an extension to tese models. Its principle is in fact to model te initial series using te univariate approac of Box & Jenkins via ARIMA or seasonal ARIMA (SARIMA) models, wile detecting, estimating and correcting as a preliminary te outliers, te missing values, te calendar effects (olidays, public olidays ) as well as structural canges, likely to disturb te estimation of te model coefficients. - SEATS program (Signal Extraction ARIMA Time Series) comes to complete TRAMO procedure by decomposing te initial series tus modelled in its components (trend, cycle, irregular and seasonality) by signal extraction, using te spectral analysis of te initial series. 2.3 Seasonal Differentiation and Seasonality Modelling Te verification of te existence of seasonal unit roots using specific tests suc as DHF (984) and HEGY (990) requires special treatment of seasonality. Te use of te filter ( L s ), suggested by Box and Jenkins (970), to differentiate te seasonal series, depends on te fact tat te variable is non-stationary at frequency 0 and at all te seasonal frequencies (Picery and Ouerfelli, 998). Te existence of seasonal unit roots leads to model te seasonal variation instead of correcting or removing it using seasonal adjustment metods. Te most largely used seasonal model is te multiplicative seasonal ARIMA model or SARIMA(p,d,q)(P,D,Q)S proposed by Box & Jenkins (970) as a generalization of ARIMA(p,d,q) models containing a seasonal part and wic is written in tis form: ( L) ( L s ) d D y ( L) ( L s ) (2) p P s t q Q t Were: S is te period of seasonality (S = 2 for montly data, S = 4 for quarterly data); s L, L,,,, are polynomials of degrees: p, P, q, Q and te s p P q Q roots are of module iger tan ; (ε t ) is a wite noise ; d and D are respectively te orders of non seasonal and seasonal differentiation. 4 We can also consider T dicotomist variables in te model and remove te constant. 5 R. Bourbonnais & Mr. Terraza (2008).

8 26 Amira Gasmi 3 Te Data Te Western-European market being te principal market transmitting tourists towards Tunisia, te empirical application is carried out based on te series of te Western- European tourist arrivals in Tunisia transformed to logaritm and subsequently noted LTOEU. Te sample covers te period from January 997 to December For te estimation, we use te data between January 997 and June 2009, te six remaining observations are used for te ex-post forecast and for te predictive performance evaluation of te various metods. Data are provided by te National office of Tunisian Tourism relating to te ministry of tourism and trade. Table 2 : Results of Seasonality tests F S F M T Value 304,959 9,797 0,2439 Decision Presence of stable seasonality Absence of moving seasonality Table 3 : Result of test DHF (984) Western-European tourist arrivals in Tunisia Test statistic Level 5% Decision Filter,66-5,84 Accepte H 0 (- L 2 ) Presence of identifiable seasonality Table 4 : Result of test HEGY (990) Western-European tourist arrivals in Tunisia Frequency Test statistic Level 5% Decision Filter 0-0,795-2,76 Accepte H 0 (- L) Π / 6 3,87 -,85 Accepte H 0 2 (- 3L + L ) Π / 3 3,2-3,25 Accepte H 0 (- L + L 2 ) Π / 2 6,080-3,25 Accepte H 0 (+ L 2 ) 2Π / 3 -,783 -,85 Accepte H 0 (+ L + L 2 ) 5Π / 6,6 -,85 Accepte H 0 2 (+ 3L + L ) Π -4,027-2,76 Rejet de H 0-4 Empirical Results Detection of te seasonality: Te presence of seasonal variation noted grapically in table is confirmed tanks to te results of te combined test wic indicates te presence of an identifiable seasonal variation, since te test statistic provides a value lower tan (see table 2). Tis is marked tanks to te results of test HEGY presented in table 4. In fact, by using te comparison of te T-statistic calculated in te table wit te critical values provided in Beaulieu and Miron (993), tis test reveals te presence of te nonseasonal unit root corresponding to te zero frequency. Tis allows us to conclude of

9 Seasonal Adjustment versus Seasonality Modelling 27 te non-stationarity of te variable. Hence, its differentiation wit te filter ( - L) is required. Furtermore, te test leads to te acceptance of te assumption H 0 of presence of unit roots at all te seasonal frequencies, except for te frequency. Consequently, te product of te filters indicated in table 4 must be applied to eliminate te seasonal and non-seasonal unit roots, tat is to say: ( L)( L L L L L ). Taking tat into account, we can conclude tat te suitability of te application of te 2 filter ( L ) to a seasonal series, as it is recommended by Box & Jenkins (970), depends on te fact tat te series is integrated at te seasonal frequency zero and at all frequencies. Tis being, tese results imply tat te automatic application of te filter of seasonal differentiation is likely to produce a specification error. Te proof presented ere indicates tat te unit roots are sometimes missing at certain seasonal frequencies, ten teir presence ave to be cecked by using te test HEGY, rater tan to impose tem a priori at all te frequencies. However, and by contrariety of simplification, and taking into account te existence of only one seasonal frequency were te assumption H 0 is rejected, we ave preferred te 2 application of te filter of seasonal differentiation ( L ) suggested by Box & Jenkins (970) and recommended by te test of Dikey, Hasza and Fuller (984) wose result arises in table 3. Comparison of te seasonal adjustment metods: Figures, 2 and 3 present te series of te Western-European tourist arrivals adjusted by te different seasonal adjustment metods considered in tis study. We propose to compare te forecasting performance. For tis purpose, we followed te forecast process of Box & Jenkins (970) and te steps of identifications, estimation and validation enabled us to retain te following forecasting models: ARIMA (2,,2), SARMA (,) (,,) 2, ARMED (,), ARIMA (2,,2), ARIMA (2,,) and ARMA(,) for eac one of tese metods of treatment of te seasonal variation, respectively: te filter of seasonal differentiation (-L 2 ) suggested by test DHF (forecasts ), te X-2-ARIMA metod (forecasts 2), te ratio-to-moving average tecnique (forecasts 3), te regression on seasonal dummies (forecasts 4) and te TRAMO-SEATS program (forecasts 5). To compare te forecasting efficiency of tese models, we retained various criteria of evaluation of te predictive precision, namely: te MAPE, te RMSE, te RMSPE and te U-Teil inequality coefficient. Te reading of table 5 makes it possible to conclude tat overall (six-monts-aead orizon), te TRAMO-SEATS seasonal adjustment metod allows to obtain te most precise forecasts since tey admit te weakest evaluation criteria, followed by te seasonal model SARIMA (second rank) and te X-2-ARIMA metod (tird rank). Terefore, modelling seasonality by te recourse to te SARIMA model (application of te filter of seasonal differentiation (-L 2 )) is more advised in terms of forecasting efficiency tan te seasonal adjustment by te X-2-ARIMA, te ratio-to-moving average and te regression on seasonal dummies metods.

10 28 Amira Gasmi la série brute la série désaisonnalisée par X-2-ARIMA la série désaisonnalisée par le ratio aux moyennes mobiles Figure : Non-parametric seasonal adjustment approaces L T O E U L T O E U _ I N D Figure 2: Seasonal adjustment wit seasonal dummies LTOEU (série brute) LTOEU_aj (série ajustée) Figure 3: Seasonal adjustment wit te TRAMO-SEATS metod Tis order is maintained for one-mont and two-mont-aead orizons. On te contrary, for te tree-mont-aead orizon, te forecasts resulting from te X-2-ARIMA metod become better tan tose obtained using te SARIMA model. In consequence, te

11 Seasonal Adjustment versus Seasonality Modelling 29 forecasting performance of te various metods can vary according to te orizon of forecast, wic corroborates wit te results found in preceding studies (Wong and al., 2007; Sen and al., 2009; Sen and al., 20). By elsewere, te empirical evidence suggests tat te tecniques of treatment of te seasonal variation affect te forecasting performance of te models, and tat differs according to stocastic or deterministic nature of te seasonal variation. In effect, te results obtained in tis empirical exercise reveal tat te best forecasts result from te TRAMO-SEATS and te X-2-ARIMA metods and also from te seasonal ARIMA model wic consider te stocastic seasonal variation (Bourbonnais and Terraza, 2008). Table 5: Forecasting performance of seasonal adjustment Metods Horizons * MAPE RMSE RMSPE U-Teil Forecasts (filter (- L 2 )) Forecasts 2 (X-2- ARIMA) Forecasts 3 (ratio-tomoving average) Forecasts 4 (regression on seasonal dummies metods) Forecasts 5 (TRAMO- SEATS) ˆ 2 X t X t X t ( Xˆ t Xt) t t t (( Xˆ X ) / X ) 2 t t t one mont (2) 0, ,4837 0,7226 0,3626% 2 monts (2) 2, ,6082 2,7568,5085% 3 monts (3) 2, ,556 2,74,275% 6 monts (2) 2,84 956,766 3,0800,4653% one mont (3) 0, ,5948 0,7527 0,3749% 2 monts (3) 2, ,824 2,7680,584% 3 monts (2), ,5787 2,0260 0,8725% 6 monts (3) 4, ,935 4,5367 2,3034% one mont (4) 2, ,443,928% 2 monts (4) 3, ,02 3,7609,7802% 3 monts (4) 8, ,649 8,2483 3,7769% 6 monts (5) 2, ,35 4,8750 4,9032% one mont (5) 4, ,842 4,6037 2,2500% 2 monts (5) 4, ,48 4,5867 2,0788% 3 monts (5) 9, ,20 9,3456 4,3768% 6 monts (4), ,436 3,48 4,370% one mont () 0, ,3498 0, ,0284% 2 monts (),69 229,9875,6724 0,8833% 3 monts (), ,527,3353 0,4448% 6 monts (), ,82,4008 0,729% t ( Xˆ X ) t 2 t ˆ 2 2 t t t t X X MAPE: Mean Absolute Percentage Error; RMSE: Root Mean Square Error; RMSPE: Root Mean Square Percentage Error. (*): Figures in brackets represent te forecasts order by orizon. 5 Conclusion In tis paper, we applied four seasonal adjustment metods: two parametric metods (TRAMO-SEATS and regression on seasonal dummies) and two non-parametric ones (te X-2-ARIMA and te ratio-to-moving average), to a montly series representing te Western-European tourist arrivals in Tunisia. We compared te forecasting performance of tese metods in particular, seasonal adjustment versus seasonality modelling. Te obtained results militate in favour of te TRAMO-SEATS metod. In fact, tis approac provides te best forecast at all te forecast orizons. Always in terms of forecasting performance, we ave been able to note tat te seasonality modelling using seasonal ARIMA (SARIMA) models may lead to better predictive results compared wit te oter tecniques of seasonal adjustment, namely: te X-2-ARIMA, te regression on seasonal dummies and te ratio-to-moving average.

12 30 Amira Gasmi Consequently, it could be sometimes more appropriate to model te seasonal variation rater tan to resort to its correction or suppression by te means of seasonal adjustment metods. Anoter conclusion tat we could draw from te results is tat te forecasting performance is influenced by te manner wit wic te seasonal variation is treated in te series, i.e. it differs according to stocastic or deterministic nature of seasonality. Indeed, te empirical results reveal tat te best predictive performance rises from te TRAMO-SEATS program, te X-2-ARIMA metod and of te SARIMA model wic consider te stocastic seasonality (Bourbonnais and Terraza, 2008). Tis is on line wit oter researces wic suggest te stocastic treatment of te seasonal variation (for example, Sen and Al, 2009). ACKNOWLEDGEMENTS: I would like to tank Mr. Regis Bourbonnais (Paris- Daupine University) for is constructive comments and elpful suggestions trougout tis work. References [] Beaulieu, J.J. and Miron, J.A., Te seasonal cycle in U.S. manufacturing, Economics Letters, 37, (99), 5 8. [2] Beaulieu, J.J. and Miron, J.A., Seasonal unit roots in aggregate U.S. data, Journal of Econometrics, 55, (993), [3] Bell, W.R. and Hillmer, S.C., Issues Involved wit te seasonal adjustment of economic time series. Journal of Business and Economic Statistics, 2, (984), [4] Bourbonnais, R. and Terraza, M., Analyse des séries temporelles, Dunod, [5] Box, G.E.P. and Jenkins, G.M., Time Series Analysis, Forecasting and Control, San Francisco: Holden-Day, 976. [6] Carpentier, A., Introduction à la téorie des processus en temps discret. Modèles ARIMA et Métode Box-Jenkins, Université Paris Daupine, [7] Clements, M.P. and Hendry, D.F., An empirical study of unit roots in forecasting. International Journal of Forecasting, 3, (997), [8] Dagum, E.B., Te XARIMA/88 Seasonal Adjustment Metod - Foundations and User s Manual, Time Series Researc and Analysis Division, Statistics Canada tecnical report, (988). [9] Darné, O. and Terraza, M., La désaisonnalisation des séries temporelles viticoles, Version préliminaire, Colloque d oenométrie, (2002). [0] Darné, O., Litago, J. and Terraza, M., Tests de racines unitaires saisonnières pour des données journalières, Revue de Statistique appliquée, 50, (2002), 7 9. [] Dickey, D.A., D.P. Hasza, and Fuller, W.A., Testing for unit roots in seasonal time series, Journal of te American Statistical Association, 79, (984), [2] Dickey, D.A., Bell, W.R. and Miller, R.B., Unit Roots in Time Series Models: Tests and Implications, Te American Statistician, 40, (986), [3] Engle, R.F., Granger, C.W.J. and Hallman, J.J., Merging sort- and long-run forecast. An application of seasonal cointegration to montly electricity sales forecasting, Journal of Econometrics, 40, (989),

13 Seasonal Adjustment versus Seasonality Modelling 3 [4] Engle, R.F., Granger, C.W.J., Hylleberg, S. and Lee, H.S., Seasonal cointegration: te Japanese consumption function, Journal of Econometrics, 55, (993), [5] Findley, D.F., B.C. Monsell, W.R. Bell, Otto, M.C. and Cen, B.C., New Capabilities of te X-2-ARIMA Seasonal Adjustment Program (wit Discussion), Journal of Business and Economic Statistics, 6, (998), [6] Franses, P.H., Seasonality, nonstationarity and te forecasting of montly time series, International Journal of Forecasting, 7, (99), [7] Franses, P.H., Recent advances in modelling seasonality, Journal of Economic Surveys, 0, (996), [8] Franses, P.H. and Hobijn, B., Critical values for unit root tests in seasonal time series, Journal of Applied Statistics, 24, (997), [9] Gysels, E., On te economics and econometrics of seasonality. In: Sims, C.A. (Ed.), Advances in Econometrics, Sixt World Congress of te Econometric Society, Cambridge University Press, Cambridge, (994). [20] Gysels, E., L analyse économétrique et la saisonnalité, L actualité économique, 70, (994), [2] Gysels, E., Lee, H.S. and No, J., Testing for unit roots in seasonal time series, Journal of Econometrics, 62, (994), [22] Gomez, V. and Maravall. A., Programs TRAMO (Time Series Regression wit Arima Noise, Missing Observations, and Outliers) and SEATS (Signal Extraction in Arima Time Series). Instructions for te User, Document de travail 9628 du Département de recerce de la Banque d Espagne, (996). [23] Gooijer, J.G. and Franses, P.H., Forecasting and seasonality, International Journal of Forecasting, 3, (997), [24] Hasza, D.P. and Fuller, W. A., Testing for non-stationary parameter specifications in seasonal time series models, Annals of Statistics, 0, (982), [25] Herwartz, H., Performance of periodic error correction models in forecasting consumption data, International Journal of Forecasting, 3, (997), [26] Hylleberg, S., Engle, R.F., Granger, C.W.J. and Yoo, B.S., Seasonal integration and cointegration, Journal of Econometrics, 44, (990), [27] Hylleberg, S., Modelling seasonal variation. In: Hargreaves, C.P. (Ed.), Nonstationary Time Series Analysis and Cointegration, Oxford University Press, Oxford, 994. [28] Hylleberg, S. and Pagan, A.R., Seasonal integration and te evolving seasonal model, International Journal of Forecasting, 3, (997), [29] Lee, H.S., Maximum likeliood inference on cointegration and seasonal cointegration, Journal of Econometrics, 54, (992), [30] Lee, H.S. and Siklos, P.L., Te influence of seasonal adjustment on te Canadian consumption function, , Journal of Economics, 26, (993), [3] Lotian, J. and Morry, M., A test for te presence of identifiable seasonality wen using te X- program, Researc paper E, Seasonal Adjustment and Time Series Staff, Statistics Canada, (978). [32] Miron, J.A., Te Economics of Seasonal Cycles, MIT Press, Cambridge USA, 996. [33] Novales, A. and Flores, R., Forecasting wit periodic models. A comparison wit time invariant coefficient models, International Journal of Forecasting, 3, (997), [34] Paap, R., Franses, P.H. and Hoek, H., Mean Sifts, Unit Roots, and Forecasting Seasonal Time Series, International Journal of Forecasting, 3, (997),

14 32 Amira Gasmi [35] Picery, M.C. and Ouerfelli, C., Juillet, La non stationnarité dans les séries saisonnières. Application au tourisme tunisien, Université de Bourgogne, Dijon, (July, 998). [36] Reimers, H.E., Forecasting of seasonal cointegrated processes, International Journal of Forecasting, 3, (997), [37] Sen, S., Li, G., and Song, H., Effect of seasonality treatment on te forecasting performance of tourism demand models, Tourism Economics, 5(4), (2009), [38] Sen, S., Li, G., and Song, H., Combination forecasts of international tourism demand, Annals of Tourism Researc, 38(), (20), [39] Siskin, J., Young, A.H. and Musgrave, J.C., Te X- variant of te census metod II seasonal adjustment, Tecnical Paper No. 5, Bureau of te Census, U.S. Department of Commerce, (967). [40] Wong, K. K. F., Song, H., Witt, S.F., and Wu, D. C., Tourism forecasting: to combine or not to combine?, Tourism Management, 28, (2007),