New innovative 3-way ANOVA a-priori test for direct vs. indirect approach in Seasonal Adjustment

Transcription

1 ISSN EWP 0 /00 Euroindicaors working papers New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen Dario Buono Enrico Infane

2 This paper was presened a he, held in Luxembourg, h - h Sepember 0. Click here for accessing he full collecion of Euroindicaors working papers Europe Direc is a service o help you find answers o your quesions abou he European Union Freephone number (*): (*) Cerain mobile elephone operaors do no allow access o numbers or hese calls may be billed. More informaion on he European Union is available on he Inerne (hp://europa.eu). Luxembourg: Publicaions Office of he European Union, 0 ISSN EWP 0 /00 Doi: 0.90/ Theme: General and regional saisics European Union, 0 Reproducion is auhorised provided he source is acknowledged. The views expressed in his publicaion are hose of he auhors, and do no necessarily reflec he official posiion of Eurosa.

3 New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen ( ) Dario Buono ( ), Enrico Infane ( 3 ) Subjec area: JEL classificaion codes: Keywords: The difference beween direc and indirec approach in seasonal adjusmen C3, C43 Seasonal adjusmen, direc vs. indirec approach, moving seasonaliy, Demera+ Table of conens Absrac Inroducion.... A 3-way ANOVA based es for direc vs. indirec approach Case sudy Simulaions Fuure research line References... 9 Absrac The Seasonal Adjused Series of an aggregae series can be calculaed by a seasonal adjusmen of he aggregae ("Direc Approach") or by aggregaing he seasonally adjused individual series ("Indirec Approach"). A hird way, he so-called "Mixed Approach", is o adjus ogeher hose series ha presen he same kind of seasonaliy, and hen re-aggregae hem for he final aggregae series. In order o undersand which series demonsrae he same kind of seasonaliy, a es based on a hree way ANOVA model is uilised. This es can be seen as a generalisaion of he Moving Seasonaliy Tes for more han one series. Addiionally, an exemplary case sudy, ogeher wih a simulaion sudy, is here presened. An auomaic procedure o choose he correc approach before obaining resuls would be helpful. The lieraure o dae has only focused upon a comparison beween he resuls achieved by he differen approaches. As such, his paper seeks o se ou an a-priori sraegy for he developmen of an effecive seasonal adjusmen of he aggregae. ( ) The views and he opinions expressed in his paper are solely of he auhors and do no necessarily reflec hose of he insiuions for which hey work. ( ) Dario Buono works as Saisical Officer a Eurosa in he uni B Qualiy, Mehodology and Research, where he deals wih seasonal adjusmen issues. Dario.BUONO@ec.europa.eu. ( 3 ) Enrico Infane works as inerim a Eurosa in he uni B Qualiy, Mehodology and Research. Enrico.INFANTE@ex.ec.europa.eu. New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen

4 . Inroducion A generic ime series Y can be he resul of an aggregaion of p series: ( X,, X, X ) Y = f, j p In mos of he common cases f is a linear (addiive or muliplicaive) funcion. This paper focuses on he case of addiive (evenually weighed) funcion, which can be generalized as follows: Y = ω p X + + ω j X j + + ω p X p = ω j j = X j As a reference example of his kind of aggregae, le s consider he European Union GDP, obained as he sum of GDPs of he 7 EU counries. To obain seasonally adjused figures, a leas hree differen approaches 4 can be applied: Direc Approach: he seasonally adjused daa are compued direcly by seasonally adjusing he aggregae Y. Indirec Approach: he seasonally adjused daa are compued indirecly by seasonally adjusing daa per each series X j. The seasonally adjused Y is hen given by he sum of he seasonally adjused componens. Mixed Approach: if i is possible o define a crierion in order o separae he series in groups, creaing sub-aggregaes (e.g. hese series have common seasonal paerns), hen i is possible o compue he seasonally adjused figures by summing he seasonal adjused daa of hese subaggregaes. The direc and he indirec approach have been discussed for many years, and here is no consensus on which is he bes approach. Usually if he series X j do no have similar paerns, indirec adjusmen is preferred. Oherwise if he series have common seasonal paerns and approximaely he same iming in heir peaks and roughs, he direc approach is preferred; in his case he aggregaion will produce a smooher series wih no loss of informaion on he seasonal paern. According o he ESS guidelines on Seasonal Adjusmen, direc approach is preferred for ransparency and accuracy, while indirec approach is preferred for consisency. The main issue wih he direc approach is he so called cancel-ou effec, ha is, if here are wo series wih opposie paerns of seasonaliy, hen he aggregaed series will possibly show no seasonaliy, e.g. he aggregaed series can show no seasonaliy even if all he sub-series have seasonaliy. An addiional ( 4 ) See also ESS guidelines on Seasonal Adjusmen, hp://ec.europa.eu/eurosa/produc?code=ks-ra &mode=view. New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen

5 issue o be considered wih he direc approach is ha here is no accouning consisency beween he aggregaes and sub-series. From a differen poin of view, he indirec approach also has some issues. Firs of all he presence of residual seasonaliy should always be carefully checked in all of he indirecly seasonally adjused aggregaes. Then, working wih an indirec approach means working wih a large number of series, and in ha way he amoun of calculaion could be quie large. The numerical resuls obained by performing he differen approaches are usually close in erms of medium and long erm evoluion (bu hey can sill presen some problems in erms of signs of he growh raes in he shor erm period), and hey may coincide if he aggregae is an algebraic sum, he decomposiion model is addiive, here are no ouliers and he filer used is he same for all he series. These condiions are rarely me in realiy. This paper focuses on he mixed approach, and proposes a new innovaive es in order o idenify which series can be aggregaed in groups and define a which level he seasonal adjusmen procedure should be run. The principal advanage of his es is ha i gives informaion abou he approach o follow before seasonally adjusing he series, so i can be considered as an a priori mehod o choose he correc approach.. A 3-way ANOVA based es for direc vs. indirec approach The classic es for moving seasonaliy is based on a wo-way ANOVA es, where he wo facors are he ime frequency (usually monhs or quarers) and he years. In order o es he presence of a moving seasonaliy beween differen series (and no beween he years of he same series, as he classic moving seasonaliy es) a es based on a hree way ANOVA model will be used. Here he hree facors are he ime frequency, he years and he series. The variable esed is he final esimaion of he unmodified Seasonal-Irregular (SI) differences 5 absolue value, if he decomposiion model is an addiive one; or he SI raio minus one absolue value, if he decomposiion model is a muliplicaive one. Wihou lack of generaliy, he procedure wih an addiive decomposiion model will be presened. For he procedure wih a muliplicaive decomposiion model jus subsiue he SI ijk wih SI ijk -. I is imporan o noe ha if a leas one series is modelled wih a nonaddiive decomposiion, hen daa mus be sandardized by he column, as o say by ime frequency facor. The model is specified as follows: SI = a + b + c + e ijk i j k ijk ( 5 ) Using he ool X- he series of SI raio is presened in he able D8. New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 3

6 This equaion implies ha he value SI ijk represens he sum of: A erm a i, i =,, M, represening he numerical conribuion due o he effec of he i-h ime frequency (usually M=, for monhly series, or M=4, for quarerly series). A erm b j, j =,, N, represening he numerical conribuion due o he effec of he j-h year. A erm c k, k =,, S, represening he numerical conribuion due o he effec of he k-h series of he aggregae. A residual componen erm e ijk, assumed o be normally disribued wih zero mean, consan variance and zero covariance. I represens he effec on he values of he Seasonal-Irregular raios of he whole se of facors no explicily aken ino accoun in he model. The es is based on he decomposiion of he variance of he observaions: S = SM + SN + SS + S R Denoing x = MNS M N S SI ijk i= j = k = he general mean, x N S i = SI ijk NS j= k= he ime frequency means, x M S j = SI ijk MS i= k = he yearly means and x M N k = SI ijk MN i= j= he series means, han i is possible o wrie: NS M SM = ( xi x) M i= measures he magniude of he seasonaliy. N MS SN = ( x j x) N j = movemen of he seasonaliy in he same series. MN S SS = ( x k x) S k = S movemen of he seasonaliy beween differen series. R = ( M )( N )( S ) variance. So he null hypohesis is he following: is he beween ime frequencies variance. I is he effec ha is he beween years variance. I is he effec ha measures he is he beween series variance. I is he effec ha measures he M N S ( SIijk xi x j x k + x) i= j = k = is he residual H c = c = = : 0 c S New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 4

7 Tha means ha here is no change in seasonaliy over he series. The es saisics is: F = T S S S R If he null hypohesis is rue, he es saisics follows a Fisher-Snedecor disribuion wih (S-) and (M- )(N-)(S-) degrees of freedom. Rejecing he null hypohesis is o say ha he pure direc approach should be avoided, and an indirec or a mixed one should be aken in consideraion. Unil now, his es is jus able o say if a direc approach is a good choice or no. In order o go for a mixed approach, if he es is rejeced, i is useful o know which S S series do no presen similar seasonal paerns. In order o avoid all he ess o compare which k = k series are he ones wih differen paerns of seasonaliy, a Tukey's range es could be used. 3. Case sudy Here a simple case sudy is presened, considering he Consrucion Producion Index of he hree Frenchspeaking European counries: France, Belgium and Luxembourg (daa are available on he EUROSTAT daabase). The ime span is from January 00 o December 00. The following is an example of a very simple aggregae: Y = FR + BE + LU Considering all he hree series, he resuls of he es are summarized in he following able: VAR Mean Square df Monhs.5003 Years Series Residual BE FR LU F-Raio = 5.8, P-Value = New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 5

8 Therefore he F-Raio is equal o 5.8 and consequenly he P-Value is equal o Thus here is no evidence of common seasonal paerns beween he series a 5 per cen level and he Direc Approach should be avoided. However, if wo of hese counries have he same seasonal paern, a Mixed Approach could be used. The es is now used for each couple of series, giving he following resuls: VAR Mean Square df Monhs.0403 Years Series 0.99 Residual FR LU F-Raio = , P-Value = VAR Mean Square df Monhs.0464 Years Series Residual FR BE F-Raio = 4.833, P-Value = VAR Mean Square df Monhs Years Series Residual LU BE F-Raio = 0.34, P-Value = New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 6

9 So he es highlighs ha common seasonal paerns beween Luxembourg and Belgium is presen a 5 per cen level. As hey have he same seasonal paerns, i is possible o Seasonally Adjus hem ogeher by using a Mixed Approach: ( Y ) = SA( FR ) + SA( BE LU ) SA + 4. Simulaions The purpose of he simulaions is o check if he es is srong enough o be used sysemaically in a large organizaion such as Eurosa. Amongs he oher possibiliies, he mos common case will be presened. Of course he analysis could be exended o a lo of differen cases. The es will be performed on marixes composed of hree series, wih a monhly frequency and wih a lengh of en years. This means ha, following he given noaion, i will be M=, N=0 and S=3. The mos common ime series model is he so-called airline model, as o say a SARIMA(0,,)(0,,). Thus, each of he hree series will follow an airline model, cenred in he value of 00, as Eurosa usually deals wih index daa. In addiion, a residual, normally disribued wih zero mean and variance equal o one, is added o each series. The MA coefficiens are beween - and, so ha he process is inverible. Such coefficiens have been generaed using a uniform disribuion. In order o simulae he seasonal peaks, wo differen groups have been creaed. In he firs group, a negaive seasonal peak in December, wih a difference of -0, has been creaed for all he series of he marix. In he second group, a negaive seasonal peak in December, wih a difference of -0, has been creaed for he firs of he hree series of he marix, while a posiive seasonal peak in Augus, wih a difference of +5, and a posiive seasonal peak in July, wih a difference of +35, have been creaed for he las wo series. By means of his emplae, he expecaions are ha in he firs group he es gives resuls of common seasonal paerns (low value of he F-raio, high p-value). On he oher hand, in he second group i gives resuls of differen seasonal paerns (high value of he F-raio, very low p-value), which are he opposie of hose in he firs group. For each group, 00 marixes will be creaed, and he es will be performed on all he 00 marixes. By defaul, here i will be esed wheher he es gives differen resuls for he groups, where in he firs group here should be common seasonal paerns, while in he second group he seasonal paerns should be differen. In order o run he es, he SI differences of he series should be obained. For his reason he series will be creaed using he sofware R 6. The seasonal adjusmen is hen performed by using Demera+ 7, choosing he auomaic procedure of X- ARIMA wihou any calendar effec, using an addiive decomposiion model and modelling each series wih a SARIMA(0,,)(0,,). Afer he SI differences are obained, he simulaions will be performed by using once more he sofware R. ( 6 ) See also hp:// ( 7 ) See also he Cros Poral, hp:// New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 7

10 The following able presens a summary of all he basic saisics for he F-raios. These are he average, he sample variance, all he quariles and he values of he confidence inerval for he average. Saisics F-Raio Average Variance Min Q Median Q Max.5684 C.I. Upper (5%).04 C.I. Lower (5%) There are jus hree marixes ha did no have common seasonal paerns a 0 per cen level. However, none of hem has evidence of common seasonal paerns a 5 per cen level. This resul, ogeher wih he band of he confidence inerval, is remarkable. Basically, i means ha he es gives good resuls when he null hypohesis is rue. In he following ables here is a summary of he analysis on he second group. Saisics F-Raio Average 6,387 Variance 0,535 Min 4,0495 Q 5,7083 Median 6,76 Q3 6,670 Max 7,936 C.I. Upper (5%) 6.79 C.I. Lower (5%) Once again, he analysis gives quie sable resuls. As for he expecaions, here are no common seasonal paerns a 5 per cen level in any of he 00 marixes. However, here is evidence of common seasonal paerns a per cen level in four of he 00 marixes esed. This resul, aken ogeher wih he confidence inerval bands, means ha, normally he es gives good resuls when he null hypohesis is false. Needless o say, he resuls obained in he analysis above need o be confirmed when he kind of seasonaliy is differen, for example when he series have differen seasonal paerns wih he same average, and when he lengh of he series is shorer. In spie of his, a firs imporan resul has been New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 8

11 given here. The es performs well for he kind of series creaed and divided in he groups described above. 5. Fuure research line Saring from his idea, more works needs o be done. A heoreical review, ogeher wih a more deailed case sudy, is already ongoing, regarding principally hree aspecs:. F-Raio: re-building he es upon he raio of he beween monhs variance and he residual variance;. Trend: an a-priori esimaion of he rend needs o be esed; 3. Co-movemens: he use of he co-movemens es as benchmarking. References [] Asolfi, R.; Ladiray, D.; Mazzi, G.L. Seasonal Adjusmen of European Aggregaes: Direc versus Indirec Approach European Communiies, 00. [] Busei, F.; Harvey, A. Seasonaliy Tess Journal of Business and Economic Saisics, Vol., No. 3, pp , Jul [3] Cenoni, M.; Cubbadda, G. Modelling Comovemens of Economic Time Series: A Selecive Survey CEIS, 0. [4] Cohen, B. Explaning Psychological Saisics (3 rd ed.), Chaper : Three-way ANOVA - New York: John Wiley & Sons, 007. [5] Crisadoro, R.; Sabbaini, R. - The Seasonal Adjusmen of he Harmonised Index of Consumer Prices for he Euro Area: a Comparison of Direc and Indirec Mehod Banca d Ialia, 000. [6] Higginson, J. An F Tes for he Presence of Moving Seasonaliy When Using Census Mehod II- X- Varian Saisics Canada, 975. [7] Hindrayano, I. - Seasonal adjusmen: direc, indirec or mulivariae mehod? Aenorm, No. 43, 004. [8] Maravall, A. An applicaion of he TRAMO-SEATS auomaic procedure; direc versus indirec approach Compuaion Saisics & Daa Analysis, 005. [9] Surradhar, B.C.; Dagum, E.B. Barle-ype modified es for moving seasonaliy wih applicaions The Saisician, Vol. 47, Par, 998. New innovaive 3-way ANOVA a-priori es for direc vs. indirec approach in Seasonal Adjusmen 9