ACCORDING TO SEASONAL SIMULTANEOUS ADDITIVE AND MULTIPLICATIVE EFFECTS

Transcription

1 he Correcon of Chronologc Seres Seasonal Flucuaons HE CORRECION OF CHRONOLOGIC SERIES SESONL FLUCUIONS CCORDING O SESONL SIULNEOUS DDIIVE ND ULIPLICIVE EFFECS bsrac R. BOURBONNIS* Ph. VLLIN** In hs sudy, we se he problem of he probable esence of an addve and mulplcave med seasonaly. In hs cone, we show by some smulaon ha he seasonaly correcon accordng o a pure addve or a pure mulplcave scheme leads o based esmaors of he coeffcens and, consequenly, of he calculaon of seasonally adjused seres whch s necessary for quanave demand analyss. he use of an analycal resoluon echnque allowng smulaneous esmaon of he rend coeffcens and he addve and mulplcave seasonal coeffcens wors perfecly f he seres s affeced by a smple lnear rend. In hs case, he esmaon gves he heorecal seasonal coeffcens. n applcaon sudy o he moble phone mare, he new produc spl n wo mares, professonal and ndvduals, allows evaluang he conrbuon of he mehodology. Keywords: seasonaly, demand analyss, me seres. JEL Classfcaon: C32, C2 he sudy of he chronologcal seres seasonaly s a prerequse for he quanave analyss of demand. When hs effec ess, s convenen o fler before beng able o analyze he oher characerscs, such as: he rend, he mpac of mareng * EURISCO, Unversy of Pars-Dauphne, Place du aréchal de Lare de assgny, Pars Cede 6, regs.bourbonnas@dauphne.fr. ** LSDE, Unversy of Pars-Dauphne, Place du aréchal de Lare de assgny, Pars Cede 6, valln@lamsade.dauphne.fr. Romanan Journal of Economc Forecasng 4/2007 5

2 Insue of Economc Forecasng m acves, ec. hs allows esmang he real effecs of he facors ha srucure he demand. In general, hs flraon proves o be ndspensable because, n mos of he cases, he scope of seasonal effec mass he mpac of he oher characerscs. he objecve of hs arcle s o pon ou ha f a chronologcal seres s no he achevemen of a jusfable process of a decompounded scheme purely addve or purely mulplcave, hen he radonal mehods of elmnang seasonaly are wea. ha s because, as a general rule of analyzng chronologc seres, one or he oher s appled afer havng deermned he mos suable decompounded scheme. If he major par of macroeconomc seres s nown for long perods of me, s no he same hng a he mcroeconomc level. he producs lfe beng shorer and shorer, companes possess sales archves over four, fve years or more. he sofware pacages for forecasng sales mos ofen used by companes, by defaul, appeal o a mulplcave scheme, hs whou any jusfcaon! We hn ha n realy, epressng he demand s rarely a leas here s no reason he resul of a pure decompounded scheme. We consder parcularly: he enerprses addressng o several mares, a publc mare and an ndusral mare. he publc mare s able o develop when he ndusral mare s able o reman sable, he wo mares havng her own proper seasonaly; for eample, he glue from panngs who nerferes n he ndusry and DIY, and he sellng of lquefed peroleum gas delvered o households and ndusres. Of course, n ceran cases, s possble o segmen he mares, bu hs s no always achevable (he absence of sascs) or desrable (because of he subsuon beween producs). For eample, he producon of ceran elecronc complaons may have as fnal desnaon he equpmens consumed on dfferen mares of very dfferen ypes, whou he producer of complaons beng able o have he basc nformaon abou he fler whch s represened by dsrbuors. he nnovaory behavor scheme, a person rapdly buyng a new produc, proposed by dgley e Dowlng (978) and presened by G. Roehrch (200). In hs scheme, wo populaons havng dfferen buyng behavors may coes: one of people wh a very nnovave aude, havng receved very favorable nformaon abou he produc, and he oher, less nnovave bu more concerned abou he caegory of producs, whch s n a favorable buyng suaon. In hs case, he seasonal mpac wll have a componen proporonal o he level of sales lned o he dssemnaon of nformaon o he frs populaon and a consan componen generaed by he second populaon. Sarng wh smulaons, we ry o demonsrae ha usng a med decompounded scheme over a shor daa hsory gves beer resuls, n oher words, a beer flraon of seasonaly han he addve or mulplcave elmnang of seasonaly. hs mprovemen of flraon hus allows for a more refned analyss of he oher characerscs of demand and, as a resul, a beer forecas. fer ha, wh he help of an analyc resoluon of a med heorecal scheme, we fnd he seasonal coeffcens of he generaed records. 6 Romanan Journal of Economc Forecasng 4/2007

3 he Correcon of Chronologc Seres Seasonal Flucuaons Fnally, an applcaon on he moble phone mare, an nnovave produc segmened n wo mares, professonals and ndvduals, allows for he evaluaon of he neres n hs mehodology. 2. Choosng he decompounded scheme 2.. Defnon of schemes I does no es a perfecly sasfyng mehod for esmang seasonal coeffcens: any mehod we choose, he rs of ncorporang flucuaons owed o errac values (called aberran or abnormal values) n seasonaly s always presen. he momen of calculang he seasonaly, s suable o do a ceran number of choces regardng he ype of seasonal coeffcens (addve/mulplcave, fed/slppng), choces whch we shall presen. he esmaed values of he seasonal coeffcens are dfferen, dependng on he mehodology used. he seasonaly of daa seres may somemes be nfluenced by he era season of/and he resdual componen. Gven he esence of hese neracons, here were derved he decompounded schemes of chronologcal seres: addve, mulplcave or complee mulplcave. - he addve scheme whch supposes he orhogonaly (ndependence). I s wren as follows: + S + R. In hs scheme, seasonaly s rgd n amplude and n perod. - he mulplcave scheme: S + R, n whch he seasonal componen s lned o he era season (fleble seasonaly wh he varance of amplude proporonal o rend). - he complee mulplcave scheme: S R n whch he seasonal componen s lned o he era season (seasonaly and he resdual componen are fleble wh he varance of amplude, n me). Currenly, s he mos used n he area of sales forecasng. hus, he dea s o compare hree mehods for elmnang seasonaly on he bass of hree schemes: an addve scheme; a mulplcave scheme; a med scheme, negrang addve and mulplcave seasonal coeffcens. We presen wo smple echnques, he frs one emprcal, for selecng he scheme he band es he band es consss n sarng from he graphc of raw seres evoluon and connecng by a shaered lne all he upward and all he downward values of he chronologcal seres. If, on he vsual eam he wo lnes seem parallel, he decomposon of he chronologcal seres can be acheved accordng o an addve Romanan Journal of Economc Forecasng 4/2007 7

4 Insue of Economc Forecasng scheme; n he oppose case, he mulplcave scheme seems more suable. Fgure and Fgure 2 show an nerpreaon of he band es. Eemple of addve scheme Fgure Eemple of mulplcave scheme Fgure he Buys-Ballo es he Buys-Ballo es (cf. Bourbonnas R. e erraza., 2004) s based on he resuls of calculang for every year he means and gaps ypes of he raw seres. he scheme s, by defnon, addve f he ype of gap and he average are ndependen; n he oppose case, s mulplcave. When he number of years s large enough, one may esmae usng he mehod of Leas Ordnary Squares (CO), he parameers of he equaon: a and a 0 8 Romanan Journal of Economc Forecasng 4/2007

5 he Correcon of Chronologc Seres Seasonal Flucuaons σ a + a 0 + ε. σ he ype of gap of cases n he year, he average of cases n he year,, N (N he number of years). In he case when he coeffcen a s no sgnfcanly dfferen from 0 (he Suden es) we accep he hypohess of an addve scheme; n he oppose case, we rejec he addve scheme and, smplfyng, we choose he mulplcave scheme. hese wo ess somemes lead o ambguous resuls. hs s he reason why he majory of auhors recommend he use of a mulplcave scheme. oreover, when he economc phenomenon s observed over a long perod of me he srucure of he seres ofen changes, passng from an addve scheme o a mulplcave one. I s hus convenen, no o confuse he hsorcal perod wh he number of observaons: s possble o have many observaons over a shor perod (he connuous socechange quoaons) when he srucure of he seres remans sable Fed or slppng coeffcens? We are also facng he choce of esmang fed or slppng seasonal coeffcens: Fed coeffcens: he coeffcens calculaed are he same whaever he analyzed year. Slppng coeffcens: he coeffcens evolve every year. seasonal movemen s repeve every year and has o repea smlarly. I seems hus mproper o calculae dfferen coeffcens every year. However, n ceran crcumsances, n whch a mareng reflecon suggess an evoluon of behavors, may be neresng o negrae a slppng seasonaly. Calculang a coeffcen for every monh, he rs of ncorporang a pece of nose n seasonaly nensfes. In fac, he dsncon beween seasonaly and he resdual componen wll be more dffcul o be done n he absence of a rgdy consran of he seasonal coeffcens. For eample, f due o weaher reasons, a year was parcularly favorable o he beer consumpon, a slppng seasonaly wll reflec hs seasonaly n he ne year whou any a pror reason, whou nowng he emperaure n advance. In hs case, a mulplcave fed seasonaly mposes. conraro, a rs deserves o be poned ou: he possble confuson beween he real seasonaly and a fconal seasonaly creaed by he company. I regards he companes mang promoons or arff varances n he same perod each year. he calculaon of he seasonal coeffcens arbues o seasonaly hese supra sales owed o he volunary polces of he frm. problem rses hus when he company modfes he dae of promoons. In hs case, usng slppng seasonal coeffcens allows negrang hs modfcaon more rapdly. Ecep for he mareng reflecons movang a predced evoluon or modfcaon n he habs of consumers, he fed seasonal coeffcens are generally used. Romanan Journal of Economc Forecasng 4/2007 9

6 Insue of Economc Forecasng 3. he mehods of elmnang seasonaly When a chronologcal seres s srucured by seasonaly, he neremporal comparsons of he phenomenon need a seres Correced for Seasonal Varances, wren down as CVS. he seasonaly of he sale of an arcle conceals he rue evoluon of sales; he sales of a raw seres are no hus able o be nerpreed. In addon, s easer o forecas sales whou he seasonal phenomena: he real rend can be calculaed; he real mpac of he eplanaory facors (publcy, promoons, ec.) can be poned ou. Elmnang seasonaly from a chronologcal seres s elmnang seasonaly whou modfyng he oher componens of he seres. I s a delcae operaon, a fac ha eplans he grea number of mehods for elmnang seasonaly. he mos used, for macroeconomc seres, s he mehod of CENSUS. However, requres hsorcal daa over long perods of me, up o 0 years. he choce of he mos approprae echnque depends on he deermnsc or random (sochasc) naure of he seasonaly of seres: when s deermnsc (n oher words, rgd, well-mared and repeve) he mehods of regresson and he usage of seasonal coeffcens dencal over he hsorc perod are adaped; when s random (he coeffcens are affeced by a random erm) he echnques of flraon by movng averages (CENSUS, for eample) mus be used. he preservaon of ranges prncple he analyss of seasonaly has as arge a new dsrbuon of he nra-annual profle of he seres whou modfyng he level acheved by annual cumulus: he annual averages of he raw and CVS seres have o be dencal. Sandardzaon allows calculang defne seasonal coeffcens. he CENSUS mehod Elmnang seasonaly by usng smple movng average s ofen nsuffcen because of varous reasons (flucuang seasonaly, era comple seasonaly). In 954, Shsn J. proposed a mehod of elmnang seasonaly usng n an erave way mulple movng averages. s CENSUS mehods are bul sarng from successve eraons of movng averages of dfferen order o beer apprehend he rend ( ), as well as he flucuaons of seasonaly, hey deermne he loss of nformaon a he fnal eremy of he seres. hs loss of nformaon s conaned by a Bo and Jenns ype of forecas, before elmnang seasonaly from he seres, by a number of pons equal o he nheren loss of nformaon when usng movng averages. 0 Romanan Journal of Economc Forecasng 4/2007

7 he Correcon of Chronologc Seres Seasonal Flucuaons In he area of sales forecasng based on a shor hsory, hs mehod s, hus, noperable. he seasonal dfferences Usng he fler proposed by Bo and Jenns (976) o seasonal dfferences n he form: ( B p ) p (p seasonal perod, p 2 for a monhly seres of annual perod) allows for mang saonary a seres nfluenced by a seasonal movemen. he ess of HEGY (Hylleberg, Engle, Granger, Yoo, 990) and Franses (990) use hs ype of fler wh polnoms ( B 4 ) for quarerly seres and ( B 2 ) for monhly seres. he compley of applyng ess for he presence of seasonal un roo maes mpossble o use hs approach n enerprses. he mehod based on relaed o rend hs mehod s he one used by us; s descrbed n Secon 4.. In fac, n he cone of an enerprse n whch sales hsory s very well nown over four years, he CENSUS mehod s noperable. ha s why we have chosen he mehod of esmang rend by movng averages and fed coeffcens, mehod whch s he mos used n enerprses. 4. ehods of elmnang seasonaly he chronologcal seres s composed of n cycles, each comprsng p perods. 4.. ehod : Classc mehods (pure addve scheme and pure mulplcave scheme) Frsly, we proceed by wo elmnaons of seasonaly followng an addve scheme and a mulplcave scheme. he mehod of elmnang seasonaly appled (cf. Bourbonnas R. and Usuner J. C. (2006), pages 42 5) s he mos wdespread n sales forecasng sofware pacages, due o he shor hsorcal daa seres avalable n companes: a) Esmaon of rend by calculang cenered movng averages over p2 monhs (he case of an annual perodcy wh monhly daa). s he average; b) he calculaon of / rao n he mulplcave scheme and he dfferences n he addve scheme; c) he seasonal raos of coeffcens are furher on calculaed by average on n years (he movng average does no allow for esmang he rend for he frs 6 and he las 6 monhs); d) las, he seasonal coeffcens, S, p; for he p perods of every cycle are calculaed for sandardzaon n order o respec he prncple of preservng ranges : p S 0 (addve scheme); Romanan Journal of Economc Forecasng 4/2007

8 p Insue of Economc Forecasng S (mulplcave scheme); relaon whch s beng used n pracce n he form: p S p, whch s he alernave used here. e) hus, he seres s whou seasonaly, due o hese seasonal coeffcens, hereby we oban wo seres correced for he seasonal values, obaned followng he wo schemes: followng an addve scheme: CVS followng a mulplcave scheme: CVS ehod 2: he leas squares esmaors for he med scheme We denoe by: n : number of cycles (of years) of common ran, p: he perodcy of seres, number of perods (of common ran ) n he cycle, : cycle ran ;,..,n, : perod ran;, 2, p: ( modulo p) +, : daa ran, (-)p+ correspondng o he perod of he cycle, : raw seres a dae (-)p+ correspondng o he perod of he cycle, : value of he real rend a dae, ˆ : lnear rend esmaed a dae, S : seasonal fed mulplcave coeffcen assocaed o he perod, S : seasonal fed addve coeffcen assocaed o he perod, E : resdual componen, he no eplcaed par of he seres. he prncple consss n deermnng smulaneously he roles of he seasonal coeffcens S and S by esmaors, havng n vew he model m. Nong ha f he rend s nown (or esmaed), he model s lnear dependng on seasonal coeffcens, we can esmae seasonal coeffcens S and S (,2,, p) by leas square esmaors, soluon of he lnear classc sysem wh 2 p unnown and 2 p equaons: S ( ) - X where: 2 Romanan Journal of Economc Forecasng 4/2007

9 he Correcon of Chronologc Seres Seasonal Flucuaons Romanan Journal of Economc Forecasng 4/ X(np ) : he column vecor for n cycles of p successve perods, n p X... X s composed by n underneah marces assocaed o cycles : p X..., n. S (2p ): he column vecor of componens S S ; p. (np 2p): he mar I I I n ; where s he dagonal mar, of order p, of elemens for cycle, varyng on p perods. I s he uny mar of order p. : ransposed. E : column mar of he resdual componens E. he esmaors of S and S are: ( ) n n n n S 2 2 ˆ S S ˆ ˆ Wh n n and n n. We mee agan he classc esmaors of he lnear model wh an eplanaory varable and a consan; here, he eplanaory varable s he esmaed rend a dae : ˆ. hs resul comes from he fac ha he esmae of he seasonal coeffcens s made by he resoluon of p ndependen blocs of n equaons. he basc model s: E S S + + or wh he wo nde noaon:

10 Insue of Economc Forecasng S + S + E In mar form, he sysem s wren as follows: X S + E () Each square mar,, n, of (p p) dmenson, he form of componen s : p hus, we reurn a p sysems of n ndependen regresson equaons, each sysem, p, beng formed by n equaons assocaed o n observaons of he cycle of ran comprsng n perods, allowng for esmang he wo parameers S and S hrough he classc resuls of he wo varables lnear regresson. he se of regresson equaons of sysem s wren as follows: S + S + E, for,2, n he second mehod: If s nversable, (hs s especally he case f ˆ a + a 0 wh a 0), he seasonal coeffcens esmaors whch mnmze he square sum of resduals are he soluon of he sysem: S ( ' ) ' X (2); he subordnae dagonal mar enables a relavely smple mar calculaon and he equaly (2) provdes he presened esmaors. hs mehod of esmang seasonal coeffcens allows for esng he sgnfcance of wo ypes of coeffcens, and hereby, for choosng one of he hree models (mulplcave, addve and med). We provde n he ne paragraph an esmaon of he seasonal coeffcens for four proposed seres. he esmaon of rend was made hrough a regresson appled o he seres of movng averages of order p 2 n order o fler he seasonal nfluences. 5. rfcal generang of daa In hs secon, we presen he mehodology used n order o mae our smulaons. We have generaed over four years (48 monhly observaons) monhly chronologcal seres havng a med decompounded scheme of he form: ( S ) S +,, 48. Wh: smulaed value of seres a momen, value of lnear rend smulaed a dae ( a 0 + a ), 4 Romanan Journal of Economc Forecasng 4/2007

11 he Correcon of Chronologc Seres Seasonal Flucuaons S mulplcave fed seasonal coeffcens assocaed o he dae, S S addve fed seasonal coeffcens, S S + 2 S + 2 Inenonally, we have no added he hazard because he goal s o prove ha a mehod no adaped (pure addve scheme and pure mulplcave scheme) does no allow for fndng he real values of coeffcens, whereas a med mehod allows for fndng eacly he coeffcens values. he daa was hus generaed over four years; he fed seasonal coeffcens havng he same roles (able ). able Seasonal coeffcens onh S S ulplcave ddve January February arch prl ay June July ugus Sepember Ocober November December he dfferen generaons accordng o he rend lne slope are: ( ) S ( S ) S ( S ) S ( ) S ( S ) S ( S ) S ( S ) S Seres denoed by, : ( ) S, Seres 2 denoed by 2, : 2, ( ) Seres 3 denoed by 3, : 3, ( ) Seres 4 denoed by 4, :, ( ) S, ( 4, + 0), ( 4, + 50) ( + 5) for,, for 3,, for 25,, for 37,, 48., 4, Romanan Journal of Economc Forecasng 4/2007 5

12 Insue of Economc Forecasng 6. Smulaon resuls 6.. ehod : Classc mehods able 2 presens he resuls of seasonal coeffcens accordng o an addve and mulplcave scheme. he esmaon of seasonal coeffcens s bad, even very far away from he real value as regards he addve scheme (whch was predcable consderng he characerscs of he smulaed seres). able 2 Resuls of classc esmaon onh heorecal values Seres Seres 2 Seres 3 Seres 4 S S S S S S S S S S January February arch prl ay June July ugus Sepember Ocober November December ehod 2: he med mehod he esmaes of seasonal coeffcens are presened n able 3. We presen as llusraon n nne II a complee eample of processng for seres 2. Resuls of med esmaons able 3 onh heorecal values Seres Seres 2 Seres 3 Seres 4 S S S S S S S S S S January February arch prl ay June July ugus Sepember Romanan Journal of Economc Forecasng 4/2007

13 he Correcon of Chronologc Seres Seasonal Flucuaons Ocober November December One may noe a very good esmaon of heorecal seasonal coeffcens (hazard does no es n he arfcal daa) for he frs hree seres. In reurn, he esmaon s less conssen n wha concerns he seasonal coeffcens of seres 4, n whch rend s lnear on fragmens over he perod. Remar: I may seem curous ha for he frs hree models, perfecly lnear and whou hazard, he coeffcens esmaed hrough he leas square mehod were no precsely equal wh he heorecal coeffcens. ha resuls from he esmaon of rend; hs esmaon eplong he movng average s slghly based by he presence of mulplcave seasonal coeffcens. If he movng average can perfecly fler compensaory addve effecs, s flraon s less effecve regardng he mulplcave coeffcens. For eample, he rend of seres 2: s esmaed a 500,66 + 9, Resuls synhess In order o evaluae each model s performance, we choose he creron whch consss n comparng he average square gap (sum of square gaps dvded by lbery degrees) beween he heorecal observed values and he adjused values wh he help of each model (rend and ype of seasonaly), as accordng o able 4. able 4 Sum of square gaps beween he observed and he adjused values Seres Seres Seres Seres Sum of square gaps addve scheme Lbery degrees verage square gap Sum of square gaps mulplcave scheme Lbery degrees verage square gap Sum of square gaps med scheme Lbery degrees verage square gap One may noe ha esmang seasonal coeffcens hrough he med mehod confers correc esmaons and domnaes very clearly he oher wo mehods, ecep when he chronologcal seres s affeced by a non-lnear rend (seres 4). noher approach wll conss n elmnang seasonaly frs accordng o an addve scheme, and aferwards over a CVS seres accordng o a mulplcave scheme. n aemp made on seres proved o be dsasrous (sum of square gaps ) due o he creaon of a new parase cycle lned o he mulplcave seasonaly. Romanan Journal of Economc Forecasng 4/2007 7

14 Insue of Economc Forecasng 7. n eample of applcaon: he case of moble elephony he moble elephony mare s a recen one, currenly hree operaors acng on he French mare: Orange, SFR, and Bouygues elecom. hs olgopolsc characer generaes a very powerful mareng acvy, wh very frequen new offers. We are suaed a he end of 999 (a mare no ye maure), a perod for whch forecasng he number of subscrbers (he buyng of Sm cards ) proved o be crucal for each operaor. he seasonaly (Graph ) ndcaes a very mporan pea n he monhs of December (gfs o ndvduals a he end of he year), because he number of subscrbers s wo mes hgher as compared o he average sales per year. wo very dsncve segmens of mare n erms of behavor and seasonaly are revealed, namely professonal and prvae. Graph Number of subscrbers (Sm cards sold) n he French moble elephony, n housands On hs sales hsory (January 995 o June 999, 54 observaons) we wll apply he hree mehods of elmnang seasonaly (able 5), alhough he addve scheme, as we see on graph, urned ou as proscrbed. he Sm card (Subscrber Ideny odule) s a card wh chp whch, nroduce n he machne allows for denfyng he subscrber. 8 Romanan Journal of Economc Forecasng 4/2007

15 he Correcon of Chronologc Seres Seasonal Flucuaons able 5 Seasonal coeffcens esmaed as accordn o he hree mehods onh Pure addve Pure ed scheme ulplcave ddve ulplcave January February arch prl ay June July ugus Sepember Ocober November December Inerpreng he esmaed seasonal coeffcens, one may noe srong dfferences. I s neresng o observe ha n he frame of he med scheme mulplcave and addve coeffcens do no have he same profle of evoluon. hs suggess us ha for he professonal mare, jusfable raher by he addve scheme whch was already sablzed n 998, he seasonal profle s no smlar o he mare for ndvduals, emphaszed by he mulplcave scheme, wh a mare n full epanson referrng o hree momens: June (before leavng for holday), Ocober (openng of schools and unverses) and December (Chrsmas and New Year s holdays). In order o evaluae he hree mehods relevance, we compare he square gaps sum (able 6) of he observed values and adjused values, wh he help of a model wh lnear rend and seasonaly. able 6 Square gaps sum of he observed and adjused values ddve scheme ulplcave scheme ed scheme Sum of square gaps Lbery degrees verage square gap Consderng he sandardzaon hypohess of gaps, we wll proceed o a Fsher es, such as: ( ) / 0,05 F * 3,5 > F ;32 2, / 32 Romanan Journal of Economc Forecasng 4/2007 9

16 Insue of Economc Forecasng hs es perms predcng a sgnfcan dfference of square gaps sum beween he mulplcave and he med schemes. s we could ancpae, he addve scheme proves o be very wea; he med scheme urns ou o be preferable o he pure mulplcave one (average square gap wo mes weaer). las, we compare (Graph 2) he resduals of esmaon beween he med scheme and he mulplcave one. Graph 2 Resduals of esmaon of mulplcave scheme and med scheme Over he perod , he mulplcave scheme provdes resdual o esmaon of equal amplude wh he resduals of he med scheme. In reurn, afer 998 he perod of emergence of he mare of ndvduals he med scheme offers an adjusmen of beer qualy, as one may see n able 7. las, one may noe ha n he case of usng hs mehod for developng forecass, nervenon varables for December 997 and December 998 should be negraed n he model. able 7 Sum of square gaps beween observed values and adjused values over he perod January 998-June 999 ulplcave scheme ed scheme Sum of square gaps he med scheme, n hs eample and all over he perod, seems preferable o a mulplcave scheme. oreover, allows for denfyng, for hs mare, mulplcave 20 Romanan Journal of Economc Forecasng 4/2007

17 he Correcon of Chronologc Seres Seasonal Flucuaons and addve seasonal coeffcens correspondng o he wo dsnc mares (for professonals and for ndvduals), whch emerged begnnng wh Concludng remars In hs sudy, we have poned ou he ssue of a probable esence of mulplcave and addve med seasonaly. In hs cone, we presened hrough smulaons ha he elmnaon of seasonaly followng a pure addve scheme or a pure mulplcave scheme nroduces a bas n he coeffcens esmae and hus, consequenly, n he calculus of he CVS seres. he qualy of demand analyss and, consequenly, of sales forecasng wll be affeced by hs. Usng a med echnque of esmaon allows for esmang smulaneously he rend coeffcens and he addve and mulplcave seasonal coeffcens funcons perfecly f daa seres s affeced by lnear rend : we regan well he heorecal seasonal coeffcens. survey on he moble elephony mare ndcaes he superory of he med mehod as compared o a mulplcave scheme and perms denfyng he esence of a double seasonaly : addve for enerprses and mulplcave for ndvduals. However, we hn ha he posulae of a lnear rend s a lm of hs mehod. hereby, a pah for research consss n sudyng he robusness of seasonal coeffcens esmaors resuled from he med mehod funcon of he error of rend esmaon. References Bourbonnas R., erraza., nalyse des séres emporelles, Dunod, Bourbonnas R., Usuner J C., Prévson des venes, Economca, 4 éd., Bo G. E. P., Jenns G.., me seres analyss: forecasng and conrol, Holdenday, 976. Dagum E. B., Fondemen des deu prncpau ypes de méhodes de désasonnalsaon e de la méhode X RI, Econome pplquée, Vol, 979. Franses P., esng for seasonal un roo n monhly daa, Economerc Insue Repor, Erasmus Unversy Roerdam, 990. Hylleberg S., Engle R., Granger C., Yoo B., Seasonal negraon and co negraon, Journal of he Economercs, 44, 990. dgley and Dowlng, Innovaveness: he concep and measuremen, Journal of Consumer Research, 4, , 978. Roehrch G., Causes de l acha d un nouveau produ: varables ndvduelles ou caracérsques perçues, Revue Françase de areng, No. 82, 200 Shsn J., Elecronc compuers and busness ndcaors, Naonal Bureau of Economc Research, Occasonal Paper, 954. Romanan Journal of Economc Forecasng 4/2007 2

18 Insue of Economc Forecasng nne Eample of esmaon of med seasonal coeffcens for seres 2, nalyc resoluon seres 2 D 2 rend S rao S rao Forecas gap CVS Year J F J J S O N D Year 2 J F J J S O N D Year 3 J F J J S O N D Year 4 J F Romanan Journal of Economc Forecasng 4/2007

19 he Correcon of Chronologc Seres Seasonal Flucuaons D 2 rend S rao S rao Forecas gap CVS J J S O N D prov. S def. 2 S prov. S def. S jan feb march aprl may june july augus sep oc nov dec Prov Provsonal seasonal coeffcens before sandardzaon. 2 Def. Defnv seasonal coeffcens afer sandardzaon (prncple of preservng areas). Romanan Journal of Economc Forecasng 4/