A multivariate Denton method for benchmarking large data sets

Transcription

1 09 A multaate Deto metho fo bechmakg lage ata sets ee Bkke, Jacco Daalmas a No Mushkua The ews epesse ths pape ae those of the autho(s) a o ot ecessaly eflect the polces of tatstcs Nethelas Dscusso pape (000) tatstcs Nethelas The ague/eele, 00

2 Eplaato of symbols. = ata ot aalable * = posoal fgue ** = ese posoal fgue = publcato pohbe (cofetal fgue) = l o less tha half of u cocee = (betwee two fgues) cluse 0 (0,0) = less tha half of u cocee blak = ot applcable = 008 to 009 cluse 008/009 = aeage of 008 up to a clug / 09 = cop yea, facal yea, school yea etc. begg 008 a eg / / 09 = cop yea, facal yea, etc. 006/ 07 to 008/ 09 cluse Due to oug, some totals may ot coespo wh the sum of the sepaate fgues. Publshe tatstcs Nethelas e Faaseef 3 49 JP The ague Pepess tatstcs Nethelas - Gafmea Coe TelDesg, otteam Ifomato Telephoe Telefa Va cotact fom: Whee to oe E-mal: ekoop@cbs.l Telefa Iteet IN: tatstcs Nethelas, The ague/eele, 00. epoucto s pemte. tatstcs Nethelas must be quote as souce X-0

3 A Multaate Deto metho fo bechmakg lage ata sets ee Bkke, Jacco Daalmas a No Mushkua ummay: Bechmakg s the pocess to achee mathematcal cosstecy betwee low-fequecy (e.g. aual) a hgh-fequecy (e.g. quately) ata. tatstcs Nethelas s gog to apply a ew bechmakg metho o Dutch atoal accouts. The ew bechmakg metho s base o a multaate Deto metho, pesete Bkke a Bujtehek (006). I oe to copoate all ecoomc elatos to ths moel, we etee wh ew methoologcal featues, such as ato costats, soft costats a equaly costats. I ths pape the ew etee multaate Deto metho s pesete. Futhemoe, the esults ae escbe of a smulato epemet o Dutch supply a use tables. The sze of these ata sets s ey lage: oles oe 0,000 tme sees, whch ae elate to each othe a costats. Befoe applyg the bechmak metho, the tables wee cleae fom a few lage scepaces. The epemet was successful, catg that the etee Deto metho woks well pactce o ata that o ot show lage scepaces. Keywos: Bechmakg, Data ecoclato, Deto metho, Natoal accouts. Itoucto The poblem that ofte ases whle complg atoal accouts s the cosstecy the souce ata. Fo eample quately ata ae usually cosstet wh aual ata a ecoomc ules that apply to oe tme peo ae olate. Maco-tegato s the pocess fo acheg cosstecy betwee ecoomc ata. The fst step of maco-tegato cossts of coectg eos, whch the lage obous scepaces ae etecte a coecte. The seco step s a balacg pocess. I ths step the tables ae coecte so that the ecoomc ules ae beg fulflle. Balacg s also ofte calle ecoclato. The leatue o ata ecoclato goes back to toe et al. (94), who pesete a costae, geealse least squaes metho. eeal othe balacg methos ae escbe Woe et al. (004, Ae A). 3

4 Ths pape focuses o a specal case of the balacg poblem, calle bechmakg. Ths s the pocess to achee cosstecy betwee ata that ae publshe at a hgh fequecy a ata that ae publshe at a low fequecy. Whout loss of geealy, s assume hee that the tme sees ae quately ata a the bechmaks ae aual fgues. Typcally, the aual ata souces poe the most elable fomato about oeall leels a the quately ata souces poe fomato about shot-tem chages. Most of the aual ata ae fe ue to ths easo. The bechmakg methos ca be boaly classfe to puely umecal methos a moel-base methos. Bloem et al. (00, Chapte VI) a Dagum a Cholette (006) ge a compehese oeew of methos. Fo the moel-base appoach we hae egesso moels, fo stace Cholette-Dagum (994), AIMA moel-base methos popose by llme a Tabels (987) a state space moels efe by Dub a Queelle (997). Closely elate to the egesso metho s the metho of Chow a L (97), whee the authos choose fo the tepolato of tme sees,.e. eg quately ata fom aual ata, although ths s ot a bechmakg metho the stct sese. The Chow a L metho may suffe fom step poblems,.e. lage gaps betwee the fouth quate of oe yea a the fst quate of the et yea. A classcal efeece to a umecal metho s Deto (97). The Deto metho s a quaatc pogammg metho a was ally popose fo uaate ata. The am of ths metho s to make the quately ata coheet wh aual totals, whle peseg all quate-to-quate chages as much as possble. Ths techque s calle the moemet peseato pcple. Because of ths popety Deto metho aos the step poblem. D Fozo a Ma (003) hae etee the Deto metho fo multaate ata. I ao to tempoal algmet, multaate ata ofte also hae to satsfy a set of cotempoaeous costats (.e. estctos betwee ffeet tme sees oe tme-peo). ubsequetly, Bkke a Bujtehek (006) hae ae elably weghts to the multaate Deto metho. Although s metoe by Bloem et al. (00) that the Deto metho s well sue fo lage scale applcatos, to the best kowlege of the authos, such applcatos hae ot bee pefome at NI s (Natoal tatstcal Istutes). The lack of aequate compute techology use to be a obstacle (Ncola, 000). tatstcs Nethelas s ow mplemetg a etee eso of the multaate bechmakg metho pesete ths pape s poucto pocess of atoal accouts. Ths applcato eques softwae that ca cope wh lage ata sets, amely oe 0,000 tme sees. The softwae s base o a state-of-the-at, commecal quaatc pogammg (QP) sole. The Bueau of Ecoomc Aalyss (BEA) uses smla softwae fo the mplemetato of a 4

5 ecoclato metho (Che, 006), but the metho s ot ame at bechmakg. I oe to copoate all ecoomc elatos to the moel, that ae specfc fo the atoal accouts, we ae ew methoologcal featues to the multaate Deto metho of Bkke a Bujtehek (006). These ole: soft, ato a equaly costats. Magus et al. (000) aleay hae copoate these featues, ecept equaly costats, to a ecoclato metho, whch s ot ectly tee fo bechmakg puposes. I the moel pesete ths pape we clue all the ules that ae specfc fo the Natoal Accouts ata, sce ou bechmakg moel wll be apple to Dutch supply a use tables. A smulato epemet was coucte at tatstcs Nethelas o ata of Dutch supply a use tables to f out whethe all eleat ecoomc costats ca be clue the moel. The epemet was successful a as metoe aboe tatstcs Nethelas s the pocess of mplemetg the metho to s poucto system. Note that a few lage scepaces wee emoe fom the ata befoe applyg the bechmakg metho. I ths pape we escbe the eteso of the multaate Deto metho a the esults of the smulato eecse. Ths pape s ogase as follows. I ecto the etee multaate Deto moel s pesete. ecto 3 eals wh the softwae use to sole the moel a escbes the esults of a smulato epemet. ecto 4 coclues a ges a outlook o futhe eseach possbles.. The moel. The uaate moel I ths subsecto we scuss the classcal Deto metho. The am of ths metho s to f a bechmake tme sees t, t =,,T, that pesees as much as possble all quate-to-quate chages of the ogal quately tme sees t, subject to aual bechmaks. Deto popose seeal measues to efe the quate-to-quate chages. We cose the ae fst-oe fucto a the popotoal fst-oe fucto. The ae fucto keeps ae coectos ( t t ) as costat as possble oe all peos. The popotoal fucto s esge to pesee the (popotoal) gowth ates of t a theefoe keeps the elate coectos ( t t )/ t as costat as possble oe all peos. I mathematcal tems: the objecte fucto of the ae Deto moel s: 5

6 T ( ( t t ) ( t t ) ) t = m ) (.) a the objecte fucto of the popotoal Deto moel s: T (( ) ( )), m ) (.) t = t t t t the popotoal moel caot be use f the ogal tme sees cotas zeoes. Note that woul be moe atual to cose the ffeeces betwee the elate chages of the ese a pelmay sees,.e. to mmse the T objecte fucto ( ( t t ) ( t t ) ) t =. owee ths olea fom s ey ffcult to hale fo lage poblems, see e.g. Öhlé (006). Both objecte fuctos ae subject to the followg costats ( a ) t= t t= 4 + ( a ) = y a, a =,, T / 4, (.3) whee a s a e of the yea a y a s a aual alue. The set of estctos epesses the algmet of fou quates to aual totals.. The multaate case The eteso of the uaate Deto moel to the multaate case of Bkke a Bujtehek (006) s staghtfowa. I the etee moel weghts ae touce, oe to escbe the elably of the tme sees. The Natoal Accouts uses a we age of souces. Natually, ffeet tme sees ae cosee moe o less elable, epeg o the souce. Vaaces ae ofte use the leatue oe to escbe the ata elably. ce pactce s almost mpossble to estmate the aaces, we efe weghts stea. Weghts ca be ewe as geealsatos of aaces,.e. they ae efe such a way that aaces ca substute the weghts. Aalogous to aaces, the weghts hae to be stctly pose a satsfy the popety that the hghe the alue, the moe eato s toleate. The multaate, ae moel s ge by N T M (.4) A = t = ( w ) (( ) ( )) such that N T = t = c = b, = K,, C (.5) 6

7 whee s the e fo the tme sees, N eotes the umbe of tme sees a A w eotes a elably weght of the th tme sees at quate t a A stas fo the ae moel. The bgge s the alues of the tme sees the hghe s weght. ow we efe the weghts wll be escbe etal subsecto.4. I (.5) s the e of the estctos a Futhe a c C s the umbe of estctos. b ae coeffcets of the estctos. Amogst othes, the aual alues, y a (.3), ae clue b. ee, supescpt stas fo ha, s use to stgush the estctos (.5) fom the soft estctos that wll be touce futhe o ths pape. The set of estctos (.5) may ole both cotempoay (.e. betwee tme sees wh oe tme peo) a tetempoal (betwee ffeet tme peos, wh oe tme sees) estctos. The eteso fo the popotoal moel s smla. I Bkke a Bujtehek (006) the popotoal a the ae moels ae combe. The use has to specfy befoeha fo each tme sees whethe the moel s popotoal o ae..3 The etee moel I ths subsecto we efe the etee multaate Deto metho. The etesos clue: soft costats, fe quates, ato costats, a equales. All of these estctos ae efe specfcally fo atoal accouts ata. At tatstcs Nethelas a lot of subject matte kowlege s use the ecoclato of atoal accouts. Fo stace: fo some peshable goos the alue of the chage of stocks, summe oe the fou quates of oe yea, shoul ot ffe much fom zeo. I oe to clue such kowlege to the moel soft costats ae eee. A set of soft lea costats s ge by T N t = = L c ~ ( b, w ), = K,, L (.6) whee L eotes the total umbe of lea costats a b s a taget alue. L The supescpt eotes that the costats ae soft. Futhe, w s a elably weght, whee the supescpt L cates that the weght belogs to a lea costat. mla otato wll be use thoughout ths secto. We use ths type of costat to clue soft aual totals the moel. The costats (.6) ae clue the moel by ag the followg pealzato tems to the objecte fucto (.4): + L ( w ) L T N b = t = = c. (.7) 7

8 Although the Deto metho focuses o the peseato of quate-to-quate, chages, some cases may be ecessay to stck to a quately alue, fo stace f that alue s aleay publshe. The we clue the followg ha estcto the moel: =. (.8) ee ae the fe quates. Lkewse, a soft costat s eote by; F ~ (,( w ) ). (.9) oft costats ae copoate the moel by ag the followg tem to the objecte fucto (.4) whee + N T = t= C, (.0) F w C s a cato fucto, that s efe by f a soft fe quate s efe o C := (.) 0 othewse. Note that (.0) s a specal case of (.7). Aothe mpotat eteso of the moel s the ato costat. May ecoomc catos ca be eote by atos of atoal accouts aables. Fo eample subject matte specalsts may hae po epectatos of the alue of the ato betwee alue ae a output of a usty. To escbe these types of elatos ha a soft ato costats ae ae to the moel, that ae ge by whee / = a / ~ (,( w ) ), (.) t t t t t eotes the umeato tme sees at quate t, t eotes the eomato tme sees at quate t, s some peeteme alue a w eotes the weght of a ato t / t. ce we ae uable to mplemet ato costats the ogal fom, we lease them fst. Followg the appoach of Magus et al. (000), the atos (.) ae lease as * = 0 a ~ (0,( w ) ). (.3) t t t t Note that the fst leazato s eact, but the seco oe s a appomato, * whch wll be scusse etal secto.4. ee, w eotes the weght of a lease ato. The epesso of s ee fom w (.33). oft lease atos ae copoate the moel, by ag the followg tem to the objecte fucto 8

9 whee N T + t t B, * (.4), t w B s a cato fucto, that s efe by f a soft costat s efe o the ato of t a t B : (.5) 0 othewse. Note that, essetally, thee s o ffeece betwee lea costats a lease ato costats. The easo fo makg the stcto the moel, s that fo the lease ato costats epessos fo weghts wll be efe a ffeet way tha fo the lea costats. Cotay to the weghts of lea costats, the weghts of the lease ato costats epe o the peeteme alue,, see subsecto.4.. Most ecoomc aables caot hae egate sgs. To copoate ths (a othe) equemet(s) the moel equaly costats ae eee. A set of equales s ge by N T a t z,..., I (.6) whee I eotes the umbe of equaly costats. If we copoate (.4) the tems efe (.7), (.0) a (.4), a a the costats efe (.8), (.3) a (.6) to (.5) the the complete, etee moel s ge by M N T A A t w N T b c L w t N T C, F t w N T, t P w N T A t B t t * w (.7) such that T N t t t c b,..., C ; (.8) 0 B,,..., N, t,..., T; (.9) 0 C,..., N, t,..., T; (.0) N T a t z,..., I. (.) 9

10 ee Futhe, A s a cato fucto, efe as follows: f the ae moel apples to tme sees A : (.) 0 f the popotoal moel apples to tme sees B a C ae bascally efe the same as the ffeece that B a C apply to ha costats, whee B B a C, wh a C ae efe fo soft costats. The fe tems the fucto (.7) eote: ae quately chages, popotoal chages, (soft) lea costats, (soft) ato costats a (soft) fe quates, espectely. The costats (.8) (.) ae: (ha) lea costats, (ha) ato costats, (ha) fe quates a equaly costats, espectely. As Bkke a Bujtehek (006) we eteme befoeha whch moel, ae o popotoal, s apple to each tme sees. Fo the ata we use the smulato epemet the popotoal moel s pefee fo most of the tme-sees. Thee ae two eceptos: ) If oe of the quately alues absolute tems s less tha some pespecfe alue. ce ou applcato the pelmay tme-sees ae tege alue, whe the al alues ae small, elate chages ae healy fluece by the peceg oug pocess a theefoe oes ot make sese to pesee them. Aothe easo fo ths ecepto s that the popotoal moel caot be use fo tme-sees that cota pelmay alues of zeo. ) If a tme-sees has both pose a egate alues. Whe the popotoal moel s use the tme-sees coul be multple by some egate umbe. Thus, all pose umbes become egate a ce esa. I pactse ths s ot the ese outcome. The poblem, efe by (.7) (.) s a staa coe quaatc pogammg (QP) poblem. The poblem s well kow the leatue, a may effcet solg techques ae aalable..4 Weghts I ths subsecto we efe the weghts use (.7) (.). I the leatue ofte aaces ae use stea. As metoe secto.3, weghts ae use as a geealsato of aaces. Theefoe ou efos ae chose such a way that they stll ema al f aaces woul substute weghts. I the fst subsecto.4. below we fomulate thee popetes of the uelyg moel. The weghts shoul be efe such a way that these 0

11 popetes ema tue. I subsecto.4. we popose epessos fo weghts that focus o these thee popetes..4. Uelyg moel popetes I the objecte fucto of the afoemetoe moel seeal k of weghts ae use (weghts of ae a popotoal chages, soft fe quates, soft ato s a soft, lea costats). These weghts hae to be eteme, so that the moel meets ceta popetes, see Öhlé (006). The fst two popetes efe hee come fom the leatue, the last oe we ae ouseles. ) Iaace of put ata (metoe by Öhlé, 006): If all put ata ae multple by the same oegate scala, the outcomes must also be chage by ths facto; ) ato symmety (metoe by Magus a Dalo, 008): The outcome wll ot chage f a ato the bechmakg moel s eplace by s ecpoke,.e. f the estcto / y ~ (, ( w / ) ) s eplace by y / ~ ( /, ( w ); y / ) 3) Iaace of moel choce: Fo costat tme sees,.e. =, the esults of the ae a popotoal moel shoul be the same. Note, that the th popety tally hols tue the uaate case. Ths popety s mpose, fo the pupose of combg the popotoal a the ae moel a multaate settg. y.4. Epessos I ths subsecto epessos wll be popose fo the weghts of popotoal a ae quately chages, fe quates, lea a ato costats. I Appe A we show that the weghts epessos efe ths subsecto, satsfy the thee popetes fom subsecto 4... Keepg m that the epessos of the weghts shoul be easy to use, we touce tug paametes that apply to a goup of smla weghts. Fo eample these paametes ae efe such a way that all weghts that ae elate to oe tme sees ca be ajuste, by chagg the alue of oe paamete oly. Fg the most appopate alues of these tug paametes s a tal a eo pocess that epes o the ese outcome of the moel, whch may be ffeet fom applcato to applcato. I ou smulato stuy the weghts fo each tme-sees wee efe globally by the subject specalst. They attache the weghts to goups of tme-sees base o the elably of the souces. Each tme afte we obtae the optmzato esults, the subject matte specalsts ewe the esults a tue the weghts. Due to the lage

12 umbe of tme-sees a estctos, optmzato a tug ha to be pefome seeal eatos. Fo popotoal quate-to-quate chages the squae weghts ae efe by P ( ) w = β, (.3) J whee > a J Ζ wh J K. The easo fo choosg the fom of (.3) s that pose alues ae guaatee. I ou smulato stuy K = 3, hece we efe see leels fo the weghts. Fo each tme sees the paamete J s efe befoeha by subject matte specalst. Fo the tme sees obtae fom the most elable souce J = 3 a ths meas that the tme sees hae the smallest weght. The optmal alues fo a J epe o the scalg of the aables the poblem. A compute caot epeset fely small ffeeces betwee umbes. The smallest ffeece betwee two umbes that a compute ca epeset s about 0E-6. Ths lm mples ceta bous o a J. P Fo tme sees wh alues s gs the tem( w ) the objecte fucto (.7) wll hae moe tha gs, whle othe tme sees may hae small alues a a small weght, leag to a tem the objecte fucto that has -4 gs beh the ecmal pot. The alues fo a J shoul be chose so that the umbe of sgfcat gs the alues the objecte fucto (.7) oes ot ecee the mamum capacy of 6 gs. The alues fo a J ca easly be eteme a smulato epemet. I ou smulato we typcally use betwee.5 a.0. The meag of the paametes s as follows: etemes the egee of aato of the weghts a J escbes the elate elables of the ffeet tme sees. These two paametes wll appea all othe weghts epessos as well. By chagg the alue of J all weghts ae ajuste that ae elate to the th tme sees,.e. the weghts of the quately chages, fe quates, a lea a ato costats whch the th tme sees appeas. The epesso of the weghts of the quately chages of the ae moel s: whee A J ( w ), = β (.4) T =, (.5) T t = whch s the aeage of the absolute alues of tme sees oe all quates. Ths aeage alue s eplace by some alue close to 0, f s below some theshol alue, sce weghts caot hae a alue of 0.

13 The epesso (.4) esembles epessos that ae popose by Beauleu a Batelsma (004). The most mpotat ffeece s that the efo oles, whee s use ou efo. Theefoe, ou appoach all weghts of the quately chages of oe tme sees ae the same. The motato fo ths pcple s that the ecoclato ajustmets of oe tme sees wll be as costat as possble oe tme, just as the ogal uaate metho of Deto (97). We popose the followg epesso fo the weghts of fe quates F ( w ) F F J = ( ) β β, α (.6) The fst compoet at the ght-ha se, F α, escbes the mpotace of the weght categoy fe quate, compaso wh the othe weght categoes, e.g. quately chages, lea a ato costats. By eceasg the alue of F α ou epemet the alue fo ajustg ths alue., all fe quates ae mae moe mpotat smultaeously. I F α was a we ot see the ecessy of F The seco compoet β escbes the elate mpotace of a specfc fe quate, compae to all othe fe quates. Aalogous to J, s assume that F s a tege-alue aable, wh F 3. J The th compoet β stas fo the elate elably of the tme sees that oles the fe quate. Fally, eotes the squae of the taget alue of the fe quate. It s eplace by some alue close to 0, f alue, because weghts caot be zeo. The epesso of the weght of lea costats s whee L ( w ) L = ( ) L T N ( ) ( c ) c t = = s smalle tha some pespecfe J α β β, (.7) N ( ) = ( c ). c (.8) = L I (.7) α eotes the mpotace of the weght categoy lea costats, compaso wh the othe categoes (quately chages, fe quates a ato costats). Note that the meag of ths compoet s smla to that of epemet was, as was the alue of F α (.6) a hee as well the alue of α touce below. L α ou 3

14 Futhe, the paamete L escbes the elate mpotace of the specfc costat compae to the othe lea costats, whch s smla to F (.6). Aalogous to J a F, s assume that L Ζ wh 3. The epesso (.7) oes ot clue two compoets that ae smla to the th a fouth compoet of (.6). The followg tem s use stea T N ( ) ( c ) c t= = J, β (.9) J.e. a weghte aeage of β, whee β quatfes the elate elably of the th tme sees a eotes the squae of the aeage, absolute leel of that tme sees. As befoe, ths alue s eplace by some alue close to zeo, f s below some pespecfe theshol alue. The aeage (.9) s take oe all tmes-sees that appea the costat. The weghts ae the squae coeffcets c of the costat a the sum of these coeffcets s eote by c, see (.8). The epesso fo the weght of a lease ato costat s J L * J J ~ ( w ) = ( ) β β β, t α (.30) whee ~ t s efe below (.34). I (.3) the epesso (.30) wll be ee fom the weght of a o-lease ato, whch wll be assume to be J J ( w ) = ( ) β β β, t t α (.3) The compoets of ths weght ae que smla to the compoets that ae metoe befoe. The fst compoet α eotes the elate mpotace of the weght categoy ato, compae to the othe weght categoes. The seco compoet escbes the elate mpotace of ato costat, compae to the othe atos. It wll be assume aga that s a tege-alue aable, whose alues ae cetee aou 0. J J The th compoet β β s the geometc mea of β a β. It stas fo the elate elably of the tme sees that appea the eomato a umeato of the ato costat. Fally, the fouth compoet eotes the squae of the taget alue of the ato. The elato betwee (.30) a (.3), the weghts of a lease a olea ato follows fom the followg J J 4

15 t t w t = t w t t t. (.3) I the left ha se of (.3) stas the oot of oe tem of the objecte fucto, coespog to a o-lease ato. The umeato of the ght ha se of (.3) escbes a lease ato. By efo, the eomato of the ght ha se s the weght of the lease ato. Thus, follows that * w = w. (.33) t t t The epesso (.33) caot be use pactce, because t eotes a aable, whose alue s ot kow po to the bechmakg. Theefoe we eplace t (.33) by ~ t t =, t + (.34) + + a weghte aeage of (008). t a t, as popose by Magus a Dalo.5 Eample Let us cose a bechmakg poblem, cosstg of quates a two tme sees a. uppose ally, each quately alue s 0 a the aual bechmaks ae aalable fo both tme sees. These ae 50, 75 a 95 fo the thee cosecute yeas. These aual fgues ae the same fo both tme sees. Now assume that the fst aual algmet s bg, wheeas the seco a the th ae ot. Ths eample s ot ey ealstc, we tetoally choose fo lage scepaces betwee the quately a aual ata oe to llustate moe ly how the bechmakg moel woks. Futhemoe, thee s oe soft ato costat, efe by /. fo t =,,. (.35) t t a the popotoal moel wll be use fo both tme sees. Note that the soft, ato costat s cosstet wh the aual fgues of both tme sees. The elate alues of the weghts of both moel compoets eteme the fluece o the moel outcome. P The paametes of the weghts ae: =, J = 0 (fo all a t), gg ( ), L L α =, C =, gg ( ) * 9.50 a ( ) w = 7.3. w = 00, w = α =, = (fo all ), gg ~ = 5

16 The esults of the bechmakg metho, epcte fgue, ae two tme sees, whose alues cease gaually oe tme. Ths cease s ue to the coecto to the aual bechmaks. Futhe ote as a esult of the ato fom the ffth quate owas t ceases moe aply tha. Dug the fst fou quates, the fluece of the ato costat s eglgble, sce the quates of both tme sees hae to stctly a up to the same aual alues. I the seco a th yea the aual algmet s soft, a theefoe the ato costat s moe mpotat tha fo the fst yea. t Tme sees Tme sees Fgue. The bechmake tme sees Table shows that the ecocle aual fgues of the seco a th yea closely appomate the taget alues. Table. Aual fgues (ecocle); J = 0 Yea Yea Yea 3 Tme ees Tme ees Tme ees / Tme ees uppose we ase the alue of J fom 0 to, a J s left utouche. As a cosequece, tme sees becomes moe mpotat compae to tme sees, amogst othes, the aual algmet of tme sees becomes moe tght. Table ee shows that the ecocle, aual fgues of tme sees appomate the taget alues moe closely, compae to Table, whle the oppose hols tue fo the seco tme sees. 6

17 Table. Aual fgues (ecocle), J = Yea Yea Yea 3 Tme ees Tme ees Tme ees / Tme ees Futhemoe, the ato costat becomes somewhat moe mpotat, compae to the case of the al moel paametes, sce the weghts of the ato costats ae posely coelate to the aeage alue of J a J. Ths ca be see by compag Table wh Table. The bechmake alue of the ato appomates s taget alues of., moe closely table. Fo stace the th yea the ato betwee the aual alues of tme sees a tme sees, s.068 Table a.073 Table. 3. The applcato of the moel 3. oftwae I oe to be useful fo pactcal mplemetato at tatstcs Nethelas, the bechmakg softwae shoul be able to cope wh ey lage ata sets. ecetly Euostat eelope ECOTIM (Bacella, 004), a softwae tool whch supples seeal uaate a multaate bechmakg a tempoal saggegato techques, clug the multaate Deto metho popose by D Fozo a Ma (003). Although ECOTIM s a useful tool, oes ot satsfy the equemets mpose by tatstcs Nethelas fo bechmakg atoal accout ata. The easos fo ths ae that ECOTIM s ot esge fo ealg wh thousas of tme sees, oes ot clue featues lke weghts, ato s, soft costats, a the possbly to combe the popotoal a ae methos of bechmakg to oe moel. tatstcs Nethelas has bult a pototype of bechmakg softwae, usg CPLEX (ILOG, 008) as sole. Ths state-of-the-at, commecal optmzato sole s able to cope wh ey lage ata sets. A bechmakg poblem wh tme sees, each cosstg of up to 3 aual, a quately alues, was taslate to a quaatc optmzato poblem wh appomately 60,000 fee aables a 75,000 costats. By usg CPLEX o a PC wh.00 GZ, Xeo E5335 wh 04 MB am, the optmal soluto was fou 45 secos. owee, oe to couct the tee bechmakg of Dutch supply a use tables, a optmzato poblem shoul be sole, wh appomately 500,000 fee aables, 0,000 equaly costats, a about 300,000 equales fo esug o-egaty. We also pefome successful tests wh the smulate ata of these lage szes. owee, to sae tme, the 7

18 costats we use these tests wee slghtly ffeet tha the oes of the supply a use tables. But we o ot epect that these ffeeces matte much. 3. A fst smulato epemet A smulato epemet was coucte o Dutch supply a use tables of The am of the epemet was to f out whethe all equemets of tatstcs Nethelas ca be copoate the moel. Thee was o teto to obta esults that fully satsfy all ecoomc elatos. Thus, we ot use eactly the same ata as the ecoclato pocess of tatstcs Nethelas, a we ot copoate all elatos amog the aables of the moel. Fo some classes of smla costats we oly woke out eamples. A mpotat featue of the supply a use tables s that they ole fe pce ata a cuet pce ata. Both ks of ata wee use the epemet. By efo, atos of cuet pce alues a fe pce alues ae pce chages. The al alues of these pce chages ha to be pesee as much as possble. The al moel paametes (lke weghts a the choce whethe a popotoal o ae o fe moel shoul be apple to some tme sees) wee eteme by atoal accout epatmet. Pats of the supply a use tables wee aggegate, to keep the moel as small as possble. As state befoe, ths moel stll oles tme sees. The outcomes of the moel wee eewe by subject matte specalsts. The commets wee taslate to chages of the moel set up, fo stace ffeet weghts, othe a / o moe costats. The esults of ths epemet ae ey goo. All emaks of subject-matte specalsts coul be popely taslate to chages of the moel set-up. owee, ths was ot always staghtfowa. The metho popose ths pape oly allows fo smple ato costats (lke A / B ), s ot possble to copoate moe complcate costats, lke A / B C / D o AB, fo stace the chage of the tae mags shoul be appomately equal to the chage of cosumpto. Theefoe some a-hoc solutos ha to be mplemete to clue all eleat ecoomc elatos the moel. Fo stace: the peseato of the al alues of the ato betwee tae mags a cosumpto fo each quate. Although oly the most mpotat ecoomc elatos wee copoate the moel, most of the outcomes closely esemble the publshe fgues of the supply a use table: the ffeece was smalle tha 0.3%-pot fo most of the ma ecoomc catos. A eplaato fo ths goo esult s that the al scepaces of most costats wee small. As the bechmak pocess stats wh comple systems of atoal accouts, thee wee o olatos of the costats wh the quates a yeas. The oly scepaces wee betwee quately a aual 8

19 fgues, but these wee maly small. owee, some eceptoal cases the al scepaces of the costats wee lage. The fomato o how to eal wh these cossteces ae ot clue the moel. ce lage al scepaces the ata ae ae, a the ese soluto s ffeet fo each poblem, s ot feasble to aapt the moel fo each case, befoeha. Istea, lage scepaces wll be etecte a ecocle by ha, befoe the Deto metho s apple. 4. Cocluso tatstcs Nethelas s plag to mplemet a multaate Deto metho s poucto pocess fo bechmakg atoal accouts ata. The metho that s tee to use was ogally popose by Bkke a Bujtehek (006). owee to satsfy all equemets mpose by tatstcs Nethelas ths metho ha to be etee wh ato costats, soft costats, equaly costats a fe quates. The etee metho s fomulate as a easly uestaable quaatc pogammg poblem. To apply ths metho to Dutch atoal accouts s cucal that ey lage ata sets ca be hale,.e. oe 00,000 aables. Eamples of such lagescale applcatos of tegato methos the leatue ae ae. A smulato eecse, by usg a state-of-the-at optmzato sole lke CPLEX showe that ths s feasble. A smulato epemet was coucte to test whethe s possble to copoate the costats of the Dutch supply a use tables the popose moel. Befoe applyg the bechmak metho, the tables wee cleae fom a few lage scepaces that caot be sole appopately by usg a mathematcal moel. The esults wee pose: commets of atoal accout epets o the fst esults coul be aequately taslate to chages of the moel. Also, the compaso of the esults wh publshe fgues of Dutch atoal accout oes ot show much (ueplaable) eato. I 009 fal tests wee cae out to poe that the tee bechmak pocess wll lea to esults that ae fully acceptable fo subject-matte specalsts. Moeoe smulato epemets ae coucte o othe pats of the atoal accouts tha the supply a use tables, such as the secto accouts a the labou accouts. I the log tem tatstcs Nethelas s plag to tegate the bechmakg pocess of these accouts. eeal methoologcal questos about the cuet Deto moel ae ope fo futhe eseach. These ae: - Deepe uestag of the elatos betwee epessos of the weghts a the popetes of the moel; 9

20 - oug: the esults of the bechmakg moel hae to be oue to tege alues. Cuetly, the oug pocess s coucte afte the bechmakg pocess. Both pocesses may be combe; - No-lea costats: A eteso of the methoology fo moe complcate o-lea costats (fo stace A / B C / D o AB ) mght mpoe the pactcal ably of the moel. Appe A Poof of the popetes of the moel I ecto.4 thee ese popetes of a bechmakg moel ae ge. ee, we show that the moel ee satsfes the popetes ) aace of put ata, ) symmety of atos a 3) aace of moel choce. Iaace of put ata Iaace of put ata meas that the multplcato of all put ata by the same oegate scala, leas to outcomes that ae chage by the same facto. The moel we popose satsfes ths popety, sce multplyg each of the aables,, b, b a z by a oegate scala oes ot chage the objecte fucto a the costats of the moel. Fo stace fo the pat of the objecte fucto that escbes the ae mutatos, hols tue that: N = t= N T T = t= ( λ λ ) ( λ λ ) ( ) ( ) J J β β λ. = (A.) The eae ca easly wok out that the othe pats of the moel also satsfy ths popety. ymmety of atos ymmety of atos meas that oes ot matte fo the esults whethe a ato s efe by /, (A.) o by s ecpoke / /. (A.3) Fo coeece, some of the subscpts of, a ae omte. The coespog tems the objecte fucto ae: 0

21 ~ a (A.4), ~ (A.5) whee. J J The popety of symmety s satsfe f (A.4) a (A.5) ae the same. Below wll be show that ths hols tue. Aalogous to the efo (.34) of ~ (A.4), the efo of ~ (A.5) s, ~ (A.6) The efo (A.6) s obtae fom (.34) by techagg a. a eplacg by /, Note that ~ ~ (A.7) By usg ths esult, follows that, ~ ~ ~ (A.8) o equaletly that (A.4) s the same as (A.5), whch fshes the poof that the symmety of ato popety s fulflle. Iaace of moel choce The aace of moel choce popety meas that the esults of the ae moel a the popotoal moel ae the same fo costat tme sees. Cose some tme sees a suppose that s al quately alues ae costat,.e. = fo all t, the oe compoet of the objecte fucto of the ae moel ca be ewte by

22 ( ) ( ) w A = w A, (A.9) a oe compoet of the objecte fucto of the popotoal moel s w P, (A.0) The aace of moel choce popety s fulflle, f (A.9) a (A.0) ae the same. That s, f: P ( w ) A = ( w ) A = ( w / ). / (A.) ce the epessos of the weghts (.3) a (.4) meet (A.), the moel popose ths pape satsfes the aace of moel choce popety. efeeces Bacella,. (994), ECOTIM: a Pogam fo Tempoal Dsaggegato of Tme ees, Pape pesete at the jot INEE a Euostat Quately Natoal Accouts Wokshop Pas o Decembe 5th a 6th, 994. Beauleu, J.J. a E.J. Batelsma (004), Itegatg Epeue a Icome Data: What to Do wh the tatstcal Dscepacy? Face a Ecoomcs Dscusso ees , Boa of Goeos of the Feeal esee ystem (U..). Bkke,.P. a. Bujtehek (006), Algmet of Quately ecto Accouts to Aual Data, CB Voobug, B487-45C7-4C3C-ACD0-oEC86E6CAFA/0/Bechmakg_QA.pf Bloem A.M.,.J. Dppelsma a N.O. Maehle (00), Quately Natoal Accouts, Maual Cocepts, Data ouces a Complato, IMF, Washgto, D.C. Che, W. (006), Balace ystem of Iusty Accouts fo the U.. a tuctual Dstbuto of tatstcal Dscepacy, Pape pesete at the NBE-CIW umme wokshop, July , Cambge. Chow, G.C. a L, A.L. (97), Best lea ubase tepolato, stbuto a etapolato of tme sees by elate sees, The eew of Ecoomcs a tatstcs, 53, Cholette, P.A. a E.B. Dagum (994), Bechmakg tme sees wh autocoelate suey eos. Iteatoal tatstcal eew, 6,

23 Dagum E.B. a P.A. Cholette (006), Bechmakg, Tempoal Dstbuto a ecoclato Methos fo Tme ees, pge, New Yok. Deto F.T. (97), Ajustmet of mothly o quately sees to aual totals: A Appoach base o quaatc mmzato, Joual of the Ameca tatstcal Assocato, 66 (333), D Fozo, T. e M. Ma (003), Bechmakg systems of seasoally ajuste tme sees accog to Deto s moemet peseato pcple. Uesy of Paoa, Wokg Pape Dub, J. a B. Queelle (997). Bechmakg by state space moels. Iteatoal tatstcal eew, 65, llme. C. a A. Tabels (987), Bechmakg of ecoomc tme sees, Joual of the Ameca tatstcal Assocato, 8, ILOG.A., CPLEX.0 efeece maual a Use s Maual, 008. Magus, J.., J.W. a Togee a A.F. e Vos, (000), Natoal Accouts Estmato Usg Icatos Aalyss, eew of Icome a Wealth (46), pp Magus, J.. a D. Dalo, (008), O the estmato of a lage spase Bayesa system: The ae pogam, Computatoal tatstcs & Data Aalyss, 5, pp Ncola, V., (000), Balacg Lage Accoutg ystems: A Applcato to the 99 Itala I-O Table, pape pesete at the 3th Iteatoal Cofeece o Iput-Output Techques, Uesy of Maceata, Italy. Öhlé,. (006), Bechmakg a easoal Ajusmet A stuy of wesh GDP tatstcs wee toe, J..N., D.G. Champeowe a J.E. Meae, (94), The Pecso of Natoal Icome Estmates, eews of Ecoomc tues, ol. 9, pp Woe, D., P. Key, U. zk a I. Weeakkoy, (998), elably a qualy catos fo Natoal Accouts Aggegates. ON (Loo). 3