A mixed splicing procedure for economic time series

Transcription

1 Esadísica Española Volumen 56, númeo 183 / 2014, pp A mixed splicing pocedue fo economic ime seies Ángel de la Fuene Moeno (*) FEDEA e Insiuo de Análisis Económico (CSIC) Absac his noe develops a flexible mehodology fo splicing economic ime seies ha avoids he exeme assumpions implici in he pocedues mos commonly used in he lieaue. I allows he use o spli he equied coecion o he olde of he seies being linked beween is levels and gowh aes on he basis of wha he knows o conjecues abou he pesisence of he facos ha accoun fo he discepancy beween he wo seies ha emeges a hei linking poin. he ime pofile of he coecion is deived fom he assumpion ha he eo in he olde seies eflecs he inadequae coveage of emeging secos o aciviies ha gow fase han he aggegae. Keywods: linking, splicing, economic seies AMS Classificaion: 62P20, 91B82 Un pocedimieno mixo de enlace paa seies empoales económicas Resumen En esa noa se desaolla una meodología flexible paa enlaza seies económicas que evia las hipóesis exemas que esán implícias en los pocedimienos más uilizados en la lieaua. Con el pocedimieno popueso, la coección que ha de inoducise a la más anigua de las seies que se enlazan puede epaise ene sus niveles y sus asas de cecimieno sobe la base de lo que el analisa sabe o conjeua sobe la pesisencia de los facoes que explican la discepancia ene las dos seies que emege en el puno de enlace. El pefil empoal de la coección se deiva del supueso de que el eo conenido en la seie más anigua efleja una inadecuada cobeua de secoes o acividades emegenes que cecen más ápido que el agegado. Palabas clave: enlace de seies económicas Clasificación AMS: 62P20, 91B82 * his pape is pa of a eseach pojec cofinanced by BBVA Reseach and Fundación Rafael del Pino. Addiional financial suppo fom he Spanish Minisy of Science and Innovaion unde pojec ECO is also gaefully acknowledged. I would like o hank he paicipans in he Economic Hisoy Wokshop a Univesidad Calos III and wo anonymous efeees of his jounal fo hei useful commens and suggesions.

2 108 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies Vol. 56 Núm. 183 / Inoducion In ode o consuc long ime seies of economic aggegaes, i is geneally necessay o piece ogehe seveal heeogeneous shoe seies. Heeogeneiy aises even in official naional accouning daa due o changes in benchmak yeas, which ae ofen accompanied by mehodological changes and by impovemens in he qualiy of pimay daa souces and in esimaion mehods. hings ae geneally even wose when we face he ask of linking unofficial seies consuced by hisoians and ohe eseaches using incomplee daa and diffeen mehodologies. he poblem has no easy soluion. Naional saisical insiues can (and someimes do) help miigae i by ecalculaing back seies of key aggegaes using cuen mehods and cieia in conjuncion wih deailed souce daa fo ealie peiods, bu even in his case hee is no sue way o know how ealie esimaes would have changed if, fo insance, new o impoved daa souces had been available ealie on. Fom he pespecive of independen eseaches, such deailed econsucions ae geneally ou of he quesion and he only feasible saegy involves he use of simple splicing o linking echniques fo pasing ogehe a se of seies on a given vaiable. he linking pocedues commonly used in he lieaue geneally involve he backwad exapolaion of he mos ecen available seies using he gowh aes of olde seies (his is wha I will call pue eopolaion fo sho) 1 o inepolaion beween he benchmak yeas of successive seies. he basic idea is simila in boh cases: we coec he olde seies so ha i maches he newe one a is saing poin while eaining some of is feaues. he naue of he coecion is, howeve, vey diffeen in each case. Reopolaion peseves he peiod-by-peiod gowh aes of he olde seies and places he enie buden of he coecion on is levels, while inepolaion peseves he saing (o benchmak) level of he olde seies and adjuss is gowh aes as needed. Boh pocedues ewie hisoy bu hey do so in vey diffeen ways. As Pados de la Escosua (2006) wans, which mehod is chosen can make a vey big diffeence, especially when we ae dealing wih long peiods of ime. 2 he linking of economic ime seies is heefoe a delicae execise ha should pobably be handled wih a bi moe cae han i ofen has been execised in he lieaue. his noe develops a new splicing pocedue ha may be a useful ool in his egad. he poposed mehod povides an inemediae opion beween he wo sandad mehods skeched above ha avoids hei ahe exeme implici assumpions and allows he eseache o disibue he equied coecion beween he levels and he gowh aes of he olde seies on he basis of wha he knows o conjecues abou he pesisence of he facos 1 As fa as I can ell, hee is no seled sandad eminology in his aea. Diffeen expessions (including backcasing, backwad pojecion, eopolaion and back calculaion) ae used o efe geneically o he backwad exapolaion of ime seies and o specific ways o go abou i. 2 his auho analyzes he implicaions of applying diffeen splicing pocedues o Spanish GDP daa coveing he peiod Accoding o his calculaions, pue eopolaion of he mos ecen seies leads o an upwad evision of oiginal GDP esimaes fo 1954 ha exceeds 30%. He obseves ha such a lage coecion would significanly ale cuen views abou Spain s elaive income level in he mid 20 h cenuy in a diecion ha does no seem eniely plausible.

3 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 109 ha accoun fo he discepancy beween he old and new seies ha emeges a hei linking poin. I also deives he ime pah of he equied coecion fom he assumpion ha such discepancies aise fom impovemens in he coveage of emeging secos o aciviies in he basic daa undelying he new seies. his may no always be he case, bu i is ceainly one of he usual suspecs. he es of he noe is oganized as follows. Secion 2 biefly descibes he simples vesions of he wo sandad linking pocedues used in he lieaue and highlighs he assumpions implici in each of hem egading he ime pofile of he "eo" conained in he olde seies. Secion 3 agues ha hese assumpions ae oo exeme o be plausible in many cases and inoduces a new pocedue ha can accommodae inemediae siuaions. Secion 4 concludes wih a bief illusaion of how he poposed pocedue has been used in he consucion of long employmen seies fo Spain. 2. Simple splicing pocedues: a quick eview Figue 1 illusaes a ypical splicing poblem. We have wo seies, X and Y, efeing o he same economic aggegae. he olde seies, X, sas a 0 and exends unil. A his ime a new and in pinciple bee seies, Y, is inoduced. As is geneally he case, he wo seies do no agee a hei linking poin,. I will denoe by D he discepancy beween he new and he old seies a he linking poin and by d = lny ln X = y x [1] he popoional o logaihmic diffeence beween hem. I is convenien (alhough no eniely accuae) o hink of d as he measuemen eo" conained in he olde seies a he linking poin. In pinciple, measuemen eo may affec all ems of he olde seies bu i can only be obseved a. Figue 1 wo seies o be spliced Vol.56. Núm. 183 / 2014

4 110 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies he poblem we face is ha of exending he moe ecen seies back o ime 0 aking as a efeence he olde one. As noed above, hee ae wo simple sandad soluions ha embody alenaive hypoheses egading he ime pofile of he "measuemen eo" conained in he olde seies: eopolaion and inepolaion. Reopolaion woks by exending he new seies backwad fom ime using he gowh aes of he old seies. As illusaed in Figue 2, he idea is o "aise" he olde seies by a consan popoion, especing is ime pofile, unil i maches he new seies a he linking poin. Using lowe case lees o indicae ha we ae woking wih logaihms, he eopolaion of Y aking X as a efeence will be given by yˆ = x + ( y x ) x + d fo 0 [2] Noice ha he spliced seies coincides wih Y a ime and peseves he gowh aes of X fo he peiod befoe he linking poin, ha is yˆ = x fo 0 and yˆ = y [3] he implici assumpion is ha he eo conained in he olde seies d = y x [4] Vol. 56 Núm. 183 / 2014 emains consan ove ime -- ha is, ha i aleady exised a ime 0 and ha is magniude, measued in popoional ems, has no changed beween 0 and. Hence, in ode o ecove he "coec" value of he magniude of inees, all we have o do is add o he olde seies (measued in logs) he popoional discepancy beween he wo seies ha is obseved a he linking poin,. d In he inepolaion mehod, he seies ae linked by focing he backwad exension of he new seies o go hough a given poin in he old one, say x o, which will geneally coespond o is base o benchmak yea. (See Figue 2). he pocedue assumes ha he eo in he olde seies has been geneaed eniely beween 0 and. In is simples fom, i also assumes ha he popoional eo inceases linealy wih ime. Unde hese assumpions, he coec value of he vaiable pio o he linking poin can be ecoveed by adding o he olde seies (in logs) a linea funcion of ime: i yˆ = x + ( y x ) x d = + [5] Poceeding in his manne, we peseve he oiginal value of he olde seies in is base yea, bu no is gowh aes, which ae aised by a consan facion of d ha depends on he lengh of he peiod beween he base yea of he olde seies and he linking poin, 1 i yˆ = x + d fo 0 and yˆo = x [6] o

5 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 111 As a esul, he ime pofile of he spliced seies can be quie diffeen fom ha of he olde seies (see Figue 2). Figue 2 Alenaive splicing pocedues I is useful o noe ha boh pocedues ae vaiaions on a common heme. In boh cases, he linked seies is consuced by adding o he olde seies an esimae of he eo conained in i ha is based on he only diec obsevaion of his magniude available o he analys: he one coesponding o he linking poin. Hence, he spliced seies obained wih pocedue j will be given by 3. A mixed splicing pocedue ˆ d = d j ˆ ˆ j 0 ˆ j y = x + d fo wih d = ˆ d = d i When should each of he splicing pocedues descibed in he pevious secion be used? In many counies, base-yea esimaes of employmen, oupu and ohe aggegaes ae buil on a subsanially moe hoough analysis han esimaes fo non-benchmak yeas. Ohe hings equal, his would be an impoan agumen in favo of he inepolaion pocedue used among ohes by Maddison (1991) and Pados de la Escosua (2003), which peseves base-yea esimaes. On he ohe hand, hings ae seldom equal fo new base yeas ae ofen accompanied by impovemens in he pimay daa and in he esimaion mehods ha ae used o consuc he naional accouns. 3 As a esul of such [7] Vol.56. Núm. 183 / An addiional souce of discepancies beween he wo seies is he inoducion of mehodological changes. he mos easonable way o eliminae his ype of disconinuiy would be o econsuc he olde seies using he new mehodology pio o splicing i wih he new seies. I am assuming his

6 112 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies impovemens, he esimaed volume of aciviy is geneally evised upwad, pesumably because bee daa and esimaion mehods allow naional accounans o measue moe accuaely emeging aciviies o secos ha wee no adequaely coveed in he olde seies. When his is he case, i seems plausible o conjecue ha i) he eo ha emeges in he new base yea aleady exised o some exen in pevious yeas and will heefoe affec he olde seies in is eniey and ii) ha he size of such eo has been gowing ove ime because coveage poblems end o be especially sevee in emeging secos ha have a gowing weigh in he aggegae. Hence, wha we may expec o be a ypical siuaion following he inoducion of a new benchmak will no fi he assumpions ha ae implici in sandad linking pocedues. his suggess ha i may be a good idea o develop an alenaive splicing mehodology ha can accommodae such siuaions. I will efe o he poposed pocedue as he mixed splicing mehod because i will occupy an inemediae posiion beween he wo sandad pocedues in he sense ha he equied coecion o he olde seies will be disibued beween is iniial level and is gowh aes. One way o descibe he diffeence beween he wo appoaches eviewed in he pevious secion is in ems of hei assumpions egading he iniial eo in he olde seies, d o. his magniude is se o zeo in inepolaion and o d in eopolaion. A naual way o poceed when neihe of hese exeme assumpions seems plausible is o paameeize he iniial eo so ha Vol. 56 Núm. 183 / 2014 whee ρ (0,1) is a fee paamee. ˆ m d o = ρ d he second change I will inoduce in he splicing pocedue has o do wih he ime pofile of he esimae of he eo conained in he olde seies. In is simples fom, inepolaion assumes ha his eo inceases linealy wih ime. Howeve, his is no he mos plausible assumpion if, as i ofen seems likely, he souce of he eo is he deficien coveage of ceain aciviies whose weigh in he aggegae inceases ove ime. In his case, he ime pah of he eo will depend on he ae of gowh of such aciviies elaive o he es of he economy. o model he evoluion of he eo in a simple way, I will assume ha he aio beween hese wo gowh aes is consan. I will assume, in paicula, ha he eo conained in he olde seies, D is a consan facion θ of he volume of a se of aciviies, Z, ha ae deficienly measued. he "eal value" of he seies of inees will hen be given by [8] Y = X + D = X +θz [9] has aleady been done o he exen ha i is possible, so ha emaining discepancies beween he seies ae due only o impovemens in pimay daa and in esimaion mehods.

7 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 113 I will also assume ha he gowh faco of X faco of G. ha is, denoing by X he gowh faco of, Z is a consan muliple of he gowh I will assume ha X +1 X = G [10] Z + 1 = µ G [11] Z whee µ is a consan whose value will be deemined lae on. Leing z = Z/X, we obain a simple diffeence equaion whose soluion is of he fom z + 1 Z 1 µ GZ Z = + = = µ = µ z [12] X G X X + 1 Dividing boh sides of (9) by, X z = z oµ [13] Y X + θz = = 1+ θ z [14] X X and aking logs of his expession, we obain a convenien appoximaion fo d : 4 Using (13), his expession implies ha Y d ln = = ln( 1+ θz) θz [15] X Vol.56. Núm. 183 / 2014 d z = zo = do θ θ µ µ Nex, we can ecove he value of µ ha is implici in ρ. Evaluaing (16) a ime and ecalling ha, by assumpion, d o = ρd we have [16] 4 As a efeee has noed, an exac soluion fo d can be easily consuced (see he Appendix). Hence, he appoximaion used in (15) can be avoided. In pacice, boh appoaches ae likely o yield vey simila esuls and he appoximae fomulae fo he spliced seies given in he ex ae maginally moe convenien.

8 114 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies whee we can solve fo µ, d do = d µ ρ µ [17] 1 1 ρµ µ = ρ Finally, we subsiue (18) ino (16) o obain he following expession fo he ime pah of he popoional eo in he olde seies: 1 [18] 1 1 d do = ρd = dρ ρ ρ Hence, he spliced seies will be given by [19] Vol. 56 Núm. 183 / 2014 m ˆ ˆm fo 0 wih ˆm y = x + d d = d ρ [20] As illusaed in Figue 3, equaion (19) descibes a ime pah fo he esimaed eo of he olde seies ha is quie diffeen fom hose assumed in he splicing pocedues descibed above, given in equaion (7). Unde he hypoheses of his secion, he eo (d ) will no be consan o a linea funcion of ime bu will gow a an inceasing ae because he weigh of he emeging aciviies ha pesumably ae no well coveed in he old seies will incease as we appoach he linking yea. his implies ha he appopiae coecion o he gowh ae of he oiginal seies will no be consan bu inceasing in ime and will be an inceasing funcion of he paamee ρ ha capues ou hypohesis egading he seveiy of he iniial eo.

9 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 115 Figue 3 Esimaed ime pah of he eo conained in he olde seies wih diffeen splicing pocedues d ˆd ˆd i ˆd m ρd 0 he one decision he analys has o make when using he mixed splicing pocedue is ha of assigning a value o he paamee ρ ha measues he seveiy of he eo in he olde seies a ime 0. While his is no geneally an obsevable magniude, he analys may have access o exenal infomaion ha can be used o appoximae is value. In his connecion, i may be useful o noe ha he mixed pocedue implicily fixes he iniial value of he spliced seies, ŷ om, as a weighed aveage of he values a ime 0 of he olde seies and he seies obained by eopolaion of he new seies: yˆ ˆ o m = xo + do m = xo + ρd = (1 + ρ ρ) xo + ρd = (1 ρ) xo + ρ( xo + d ) = (1 ρ) xo + ρ yˆ o Hence, a compaison of he iniial values of hese wo seies wih some ouside esimae of he elevan magniude a 0 o some neaby yea may allow us o assess he elaive plausibiliy of hese wo esimaes and help us se he value of ρ. In he absence of ouside infomaion, i may sill be possible o fomulae a plausible conjecue abou he value of ρ on he basis of whaeve is known abou he facos ha accoun fo he discepancy beween he wo seies a he linking poin. While his is no an ideal siuaion, i seems pefeable o having o choose beween he wo exeme assumpions implici in he pocedues discussed above. A he vey leas, he poposed pocedue bings ou ino he open a poblem ha boh of he sandad appoaches hide ha we don' know he ime pofile of he eo in he olde seies and allows us o y o make a easonable guess on he basis of whaeve infomaion may be available. [21] Vol.56. Núm. 183 / 2014

10 116 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies An alenaive paameeizaion An alenaive paameeizaion of he mixed pocedue can also be useful a imes. Insead of seing ρ diecly, we can wok in ems of a paamee H ha measues he pesisence ove ime of he measuemen eo ha emeges a. H is defined as he half-life, going backwad in ime, of he measuemen eo, i. e as he ime equied fo d o dop o half he value we obseve a. By definiion, H mus saisfy he following expession: d H = 0.5* d Using (19) evaluaed a -H in his expession, we have [22] fom whee ( H ) d H = dρ = 0.5 * d [23] ρ H and subsiuing his back ino (19) H = ρ = = 2 H [24] Vol. 56 Núm. 183 / H 1 d = dρ = d = d 2 2 which gives us he ime pah of he coecion o he olde seies as a funcion of H ahe han ρ. Figue 4 shows he pofile of gadual adjusmen ha is geneaed by equaion (25) wih wo alenaive values of H. Wih a half-life of en yeas, fo insance, a discepancy of 10% beween he wo seies a he linking poin implies a coecion of 5% o he olde seies a -10, 2.5% a -20 and 0.625% a -40. H [25]

11 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 117 Figue 4 ime pofile of he coecion o he olde seies geneaed by he mixed splicing pocedue facion of he disconinuiy a ha pesiss yeas fom he linking poin (going backwad in ime) H = 10 H=20 When no ouside infomaion is available o help us fix he iniial value of he linked seies, i may be moe naual o make some assumpion egading he half-life of he measuemen eo. Woking wih his paamee may be helpful in ensuing he consisency of ou assumpions when he consucion of he spliced seies equies he linking of seveal segmens of diffeen size. 4. Some illusaions A few commens on a specific applicaion may pehaps be useful as an illusaion of how he poposed pocedue may be applied and of is poenial advanages. he Spanish Saisical Insiue (INE) has published seies of naional and egional accouns ha ake as a benchmak he yea 2000 (INE00) and a spliced seies ha exends his seies back o 1995, which was he base yea fo he pevious official seies. 5 he spliced seies has been consuced (essenially) by inepolaion in ode o espec he oiginal esimae fo 1995, which he INE consides quie eliable due o is "sucual chaace" (INE, 2007). Howeve, compaison of he wo seies in hei common yea of 2000 eveals a sizable upwad evision of employmen, wih an incease of 7.55% in he esimaed numbe of jobs. Since his discepancy is disibued by he INE ove only 5 yeas, he ime pofile of is spliced employmen seies is vey diffeen fom ha of he olde one and no eniely plausible. As shown in Figue 5, fo insance, INE s spliced seies implies a educion in aveage labo poduciviy beween 1995 and 2000, while he pevious official seies (INE95) showed a posiive, alhough modes, incease in his vaiable duing he same peiod. Vol.56. Núm. 183 / Boh seies ae available a INE (2009).

12 118 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies Figue 5 Aveage labo poduciviy in Spain consan pices of 2000 (100 = 2000 in INE00) INE00 INE95 INE's splice alenaive splice Vol. 56 Núm. 183 / 2014 he upwad evision of employmen seems o be due mosly o impovemens in he design of he Spanish Labo Foce Suvey (EPA) ha have allowed a moe accuae measuemen of he numbe of pa-ime wokes. 6 I seems had o ague ha poblems wih he measuemen of pa-ime jobs did no aise unil 1995, bu i is pobably ue ha he seveiy of he poblem has inceased ove ime as pa-ime wok has become moe fequen in Spanish sociey. Hence, neihe of he exeme assumpions made by he sandad linking pocedues discussed in secion 2 seems o be appopiae in his case. As fa as I know, hee ae no had daa available ha can be used o se he value of ρ. I seems likely, howeve, ha he eo due o his poblem was only maginally smalle in 1995 han in 2000, which poins o a value of ρ only slighly below 1. Figue 5 compaes INE s spliced poduciviy seies (INE s splice) wih an alenaive one ha has been consuced by he mixed pocedue wih a ρ of 0.90 (see De la Fuene, 2010). While hee is absoluely no guaanee ha his is he coec value of he paamee, he coecion does a leas yield a spliced seies wih a ahe moe plausible ime pofile. A second illusaion involves he pojecion of he Spanish employmen seies back o he mid 1950s. Pue eopolaion of my spliced seies, using as a efeence he hisoical seies consuced by Maluque and Llonch (M&L, 2005) woking wih naional accouns daa wih benchmaks in 1986 and ealie yeas, would lead o a 13% upwad evision of hese auhos oiginal esimae of Spanish employmen in 1955, as illusaed in Figue 6. 6 See fo insance Albacee and Laboda (2005).

13 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 119 Figue 6 Employmen seies fo Spain logaihmic scale In ode o deemine whehe such a lage evision may be plausible, we need o analyze he souces of he discepancy ha exised beween he wo seies in 1995 (which is caied back unchanged o 1955 by he eopolaion pocedue). I uns ou ha he beak in he seies in 1995 is due o wo facos of appoximaely equal weigh. One is he aleady menioned change in he mehodology of he Labo Foce Suvey ha has led o he emegence in 2000 of a lage numbe of pa-ime jobs held mosly by women-- 90% of which have been caied back o 1995 in my spliced seies. he ohe is a change in he concep of employmen used in he Spanish Naional Accouns, which supplied daa on he numbe of employed wokes unil he 1995 base was inoduced and swiched a his poin o he numbe of jobs. o se he value of ρ we need o y o esablish how elevan hese wo facos (pa ime employmen and wokes holding moe han one job) wee in he mid 50s. Founaely, hee ae some ouside souces ha can be used o shed some ligh on his quesion. One is he 1950 Census, which ells us ha only 0.55% of acive wokes declaed o have a seconday occupaion on op of hei pimay job (INE, 1950, volume II, able V). he second is an independen esimae of he numbe of jobs in he yea of inees consuced by he eseach depamen of a lage Spanish bank (FBBV, 1999), which is consideably close o M&L s oiginal esimae han o he eopolaed seies. Hence, he available infomaion suggess ha mos of he eo in he olde seies was no hee in 1955 and poins heefoe o a low value of ρ. Moe infomal evidence also poins in he same diecion, as we know ha female labo foce paicipaion was much lowe a few decades back and ha he use of pa-ime conacs was much less common han i is now. On he basis of hese consideaions, I have chosen a value of 0.10 fo ρ, obaining he spliced seies ha is shown in Figue 6. 7 Vol.56. Núm. 183 / See De la Fuene, A. (2009) fo addiional deails.

14 120 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies Appendix: An exac soluion As a efeee has poined ou, an exac soluion fo ˆd m is easily obained and is only maginally less convenien han he appoximae soluion given in he ex fo he spliced seies unde he mixed pocedue. Insead of using an appoximaion, we can simply subsiue equaion (13) in he ex ino equaion (15) o obain Evaluaing his expession a = 0 hen yields Y d 1 = n = 1n 1+ z = 1n 1+ zo [A.1] X ( θ ) and subsiuing (A.2) back ino (A.1): ( θ ) ( θ µ ) d o d = ln 1+ z θz = e 1 o o o d ( µ ( e d o )) [A.2] = ln [A.3] Nex, we can ecove he value of µ ha is implici in ρ. Evaluaing (A.3) a ime and ecalling ha, by assumpion, d o = ρd we have d ( µ ( e ρd )) = ln [A.4] Vol. 56 Núm. 183 / 2014 whee we can solve fo µ, 1 e d 1 ρ d µ = e 1 [A.5] Finally, we subsiue (A.5) ino (A.3) o obain he following expession fo he ime pah of he popoional eo in he olde seies: d = ln 1 + e 1 = ln 1+ e 1 e 1 e 1 e 1 e 1 d e 1 ρd 1 1 d ρd d ρd ρd ( ) ( ) ( ) ( ) ( ) which in un yields he following exac expession fo he spliced seies d Y 1 X ( e 1) ( e ρ = + 1) m d 1 [A.6] [A.7]

15 Ángel de la Fuene A mixed splicing pocedue fo economic ime seies 121 Refeences ALBACEE, R. AND A. LABORDA (2005). «Cambios en la Encuesa de Población Aciva y en la Conabilidad Nacional.» Cuadenos de Infomación Económica 186, mayo-junio, pp DE LA FUENE, A. (2010). "Un enlace alenaivo de los agegados de VAB y empleo ene las bases 1995 y 2000 de la Conabilidad Nacional y Regional." Esadísica Española 52 (173), pp DE LA FUENE, A. (2009). Seies enlazadas de algunos agegados económicos nacionales y egionales, Vesión 2.1. Mimeo, Insiuo de Análisis Económico, CSIC, Bacelona. hp:// FUNDACIÓN BBV (FBBV, 1999). «Rena nacional de España y su disibución povincial. Seie homogénea». Años 1955 a 1993 y avances 1994 a Bilbao. INSIUO NACIONAL DE ESADÍSICA (INE, 1950). «Censo de la Población de España y eioios de su sobeanía y poecoado según el empadonamieno ealizado el 31 de diciembe de 1950». In INEbase, eleconic daabase: Demogafía y población. Cifas de población y censos demogáficos. Censos de población desde hp:// INSIUO NACIONAL DE ESADÍSICA (INE, 2007). «Conabilidad egional de España, base Seie homogénea Noa meodológica.» In INEbase eleconic daabase: Economía: Cuenas Económicas: Conabilidad Regional de España. Madid. hp:// INSIUO NACIONAL DE ESADÍSICA (INE, 2009). «Conabilidad Regional de España». In INEbase eleconic daabase: Economía: Cuenas Económicas: Conabilidad Regional de España. Madid. hp:// MADDISON, A. (1991). «A Revised Esimae of Ialian Economic Gowh » Banca Nazionale del Lavoo Quaely Review, pp MALUQUER, J. AND M. LLONCH (2005). «abajo y elaciones laboales». In A. Caeas y X. afunell, coodinadoes. Esadísicas hisóicas de España, siglos XIX-XX, second ediion. Fundación BBVA, Bilbao, pp PRADOS DE LA ESCOSURA, L. (2003). «El pogeso económico de España, ». Bilbao: Fundación BBVA. PRADOS DE LA ESCOSURA, L. (2006). «Assessing he bias in spliced GDP seies: evidence fom Spain s Naional Accouns, » Mimeo, Univesidad Calos III de Madid. Vol.56. Núm. 183 / 2014