Do Spanish regions converge? A time-series approach using fractional cointegration

Full Te ms & Condi ions o access and use can be ound a h ps://www. and online.com/ac ion/jou nalIn o ma ion?jou nalCode= aec20 Applied Economics ISSN: (P in ) (Online) Jou nal homepage: h ps://www. and online.com/loi/ aec20 Do Spanish egions con e ge? A ime-se ies app oach using ac ional coin eg a ion Ma iam Kamal & Josu A eche To ci e his a icle: Ma iam Kamal & Josu A eche (02 Jan 2024): Do Spanish egions con e ge? A ime-se ies app oach using ac ional coin eg a ion, Applied Economics, DOI: 10.1080/00036846.2023.2293089 To link o his a icle: h ps://doi.o g/10.1080/00036846.2023.2293089 © 2024 The Au ho (s). Published by In o ma UK Limi ed, ading as Taylo & F ancis G oup. View supplemen a y ma e ial Published online: 02 Jan 2024. Submi you a icle o his jou nal View ela ed a icles View C ossma k da a Do Spanish egions con e ge? A ime-se ies app oach using ac ional coin eg a ion Ma iam Kamal and Josu A eche Depa men o Quan i a i e Me hods, Uni e si y o he Basque Coun y, Bilbao, Spain ABSTRACT This a icle in es iga es economic con e gence in e ms o eal income pe capi a be ween he au onomous egions o Spain o e he pe iod 1955–2020. In o de o con e ge, he se ies should be coin eg a ed. This necessa y condi ion is checked using wo es ing s a egies ecen ly p o- posed o ac ional coin eg a ion, inding no e idence o coin eg a ion, which ules ou he possibili y o con e gence be ween all o some o he Spanish egions. As an addi ional con ibu- ion, an ex ension o he c i ical alues o Nielsen’s (2010) es o ac ional coin eg a ion is p o ided o a di e en numbe o a iables and sample sizes om hose o iginally p o ided by he au ho , i ing hose conside ed in his a icle. KEYWORDS F ac ional in eg a ion; ac ional coin eg a ion; long memo y; pe sis ence JEL CLASSIFICATION C12; C22; C32 I. In oduc ion Economic con e gence has been one o he main ocal poin s o he empi ical li e a u e on economic g ow h. I implies ha income gaps be ween coun- ies/ egions end o disappea , hence in ol ing con e gence o a single s eady s a e (equilib ium). The Eu opean economy has become mo e in e- g a ed in ecen decades, wi h he s a es ollowing a con e ging pa h due o economic, poli ical, and ins i u ional ac o s, such as, o example, he exchange a e mechanism in 1979, and he in oduc- ion o he eu o in 2001. Nume ous empi ical s udies ha e p o ided e idence o his in eg a ion in he Eu opean Union (Beck ield 2006; Capo aso and Pelowski 1971; Ma in and Ross 2004). Howe e , his in eg a ion among s a es can come oge he wi h economic dispa i ies be ween he di e en egions, which may cause non-con e gence wi hin a coun y. This a icle analyses his possibili y in Spain. Se e al coun ies ha e ypically employed egio- nal policies o add ess s uc u al dispa i ies among hei geog aphical a eas. Spain, o example, began implemen ing egional policies in he ea ly 1960s. Howe e , since 1986, Spanish egional policies ha e unde gone signi ican changes due o i s inclusion in he Eu opean Union (EU), which has been pa icula ly impo an in p o iding egional go e nmen s wi h oppo uni ies o engage in Eu opean ne wo ks, acili a ing he exchange o in e es s, knowledge, and alues. As a gued by A egui (2020), Spain is likely o be one o he membe s whe e some s a e es uc u ing has aken place, bo h a na ional and egional le el. This ans o ma ion has been in luenced by bo h Eu opean in eg a ion and he decen aliza ion o poli ical powe . These wo p ocesses ha e mu ually ein o ced each o he , and Spain’s EU membe ship has solidi ied he ole o Spanish Au onomous Communi ies in shaping and implemen ing poli- cies in c ucial a eas such as en i onmen , ag icul- u e, o ishing policies (A egui 2020). Spanish egions a e di ided in o 17 Au onomous Communi ies. Some o hese egions a e iche han o he s due o hei economic o sec o specializa ion and disagg ega ion acco ding o b anches o ac i i y. The income o each Au onomous Communi y depends on he economic specializa ion o ha egion, wi h some specializa ions gene a ing low incomes, while o he s gene a e signi ican ly highe incomes. Table 1 shows he high-sec o he e ogene- i y p esen ed by he di e en Spanish egional economies. These egions span om hose expe ien- cing subs an ial g ow h d i en by ou ism- ela ed CONTACT Ma iam Kamal [email p o ec ed] Depa men o Quan i a i e Me hods a he Uni e si y o he Basque Coun y, Bilbao 48015, Spain Supplemen al da a o his a icle can be accessed online a h ps://doi.o g/10.1080/00036846.2023.2293089 APPLIED ECONOMICS h ps://doi.o g/10.1080/00036846.2023.2293089 © 2024 The Au ho (s). Published by In o ma UK Limi ed, ading as Taylo & F ancis G oup. This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es License (h p://c ea i ecommons.o g/licenses/by-nc- nd/4.0/), which pe mi s non-comme cial e-use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed, and is no al e ed, ans o med, o buil upon in any way. The e ms on which his a icle has been published allow he pos ing o he Accep ed Manusc ip in a eposi o y by he au ho (s) o wi h hei consen . ac i i ies (e.g. Balea ic Islands, Cana y Islands) o hose whe e economic ac i i y is s ill la gely depen- den on p ima y sec o s (e.g. Galicia, La Rioja, Mu cia, Andalucía). This he e ogenei y in he egio- nal economic s uc u es and sou ces o income gen- e a es egional dispa i ies ha may p e en con e gence, as each egion’s g ow h may be d i en by di e en ac o s and indus ies. The di e en ways o gene a ing income among di e en egions ha e p oduced he cu en egio- nal di e ences in p oduc i i y and income. Hence, i migh be unsu p ising o ind dispa i ies be ween he 17 Au onomous Communi ies. This p omp s us o ques ion whe he economic con e gence among he 17 Au onomous Communi ies could no occu , which is a ques ion ha d i es he cu - en esea ch. In addi ion, we also aim o de e mine whe he some con e ging subg oups can be iden- i ied, o example, among de eloped o less de el- oped egions, which may be used o de ine mo e e icien egional policies delimi ing hei geog a- phical impac . As a i s example o his he e ogenei y ac oss he Spanish egions, Figu e 1 shows he e olu ion o he c oss-sec ional s anda d de ia ions o all he logs o pe capi a income in he 17 Au onomous Communi ies in Spain om 1955 o 2020. The dispe sion begins in 1955 a a ound 0.91 and ises and declines o e ime ending a a ound 0.94 in 2020, con i ming he absence o sigma con e gence. The la ge dispe sion shown in he g aph con- i ms ou p e ious suspicion o possible he e oge- nei y and non-con e gence. No e ha he dispe sions in he decade o 1970 and 2010 a e he highes . Howe e , he he e ogenei y in Spain emains in he sample be o e and a e hese peaks, which may hinde egional con e gence. In he li e a u e on economic g ow h, he e a e h ee main de ini ions o con e gence: (i) be a con- e gence, (ii) sigma con e gence and (iii) s ochas- ic con e gence based on ime-se ies analysis. As he i s wo ha e se e al s a is ical p oblems (see Du lau 2000; F iedman 1992; Quah 1993), we will ocus on he ime-se ies app oach on coin eg a ion, as sugges ed by Be na d and Du lau (1995). The use o di e en echniques has usually led o di - e en conclusions abou he exis ence o con e - gence (see, o example, Du lau 2000). We ollow a ime-se ies app oach o es o ou pu Table 1. Cha ac e is ics and p oduc i e s uc u es o he Spanish egions. Andalucia Ag i- ood sec o T anspo and logis ics sec o Ex emadu a Ag i- ood sec o . Li es ock a ming Food Indus y A agón Au omo i e indus y T anspo and logis ics sec o Galicia Tex ile and au omo i e sec o Ag i- ood sec o As u ias Me al and mining sec o Mad id Biomedical and pha maceu ical companies In o ma ion and Communica ion Technology sec o (ICT) Logis ics and anspo a ion Ae ospace indus y Balea ic Islands Tou ism sec o Food and ca e ing indus y Fashion indus y Mu cia Ag icul u al sec o Plas ic sec o Cana y Islands Tou ism Sec o Cul u al indus ies Logis ics sec o Na a e Au omo i e sec o Biomedical clus e ICT sec o Can ab ia Ag i- ood sec o Au omo i e componen s Bio echnology and heal h Basque Coun y Ene gy sec o Au omo i e and ae onau ic sec o Ma i ime indus y ICT sec o Bio-heal h sec o Se ice sec o Ca alonia Bio echnology Pe ochemical sec o Au omo i e sec o Ag icul u al sec o La Rioja Ag i- ood sec o Foo wea sec o Au omo i e sec o Se ice sec o Cas illa-La Mancha Ag i- ood sec o Wine p oduc ion Valencia Au omo i e and capi al goods sec o Ag i- ood sec o ICT and se ices sec o Chemical and pha maceu ical sec o Plas ic sec o Cas illa-León Ag i- ood sec o Chemical-Pha maceu ical sec o 2M. KAMAL AND J. ARTECHE con e gence, paying pa icula a en ion o he analysis o coin eg a ion, which p o ides a na u al se ing o es ing ela ions be ween a iables (see Be na d and Du lau 1995, 1996; Du lau 2000; E ans 1996; Quah 1993). Acco ding o he ime-se ies app oach o Be na d and Du lau , 1995, 1996), wo se ies con e ge i he ollowing condi ions a e sa is- ied: (i) The a iables a e coin eg a ed, (ii) he coin eg a ing ec o is (1, −1), and (iii) he di e ence be ween he se ies is a s ochas ic a i- able wi h ze o mean. Based on hese condi ions, he no ion o con e gence can be di ided in o s ong and weak con e gence (de ined as ca ch- ing-up in he con e gence li e a u e). I condi- ions (i) and (ii) a e ul illed, he se ies a e coin eg a ed wi h coin eg a ing ec o ½1;1], bu he di e ence be ween hem is a s ochas ic a iable wi h a mean di e en om ze o, which sugges s ha he de ia ion be ween he se ies is expec ed o dec ease, bu no o disappea . This is weak con e gence, i.e. ca ching up, which e e s o he si ua ion in which na owing o he di e ences be ween he a iables is obse ed o e ime, bu he con e gence p ocess has ye o be comple e. I all condi ions (i), (ii), and (iii) a e ul illed he e is s ong con e gence because he di e ence be ween he a iables anishes. The e o e, i he e is no coin eg a ion, con e - gence does no occu , nei he weak no s ong. The es o he a icle is o ganized as ollows. Sec ion II p o ides a li e a u e e iew on con e - gence in Spain. Sec ion III explains he me hodol- ogies used in ou analysis. Sec ion IV con ains he da a and p esen s he esul s, and inally, Sec ion 5 p esen s he conclusions. II. Li e a u e e iew The e a e ew s udies ha speci ically examine ou - pu con e gence be ween Spanish egions. The majo i y o hem use c oss- egional analysis app oaches o es ima e be a con e gence. Meanwhile, hose using a ime-se ies app oach op o uni oo es as he Augmen ed Dickey Fulle es in a non- ac ionally in eg a ed con ex , which lacks he powe and lexibili y needed o a comp ehensi e analysis. Some au ho s ha e ound esul s indica ing non-con e gence in he Spanish egions in ag eemen wi h ou esul s. Ma ínez-A güelles and Rubie a-Mo ollón (1998), ocusing solely on he se ice sec o , iden i y dis inc egional g ow h pa e ns wi hin his sec o using in ege coin eg a ion echniques. Cuad ado-Rou a e al. (1999) in es iga e he e olu ion o egional di - e ences in Spain and use an analysis o be a and sigma con e gence o conclude ha he p ima y sou ce o con e gence in obse ed p oduc i i y is he alignmen o egional sec o ial s uc u es. Lamo (2000) examines ou pu con e gence ac oss Spanish egions using c oss-sec ional dis- ibu ion dynamics. She inds no e idence o income con e gence. Maza (2006) examines he phenomenon o egional con e gence in pe capi a income in Spain and s udies wha ac o s in luence mig a ion pa e ns wi hin hese egions. Using a be a con e gence analysis, he Figu e 1. C oss sec ional s anda d de ia ion o he log o pe capi a income. APPLIED ECONOMICS 3 concludes ha he e is no con e gence among Spanish egions because mig an s end o mo e owa ds egions wi h highe pe capi a income, inducing a slowe pace o egional con e gence in Spain. A oyo e al. (2013) examine he pai - wise con e gence hypo hesis among he 17 Spanish egions using he Augmen ed Dickey and Fulle (1979) es o uni oo s. The ind- ings e eal incomple e ca ching-up in many ins ances, wi h only ou con e ging egions (Andalucia, Ex emadu a, Cas illa-La Mancha, and Galicia) and jus one (Balea es) con e ging wi h he Eu opean Union. Puen e (2017) uses adi ional g ow h eg essions o analyse he exis ence o be a-con e gence. The esul s indi- ca e ha labou p oduc i i y con e gence s ands ou as he p ima y d i e in na owing egional income dispa i ies. On he con a y, labou ma - ke a iables such as employmen and unem- ploymen , as well as o al ac o p oduc i i y, do no ha e a subs an ial impac on diminishing egional dispa i ies du ing he pe iod unde analysis. O he s udies ha e ocused on he con e gence among p o inces a he han Au onomous Communi ies. Dolado e al. (1994) examine he g ow h and dispa i ies ac oss Spanish p o inces. They use adi ional c oss- egional analysis o es ima e be a con e gence, and ind e idence o p o incial con e gence, al hough wi h some signs o ins abili y in he speed o con e gence du ing speci ic subpe iods. Ga deazábal (1996) analyses he dynamic e olu ion o income dis ibu ion among Spanish p o inces. Using Ma ko p o- cesses, he concludes ha pe capi a incomes among Spanish p o inces con e ge owa ds equi- lib ium. Villa e de (2005) examines he exis ence o be a con e gence in labou p oduc i i y in he p o inces o Spain. He concludes ha Spanish p o inces wi h low (high) ela i e p oduc i i y end o be geog aphically close o each o he , indica ing a concen a ion o p oduc i i y. The con e gence p ocess does occu , bu a a sligh ly slowe pace han in he classical model, and he e is a gap ha sepa a es he p o inces om hei s eady s a e. Hie o and Maza (2010) in es iga e he ole played by in e nal mig a ion o o eign indi iduals in he income con e gence o p o- inces in Spain be ween 1996 and 2005. Thei esul s e u e he hypo hesis ha in e nal mig a- ion o he o eign-bo n in luences income con- e gence. Mon añés e al. (2018) in es iga e con e gence be ween Spanish p o inces, wi h a pa icula ocus on he impac o he ecen in e na ional c isis. Thei esul s indica e he o - ma ion o se e al con e gence clubs, he pa e ns o which we e al e ed by he 2007 c isis. Tapia and Gala aga (2020) in es iga e he empi ical con- nec ion be ween economic g ow h and inequali y, quan i ying he dispa i ies be ween Spanish p o- inces o a ious e e ence yea s spanning om 1860 o 1930 and concluding ha he g ow h o income did no di ec ly lead o a educ ion in inequali y. This a icle analyses con e gence in annual eal ou pu pe capi a o he 17 Au onomous Communi ies in Spain om 1955 o 2020. We con ibu e o he exis ing empi ical li e a u e in h ee main dimensions: ●We es coin eg a ion in an economic amewo k o con e gence ha ollows he Be na d and Du lau (1995, 1996) de ini ion o ime-se ies con e gence. ●We use semipa ame ic and nonpa ame ic echniques, which ha e ne e been used be o e o analyse egional con e gence, o es o ac ional coin eg a ion: he s a e- gies p oposed by Robinson (2008), Hualde (2012) and Nielsen (2010). These ac ional in eg a ion and coin eg a ion echniques a e used o a oid he low powe o adi- ional uni oo and coin eg a ion es s agains ac ional al e na i es and a e mo e eliable o explo e economic con e gence. ●We complemen Nielsen (2010) es wi h a new se o c i ical alues o independen in e es o p ac i ione s. In pa icula , we p o ide c i ical alues o up o 17 a iables and h ee di e en sample sizes T = 66, T = 150 and T = 1000 o wo di e en alues o he memo y pa ame e o he o iginal se ies: d = 1 and d = 1.4. The la e co esponds o he alues ound in he se ies analysed he e, while he o me (in he supplemen a y ma e ial) co esponds o he adi ional uni oo case. 4M. KAMAL AND J. ARTECHE III. Me hodology The me hodology used in his a icle is based on he concep s o ac ional in eg a ion and ac ional coin eg a ion. F ac ional in eg a ion and coin eg a ion The idea o ac ional in eg a ion was in oduced by G ange and Joyeux (1980), G ange (1980, 1981) and Hosking (1981) allowing a con inuous ansi- ion om non-uni o uni oo beha iou s, o e ing a mo e lexible con ex o he modelling o long- un pe sis ence. A ime-se ies {y ; ¼1;2;3;. . .gis ( ac ionally) in eg a ed o o de d, I dð Þ, i i sa is ies: 1Lð Þdy ¼u ; ¼0;�1;. . . ;(1) whe e d is he memo y pa ame e and u eI0ð Þ, meaning ha u has a ini e a iance and a spec al densi y unc ion wð Þ, sa is ying 0< wð Þ<1. I d¼0;y ¼u and y is sho memo y; i 0<d<1 2, y is said o be long memo y. Finally, i 1 2<d<0;y p esen s an i-pe sis ence. Also, i d<0:5;y is co a iance s a iona y. Howe e , a alue d�0:5 implies non-s a iona i y, bu i d<1; he se ies is mean e e ing. In addi ion, i d¼1, he se ies has a uni oo . I d<1 he e ec s o he shocks disappea in he long- un and i d�1 he shocks pe sis inde ini ely. No e ha u in (1) may include some ype o weak dependence in he o m o , o example, a s a iona y and in e ible au o eg essi e mo ing a e age (ARMA) p ocess: Φ Lð Þu ¼θ Lð Þε ; ¼0;�1;...;(2) whe e ε is an independen and iden ically dis ib- u ed (iid) sequence. In his case;y in (1) is an Au o-Reg essi e F ac ionally In eg a ed Mo ing A e age (ARFIMA) p ocess: Φ Lð Þ 1Lð Þdy ¼θ Lð Þε ; ¼0;�1;(3) Engle and G ange (1987) de ined coin eg a ion as ollows: “A ec o y is said o be co-in eg a ed o o de d, b, deno ed y ~ CI d;bð Þ, i he componen s o y a e I dð Þand he e exis s a ec o α �0ð Þsuch ha z ¼α0y ~ I d bð Þ;b>0:The ec o α is called he co-in eg a ing ec o and b deno es he deg ee o coin eg a ion’’. The o iginal es ing s a egies p oposed o coin eg a ion we e only sui able o bi a ia e se - ings, and hey hus could only iden i y one coin- eg a ion ec o . Johansen (1988, 1991, 1995) de eloped a maximum likelihood app oach o es ing coin eg a ion in a mul i a ia e se ing, allowing o se e al ela ions and de e mining he ank o coin eg a ion. Following hese pio- nee ing au ho s, o he s anda d echniques we e de eloped by Phillips and Oulia is (1990), Ha is (1997), Bie ens (1997), and B ei ung (2002), among o he s. The gene aliza ion o he adi- ional Johansen es o a ac ional con ex was p oposed by Johansen (2008) and Johansen and Nielsen (2010, 2012, 2014) wi h he ac ionally coin eg a ed ec o au o eg essi e (FCVAR) model. S anda d adi ional coin eg a ion is jus one pa icula case o ac ional coin eg a ion whe e he memo y pa ame e s d and he deg ee o coin- eg a ion b a e es ic ed o be in ege alues. F ac ional alues o d and b allow mo e lexibili y and a e good al e na i es because many economic se ies a e known o exhibi non-s a iona y beha- iou s ha may no be exac ly I(1), and he e is also no need o assume ha he equilib ium ela ion is exac ly I(0). Tes ing o ac ional coin eg a ion The s a egy we ollow is based on he es ima ion o he coin eg a ion ank in a ac ional se ing using wo di e en and lexible echniques wi h good asymp o ic p ope ies unde mild condi ions. Fi s , he me hodology p oposed by Nielsen (2010) has he ollowing ad an ages o e o he coin eg a ion es s: (i) The es s a is ic is compu ed wi hou p io knowledge o he o de o in eg a ion o he se ies. (ii) Since he es is nonpa ame ic, i does no equi e speci ica ion o a pa icula model and is in a ian o sho - un dynamics. This is impo an because mis-speci ied sho - un dynamics may lead o inconsis en es ima ion and hence o e oneous in e ence ega ding he coin e- g a ion ank in o he pa ame ic echniques. (iii) The p oposed es has good powe o la ge and small samples. APPLIED ECONOMICS 5 Second, he me hodology o e ed by Hualde (2012), oge he wi h he es ing s a egy used by Robinson (2008), is cha ac e ized by he ollowing bene i s: (i) The es ing s a egies in Robinson (2008) do no equi e es ima ion o any coin eg a - ing ela ions o p io selec ion o any uning num- be s beyond one bandwid h pa ame e . (ii) Hualde (2012) p oposes a p ocedu e o es ima e he ank o coin eg a ion in mul i a ia e ac ional se ies, and he e o e his can be implemen ed oge he wi h he p ocedu e in Robinson (2008) o in e he dimension o he possible coin eg a ing subspaces. (iii) The combina ion o bo h echniques allows o p ecise de ec ion o he common ends. Robinson (2008) and Hualde’s (2012) app oaches We i s conside he es p oposed by Robinson (2008) combined wi h he s a egy in Hualde (2012). Hualde’s p ocedu e has he ad an age o e o he ac ional coin eg a ion app oaches o p o iding an au oma ic me hod o in e ing coin- eg a ing ela ionships wi hou any p io in o ma- ion abou he a iables. He de ines he possibili y o coin eg a ion as a si ua ion in which a linea combina ion o ac ional p ocesses is in eg a ed o a s ic ly smalle o de han he maximum o de o he elemen s o he linea combina ion. Fo example, i one o he a iables has an in eg a- ion o de ha is s ic ly g ea e han he es o he a iables, hen any linea combina ion wi h ze o weigh on his pa icula a iable is conside ed o be a i ial coin eg a ing ela ion. Unde his de i- ni ion, he a iables can ha e di e en in eg a ion o de s. Howe e , when all he a iables ha e he same in eg a ion o de , his de ini ion coincides wi h ha o iginally p o ided by Engle and G ange (1987). The p oposal by Hualde (2012) is based on an es ima o o he coin eg a ing ank, , ob ained by applying sequen ially he p ocedu e discussed in his Theo em 1, which we ew i e he e: Theo em 1 (Hualde 2012). y has coin eg a ing ank 21;...;p1 g whe e p is he numbe o a iables in he ec o y , i (i) and (ii) a e sa is ied, whe e: (i) The e exis s a p  ð Þ dimensional sub- ec o o y , deno ed as ybð Þ , whose indi idual componen s a e common ends, deno ed as CT. (ii) All sub ec o s o y o dimension la ge han p con aining y bð Þ coin eg a e. The p ocedu e o es ima e he ank o coin eg a- ion is based on he ollowing s eps. Fi s , he es ima es o he in eg a ion o de s (memo y pa a- me e s), ^ di;i¼1;...;p a e ob ained o de ine he CT as he se ies wi h he highes o de o in eg a- ion. Then, he ollowing hypo hesis is es ed: Hj1;...;jk ð Þ :yj1 ;yj2 ;...;yjk a e no coin eg a ed � �; agains � Hj1;...;jk ð Þ :Hj1;...;jk ð Þ is no ue; whe e j1. . . ;jk21;. . . ;p g;k�p;is sequen ially es ed. In o de o es ima e he memo y pa a- me e s, he uni a ia e local Whi le es ima o ^ di, p oposed by Robinson (1995a) is used. Nex , he hypo heses a e es ed using he s a is ic X* p o- posed by Robinson (2008), de ined as: X�¼ms�ð~ dÞ2=p2 ^ R�A^ R�A � �p n o (4) whe e m is he bandwid h s�~ d ��¼ ^ G�~ d ��1^ H�~ d �� � � ^ G�dð Þ ¼ 1 mX m j¼1 Iyλj  �λ2d j ^ H�dð Þ ¼ 1 mX m j¼1 jIyλj  �λ2d j j¼log j1 mX m i¼1 log i ^ R�¼^ D1=2^ G�~ d ��^ D1=2 ^ D¼diag ^ g11;...;^ gpp n o, whe e ^ gii is he i h diago- nal elemen o ^ G�~ d �� A¼diag a1;...;ap � � 6M. KAMAL AND J. ARTECHE whe e Iyλj  �is he pe iodog am ma ix o y al equency λj, ~ d¼P p i¼1 ai^ di and he ai a e a bi a ily chosen weigh s sa is ying ha P p i¼1 ai¼1:Fo ins ance, Robinson (2008) akes ai;1=p, so he a i hme ic mean o he ^ di is used. Ano he op ion is using aj¼1;ai¼0;i�j some j. In ou case, we use he i s op ion as ecommended by Robinson (2008). Unde he null hypo hesis o non-coin eg a ion and s a iona i y o he se ies, which implies ha all he memo y pa ame e s a e smalle han 0.5, X�! dX2 1as T! 1:(5) The me hodology o es ima e he coin eg a ion ank is cha ac e ized by he ollowing s eps: S ep 1. Es ima e he indi idual in eg a ion o de s, di, by ^ di;i¼1;...;p. Then choose a possible CT yc1 as he a iable wi h he highes es ima ed o de , such ha c121;. . . ;p g. Nex , eo de he a i- ables in y so ha yp ¼yc1 in he new o de ing. Finally, gi en he possible CT i.e. yp , we es he ollowing hypo heses: H1ð Þ :[ p1 i¼1Hp;i e sus � H1ð Þ : p1 i¼1 � Hp;i No e ha H 1ð Þ means non-coin eg a ion in pai s o each a iable wi h he CT, and � H(1) means ha H 1ð Þ is no ue. The p ocess ends i H1ð Þis ejec ed, and so i is concluded ha ^ ¼p1. O he wise, i is no ejec ed, and he p ocess con inues o S ep 2. Consequen ly, ollowing Theo em 1, he hypo h- eses a e equi alen o <p1 and ¼p1 espec i ely. S ep 2. I H1ð Þ is no ejec ed, choose a second possible CT as he a iable wi h he smalles s a is ic X* i.e. yc2 ;c221;. . . ;p1 g. The e will be wo possible CTs al oge he . These CTs a e deno ed as yp and yc2 espec i ely. Then, we eo de again he a iables so ha yp ¼yc1 and yp1; ¼yc2 in he new o de ing. Finally, gi en he possible CTs, i.e. yp ;yp1; , we es he ollowing hypo heses: H2ð Þ :[ p2 i¼1Hp;p1;i e sus � H2ð Þ : p2 i¼1 � Hp;p1;i. No e ha H 2ð Þ means non-coin eg a ion o any se o h ee a iables con aining he CTs yp ;yp1; , and � Hð2) means ha H 2ð Þis no ue. Then, he hypo h- eses a e equi alen o <p2 and ¼p2 espec i ely and he p ocess ends i H 2ð Þis ejec ed. S ep k ( o k ¼2;. . . ;p1). I H k 1ð Þ is no ejec ed, chooseck. So he a iables so ha yp ¼yc1 ;...;ypkþ2; ¼yck1; and choose he pos- sible CTs, as p e iously. Finally, es he ollowing hypo hesis: H(k): ∪ p-k i-1 H p,p-1,. . .,p-k+1,i e sus — H(k): ∩ p-k i-1 — H p, p-1,. . .,p-k,i and i he e is coin eg a ion, he es i- ma ion will be ^ ¼pk. Howe e , i we each he las s ep k¼p1 his means ha ^ ¼0 and H ið Þ;i¼1;2;3;...;p1;a e no ejec ed: The es ing p ocedu e based on he s a is ic X* has low powe wi h small sample sizes, which can sig- ni ican ly in luence he esul s ob ained when ana- lysing he possibili y o coin eg a ion in he Spanish egions. To complemen he esul s ob ained we also conside he es p oposed by Nielsen (2010), which has highe powe o small samples (see Nielsen’s Mon e Ca lo). Nielsen’s (2010) app oach and c i ical alues ex ension The es s a is ic is de ined as ollows: Λp; d1 ð Þ ¼ T2d1Xp j¼1#j; ¼0;. . . ;p1 (6) whe e #j;j¼1;...;p;a e he eigen alues o #BTAT j j ¼ 0, o AT¼P T ¼1 Z Z 0;BT¼P T ¼1 ~ Z ~ Z 0, BT¼PT ¼1~ Z ~ Z0 , ~ Z ¼Δd1 1Z wi h d1>0; ¼1;2;...;T;and Z is he p- ec o o ime se ies unde analysis (pe haps a e ex ac ing de e minis ic e ms), which is ac ionally in e- g a ed o o de d, whe e d is a ec o con aining he indi idual o de s o in eg a ion o he elemen s in Z ;which possibly di e om each o he . No e ha (6) de ines a amily o es s indexed by he ac ional in eg a ion pa ame e , d1. Nielsen (2010) a gues in a ou o using d1 = 0.1 based on an asymp o ic local powe analysis and on simula- ions. Fo his eason, we use his alue in he empi ical applica ion. La ge alues o Λp; 0d1 ð Þ a e associa ed wi h he ejec ion o he null hypo hesis H0: ¼ 0 e susH1: > 0: APPLIED ECONOMICS 7 Nielsen’s p ocedu e has he ad an age o no equi ing knowledge o he ac ional in eg a ion and coin eg a ion o de s d and b as long as he se ies a e non-s a iona y, implying memo y pa a- me e s g ea e han 0.5. Howe e , i s asymp o ic dis ibu ion is non-s anda d, bu Nielsen (2010) simula ed c i ical alues o p < 8 a iables wi h a sample size o 1000 obse a ions o acili a e i s applica ion. We complemen Nielsen’s (2010) ables by p o iding mo e c i ical alues o co e up o 17 a iables o all models and h ee di e en sample sizes. The obse ed ime se ies Y gT ¼1conside ed by Nielsen (2010) a e gene a ed by Y ¼α0δ þZ ; ¼1;2;. . . (7) whe e δ may con ain de e minis ic e ms. Th ee di e en cases a e analysed: δ ¼0 when he e a e no de e minis ic e ms, δ ¼1 when he e is a non- ze o mean, and δ ¼1; ½ �0when he e is a de e minis ic end. The c i ical alues in Nielsen (2010) a e he e ex ended o hese h ee cases up o 17 a iables, sample sizes T = 1000, 150 and 66, and wo di e en alues o he memo y pa ame e o he o iginal se ies d = 1 and d = 1.4, he la e co e- sponding o he alues ound in he se ies he e analysed. All ables a e based on 100,000 eplica- ions. Tables 2, 3 and 4 show hese new c i ical alues o a memo y pa ame e equal o d = 1.4. Tables 24, 25 and 26 in he Supplemen a y Ma e ial show he same se o c i ical alues o d = 1, which a e o independen in e es o p ac i ione s. IV. Empi ical analysis P elimina y analysis o he a iables The da a analysed a e he loga i hms o he annual eal GDP pe capi a (cons an 2010 €) o he 17 Au onomous Communi ies in Spain (omi ing he au onomous ci ies) in housands o eu os om 1955 o 2020 o a o al o T = 66 obse a ions. In o de o apply he semipa ame ic coin eg a ion analysis by Robinson (2008), he se ies a e di e enced o ob ain g ow h a es ha a e s a iona y, whe eas o he non- pa ame ic coin eg a ion analysis by Nielsen (2010), he se ies a e aw se ies (non-s a iona y). The a iables in loga i hms a e deno ed by he name o he Au onomous Communi y, and he g ow h a es a e deno ed by he abb e ia ion o hese Au onomous Communi ies (in pa en hesis he no a ion o he g ow h a es): Andalucia (andal), A agon (a a), As u ias (as ), Balea ic Islands (bal), Cana y Islands (can), Can ab ia (can ), Ca alonia (ca ), Cas illa- LaMancha (clm), Cas illa-Leon (cyl), Ex emadu a (ex ), Galicia (gal), Mad id (mad), Mu cia (mu ), Na a e (na ), Basque Coun y (p ), Rioja ( io) and Valencia ( al). All da a we e p o ided by FEDEA (Founda ion o he S udy o Applied Economics) and INE (Spanish Na ional S a is ics Ins i u e). Figu es 2 and 3 show hese se ies. To shed mo e ligh on he pe sis ence o he se ies, he Exac Local Whi le (ELW) es ima o p oposed by Shimo su, e al., 2005, which is consis en and asymp- o ically no mal o any alue o d, was applied. The ELW es ima es and hei 95% con idence in e als a e shown in Table 5. They con i m he non-s a iona i y o he se ies, which is a equi emen o he applic- abili y o Nielsen’s p ocedu e. No e also ha a alue o d = 1.4 is no ejec ed o any o he se ies, alling wi hin all he con idence in e als, which jus i ies he use o his alue in he cons uc ion o he s a is ic o Nielsen’s es . The es ima ion o he memo y pa ame e o he g ow h a es, equi ed o he applica ion o Robinson’s (2008) es is, howe e , ob ained using he Local Whi le (LW) es ima o o Robinson (Robinson 1995) as sugges ed in Robinson (2008), which is consis en o d < 1 and asymp o ically no mal o d < 0.75. All he LW es ima es, shown in Table 6, a e be ween 0.2 and 0.5, indica ing ha he g ow h a e se ies can be conside ed s a ion- a y (d<0:5Þ. Robinson (2008) and Hualde (2012) coin eg a ion esul s In o de o analyse he obus ness o he esul s o he selec ion o he bandwid h, he en i e analysis has been implemen ed using h ee di e en bandwid hs, m = 18, m = 23 and m = 28. Acco ding o he esul s o he LW es ima ed mem- o y pa ame e s, he possible CT in S ep 1 ( a iable wi h he la ges es ima ed d) is Mu cia (mu ) o all 8M. KAMAL AND J. ARTECHE