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Applied Economics
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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.
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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:
1Lð 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 1Lð 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;...;p1 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 j1
mX
m
i¼1
log i
^
R�¼^
D1=2^
G�~
d
��^
D1=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 :[
p1
i¼1Hp;i e sus �
H1ð Þ :
p1
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 ^
¼p1. 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 <p1 and ¼p1
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;. . . ;p1 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 yp1; ¼yc2 in he
new o de ing. Finally, gi en he possible CTs, i.e.
yp ;yp1; , we es he ollowing hypo heses:
H2ð Þ :[
p2
i¼1Hp;p1;i e sus �
H2ð Þ :
p2
i¼1
�
Hp;p1;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 ;yp1; , and
�
Hð2) means ha H 2ð Þis no ue. Then, he hypo h-
eses a e equi alen o <p2 and ¼p2
espec i ely and he p ocess ends i H 2ð Þis ejec ed.
S ep k ( o k ¼2;. . . ;p1). I H k 1ð Þ is no
ejec ed, chooseck. So he a iables so ha
yp ¼yc1 ;...;ypkþ2; ¼yck1; 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 ^
¼pk. Howe e , i we each
he las s ep k¼p1 his means ha ^
¼0
and H ið Þ;i¼1;2;3;...;p1;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;. . . ;p1 (6)
whe e #j;j¼1;...;p;a e he eigen alues o
#BTAT
j j ¼ 0, o AT¼P
T
¼1
Z Z 0;BT¼P
T
¼1
~
Z ~
Z 0,
BT¼PT
¼1~
Z ~
Z0
, ~
Z ¼Δd1
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
