Gál ez-Rod íguez, José F.; Manzano-Hidalgo, Miguel; Ga cía-Luengo, Amelia V.
A icle
P edic ing con e gence o pe capi a income in Spain: A
Ma ko and clus e app oach
Economies
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Ci a ion: Gál ez-Rod íguez, J.F.,
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A.V. (2025). P edic ing Con e gence o
Pe Capi a Income in Spain: A Ma ko
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A icle
P edic ing Con e gence o Pe Capi a Income in Spain:
A Ma ko and Clus e App oach
José F. Gál ez-Rod íguez * , Miguel Manzano-Hidalgo and Amelia V. Ga cía-Luengo *
Depa men o Ma hema ics, Uni e si y o Alme ía, 04120 Alme ía, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed] (J.F.G.-R.); amga [email p o ec ed] (A.V.G.-L.)
Abs ac : In his wo k we analyze he e olu ion o p oduc i i y, in e ms o he con e gence
o pe capi a income, o all he Spanish p o inces, based on da a om he p e ious decade.
On he one hand, a clus e analysis allows us o g oup he Spanish p o inces acco ding o
ou income le els (low, medium-low, medium-high and high), which can be de e mined
om he qua iles o he dis ibu ion, and, on he o he hand, Ma ko chains make i
possible o s udy he long- e m e olu ion o p oduc i i y and con e gence be ween he
p o inces, as well as he speed o con e gence owa ds he equilib ium si ua ion. Mo eo e ,
we can ob ain he a e age ime o e u n o an income le el in which a p o ince was
p e iously. Wi h he abo e, p edic ions o u u e income le els a e made o he p o inces,
bo h in he cu en si ua ion, and i he pandemic caused by COVID-19 had no exis ed,
which leads us o e alua e he impac o he heal h eme gency.
Keywo ds: clus e analysis; Ma ko chain; pe capi a income
1. In oduc ion
Ma ko chains Takács (1960) a e an impo an ma hema ical ool in he analysis o
ime se ies and ha e been used by se e al au ho s when s udying he economic g ow h
and o he phenomena, e en in o he disciplines. Ma ko analysis s a s om he p esen
alue o a ce ain a iable in o de o p edic i s u u e. Fo example, Quah (1993) uses
Ma ko chains o analyze how coun ies exchange posi ions o e ime in e ms o hei
GDP pe capi a. Each ini e and homogeneous Ma ko chain is cha ac e ized by a ma ix
which con ains di e en p obabili ies. In he s udy ci ed abo e, he p obabili ies ha a
coun y emains in i s cu en income g oup o mo es o a highe o lowe income g oup
a e calcula ed and collec ed in his ma ix. I is concluded ha , in gene al, coun ies end
o con e ge in e ms o hei GDP pe capi a o e ime, bu he a e o con e gence a ies
depending on he egion and he ime pe iod s udied. This app oach has been used by
o he au ho s in simila analyses in a wide a ie y o con ex s. Some ele an examples o
s udies in which he Ma ko chain me hodology has been used o analyze he dynamics o
g ow h and con e gence a e he ollowing:
•
Fingle on (1997) uses he e idence gi en by da a om 1975 o 1993 in o de o jus i y
ha di e en egions o he Eu opean Union seem o be con e ging owa ds s able
p opo ions in e ms o pe capi a income le els.
•
Bode and Nunnenkamp (2011) analyze he impac o o eign di ec in es men on
pe capi a income and g ow h, in gene al, in he Uni ed S a es, since he mid-1970s,
demons a ing ha he in es men in employmen has a o ed he inc ease in income,
al hough he in es men des ined o capi al has no had he same impac on he s a es
wi h he g ea es po e y, whe e i has been mos in ensi e.
Economies 2025,13, 17 h ps://doi.o g/10.3390/economies13010017
Economies 2025,13, 17 2 o 16
•
Lip ák (2011) examines he e olu ion o unemploymen in he Hunga ian labo ma ke
du ing he pe iod 1992–2009.
Despi e being a me hodology exposed a he end o he p e ious cen u y, Ma ko
chains keep on appea ing in he nowadays li e a u e o s udy he e olu ion o economic
a iables. We e e he eade o he wo ks by Wenxuan (2023), Rey (2023), Papanikolaou
(2020), A eola and Mon iel (2024), Ke kouch e al. (2024), Chen e al. (2022), Ka ahasan
(2020) and Halle e al. (2020).
Howe e , knowing he p obabili y o ansi ion be ween s a es along a pe iod o ime
is no enough, since we s ill do no know which egion con inues in he same si ua ion o
change i a e se e al pe iods o ime. Tha is he eason why conside ing a clus e analysis
can be o some in e es in o de o know, also, he elemen s o each g oup wi h espec
o he pas da a. Clus e analysis, o wha a good e e ence is E e i e al. (2011), is a
s a is ical echnique whose p ima y objec i e is o g oup da a acco ding o hei simila i ies
and di e ences based on ce ain cha ac e is ics. I is a widely used me hod in Economics
and Finance, o analyze he s uc u e o ma ke s, iden i y pa e ns and ends, and classi y
coun ies o companies in o di e en g oups in acco dance wi h hei economic and inancial
p ope ies. A pa icula example o a s udy wi h clus e analysis is he one ca ied ou by
Yang and Hu (2008), who analyze da a om he China Human De elopmen Index in 1982,
1995, 1999 and 2003, in o de o classi y i s p o inces in o ou le els o income based on
he h ee basic aspec s con empla ed in ha index. We can also e e o o he ecen s udies
which show he in e es o clus e ing in analyzing some economic a iables. Fo example,
Gos kowski e al. (2021) s udy he ela ionship be ween he na ional le el o economic
de elopmen and ene gy consump ion in he main sec o s o indus y o he coun ies ha
belong o he Viseg ad G oup; He e al. (2021) analyze he socio-economic spa ial s uc u e o
u ban agglome a ion in China by using clus e ing, and
Za ikas e al. (2020)
use clus e ing o
g oup coun ies wi h espec o ac i e cases, ac i e cases pe popula ion and ac i e cases pe
popula ion and pe a ea, so ha he impac o he COVID-19 pandemic can be explo ed. In
his sense, di e en economic a iables such as GDP pe capi a, economic g ow h a e, o eign
in es men , in e na ional ade, among o he s, can be used o g oup coun ies in o di e en
ca ego ies based on hei le el o economic de elopmen and i s g ow h dynamics. Fo
example, clus e analysis can be used o iden i y g oups o coun ies ha con e ge wi h each
o he , ha is, which ha e simila le els o economic de elopmen and a e g owing a a simila
a e. In his way, as we will de elop la e , pa e ns o con e gence o di e gence be ween
coun ies can be iden i ied and he unde lying causes o hese ends can be analyzed.
We can conclude om his ha clus e analysis is a use ul ool o analyzing he
s uc u e and dynamics o coun ies, and can be used in combina ion wi h o he s a is ical
echniques, such as ime se ies analysis and Ma ko chains, o ob ain a mo e comple e and
accu a e iew o economic con e gence be ween coun ies.
The main goal o his wo k is he applica ion o he p e iously men ioned me hodolo-
gies o he s udy o he con e gence o pe capi a income o he Spanish p o inces (in ac ,
we conside he i y p o inces oge he wi h he wo au onomous ci ies in Spain) based
on he GDP pe capi a o each o hem in he yea s o he las decade. Mo eo e , we can
e alua e he impac o COVID-19 pandemic on his con e gence and s udy i s speed o
his con e gence, bo h wi h and wi hou COVID-19. Indeed, he e a e some ecen wo ks
in which he au ho s ace COVID-19 o income inequali y, such as he one ca ied ou by
Dea on (2021). Addi ionally, a clus e analysis can le us ha e an idea o he possible
g oups o income in he u u e. Hence, he s uc u e o he wo k is as ollows: Sec ion 2
p o ides some concep s and esul s ha he eade should ecall, which ha e o do wi h
hese wo s a is ical me hodologies as well as some easons o chose bo h o hem in he
s udy. Mo eo e , Sec ion 3includes he applica ion o hem o he s udy o he e olu ion o
Economies 2025,13, 17 3 o 16
pe capi a income in Spain, by p o inces, based on his o ical da a, analyzing he beha io o
he Spanish economy in he long e m, bo h in he cu en si ua ion and in case he pandemic
gene a ed by COVID-19 had no exis ed. Fu he mo e, in his sec ion, a discussion is gi en
oge he wi h he esul s. Finally, Sec ion 4collec s he main conclusions o he wo k.
P o incial Income Dispa i ies in Spain: A Li e a u e O e iew
As s a ed a he beginning o his sec ion, Ma ko chains ha e become an in e es ing
app oach o deal wi h he e olu ion o economic a iables and, in pa icula , o pe capi a
income o e he yea s, hanks o many wo ks in he economic li e a u e. In his subsec ion,
we ocus on Spanish s udies which a e ela ed o his opic. On he one hand, i is wo h
men ioning he wo k by Le Gallo and Chasco (2008), who s udy he e olu ion o he
popula ion g ow h among he g oup o 722 municipali ies included in he Spanish u ban
a eas o e he pe iod 1900–2001. Fu he mo e, Ayuda e al. (2010) analyze he dispa i ies
in long- un egional popula ion g ow h in con inen al Eu ope, concluding ha he e is a
common pa e n o di e gence in economic g ow h o Eu ope. Ma ko chains a e used in
his pape and hey also conside Spain as a pa icula case in he s udy, o wha a clus e
analysis is use ul. On he o he hand, Ga deazábal (1996) s udies he Spanish p o inces
dynamics, in e ms o hei income, in he ime pe iod be ween 1967 and 1991, concluding
ha hey end o he equilib ium dis ibu ion, being mo e concen a ed in medium le els
o income. Howe e , Ti ado e al. (2016) explo e pe -capi a GDP dispa i ies ac oss Spanish
p o inces om 1860 o 2010. The p e ious ci ed wo ks sugges us conside ing Ma ko
chains and clus e analysis in o de o s udy he e olu ion o pe capi a income in Spain o
he las decade, which has no been analyzed in he li e a u e ye .
2. Me hodology
In his sec ion, we collec some concep s and esul s ela ed o Ma ko chains and
clus e analysis, which will be used in he ollowing sec ion o ge he esul s o he main
s udy. Mo eo e , we in oduce he da a we will wo k wi h and discuss he hypo hesis on
he Ma ko chains app oach and he sui abili y o he ype o clus e ing used in he s udy.
2.1. Ma ko Chains
We e e he eade o Appendix Ain o de o ecall some basic p elimina ies on
Ma ko chains. In his pape , we will wo k wi h ini e and homogeneous disc e e- ime
Ma ko chains. The Ma ko chain mus be ini e because we wan o g oup he p o inces
in o a ini e numbe o income s a es, as usual in he ela ed li e a u e. Finally, homogenei y
assump ion has o do wi h he idea o ge ing a unique ma ix which ga he s all he
in o ma ion acco ding o he collec ed da a.
2.1.1. Long-Te m Beha io
Dynamic models which a e based on Ma ko chains ha e, as a poin o in e es , he
analysis o he con e gence o he ansi ion p obabili ies when he ime ends o in ini y.
This leads us o s udy he long- e m beha io , also known as s a iona y o limi , and
which is undamen al in his wo k o ca y ou a dynamic analysis o economic aspec s.
Pa icula ly, i
{Xn:n∈N}
is a homogeneous and ini e Ma ko chain, wi h
k
possible
s a es, he limi ing dis ibu ion is he p obabili y dis ibu ion gi en by
lim
n→∞P1(1)P2(1). . . Pk(1)
p11 p12 . . . p1k
p21 p22 . . . p2k
.
.
..
.
.....
.
.
pk1pk2. . . pkk
n
.
Economies 2025,13, 17 4 o 16
Ge ing he limi ing dis ibu ion in he s udy o his pape means ha we can know
wha is likely o happen in he long e m acco ding o he dis ibu ion o he pe capi a
income. Roughly speaking, he Spanish p o inces end o g oup in o some s a es acco ding
o he p opo ions gi en by he limi ing dis ibu ion.
Howe e , he limi ing dis ibu ion does no ha e o exis and, i i exis s, i may no
be unique, since i will depend on he ini ial dis ibu ion. In case he p e ious limi does
no change o any ini ial dis ibu ion, we say ha i is he equilib ium dis ibu ion o he
chain. Fu he mo e, he s a iona y dis ibu ion is he one ha does no change a e each
ansi ion o he chain acco ding o he ansi ion p obabili y ma ix, ha is, i is a ow
ma ix
p
such ha
pP =p
. I should be aken in o accoun ha when he equilib ium
dis ibu ion exis s and is unique, hen i mee s he s a iona y one.
2.1.2. S a es Classi ica ion
A i s classi ica ion o he s a es o a Ma ko chain has o do wi h he access be-
ween hem:
•
We will say ha a s a e
xj
is accessible om ano he s a e
xi
when
pij >
0 a some
ins an o ime. Fu he mo e, when i is p obable o go om one s a e o ano he in
bo h di ec ions, we will say ha bo h s a es communica e. I all he s a es o a Ma ko
chain communica e, we say ha he Ma ko chain is i educible.
•
Howe e , i he e is a s a e ha canno be eached om any o he in he Ma ko chain,
we will say ha i is epheme al.
•
On he o he hand, i he e is a s a e om which we canno each any o he one, we
say ha i is abso bing. Ma hema ically, he s a e xiwill be abso bing i pii =1.
We can conside ano he c i e ion o classi y he s a es o a Ma ko chain, which has
o do wi h he p obabili y o coming back o a ce ain s a e a some poin o ime. I his
p obabili y is 1, we say ha he s a e is ecu en , while i i is less han 1, we will say
ha i is ansi o y. A well-known ma hema ical esul es ablishes ha wo s a es ha
communica e a e bo h ecu en o bo h ansi o y. Addi ionally, i we conside a ecu en
s a e and de ine he andom a iable ha desc ibes he numbe o ansi ions needed o
come back o ha s a e, we can ind i s expec ed alue, which will p o ide us he mean
ecu ence ime, ha is, he mean ime since he s a e is le un il he Ma ko chain e u ns
o i . In pa icula , we alk abou a posi i e ecu en s a e i i s mean ecu ence ime is
ini e. On he o he hand, a s a e is said o be pe iodic i , s a ing om i , i is only possible
o e u n o i in a numbe o s ages mul iple o an in ege g ea e han 1. I will be ape iodic
i we can e u n o i a e each ansi ion. In his case, he pe iod is 1.
Table 1summa izes he classi ica ion o he s a es o a Ma ko chain acco ding o
bo h c i e ia.
Table 1. S a es classi ica ion.
Type o S a e Condi ions
xjaccesible om xipij >0
xjepheme al pij =0 o each i
xiabso bing pii =1
Recu en P obabili y o coming back o i =1
Posi i e ecu en Fini e mean ecu ence ime
T ansi o y P obabili y o coming back o i <1
Ape iodic Pe iod =1
Economies 2025,13, 17 5 o 16
The nex esul is one o he keys o his wo k:
Theo em 1. I
{Xn:n∈N}
is a homogeneous, ini e, i educible and ape iodic Ma ko chain,
hen he s a iona y dis ibu ion exis s, is unique and mee s he equilib ium dis ibu ion. Mo eo e ,
he mean ecu ence ime o each s a e is gi en by he in e se o he espec i e p obabili y in he
equilib ium dis ibu ion.
This heo em gi es special emphasis o homogeneous, ini e and disc e e- ime Ma ko
chains. I is wo h no ing ha he so wa e R has a package which is especially ocused on
he s udy o hese s ochas ic p ocesses. I is called “ma ko chain” and can simpli y some
calcula ions, so ha we can ge conclusions om a esea ch wo k based on he applica ion
o dynamic p obabilis ic models. In o de o show his me hodology, we e e he eade o
Appendix B.1 so ha he basic codes in his so wa e can be seen.
2.1.3. Es ima ing he T ansi ion P obabili y Ma ix
Suppose ha
{Xn:n∈N}
is a homogeneous and ini e Ma ko chain, wi h
k
possible
s a es. Then he ansi ion p obabili y ma ix is cons an . I he ansi ion p obabili ies a e
no known in ad ance, we can es ima e hem i we ha e da a o indi idual ansi ions
be ween wo consecu i e ins an s o ime. In o he wo ds, i
nij
is he numbe o indi iduals
ha we e in s a e
xi
a ime
, and each s a e
xj
a ime
+
1, hen he maximum likelihood
es ima o o he ansi ion p obabili y
pij
is gi en, acco ding o Ande son and Goodman
(1957), by
b
pij =nij
∑k
j=1nij
.
The e o e, he p obabili y o mo ing om s a e
xi
o s a e
xj
can be calcula ed as he
p opo ion o indi iduals who, being in he s a e
xi
in a ce ain ins an o ime, each he
s a e
xj
in he ollowing pe iod. In ac , his es ima o is jus i ied o be consis en , ha
is, he la ge he sample size, he be e he es ima ion made. Mo eo e , i is known ha ,
al hough his es ima o is biased, i s bias dec esases as he numbe o indi iduals unde
s udy inc eases.
2.2. Da a and T ea men
Nex s ep is choosing he da a we a e going o wo k wi h. We collec he GDP pe
capi a, in eu os, o he i y Spanish p o inces oge he wi h he wo au onomous ci ies, in
he ime pe iod be ween 2010 and 2020. We ha e no conside ed mo e yea s, since he idea
is o compa e he u u e es ima ion wi h and wi hou COVID-19 pandemic. The sou ce
o hese da a is he Na ional S a is ics Ins i u e (2023) (Spain). The da a a e collec ed in
Table 2, in which he numbe s ha e been ounded o h ee decimal places.
Table 2. GDP pe capi a ela i e o he Spanish a e age o each Spanish p o ince/ au onomous ci y
and yea (2010–2020).
P o ince/Au on. Ci y 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
Alme ía 0.802 0.754 0.767 0.760 0.783 0.801 0.838 0.857 0.839 0.841 0.862
Cádiz 0.735 0.736 0.729 0.717 0.699 0.691 0.691 0.695 0.693 0.700 0.679
Có doba 0.717 0.715 0.696 0.711 0.706 0.717 0.709 0.711 0.703 0.681 0.707
G anada 0.708 0.713 0.719 0.719 0.732 0.737 0.715 0.707 0.705 0.713 0.725
Huel a 0.747 0.779 0.781 0.732 0.719 0.729 0.735 0.763 0.781 0.757 0.767
Jaén 0.706 0.713 0.661 0.715 0.675 0.731 0.699 0.691 0.709 0.670 0.718
Economies 2025,13, 17 6 o 16
Table 2. Con .
P o ince/Au on. Ci y 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
Málaga 0.758 0.748 0.731 0.724 0.730 0.725 0.715 0.720 0.726 0.729 0.710
Se illa 0.814 0.815 0.818 0.799 0.803 0.790 0.780 0.782 0.783 0.785 0.795
Huesca 1.124 1.132 1.118 1.164 1.139 1.105 1.168 1.136 1.117 1.119 1.216
Te uel 1.045 1.045 1.062 1.085 1.084 1.023 0.994 0.955 0.977 0.965 0.994
Za agoza 1.093 1.087 1.076 1.085 1.086 1.071 1.076 1.091 1.096 1.096 1.127
As u ias 0.917 0.914 0.905 0.892 0.883 0.882 0.872 0.878 0.880 0.879 0.887
Balea es 1.059 1.059 1.067 1.065 1.077 1.077 1.087 1.085 1.081 1.071 0.913
Las Palmas 0.838 0.836 0.824 0.833 0.820 0.801 0.811 0.810 0.810 0.800 0.716
S a. C uz de Tene i e 0.890 0.886 0.878 0.860 0.851 0.843 0.824 0.827 0.816 0.807 0.742
Can ab ia 0.945 0.937 0.934 0.921 0.927 0.910 0.913 0.912 0.918 0.922 0.934
Á ila 0.788 0.801 0.818 0.808 0.802 0.788 0.775 0.776 0.784 0.791 0.828
Bu gos 1.111 1.132 1.156 1.129 1.112 1.100 1.112 1.128 1.145 1.128 1.144
León 0.870 0.867 0.874 0.856 0.848 0.838 0.818 0.818 0.823 0.829 0.867
Palencia 1.022 1.043 1.024 1.028 1.010 1.027 1.062 1.004 1.054 1.041 1.069
Salamanca 0.804 0.810 0.810 0.798 0.796 0.797 0.810 0.808 0.811 0.822 0.855
Sego ia 0.939 0.930 0.921 0.917 0.926 0.931 0.907 0.850 0.858 0.860 0.887
So ia 1.004 1.015 0.993 1.016 1.022 1.018 0.999 0.986 1.082 1.066 1.074
Valladolid 1.026 1.020 1.018 1.020 1.025 1.022 1.045 1.057 1.074 1.064 1.092
Zamo a 0.798 0.823 0.851 0.832 0.819 0.819 0.810 0.743 0.756 0.770 0.807
Albace e 0.801 0.793 0.798 0.801 0.783 0.798 0.793 0.806 0.815 0.823 0.851
Ciudad Real 0.835 0.837 0.841 0.826 0.798 0.830 0.834 0.836 0.839 0.824 0.857
Cuenca 0.839 0.859 0.870 0.875 0.852 0.868 0.866 0.867 0.882 0.856 0.891
Guadalaja a 0.832 0.836 0.830 0.814 0.770 0.742 0.757 0.776 0.789 0.794 0.816
Toledo 0.760 0.743 0.730 0.731 0.718 0.716 0.721 0.716 0.724 0.717 0.746
Ba celona 1.172 1.167 1.172 1.181 1.193 1.193 1.201 1.208 1.205 1.208 1.201
Ge ona 1.156 1.143 1.151 1.144 1.149 1.143 1.154 1.093 1.080 1.084 1.083
Lé ida 1.191 1.190 1.219 1.247 1.241 1.240 1.171 1.091 1.091 1.105 1.119
Ta agona 1.168 1.156 1.153 1.157 1.170 1.190 1.206 1.211 1.175 1.154 1.113
Alican e 0.769 0.749 0.737 0.738 0.749 0.747 0.758 0.762 0.756 0.754 0.762
Cas ellón 0.970 0.998 0.972 0.989 0.992 1.021 1.037 1.090 1.067 1.062 1.048
Valencia 0.940 0.939 0.930 0.935 0.944 0.934 0.920 0.909 0.922 0.921 0.929
Badajoz 0.716 0.709 0.694 0.702 0.690 0.702 0.704 0.715 0.711 0.704 0.738
Cáce es 0.715 0.701 0.716 0.723 0.720 0.721 0.730 0.752 0.765 0.771 0.786
La Co uña 0.949 0.938 0.931 0.940 0.924 0.935 0.941 0.930 0.940 0.935 0.952
Lugo 0.865 0.880 0.899 0.918 0.936 0.950 0.924 0.892 0.910 0.893 0.889
O ense 0.803 0.824 0.842 0.839 0.829 0.825 0.838 0.839 0.852 0.875 0.890
Pon e ed a 0.856 0.842 0.840 0.852 0.856 0.854 0.850 0.871 0.859 0.869 0.904
Mad id 1.340 1.360 1.377 1.372 1.372 1.373 1.373 1.370 1.364 1.369 1.370
Mu cia 0.832 0.819 0.823 0.831 0.822 0.838 0.833 0.830 0.816 0.818 0.834
Economies 2025,13, 17 7 o 16
Table 2. Con .
P o ince/Au on. Ci y 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
Na a a 1.229 1.234 1.226 1.236 1.239 1.228 1.224 1.220 1.205 1.210 1.221
Ála a 1.474 1.496 1.507 1.532 1.550 1.515 1.546 1.524 1.515 1.475 1.506
Vizcaya 1.235 1.227 1.239 1.231 1.245 1.246 1.237 1.212 1.217 1.221 1.220
Guipúzcoa 1.289 1.299 1.311 1.298 1.286 1.271 1.264 1.297 1.289 1.297 1.288
La Rioja 1.082 1.082 1.082 1.087 1.102 1.096 1.068 1.063 1.068 1.061 1.087
Ceu a 0.850 0.835 0.822 0.839 0.821 0.814 0.805 0.782 0.786 0.795 0.839
Melilla 0.794 0.777 0.752 0.759 0.751 0.743 0.741 0.717 0.724 0.728 0.765
The main idea is o ollow he me hodology p oposed by Quah (1993), o analyze
o he e olu ion o GDP pe capi a based on da a om he las decade. This analysis,
wi h a Ma ko ian app oach, will ha e a double goal: e alua e he impac o COVID-19
pandemic on economic con e gence by p o inces in Spain; and s udy he speed o his
con e gence, bo h wi h and wi hou COVID-19. Speci ically, we s a om a homogeneous
and ini e Ma ko chain, wi h which o s udy he ansi ions be ween p oduc i i y s a es.
P oduc i i y will be g ouped in o ou s a es, “Low income”, “Medium-low income”,
“Medium-high income” and “High income”, which will be gi en by he qua iles o he
o e all GDP pe capi a dis ibu ion o he p o inces ela i e o he na ional a e age o each
yea , ha is, aking as da a he esul o di iding he GDP pe capi a o each p o ince by
he Spanish GDP pe capi a o he co esponding yea .
Nex s ep is cons uc ing he Ma ko chain ansi ion p obabili y ma ix. Wi h ha pu -
pose, we ob ain a ansi ion p obabili y ma ix be ween each pai o yea s o he conside ed
pe iod, which will gi e us a o al o en. To ob ain each o hem, he maximum likelihood
es ima o is used (see Sec ion 2.1.3), acco ding o which each ansi ion p obabili y can be
ob ained as he p opo ion o p o inces ha , being in a ce ain s a e in yea
, change o a
ce ain s a e in yea
+
1. The gene al ma ix is he esul o a e aging he numbe s in he
same posi ion o each o he en ma ix we ha e cons uc ed be o e. Then we check i he
ansi ion p obabili y ma ix sa is ies condi ions o Theo em 1, so ha he s a iona y and
equilib ium dis ibu ion can be ound and mee , as well as he mean ecu ence ime o
each s a e.
Finally, once we know he s a iona y dis ibu ion o he Ma ko chain, we y o ge he
con e gence speed o he p o inces owa ds his si ua ion. Fo his pu pose, Sho ocks (1978)
p oposes an index o analyze he mobili y be ween s a es gi en by he ansi ion p obabili y
ma ix in which he elemen s o he main diagonal a e each g ea e han o equal o he en ies
o he ma ix ha a e loca ed in he emaining posi ions. Speci ically, his index is
I1=n− (P)
n−1,
whe e
(P)
deno es he ace o he ma ix
P
and
n
is he numbe o s a es o he Ma ko
chain. This index gi es us alues be ween 0 and 1: mobili y is null when
I1=
1, because
in his case
(P) = n
and, consequen ly, all s a es a e abso ben ; 0 means pe ec mobili y.
Addi ionally, Somme s and Conlisk (1979) gi e ano he index wi h which o know he
speed a which he Ma ko chain eaches he s eady s a e, and i is
I2=1− |λ2|,
Economies 2025,13, 17 8 o 16
whe e
λ2
is he eigen alue o he ansi ion p obabili y ma ix wi h he second highes
modulus (in ac , in each Ma ko chain,
λ1=
1 is always an eigen alue, and i is he one
ha ing he g ea es modulus). Fo u he e e ence abou measu es o mobili y, see, o
example, Fo mby e al. (2004).
2.3. Clus e Analysis
Clus e analysis is a da a analysis echnique whose main goal is o g oup he da a in a
homogeneous way, which means ha he elemen s o he same g oup a e simila o each
o he in e ms o he cha ac e is ic which has been analyzed, as long as he disc epancies
be ween indi iduals om di e en g oups a e signi ican . In o he wo ds, his echnique
ies o minimize he in a-g oup a iabili y while ying o maximize he in e -g oup one.
In o de o de e mine which indi iduals in he sample ha e a ce ain simila i y, dis ances
a e gene ally used and, in his wo k, we will ope a e wi h he mos classic dis ance: he
Euclidean one. This dis ance, in essence, measu es he longi udinal magni ude in a s aigh
line om one poin o ano he . The e a e o he dis ances used in he cons uc ion o a
clus e , such as he Manha an dis ance, he Mahalanobis one o he maximum one. We
will ha e o minimize his dis ance be ween indi iduals o conclude which o hem a e he
mos simila .
Indi idual g ouping me hods a e classi ied in o wo g oups: hie a chical and non-
hie a chical clus e ing. In he i s case, we can gi e a ee-based ep esen a ion (called
dend og am) so ha in each i e a ion, an o de is ollowed and he s uc u e o c ea e he
g oups is kep . Mo eo e , hey can be classi ied in o wo g oups:
•
Agglome a i e: hey s a om simple g oups which become mo e sophis ica ed as
mo e i e a ions a e aken. I is, he e o e, an ascending app oach be ween indi iduals.
•
Di isi e: we s a om he sample as a g oup and, a each s ep, smalle g oups a e buil
un il he desi ed numbe o clus e s is achie ed. I is, he e o e, a descending app oach.
In non-hie a chical clus e ing, he numbe o g oups is chosen and, subsequen ly,
indi iduals a e included in each g oup, being able o mo e om one g oup o ano he a
each s ep, un il a ce ain op imali y c i e ion is go .
In his wo k we will ocus on he agglome a i e hie a chical clus e ing based on Wa d’s
me hod, using he Euclidean dis ance o ind he dis ance be ween elemen s. The me hod
used has o do wi h he way o calcula ing he dis ance be ween g oups. I does make sense
o conside he hie a chical clus e ing in his wo k, since i s ep esen a ion (dend og am)
is qui e use ul o he eade in o de o ha e a quick idea o he ela ionship be ween
p o inces in e ms o hei pe capi a income. Wha is mo e, hanks o his ep esen a ion,
one can g oup p o inces in o a di e en desi ed numbe o clus e s. Pa icula ly, we ha e
chosen he agglome a i e one, which is he mos common ype o hie a chical clus e ing,
indeed Kassamba a (2017). Since we wan o g oup p o inces in e ms o hei pe capi a
income, we s a by ea ing each p o ince as a single on clus e and, nex , pai s o clus e s
a e successi ely me ged un il all o hem belong o a single clus e , con aining all p o inces.
Finally, i is wo h no ing ha we ha e conside ed ou clus e s as ecommended by he
Elbow me hod and in o de o mee he numbe o s a es in he Ma ko chains analysis.
Fo ha pu pose, so wa e R is used in o de o ge he clus e ing. In o de o ge he
inal g oups, we ha e added Appendix B.2, in which he used code is explained.
3. Resul s and Discussion
3.1. E olu ion o pe Capi a Income in P esence o COVID-19
Le us conside he dis ibu ion o he pe capi a income o he i y p o inces and
wo au onomous ci ies in Spain o all yea s be ween 2010 and 2020 ela i e o he annual
Economies 2025,13, 17 15 o 16
s eadyS a es(mc)
Mo eo e , we can check i i is ape iodic by using he command
pe iod(mc)
I he esul is 1, he Ma ko chain is ape iodic. We can also ge he mean ecu ence
ime o each s a e:
meanRecu enceTime(mc)
Finally, inding ou i i is i educible is possible by w i ing
is.i educible(mc)
and checking i he esul is “TRUE” o “FALSE”. As a complemen , he command
plo (mc)
gi es us he ansi ion diag am, which is a plo whe e we can see he connec ion be ween
s a es h ough he ansi ion p obabili ies.
Appendix B.2. Clus e Analysis
Nex , we expose he s eps ha ha e been ollowed in o de o ge he inal g oups
when clus e ing (agglome a i e hie a chical clus e ing) p o inces acco ding o hei income
in he pe iod 2010–2020:
1.
S anda ize he a iables, ha is, sub ac he mean om he alue o each one and
di ide he esul by he s anda d de ia ion o he alues o he a iable. I he da a
con ain he in o ma ion o be p ocessed, we mus implemen
d =as.da a. ame(scale(da a))
2. Calcula e he p oximi y ma ix by using he Euclidean dis ance:
d_eu <-dis (d , me hod =’euclidean’ )
3. Find he agglome a i e hie a chical clus e wi h Wa d’s me hod:
clus e <- hclus (d_eu, me hod = ’wa d.D’)
4. D aw he dend og am:
plo (as.dend og am(clus e ))
5. D aw ec angles ha g oup a ce ain numbe , k, o he indi iduals in he sample:
ec .hclus (clus e , k = 4)
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