Modelling Abs en ion Ra e using Spa ial
Reg ession
A onso Mani a San os Mo a
Disse a ion p esen ed as pa ial equi emen o ob aining
he Mas e ’s deg ee in S a is ics and In o ma ion
Managemen , wi h specializa ion in In o ma ion Analysis and
Managemen .
A onso Mani a San os Mo a
ing Spa ial Reg ession
A onso Mani a San os Mo a
A onso Mani a San os Mo a
Disse a ion p esen ed as pa ial equi emen o ob aining
he Mas e ’s deg ee in S a is ics and In o ma
ion Managemen , wi h specializa ion in In o ma ion Analysis
and Managemen .
Disse a ion p esen ed as pa ial equi emen o ob aining
he Mas e ’s deg ee in S a is ics and In o ma ion
Managemen , wi h specializa ion in In o ma ion Analysis and
Managemen .
2018
Modelling Abs en ion Ra e using Spa ial Reg ession
A onso Mani a San os Mo a
MEGI
2018
Modelling Abs en ion Ra e using Spa ial Reg ession
A onso Mani a San os Mo a
MEGI
i
i
NOVA In o ma ion Managemen School
Ins i u o Supe io de Es a ís ica e Ges ão de In o mação
Uni e sidade No a de Lisboa
MODELLING ABSTENTION RATE USING SPATIAL REGRESSION
by
A onso Mani a San os Mo a
Disse a ion p esen ed as
ii
pa ial equi emen o ob

iii
aining he Mas e ’s deg ee in S a is ics and In o ma ion Managemen , wi h specializa ion in
In o ma ion Analysis and Managemen .
Ad iso Ana C is ina Ma inho da Cos a, PhD
No embe 2018
ABSTRACT
Du ing he las ew elec ions ha we e held in Po ugal, he e ha e been e y low
pe cen ages o o e u nou . This will ob iously impac he esul o hose elec ions and can
maybe be ela ed o he gene al disenchan men o he popula ion ega ding he coun y’s
ecen poli ical en i onmen .
This s udy aims o con ibu e o a be e unde s anding o he pa e ns in he abs en ion
a e o he las elec ions in Po ugal. Sociological and economic a iables such as age,
unemploymen a e, educa ion le el and many o he s will be used in ying o ind ou i
hey in luence he abs en ion a e. I is logical o assume ha he abs en ion a e in a ce ain
municipali y will be ela ed o he abs en ion in neighbo ing municipali ies. The e o e, he
s udy also in es iga es i he e is spa ial au oco ela ion in he abs en ion a es.
Modeling a phenomenon like his wi h a simple linea eg ession model, es ima ed by
O dina y Leas Squa es (OLS), will ende less e icien and biased esul s because o he
spa ial co ela ion o he obse a ions and possible spa ial clus e ing o alues. Spa ial
eg ession me hods ha e been p oposed o o e come hese d awbacks, pa icula ly he
i
Geog aphically Weigh ed Reg ession (GWR). This me hod will ake in o accoun possible
local in luences, allowing he coe icien s o he model o a y depending on he geog aphic
loca ion, possibly ob aining a mo e app op ia e i . Many di e en OLS and GWR models
we e in es iga ed by conside ing di e en combina ions o explana o y a iables and
diagnosing hei esul s h ough s a is ical es s and goodness-o - i measu es.
Resul s show ha indeed he da a exhibi s a non- andom spa ial pa e n, and ha a GWR
model is a be e app oach in modeling abs en ion a es, when compa ed o an OLS model.
Hence, he pe cen age o o e u nou in a municipali y is likely o be be e modelled
aking in o accoun i s geog aphic loca ion.
KEYWORDS
Vo e u nou ; Abs en ion a e; Sociological Va iables; Economic Va iables; Spa ial Analysis;
Geog aphically Weigh ed Reg ession; Spa ial Non-S a iona i y.
INDEX
1. In oduc ion .................................................................................................................. 1
1.1. S udy Rele ance and Impo ance .......................................................................... 1
1.2. S udy Objec i es .................................................................................................... 1
2. Li e a u e e iew .......................................................................................................... 3
2.1. Vo e Tu nou ........................................................................................................ 3
2.2. Theo e ical F amewo k ......................................................................................... 5
2.2.1. Explo a o y Spa ial Da a Analysis ................................................................... 5
2.2.2. O dina y Leas Squa es Reg ession ................................................................ 5
2.2.3. Geog aphically Weigh ed Reg ession ............................................................ 6
3. Me hodology ................................................................................................................ 9
3.1. Da a Collec ion ...................................................................................................... 9
3.2. Explo a o y Spa ial Da a Analysis ........................................................................ 11
3.2.1. Global Mo an’s I s a is ic .............................................................................. 11
3.2.2. Local Mo an’s I s a is ic ................................................................................ 12
3.2.3. Ge is-O d Gene al G s a is ic ........................................................................ 13
3.2.4. Ge is-O d Gi* s a is ic .................................................................................. 14
3.3. O dina y Leas Squa es Models ........................................................................... 15
3.4. Geog aphically Weigh ed Reg ession Model ...................................................... 16
3.5. Compa ing Models .............................................................................................. 19
4. Resul s and discussion ................................................................................................ 20
4.1. Explo a o y Spa ial Da a Analysis ........................................................................ 20
4.2. O dina y Leas Squa es Models ........................................................................... 23
4.3. Geog aphically Weigh ed Reg ession Model ...................................................... 25
4.4. Compa ing Models .............................................................................................. 31
5. Conclusion .................................................................................................................. 33
5.1. Limi a ions ........................................................................................................... 33
5.2. Fu u e wo k ......................................................................................................... 34
6. REFERENCES ................................................................................................................ 35
i
LIST OF TABLES
Table 1 – Ini ial a iables o he s udy ..................................................................................... 11
Table 2 – Global Mo an’s Analysis ........................................................................................... 21
Table 3 – Ge is-O d Gene al G Analysis ................................................................................... 21
Table 4 – Explana o y a iables used o he bes GWR models o each numbe o a iables
.......................................................................................................................................... 25
Table 5 – Resul s o he GWR models ..................................................................................... 26
5
2.2. THEORETICAL FRAMEWORK
2.2.1. Explo a o y Spa ial Da a Analysis
Acco ding o Anselin (1998), Explo a o y Spa ial Da a Analysis (ESDA) is a collec ion o
echniques o:
▪ Desc ibe and isualize spa ial dis ibu ions, by mapping he alues o each a iable
and examining he pa e ns i displays;
▪ Iden i y spa ial ou lie s and disco e pa e ns o spa ial associa ion such as local
clus e . Which can be done by analyzing he alues o he Local Mo an’s I s a is ic;
▪ Iden i y a iables ha exhibi high o low alue clus e s (Ho spo Analysis), by
examining he alue o he Ge is-o d Gi* s a is ic;
▪ Sugges spa ial egimes o o he o ms o spa ial he e ogenei y which happens when
a a iable has dis inc dis ibu ions o di e en geog aphic sub egions.
The objec i e o his analysis is o ge an ini ial isualiza ion o how he a iables a e
dis ibu ed h oughou he s udy egion and hope ully iden i ying spa ial au oco ela ion in
he da ase . Spa ial au oco ela ion measu es i a a iable is co ela ed wi h i sel in
neighbo ing loca ions. I i has high posi i e alues, simila alues occu close o one ano he ,
i i has high nega i e alues, we obse e e y di e en alues occu ing close o one
ano he .
2.2.2. O dina y Leas Squa es Reg ession
Simple linea eg ession is he mos used eg ession me hod. This me hod helps in es iga e
bi a ia e and mul i a ia e ela ionships be ween a iables, whe e we hypo hesize ha he e
is one p edic ed a iable ha depends on a combina ion o o he a iables. This model
c ea es an es ima e o he coe icien s o each a iable using he O dina y Leas Squa es
(OLS) es ima o , which is p o en o be he bes linea unbiased es ima o gi en he ollowing
assump ions as explained by Poole & O’Fa ell, 1971 and by Hayashi, 2000:
1. Linea i y (i.e. he model is co ec ly speci ied);

6
2. Random sampling (i.e. obse a ions a e independen o each o he , hus no
au oco ela ion o he esiduals);
3. S ic exogenei y (expec ed alue o he esiduals equal o ze o);
4. No mul icollinea i y;
5. Sphe ical e o a iance (homoscedas ici y o he esiduals).
6. No mally dis ibu ed esiduals;
To es i hese assump ions a e me in a gi en da ase , se e al diagnos ics ha e o be made:
▪ Examine he sca e plo s o he dependen a iable wi h each explana o y a iable
(Assump ion 1);
▪ Du bin-Wa son es o assess he empo al au oco ela ion o he esiduals
(Assump ion 2);
▪ S uden ’s T- es (Assump ion 3);
▪ Examine he Va iance In la ion Fac o (Assump ion 4);
▪ Examine he plo o esiduals e sus p edic ed alues, and es homoscedas ici y o
example wi h he B eush-Pagan es (Assump ion 5)
▪ Tes esiduals no mali y, o example wi h he Shapi o-Wilk’s es o Ja que-Be a es
(Assump ion 6)
2.2.3. Geog aphically Weigh ed Reg ession
When he esiduals o an OLS model ha e spa ial au oco ela ion o exhibi spa ial
he e ogenei y such as clus e s (i.e. spa ial non-s a iona i y), spa ial eg ession models should
be used.
The Geog aphically Weigh ed Reg ession (GWR) models a e equen ly used in geog aphical
analysis. GWR was de eloped ini ially by Fo he ingham e al. (1997) and mo e ully de ailed
again by Fo he ingham e al. (2002). In hese pape s, GWR is explained as a me hod o
c ea e local summa y s a is ics om geog aphically weigh ed poin da a. A e wa ds hese
s a is ics a e mapped and used o iden i y possible a ia ions in he dis ibu ion o he
a iable o in e es om loca ion o loca ion. This me hod has been used o model a ious
si ua ions om a ious di e en a eas, such as Ag icul u e (Xu & Lin, 2017), Heal h,
7
En i onmen (Wu, Yang, Guo, & Han, 2017), Economy (Benassi & Nacca a o, 2017) and
T anspo s (Chiou, Jou, & Yang, 2015).
GWR is a spa ial eg ession me hod ha is pa icula ly used when spa ial non-s a iona i y is
ound in he s udy da a. Since s able a iance in he da a is equi ed in o de o assume ha
he O dina y Leas Squa es Reg ession p o ides he bes linea unbiased es ima o , he GWR
model will di e om he OLS model by es ima ing coe icien s ha depend on he spa ial
loca ion o each obse a ion. The e o e, unlike he OLS model, GWR is a local model.
This me hod akes in o accoun he a iabili y o he da a h oughou he s udy a ea. And, o
do so, i es ima es a se o coe icien s, one o each a iable, a any gi en loca ion (poin o
polygon, gene ally e e ed o as ea u e). GWR makes a poin -wise calib a ion o he
coe icien s by assuming ha obse a ions which a e close o he eg ession ea u e will
ha e a bigge in luence in es ima ing ha se o coe icien s when compa ed o he
obse a ions ha a e a he away (B unsdon, Fo he ingham, Cha l on, B unsdon , &
Cha i on, 1998). Basically, GWR models ela ionships a ound each loca ion in he da a se ,
es ima ing he eg ession coe icien s by weigh ed leas squa es using a spa ial weigh s
ma ix. Fo each loca ion, he da a will be weigh ed di e en ly so ha he esul s o any one
calib a ion a e unique o a pa icula loca ion.
The Weigh ing ma ix can be calcula ed wi h many di e en echniques, bu he one which is
mos used is he ‘Gaussian-like’ ke nel me hod. The ke nel bandwid h is a pa ame e ha
needs o be speci ied ei he by a ixed numbe o nea es neighbou s o by a ixed dis ance.
When hese alues a e he same o all ea u es in he da a se he p ocedu e is named as
ixed spa ial ke nel.
Fixed spa ial ke nels ha e a ew po en ial d awbacks. Whe e da a poin s a e spa se he local
models migh be calib a ed on e y ew da a poin s, hus he coe icien es ima es a e less
eliable (i.e. ha e la ge s anda d e o s). Whe e da a poin s a e dense he coe icien
es ima es a e mo e likely o be biased, because he e is mo e scope o examining changes
in ela ionships o e ela i ely small dis ances and such changes migh be missed wi h la ge
ke nels.
8
I he ea u es a e easonably egula ly spaced in he s udy egion, hen a ixed spa ial ke nel
is app op ia e o modelling. Howe e , acco ding o he p e ious discussion, he use o
adap i e spa ial ke nels is ecommended o i egula ly sampled da a (A. S. S. Fo he ingham
e al., 2002). Adap i e ke nels inc ease he bandwid h size when he sample poin s a e
spa se and dec ease i s size when he sample poin s a e dense .
The choice o dis ance me ic is impo an o s udy hese phenomena, and usually GWR uses
Euclidean Dis ance o measu e he “geog aphical p oximi y” o wo di e en obse a ions.
On he o he hand, he e ha e been se e al a emp s o use non-Euclidean Dis ance such as:
a modi ied wa d- o-wa d dis ance ma ix (Shu lewo h & Lloyd, 2005), o a spa ial- empo al
dis ance ha akes in o accoun no only he geog aphical loca ion o an obse a ion bu
also he ime a which i was eco ded. Besides his, Longley e al. showed in 2005 ha a
dis ance me ic can depend on a numbe o di e en ac o s such as he p esence o i e s
be ween wo loca ions, he quali y o he oad in as uc u e, p esence o no o ious public
spaces, e c. In conclusion, in o de o choose a dis ance ma ix, we need o ake in o accoun
he s udy a ea’s spa ial con ex o see i a non-Euclidean Dis ance is app op ia e.
The e a e also a ew p oblems associa ed wi h his model. Fi s o all, i he e is global
mul icollinea i y in he da a, as i could be expec ed, bo h OLS and GWR models will no be
able o p oduce eliable es ima es. Mo e likely han his is o exis local mul icollinea i y in
he da a, which is also a p oblem when ying o implemen a GWR model. This
cha ac e is ic in he spa ial da a can p e en he Akaike’s In o ma ion C i e ion (AIC) and
C oss-Valida ion (CV Bandwid h) me hod in A cGIS so wa e om disco e ing he op imal
dis ance o numbe o neighbou s o he bandwid h, which may esul in a w ong
in e p e a ion o he spa ial pa e ns in he da a.
9
3. METHODOLOGY
To s udy he pe cen age o o e u nou in municipali ies o Po ugal, we will use he A cGIS
so wa e o p oduce he analyses o he ollowing s ages:
1. Da a Collec ion;
2. ESDA;
3. OLS models;
4. GWR model;
5. Compa ing esul s.
Each o hese s ages is de ailed in he ollowing sec ions.
3.1. DATA COLLECTION
The dependan a iable o ou s udy will be he abs en ion a e o he 2013 Po uguese
municipal elec ions.
Taking in o accoun di e en s udies on his ma e , we chose a iables ha ha e been
shown o ha e some explana o y alue ega ding o e u nou and / o abs en ion a es.
The e o e, using he websi es PORDATA (h p://www.po da a.p ) and S a is ics Po ugal
(h ps://www.ine.p ), which con ain da a om a ious di e en subjec s om se e al le els
(Eu opean, Po uguese and Municipali y), we we e able o ge all he a iables ha will be
necessa y o conduc he s udy. Taking in o accoun ha no all o he a iables will be
chosen o he inal model, he ini ial se o a iables ha we e chosen o he ini ial
Explo a o y Spa ial Da a Analysis a e desc ibed in Table 1. All da a ha e been collec ed a
municipali y le el o con inen al Po ugal.
Va iable
Uni Measu e
Desc ip ion
Tx_abs
%
Abs en ion a e
Dens_pop
hab/km2
Popula ion densi y
Tx_desemp
%
Unemploymen a e
10
Va iable
Uni Measu e
Desc ip ion
C imes
#/1000hab
Numbe o c imes commi ed by 1000 esiden s
Sal_med
€
A e age Sala y
Dim_ am
#
A e age amily size
Pe _Di
%
Di o ce a e
Inac i os
%
Pe cen age o inac i e esiden s compa ed o he ac i e esiden s
Inac i os_hom
%
Pe cen age o inac i e male esiden s compa ed o he ac i e male esiden s
Inac i os_mulh
%
Pe cen age o inac i e emale esiden s compa ed o he ac i e emale
esiden s
Ind_en
%
Age index (Numbe o people o e 65 o e e y 100 people unde 15)
Aloj_km2
#/km2
A e age numbe o inhabi ed houses pe km2
Pode _comp a
%
Pu chase powe o each municipali y ega ding he coun y
Es _UE
#/km2
A e age numbe o EU o eigne s pe km2
Es _ou o
#/km2
A e age numbe o non-EU o eigne s pe km2
Homens
#/km2
A e age numbe o men pe km2
Mulhe es
#/km2
A e age numbe o women pe km2
Recei as
%
Re enue o he municipali y di ided by he expenses
Sem_escola idade
%
Pe cen age o esiden s wi hou any educa ion
Ciclo1
%
Pe cen age o esiden s wi h a P ima y Educa ion
Ciclo2
%
Pe cen age o esiden s ha comple ed he 2nd Cycle
Ciclo3
%
Pe cen age o esiden s wi h a Basic Educa ion
Secunda io
%
Pe cen age o esiden s wi h a Secunda y Educa ion
Supe io
%
Pe cen age o esiden s wi h a Highe Educa ion
Sec 1
%
Pe cen age o esiden s wo king on he 1s Sec o
Sec 2
%
Pe cen age o esiden s wo king on he 2nd Sec o
Sec 3
%
Pe cen age o esiden s wo king on he 3 dSec o
Hab18_29
%
Pe cen age o esiden s o e 18 and unde 29
Hab30_49
%
Pe cen age o esiden s o e 30 and unde 49

11
Va iable
Uni Measu e
Desc ip ion
Hab50_69
%
Pe cen age o esiden s o e 50 and unde 69
Hab70_
%
Pe cen age o esiden s o e 70
Table 1 – Ini ial a iables o he s udy
3.2. EXPLORATORY SPATIAL DATA ANALYSIS
A e con e ing ou da a in o de o i o be compa ible wi h he A cGIS o ma s we make
an ESDA. This analysis s a s wi h co ela ion analysis and g aphical and isual me hods, such
as using mapped da a o y o ind pa e ns, ou lie s, o clus e s in some egion o he map.
A e wa ds, s a is ics like he Global and Local Mo an’s I s a is ics, and he Ge is-O d Gene al
G and Ge is-o Gi* s a is ic a e used o e alua e i he e is spa ial au oco ela ion and non-
s a iona i y in he conside ed a iables.
3.2.1. Global Mo an’s I s a is ic
The Global Mo an’s I s a is ic is used o measu e spa ial au oco ela ion. Gi en a se o
ea u es associa ed o a ce ain a ibu e, i will help de e mine i he pa e n in he da a is
clus e ed, dispe sed o andom.
In he so wa e used o pe o m his s udy, he e is a “Spa ial Au oco ela ion (Global
Mo an’s I” ool ha will calcula e his s a is ic in he ollowing manne :
Wi h being he dis ance om he alue o a a iable o a gi en loca ion o i s mean,
being he spa ial weigh be ween loca ion and , being he o al numbe o
loca ions in he da a and being he sum o all spa ial weigh s.
This ool will also calcula e he Mo an’s Index p- alue and es s a is ic ( -sco e) as:
12
Wi h and .
A e wa ds we will in e p e , o each a iable, he p- alues and -sco e wi h he ollowing
c i e ia, conside ing a 5% signi icance le el:
• I he p- alue is no s a is ically signi ican we canno ejec he null hypo hesis. So i
is qui e possible ha his a iable has a andom spa ial dis ibu ion;
• I he p- alue is s a is ically signi ican and he -sco e is posi i e, hen we ejec he
null hypo hesis and can s a e ha he da a exhibi s a clus e ed spa ial dis ibu ion.
This means ha high [low] alues o he gi en a iable in a ce ain loca ion, will be
associa ed wi h high [low] alues on neighbou ing loca ions;
• I he p- alue is s a is ically signi ican and he -sco e is nega i e, hen we ejec he
null hypo hesis and can s a e ha he da a exhibi s a dispe sed spa ial dis ibu ion.
This means ha high alues o he gi en a iable in a ce ain loca ion, will be
associa ed wi h low alues on neighbou ing loca ions, o ice- e sa.
3.2.2. Local Mo an’s I s a is ic
The Local Mo an’s I s a is ic is simila o he Global Mo an’s I s a is ic in he sense ha bo h
assess spa ial au oco ela ion in he da a. While he Global Mo an’s I s a is ic allows d awing
a conclusion o he spa ial pa e n o he whole s udy a ea, he Local Mo an’s I s a is ic
e alua es local pa e ns. Hence, he Local Mo an’s I s a is ic iden i ies spa ial clus e s whe e
a iables ha e high o low alues (posi i e spa ial au oco ela ion), and spa ial ou lie s
whe e high alues co ela e wi h low neighbo ing alues and ice e sa (nega i e spa ial
au oco ela ion).
In he so wa e used o pe o m his s udy, he e is a “Clus e and Ou lie Analysis” ool ha
will calcula e his s a is ic in he ollowing manne :
13
Wi h being he alue o a a iable o a gi en loca ion , being he mean o said a iable,
being he spa ial weigh be ween loca ion and , being he o al numbe o
loca ions in he da a and being:
This ool will also calcula e he local Mo an’s Index p- alue and es s a is ic ( -sco e) as:
Wi h and .
A e wa ds we will in e p e , o each a iable, he p- alues and -sco e wi h he ollowing
c i e ia:
• I he p- alue is no s a is ically signi ican we canno ejec he null hypo hesis. So i
is qui e possible ha his loca ion is nei he an ou lie no pa o a clus e ;
• I he p- alue is s a is ically signi ican and he -sco e is posi i e, hen we ejec he
null hypo hesis and can s a e ha loca ion is pa o a clus e o ei he high o low
alues;
• I he p- alue is s a is ically signi ican and he -sco e is nega i e, hen we ejec he
null hypo hesis and can s a e ha loca ion is a spa ial ou lie (i.e. dissimila alues
clus e oge he ).
3.2.3. Ge is-O d Gene al G s a is ic
The Ge is-O d Gene al G s a is ic is used o measu e he concen a ion o high o low alues
in a da ase o a gi en a iable. Gi en a se o ea u es associa ed o a ce ain a ibu e, i
will help de e mine i he pa e n in he da a has clus e o high o low ela i e alues.
In he so wa e used o pe o m his s udy, he e is a “High/Low Clus e ing (Ge is-O d
Gene al G)” ool ha will calcula e his s a is ic in he ollowing manne :
14
Wi h and being he alue o a a iable in loca ions and , being he spa ial weigh
be ween loca ion and , being he o al numbe o loca ions in he da a.
This ool will also calcula e he Ge is-O d Gene al G p- alue and es s a is ic ( -sco e) as:
Wi h and .
A e wa ds we will in e p e , o each a iable, he p- alues and -sco e wi h he ollowing
c i e ia:
• I he p- alue is no s a is ically signi ican we canno ejec he null hypo hesis. So i
is qui e possible ha we canno dis inguish he possible clus e s o hese a iables
om being o high o low alues;
• I he p- alue is s a is ically signi ican and he -sco e is posi i e, hen we ejec he
null hypo hesis and can s a e ha his a iable exhibi s a clus e ed spa ial
dis ibu ion o high alues;
• I he p- alue is s a is ically signi ican and he -sco e is nega i e, hen we ejec
he null hypo hesis and can s a e ha his a iable exhibi s a clus e ed spa ial
dis ibu ion o low alues.
3.2.4. Ge is-O d Gi* s a is ic
The Ge is-O d Gi* s a is ic is used o ind a gi en loca ion in he da ase belongs o a high o
low alue clus e o ho spo .
In he so wa e used o pe o m his s udy, he e is a “Ho Spo Analysis” ool ha will
calcula e his s a is ic in he ollowing manne :
21
• Va iable Dim_ am is hea ily co ela ed wi h Ciclo2 and since i has a highe
co ela ion wi h ou dependen a iable, we will keep a iable Dim_ am and exclude
Ciclo2 om ou u he analysis;
A e wa ds we compu ed he Global Mo ans’ I and he Ge is-O d Gene al G s a is ics which
ende ed he ollowing p- alues and z-sco es:
Va iables
Z-sco e
P- alue
Tx_abs
24,9
< 0,0001
Dens_pop
34,2
< 0,0001
Tx_desemp
18,0
< 0,0001
C imes
13,1
< 0,0001
Sal_Med
23,2
< 0,0001
Dim_ am
52,7
< 0,0001
Pe _Di
10,2
< 0,0001
Pode _comp a
24,4
< 0,0001
Es _UE
20,8
< 0,0001
Es _ou o
34,1
< 0,0001
Recei as
9,5
< 0,0001
Ciclo3
27,1
< 0,0001
Secunda io
37,8
< 0,0001
Supe io
18,6
< 0,0001
Sec 1
30,6
< 0,0001
Sec 2
41,8
< 0,0001
Sec 3
40,2
< 0,0001
Hab18_29
27,2
< 0,0001
Hab30_49
27,6
< 0,0001
Hab50_69
27,1
< 0,0001
Hab70_
25,1
< 0,0001
Table 2 – Global Mo an’s Analysis
Va iables
Z-sco e
P- alue
Tx_abs
2,0
0,0492
Dens_pop
25,4
< 0,0001
Tx_desemp
1,3
0,1810
C imes
-4,8
< 0,0001
Sal_Med
2,5
0,0134
Dim_ am
10,8
< 0,0001
Pe _Di
-0,8
0,3977
Pode _comp a
20,4
< 0,0001
Es _UE
19,2
< 0,0001
Es _ou o
34,2
< 0,0001
Recei as
3,7
0,0002
Ciclo3
0,0
0,9905
Secunda io
-2,5
0,0133
Supe io
5,0
< 0,0001
Sec 1
2,7
0,0069
Sec 2
13,0
< 0,0001
Sec 3
-5,8
< 0,0001
Hab18_29
21,1
< 0,0001
Hab30_49
21,2
< 0,0001
Hab50_69
20,3
< 0,0001
Hab70_
18,1
< 0,0001
Table 3 – Ge is-O d Gene al G Analysis
F om hese esul s we can d aw he ollowing conclusions:
• Fo he Global Mo an’s I s a is ics, since all p- alues a e s a is ically signi ican , we
can conside wi h a e y high le el o ce ain y ha all he a iables do no exhibi a
andom spa ial dis ibu ion. And, because all z-sco es a e posi i e, we may conclude
ha he spa ial dis ibu ion o high alues and/o low alues in all a iables is
spa ially clus e ed.
• Fo he Ge is-O d Gene al G s a is ics, we can conclude wi h a con idence o 95% ha
nea ly e e y a iable (e e y one wi h a p- alue lowe han 5%) exhibi s a spa ially
clus e ed pa e n wi h clus e s being o ei he high o low laues. Ac ually, we canno

22
ejec he null hypo hesis only o Tx_desemp, Pe _Di and Ciclo 3. Fo all o he
a iables, hose ha ha e posi i e z-sco es ha e clus e s o high alues and, on he
o he hand, hose ha ha e nega i e z-sco es ha e clus e s o low alues.
Now, we will examine o each municipali y wha kind o pa e n is exhibi ed in each
alue’s neighbo hood. In o de o do ha , we compu ed he Local Mo an’s I Index and
he Ge is-O d Gi* s a is ic o e e y municipali y in ou da ase . Fo he dependen
a iable Tx_abs , we had he ollowing esul s:
Figu e 2 – Local Mo an’s I o abs en ion a e
Figu e 3 – Ge is-O d Gi* o abs en ion a e
So, aking in o accoun ha colo ed municipali ies a e he ones ha e u ned a s a is ically
signi ican p- alue o he co esponding es , we can conclude he ollowing:
• F om he Local Mo an’s I, we can conclude ha he b igh ly ed colo ed
municipali ies (Ab an es, Beja, El as, Mou a, Penedono, Peso da Régua and
Reguengos de Monsa az) co espond o spa ial ou lie s o high abs en ion a es
su ounded by municipali ies o low a es (signi ican nega i e spa ial
23
au oco ela ion). We also conclude ha he blue b igh ly colo ed municipali ies
(Espinho and Monchique) co espond o spa ial ou lie s o low abs en ion a es
su ounded by municipali ies o high a es (signi ican nega i e spa ial
au oco ela ion). Municipali ies wi h so ed o so blue colo s a e spa ial clus e s,
hus ha ing a local pa e n o signi ican posi i e spa ial au oco ela ion. Coas al
municipali ies colo ed wi h so ed ha e a high alue su ounded p ima ily by high
alues, whe eas hose inland ones colo ed wi h so blue ha e a low alue
su ounded p ima ily by low alues.
• F om he Ge is-O d Gi*, we conclude ha municipali ies ha a e colo ed in
o ange/ ed (A ei o, Fa o, Lisboa and Se úbal), a e ho spo s o high abs en ion a es.
On he o he hand, municipali ies ha a e colo ed blue (Beja, B agança, Cas elo
B anco, Gua da, Po aleg e and Po o) a e coldspo s o low abs en ion a es.
These analyses helped us o ha e a be e unde s anding o he spa ial pa e ns in he
da ase , which will be help ul in he nex s eps o he s udy and he in e p e a ion o u u e
esul s.
4.2. ORDINARY LEAST SQUARES MODELS
In his s ep, we made explo a o y eg essions using he “Explo a o y Reg ession” ool in
A cGIS. Using an i e a i e p ocess, we chose subse s o a iables in o de o ind which
a iables would p o ide a be e model, no only conside ing he model’s compa ison me ic
(AICc), bu also he esul s o he diagnosing es s as desc ibed in he Me hodology sec ion.
The a iables ha had be e explana o y powe ega ding he dependen a iable we e:
Posi i ely signi ican :
• Pe cen age o esiden s wi h a Secunda y Educa ion (Secunda io)
• Numbe o c imes commi ed by 1000 esiden s (C imes)
• Re enue o he municipali y di ided by he expenses (Recei as)
• Pe cen age o esiden s wi h a Highe Educa ion (Supe io )
• Pe cen age o esiden s o e 50 and unde 69 (Hab50_69)
• Di o ce a e (Pe _Di )
24
• A e age numbe o non-EU o eigne s pe km2 (Es _ou o);
Nega i ely signi ican :
• Age index: numbe o people o e 65 o e e y 100 people unde 15 (Ind_en )
• Pe cen age o esiden s wi hou any educa ion (Sem_escola idade)
This means ha , o mos o he models c ea ed du ing each i e a ion o he explo a o y
analysis, an inc ease in he alues o he i s subse o a iables would cause a signi ican
inc ease in he abs en ion a e. On he o he hand, an inc ease in he alues o he
nega i ely signi ican a iables would cause a signi ican dec ease in he abs en ion a e.
O all he models ha we e es ed, one s ood ou ega ding he c i e ia ha we e s a ed
be o e. The model was he ollowing:
Tx_abs = 10,84 + 0,14 * C imes + 0,03 * Pe _Di – 0,02 * Ind_en + 0,09 * Recei as +
+ 0,91 * Secundá io + 0,45 * Supe io
• Explana o y a iables:
> Numbe o c imes commi ed by 1000 esiden s (C imes)
> Di o ce a e (Pe _Di )
> Age index: numbe o people o e 65 o e e y 100 people unde 15 (Ind_en )
> Re enue o he municipali y di ided by he expenses (Recei as)
> Pe cen age o esiden s wi h a Secunda y Educa ion (Secunda io)
> Pe cen age o esiden s wi h a Highe Educa ion (Supe io )
• S a is ic/p- alues:
> Adjus ed R2: 50%
> Co ec ed Akaike´s In o ma ion C i e ion (AICc): 1.837,45
> Va iance In la ion Fac o (VIF): 2,98
> Ja que-Be a p- alue: 0,37
> Koenke (BP) s a is ic p- alue: 0,07
> Global Mo an’s I p- alue: 0,00
25
Since he VIF alue was smalle han 7,5 he e is no mul icollinea i y. The Ja que-Be a es
allows o conclude ha he e is e idence ha he esiduals o his model a e no mally
dis ibu ed, o he usual signi icance le els. The p- alue o he Koenke (BP) s a is ic
indica es ha he e is e idence o he e oscedas ici y a signi icance le els g ea e han 7%.
The Global Mo an’s I es allows o conclude ha he esiduals ha e spa ial au oco ela ion,
a any signi icance le el.
So, ollowing his analysis, since we canno us he esul s o OLS due o he spa ial
au oco ela ion and non-s a iona i y in ou da a, we will in es iga e he GWR model ha
uses hese explana o y a iables.
4.3. GEOGRAPHICALLY WEIGHTED REGRESSION MODEL
In his s ep o he s udy, we used he a iables om he bes OLS model ound and,
i e a i ely, ied e e y possible subse o a iables as he se o explana o y a iables o a
GWR model. A e all possible combina ions, we egis e ed in Table 4 he six bes models o
each possible numbe o explana o y a iables ( om 1 o 6), and hei esul s a e de ailed in
Table 5.
Explana o y
Va iables
C imes
Pe _Di
Ind_en
Recei as
Secundá io
Supe io
Model 1
X
Model 2
X
X
Model 3
X
X
X
Model 4
X
X
X
X
Model 5
X
X
X
X
X
Model 6
X
X
X
X
X
X
Table 4 – Explana o y a iables used o he bes GWR models o each numbe o a iables
26
S a is ics
Neighbou s
Sum o
Residual
Squa es
E ec i e
Numbe o
Coe icien s
Sigma
AICc
Adjus ed R2
Model 1
31
6.558
57
5,45
1.774,58
64%
Model 2
35
5.742
67
5,21
1.760,62
67%
Model 3
58
6.349
57
5,35
1.766,36
66%
Model 4
98
7.178
43
5,53
1.771,12
63%
Model 5
272
10.278
12
6,20
1.815,27
53%
Model 6
Local Mul icolinea i y E o
Table 5 – Resul s o he GWR models
So, i s o all we can conclude ha a iable Recei as exhibi s Local Mul icollinea i y, which
makes i impossible o c ea e a GWR model using his a iable. Then, examining he AICc o
he models we see ha he model which is a be e i o ou da a is Model 2 wi h jus 2
explana o y a iables (Ind_en and Supe io ). As expec ed his model also has he lowes
Sum o Residual Squa es and Sigma, and he highes Adjus ed R2.
To analyse he S anda dized Residuals o his model, we s a ed by mapping hem:

27
Figu e 4 – S anda dized Residuals o he GWR model
As we can see, he e a e no clea a eas wi h clus e ed high o low alues, so he e is
e idence ha he esiduals ha e a andom pa e n, hus he model is well-speci ied and i is
no missing any key explana o y a iables. To make su e ha his conclusion was alid, we
measu ed he Global Mo an’s s a is ic o he esiduals and had he ollowing esul s:
• Mo an’s Index: 0,0132
• Z-sco e: 0,58
• P- alue: 0,56
So we can in ac say ha he esiduals exhibi a andom spa ial pa e n h oughou he da a
which sugges s ha his model is able o p edic he spa ial pa e n o he dependen
a iable.
Figu e 5, shows he Local R2 o each municipali y, which is a measu e o he a iabili y o he
abs en ion a e ha is explained by he local model. We can use his o unde s and whe e
he model will ende mo e explana o y powe .
28
Figu e 5 – Local R2
So, we can conclude ha o he egions o Alga e and Alen ejo, and o dis ic s such as
Cas elo B anco, Coimb a, Lei ia, Po o, B aga and Viana do Cas elo, he local models will
be e explain he alue o he abs en ion a e. On he o he hand, he a iabili y o he
abs en ion a e explained by he local models in municipali ies such as Lisbon, Gua da,
Viseu, Vila Real and B agança is much smalle .
Now, we will examine how each explana o y a iables in luences he abs en ion a e by
mapping he eg ession coe icien s o each a iable:
29
Figu e 6 – Coe icien s o Ind_en
Figu e 7 – S anda d E o o he Coe icien s
o Ind_en
Figu e 8 – Coe icien s o Supe io
Figu e 9 - S anda d E o o he Coe icien s
o Supe io
30
Fi s o all, looking a he coe icien s o Ind_en (Age index: numbe o people o e 65 o
e e y 100 people unde 15), we can see ha in he no he n dis ic s o Po ugal, excluding
Coimb a, an olde popula ion will esul in a highe abs en ion a e. On he o he hand, in
he es o he coun y, and especially in he dis ic s o Se úbal and Fa o, an olde
popula ion will esul in a lowe abs en ion a e. Adding o his, we can say ha , acco ding o
he S anda d e o o he coe icien s o Ind_en , pa ame e es ima es o loca ions o he
no h o Lisbon will be mo e eliable han hose o he sou h.
Looking a he coe icien s o Supe io (Pe cen age o esiden s wi h a Highe Educa ion),
we can see ha in a highe pe cen age o he popula ion wi h a Uni e si y deg ee will
gene ally esul in a highe abs en ion a e, pa icula ly in he dis ic s o Beja, Cas elo
B anco, É o a, Fa o, Po aleg e and Viseu. We can also say ha , looking a he S anda d
e o o he coe icien s o Supe io , pa ame e es ima es o municipali ies close o Lisbon
and Po o will be mo e eliable when compa ed o o he municipali ies.
Finally, we mapped he alues o he abs en ion a e p edic ed by he model and compa e
hem o he ac ual alues o e alua e he goodness o i o he model (Figu es 6 and 7).
37
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7. APPENDIX
Appendix A – Co ela ion Ma ix