Modelling abstention rate using spatial regression

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 Paldam, M. (2008). Vo e and popula i y unc ions. In Readings in Public Choice and Cons i u ional Poli ical Economy (pp. 533–550). h ps://doi.o g/10.1007/978-0-387-75870-1_29 Poole, M. A., & O’Fa ell, P. N. (1971). The Assump ions o he Linea Reg ession Model. 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Spa ial pa e ns and o igins o hea y me als in Sheyang Ri e ca chmen in Jiangsu, China based on geog aphically weigh ed eg ession. Science o he To al En i onmen , 580, 1518–1529. h ps://doi.o g/10.1016/j.sci o en .2016.12.137 Xu, B., & Lin, B. (2017). Fac o s a ec ing CO2 emissions in China’s ag icul u e sec o : E idence om geog aphically weigh ed eg ession model. Ene gy Policy, 104(July 2016), 404–414. h ps://doi.o g/10.1016/j.enpol.2017.02.011 38 7. APPENDIX Appendix A – Co ela ion Ma ix