Modelling abstention rate using spatial regression

Modelling Abstention Rate using Spatial Regression Afonso Manita Santos Mota Dissertation presented as partial requirement for obtaining the Master’s degree in Statistics and Information Management, with specialization in Information Analysis and Management. Afonso Manita Santos Mota ing Spatial Regression Afonso Manita Santos Mota Afonso Manita Santos Mota Dissertation presented as partial requirement for obtaining the Master’s degree in Statistics and Informa ion Management, with specialization in Information Analysis and Management. Dissertation presented as partial requirement for obtaining the Master’s degree in Statistics and Information Management, with specialization in Information Analysis and Management. 2018 Modelling Abstention Rate using Spatial Regression Afonso Manita Santos Mota MEGI 2018 Modelling Abstention Rate using Spatial Regression Afonso Manita Santos Mota MEGI i i NOVA Information Management School Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa MODELLING ABSTENTION RATE USING SPATIAL REGRESSION by Afonso Manita Santos Mota Dissertation presented as ii partial requirement for obt iii aining the Master’s degree in Statistics and Information Management, with specialization in Information Analysis and Management. Advisor Ana Cristina Marinho da Costa, PhD November 2018 ABSTRACT During the last few elections that were held in Portugal, there have been very low percentages of voter turnout. This will obviously impact the result of those elections and can maybe be related to the general disenchantment of the population regarding the country’s recent political environment. This study aims to contribute to a better understanding of the patterns in the abstention rate of the last elections in Portugal. Sociological and economic variables such as age, unemployment rate, education level and many others will be used in trying to find out if they influence the abstention rate. It is logical to assume that the abstention rate in a certain municipality will be related to the abstention in neighboring municipalities. Therefore, the study also investigates if there is spatial autocorrelation in the abstention rates. Modeling a phenomenon like this with a simple linear regression model, estimated by Ordinary Least Squares (OLS), will render less efficient and biased results because of the spatial correlation of the observations and possible spatial clustering of values. Spatial regression methods have been proposed to overcome these drawbacks, particularly the iv Geographically Weighted Regression (GWR). This method will take into account possible local influences, allowing the coefficients of the model to vary depending on the geographic location, possibly obtaining a more appropriate fit. Many different OLS and GWR models were investigated by considering different combinations of explanatory variables and diagnosing their results through statistical tests and goodness-of-fit measures. Results show that indeed the data exhibits a non-random spatial pattern, and that a GWR model is a better approach in modeling abstention rates, when compared to an OLS model. Hence, the percentage of voter turnout in a municipality is likely to be better modelled taking into account its geographic location. KEYWORDS Voter turnout; Abstention rate; Sociological Variables; Economic Variables; Spatial Analysis; Geographically Weighted Regression; Spatial Non-Stationarity. v INDEX 1. Introduction .................................................................................................................. 1 1.1. Study Relevance and Importance .......................................................................... 1 1.2. Study Objectives .................................................................................................... 1 2. Literature review .......................................................................................................... 3 2.1. Voter Turnout ........................................................................................................ 3 2.2. Theoretical Framework ......................................................................................... 5 2.2.1. Exploratory Spatial Data Analysis ................................................................... 5 2.2.2. Ordinary Least Squares Regression ................................................................ 5 2.2.3. Geographically Weighted Regression ............................................................ 6 3. Methodology ................................................................................................................ 9 3.1. Data Collection ...................................................................................................... 9 3.2. Exploratory Spatial Data Analysis ........................................................................ 11 3.2.1. Global Moran’s I statistic .............................................................................. 11 3.2.2. Local Moran’s I statistic ................................................................................ 12 3.2.3. Getis-Ord General G statistic ........................................................................ 13 3.2.4. Getis-Ord Gi* statistic .................................................................................. 14 3.3. Ordinary Least Squares Models ........................................................................... 15 3.4. Geographically Weighted Regression Model ...................................................... 16 3.5. Comparing Models .............................................................................................. 19 4. Results and discussion ................................................................................................ 20 4.1. Exploratory Spatial Data Analysis ........................................................................ 20 4.2. Ordinary Least Squares Models ........................................................................... 23 4.3. Geographically Weighted Regression Model ...................................................... 25 4.4. Comparing Models .............................................................................................. 31 5. Conclusion .................................................................................................................. 33 5.1. Limitations ........................................................................................................... 33 5.2. Future work ......................................................................................................... 34 6. REFERENCES ................................................................................................................ 35 vi LIST OF TABLES Table 1 – Initial variables of the study ..................................................................................... 11 Table 2 – Global Moran’s Analysis ........................................................................................... 21 Table 3 – Getis-Ord General G Analysis ................................................................................... 21 Table 4 – Explanatory variables used for the best GWR models for each number of variables .......................................................................................................................................... 25 Table 5 – Results for the GWR models ..................................................................................... 26 5 2.2. THEORETICAL FRAMEWORK 2.2.1. Exploratory Spatial Data Analysis According to Anselin (1998), Exploratory Spatial Data Analysis (ESDA) is a collection of techniques to: ▪ Describe and visualize spatial distributions, by mapping the values of each variable and examining the patterns it displays; ▪ Identify spatial outliers and discover patterns of spatial association such as local cluster. Which can be done by analyzing the values of the Local Moran’s I statistic; ▪ Identify variables that exhibit high or low value clusters (Hotspot Analysis), by examining the value of the Getis-ord Gi* statistic; ▪ Suggest spatial regimes or other forms of spatial heterogeneity which happens when a variable has distinct distributions for different geographic sub regions. The objective of this analysis is to get an initial visualization of how the variables are distributed throughout the study region and hopefully identifying spatial autocorrelation in the dataset. Spatial autocorrelation measures if a variable is correlated with itself in neighboring locations. If it has high positive values, similar values occur close to one another, if it has high negative values, we observe very different values occurring close to one another. 2.2.2. Ordinary Least Squares Regression Simple linear regression is the most used regression method. This method helps investigate bivariate and multivariate relationships between variables, where we hypothesize that there is one predicted variable that depends on a combination of other variables. This model creates an estimate for the coefficients of each variable using the Ordinary Least Squares (OLS) estimator, which is proven to be the best linear unbiased estimator given the following assumptions as explained by Poole & O’Farrell, 1971 and by Hayashi, 2000: 1. Linearity (i.e. the model is correctly specified); 6 2. Random sampling (i.e. observations are independent of each other, thus no autocorrelation of the residuals); 3. Strict exogeneity (expected value of the residuals equal to zero); 4. No multicollinearity; 5. Spherical error variance (homoscedasticity of the residuals). 6. Normally distributed residuals; To test if these assumptions are met in a given dataset, several diagnostics have to be made: ▪ Examine the scatter plots of the dependent variable with each explanatory variable (Assumption 1); ▪ Durbin-Watson test to assess the temporal autocorrelation of the residuals (Assumption 2); ▪ Student’s T-test (Assumption 3); ▪ Examine the Variance Inflation Factor (Assumption 4); ▪ Examine the plot of residuals versus predicted values, and test homoscedasticity for example with the Breush-Pagan test (Assumption 5) ▪ Test residuals normality, for example with the Shapiro-Wilk’s test or Jarque-Bera test (Assumption 6) 2.2.3. Geographically Weighted Regression When the residuals of an OLS model have spatial autocorrelation or exhibit spatial heterogeneity such as clusters (i.e. spatial non-stationarity), spatial regression models should be used. The Geographically Weighted Regression (GWR) models are frequently used in geographical analysis. GWR was developed initially by Fotheringham et al. (1997) and more fully detailed again by Fotheringham et al. (2002). In these papers, GWR is explained as a method to create local summary statistics from geographically weighted point data. Afterwards these statistics are mapped and used to identify possible variations in the distribution of the variable of interest from location to location. This method has been used to model various situations from various different areas, such as Agriculture (Xu & Lin, 2017), Health, 7 Environment (Wu, Yang, Guo, & Han, 2017), Economy (Benassi & Naccarato, 2017) and Transports (Chiou, Jou, & Yang, 2015). GWR is a spatial regression method that is particularly used when spatial non-stationarity is found in the study data. Since stable variance in the data is required in order to assume that the Ordinary Least Squares Regression provides the best linear unbiased estimator, the GWR model will differ from the OLS model by estimating coefficients that depend on the spatial location of each observation. Therefore, unlike the OLS model, GWR is a local model. This method takes into account the variability of the data throughout the study area. And, to do so, it estimates a set of coefficients, one for each variable, at any given location (point or polygon, generally referred to as feature). GWR makes a point-wise calibration of the coefficients by assuming that observations which are closer to the regression feature will have a bigger influence in estimating that set of coefficients when compared to the observations that are farther away (Brunsdon, Fotheringham, Charlton, Brunsdont, & Chariton, 1998). Basically, GWR models relationships around each location in the data set, estimating the regression coefficients by weighted least squares using a spatial weights matrix. For each location, the data will be weighted differently so that the results of any one calibration are unique to a particular location. The Weighting matrix can be calculated with many different techniques, but the one which is most used is the ‘Gaussian-like’ kernel method. The kernel bandwidth is a parameter that needs to be specified either by a fixed number of nearest neighbours or by a fixed distance. When these values are the same for all features in the data set the procedure is named as fixed spatial kernel. Fixed spatial kernels have a few potential drawbacks. Where data points are sparse the local models might be calibrated on very few data points, thus the coefficient estimates are less reliable (i.e. have large standard errors). Where data points are dense the coefficient estimates are more likely to be biased, because there is more scope for examining changes in relationships over relatively small distances and such changes might be missed with larger kernels. 8 If the features are reasonably regularly spaced in the study region, then a fixed spatial kernel is appropriate for modelling. However, according to the previous discussion, the use of adaptive spatial kernels is recommended for irregularly sampled data (A. S. S. Fotheringham et al., 2002). Adaptive kernels increase the bandwidth size when the sample points are sparser and decrease its size when the sample points are denser. The choice of distance metric is important to study these phenomena, and usually GWR uses Euclidean Distance to measure the “geographical proximity” of two different observations. On the other hand, there have been several attempts to use non-Euclidean Distance such as: a modified ward-to-ward distance matrix (Shuttleworth & Lloyd, 2005), or a spatial-temporal distance that takes into account not only the geographical location of an observation but also the time at which it was recorded. Besides this, Longley et al. showed in 2005 that a distance metric can depend on a number of different factors such as the presence of rivers between two locations, the quality of the road infrastructure, presence of notorious public spaces, etc. In conclusion, in order to choose a distance matrix, we need to take into account the study area’s spatial context to see if a non-Euclidean Distance is appropriate. There are also a few problems associated with this model. First of all, if there is global multicollinearity in the data, as it could be expected, both OLS and GWR models will not be able to produce reliable estimates. More likely than this is to exist local multicollinearity in the data, which is also a problem when trying to implement a GWR model. This characteristic in the spatial data can prevent the Akaike’s Information Criterion (AIC) and Cross-Validation (CV Bandwidth) method in ArcGIS software from discovering the optimal distance or number of neighbours for the bandwidth, which may result in a wrong interpretation of the spatial patterns in the data. 9 3. METHODOLOGY To study the percentage of voter turnout in municipalities of Portugal, we will use the ArcGIS software to produce the analyses of the following stages: 1. Data Collection; 2. ESDA; 3. OLS models; 4. GWR model; 5. Comparing results. Each of these stages is detailed in the following sections. 3.1. DATA COLLECTION The dependant variable of our study will be the abstention rate of the 2013 Portuguese municipal elections. Taking into account different studies on this matter, we chose variables that have been shown to have some explanatory value regarding voter turnout and / or abstention rates. Therefore, using the websites PORDATA (http://www.pordata.pt) and Statistics Portugal (https://www.ine.pt), which contain data from various different subjects from several levels (European, Portuguese and Municipality), we were able to get all the variables that will be necessary to conduct the study. Taking into account that not all of the variables will be chosen for the final model, the initial set of variables that were chosen for the initial Exploratory Spatial Data Analysis are described in Table 1. All data have been collected at municipality level for continental Portugal. Variable Unit Measure Description Tx_abst % Abstention rate Dens_pop hab/km2 Population density Tx_desemp % Unemployment rate 10 Variable Unit Measure Description Crimes #/1000hab Number of crimes commited by 1000 residents Sal_med € Average Salary Dim_fam # Average family size Per_Div % Divorce rate Inactivos % Percentage of inactive residents compared to the active residents Inactivos_hom % Percentage of inactive male residents compared to the active male residents Inactivos_mulh % Percentage of inactive female residents compared to the active female residents Ind_env % Age index (Number of people over 65 for every 100 people under 15) Aloj_km2 #/km2 Average number of inhabited houses per km2 Poder_compra % Purchase power of each municipality regarding the country Estr_UE #/km2 Average number of EU foreigners per km2 Estr_outro #/km2 Average number of non-EU foreigners per km2 Homens #/km2 Average number of men per km2 Mulheres #/km2 Average number of women per km2 Receitas % Revenue of the municipality divided by the expenses Sem_escolaridade % Percentage of residents without any education Ciclo1 % Percentage of residents with a Primary Education Ciclo2 % Percentage of residents that completed the 2nd Cycle Ciclo3 % Percentage of residents with a Basic Education Secundario % Percentage of residents with a Secundary Education Superior % Percentage of residents with a Higher Education Sect1 % Percentage of residents working on the 1st Sector Sect2 % Percentage of residents working on the 2nd Sector Sect3 % Percentage of residents working on the 3rdSector Hab18_29 % Percentage of residents over 18 and under 29 Hab30_49 % Percentage of residents over 30 and under 49 11 Variable Unit Measure Description Hab50_69 % Percentage of residents over 50 and under 69 Hab70_ % Percentage of residents over 70 Table 1 – Initial variables of the study 3.2. EXPLORATORY SPATIAL DATA ANALYSIS After converting our data in order for it to be compatible with the ArcGIS formats we make an ESDA. This analysis starts with correlation analysis and graphical and visual methods, such as using mapped data to try to find patterns, outliers, or clusters in some region of the map. Afterwards, statistics like the Global and Local Moran’s I statistics, and the Getis-Ord General G and Getis-or Gi* statistic are used to evaluate if there is spatial autocorrelation and nonstationarity in the considered variables. 3.2.1. Global Moran’s I statistic The Global Moran’s I statistic is used to measure spatial autocorrelation. Given a set of features associated to a certain attribute, it will help determine if the pattern in the data is clustered, dispersed or random. In the software used to perform this study, there is a “Spatial Autocorrelation (Global Moran’s I” tool that will calculate this statistic in the following manner: With being the distance from the value of a variable for a given location to its mean, being the spatial weight between location and , being the total number of locations in the data and being the sum of all spatial weights. This tool will also calculate the Moran’s Index p-value and test statistic ( -score) as: 12 With and . Afterwards we will interpret, for each variable, the p-values and -score with the following criteria, considering a 5% significance level: • If the p-value is not statistically significant we cannot reject the null hypothesis. So it is quite possible that this variable has a random spatial distribution; • If the p-value is statistically significant and the -score is positive, then we reject the null hypothesis and can state that the data exhibits a clustered spatial distribution. This means that high [low] values of the given variable in a certain location, will be associated with high [low] values on neighbouring locations; • If the p-value is statistically significant and the -score is negative, then we reject the null hypothesis and can state that the data exhibits a dispersed spatial distribution. This means that high values of the given variable in a certain location, will be associated with low values on neighbouring locations, or vice-versa. 3.2.2. Local Moran’s I statistic The Local Moran’s I statistic is similar to the Global Moran’s I statistic in the sense that both assess spatial autocorrelation in the data. While the Global Moran’s I statistic allows drawing a conclusion for the spatial pattern of the whole study area, the Local Moran’s I statistic evaluates local patterns. Hence, the Local Moran’s I statistic identifies spatial clusters where variables have high or low values (positive spatial autocorrelation), and spatial outliers where high values correlate with low neighboring values and vice versa (negative spatial autocorrelation). In the software used to perform this study, there is a “Cluster and Outlier Analysis” tool that will calculate this statistic in the following manner: 13 With being the value of a variable for a given location , being the mean of said variable, being the spatial weight between location and , being the total number of locations in the data and being: This tool will also calculate the local Moran’s Index p-value and test statistic ( -score) as: With and . Afterwards we will interpret, for each variable, the p-values and -score with the following criteria: • If the p-value is not statistically significant we cannot reject the null hypothesis. So it is quite possible that this location is neither an outlier nor part of a cluster; • If the p-value is statistically significant and the -score is positive, then we reject the null hypothesis and can state that location is part of a cluster of either high or low values; • If the p-value is statistically significant and the -score is negative, then we reject the null hypothesis and can state that location is a spatial outlier (i.e. dissimilar values cluster together). 3.2.3. Getis-Ord General G statistic The Getis-Ord General G statistic is used to measure the concentration of high or low values in a dataset for a given variable. Given a set of features associated to a certain attribute, it will help determine if the pattern in the data has cluster of high or low relative values. In the software used to perform this study, there is a “High/Low Clustering (Getis-Ord General G)” tool that will calculate this statistic in the following manner: 14 With and being the value of a variable in locations and , being the spatial weight between location and , being the total number of locations in the data. This tool will also calculate the Getis-Ord General G p-value and test statistic ( -score) as: With and . Afterwards we will interpret, for each variable, the p-values and -score with the following criteria: • If the p-value is not statistically significant we cannot reject the null hypothesis. So it is quite possible that we cannot distinguish the possible clusters of these variables from being of high or low values; • If the p-value is statistically significant and the -score is positive, then we reject the null hypothesis and can state that this variable exhibits a clustered spatial distribution of high values; • If the p-value is statistically significant and the -score is negative, then we reject the null hypothesis and can state that this variable exhibits a clustered spatial distribution of low values. 3.2.4. Getis-Ord Gi* statistic The Getis-Ord Gi* statistic is used to find a given location in the dataset belongs to a high or low value cluster or hotspot. In the software used to perform this study, there is a “Hot Spot Analysis” tool that will calculate this statistic in the following manner: 21 • Variable Dim_fam is heavily correlated with Ciclo2 and since it has a higher correlation with our dependent variable, we will keep variable Dim_fam and exclude Ciclo2 from our further analysis; Afterwards we computed the Global Morans’ I and the Getis-Ord General G statistics which rendered the following p-values and z-scores: Variables Z-score P-value Tx_abst 24,9 < 0,0001 Dens_pop 34,2 < 0,0001 Tx_desemp 18,0 < 0,0001 Crimes 13,1 < 0,0001 Sal_Med 23,2 < 0,0001 Dim_fam 52,7 < 0,0001 Per_Div 10,2 < 0,0001 Poder_compra 24,4 < 0,0001 Estr_UE 20,8 < 0,0001 Estr_outro 34,1 < 0,0001 Receitas 9,5 < 0,0001 Ciclo3 27,1 < 0,0001 Secundario 37,8 < 0,0001 Superior 18,6 < 0,0001 Sect1 30,6 < 0,0001 Sect2 41,8 < 0,0001 Sect3 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 Moran’s Analysis Variables Z-score P-value Tx_abst 2,0 0,0492 Dens_pop 25,4 < 0,0001 Tx_desemp 1,3 0,1810 Crimes -4,8 < 0,0001 Sal_Med 2,5 0,0134 Dim_fam 10,8 < 0,0001 Per_Div -0,8 0,3977 Poder_compra 20,4 < 0,0001 Estr_UE 19,2 < 0,0001 Estr_outro 34,2 < 0,0001 Receitas 3,7 0,0002 Ciclo3 0,0 0,9905 Secundario -2,5 0,0133 Superior 5,0 < 0,0001 Sect1 2,7 0,0069 Sect2 13,0 < 0,0001 Sect3 -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 – Getis-Ord General G Analysis From these results we can draw the following conclusions: • For the Global Moran’s I statistics, since all p-values are statistically significant, we can consider with a very high level of certainty that all the variables do not exhibit a random spatial distribution. And, because all z-scores are positive, we may conclude that the spatial distribution of high values and/or low values in all variables is spatially clustered. • For the Getis-Ord General G statistics, we can conclude with a confidence of 95% that nearly every variable (every one with a p-value lower than 5%) exhibits a spatially clustered pattern with clusters being of either high or low vlaues. Actually, we cannot 22 reject the null hypothesis only for Tx_desemp, Per_Div and Ciclo 3. For all other variables, those that have positive z-scores have clusters of high values and, on the other hand, those that have negative z-scores have clusters of low values. Now, we will examine for each municipality what kind of pattern is exhibited in each value’s neighborhood. In order to do that, we computed the Local Moran’s I Index and the Getis-Ord Gi* statistic for every municipality in our dataset. For the dependent variable Tx_abst, we had the following results: Figure 2 – Local Moran’s I of abstention rate Figure 3 – Getis-Ord Gi* of abstention rate So, taking into account that colored municipalities are the ones that returned a statistically significant p-value for the corresponding test, we can conclude the following: • From the Local Moran’s I, we can conclude that the brightly red colored municipalities (Abrantes, Beja, Elvas, Moura, Penedono, Peso da Régua and Reguengos de Monsaraz) correspond to spatial outliers of high abstention rates surrounded by municipalities of low rates (significant negative spatial 23 autocorrelation). We also conclude that the blue brightly colored municipalities (Espinho and Monchique) correspond to spatial outliers of low abstention rates surrounded by municipalities of high rates (significant negative spatial autocorrelation). Municipalities with soft red or soft blue colors are spatial clusters, thus having a local pattern of significant positive spatial autocorrelation. Coastal municipalities colored with soft red have a high value surrounded primarily by high values, whereas those inland ones colored with soft blue have a low value surrounded primarily by low values. • From the Getis-Ord Gi*, we conclude that municipalities that are colored in orange/red (Aveiro, Faro, Lisboa and Setúbal), are hotspots of high abstention rates. On the other hand, municipalities that are colored blue (Beja, Bragança, Castelo Branco, Guarda, Portalegre and Porto) are coldspots of low abstention rates. These analyses helped us to have a better understanding of the spatial patterns in the dataset, which will be helpful in the next steps of the study and the interpretation of future results. 4.2. ORDINARY LEAST SQUARES MODELS In this step, we made exploratory regressions using the “Exploratory Regression” tool in ArcGIS. Using an iterative process, we chose subsets of variables in order to find which variables would provide a better model, not only considering the model’s comparison metric (AICc), but also the results of the diagnosing tests as described in the Methodology section. The variables that had better explanatory power regarding the dependent variable were: Positively significant: • Percentage of residents with a Secundary Education (Secundario) • Number of crimes commited by 1000 residents (Crimes) • Revenue of the municipality divided by the expenses (Receitas) • Percentage of residents with a Higher Education (Superior) • Percentage of residents over 50 and under 69 (Hab50_69) • Divorce rate (Per_Div) 24 • Average number of non-EU foreigners per km2 (Estr_outro); Negatively significant: • Age index: number of people over 65 for every 100 people under 15 (Ind_env) • Percentage of residents without any education (Sem_escolaridade) This means that, for most of the models created during each iteration of the exploratory analysis, an increase in the values of the first subset of variables would cause a significant increase in the abstention rate. On the other hand, an increase in the values of the negatively significant variables would cause a significant decrease in the abstention rate. Of all the models that were tested, one stood out regarding the criteria that were stated before. The model was the following: Tx_abst = 10,84 + 0,14 * Crimes + 0,03 * Per_Div – 0,02 * Ind_env + 0,09 * Receitas + + 0,91 * Secundário + 0,45 * Superior • Explanatory variables: > Number of crimes commited by 1000 residents (Crimes) > Divorce rate (Per_Div) > Age index: number of people over 65 for every 100 people under 15 (Ind_env) > Revenue of the municipality divided by the expenses (Receitas) > Percentage of residents with a Secundary Education (Secundario) > Percentage of residents with a Higher Education (Superior) • Statistic/p-values: > Adjusted R2: 50% > Corrected Akaike´s Information Criterion (AICc): 1.837,45 > Variance Inflation Factor (VIF): 2,98 > Jarque-Bera p-value: 0,37 > Koenker (BP) statistic p-value: 0,07 > Global Moran’s I p-value: 0,00 25 Since the VIF value was smaller than 7,5 there is no multicollinearity. The Jarque-Bera test allows to conclude that there is evidence that the residuals of this model are normally distributed, for the usual significance levels. The p-value of the Koenker (BP) statistic indicates that there is evidence of heteroscedasticity at significance levels greater than 7%. The Global Moran’s I test allows to conclude that the residuals have spatial autocorrelation, at any significance level. So, following this analysis, since we cannot trust the results of OLS due to the spatial autocorrelation and non-stationarity in our data, we will investigate the GWR model that uses these explanatory variables. 4.3. GEOGRAPHICALLY WEIGHTED REGRESSION MODEL In this step of the study, we used the variables from the best OLS model found and, iteratively, tried every possible subset of variables as the set of explanatory variables for a GWR model. After all possible combinations, we registered in Table 4 the six best models for each possible number of explanatory variables (from 1 to 6), and their results are detailed in Table 5. Explanatory Variables Crimes Per_Div Ind_env Receitas Secundário Superior 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 – Explanatory variables used for the best GWR models for each number of variables 26 Statistics Neighbours Sum of Residual Squares Effective Number of Coefficients Sigma AICc Adjusted 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 Multicolinearity Error Table 5 – Results for the GWR models So, first of all we can conclude that variable Receitas exhibits Local Multicollinearity, which makes it impossible to create a GWR model using this variable. Then, examining the AICc of the models we see that the model which is a better fit for our data is Model 2 with just 2 explanatory variables (Ind_env and Superior). As expected this model also has the lowest Sum of Residual Squares and Sigma, and the highest Adjusted R2. To analyse the Standardized Residuals of this model, we started by mapping them: 27 Figure 4 – Standardized Residuals of the GWR model As we can see, there are no clear areas with clustered high or low values, so there is evidence that the residuals have a random pattern, thus the model is well-specified and it is not missing any key explanatory variables. To make sure that this conclusion was valid, we measured the Global Moran’s statistic for the residuals and had the following results: • Moran’s Index: 0,0132 • Z-score: 0,58 • P-value: 0,56 So we can in fact say that the residuals exhibit a random spatial pattern throughout the data which suggests that this model is able to predict the spatial pattern of the dependent variable. Figure 5, shows the Local R2 for each municipality, which is a measure of the variability of the abstention rate that is explained by the local model. We can use this to understand where the model will render more explanatory power. 28 Figure 5 – Local R2 So, we can conclude that for the regions of Algarve and Alentejo, and for districts such as Castelo Branco, Coimbra, Leiria, Porto, Braga and Viana do Castelo, the local models will better explain the value of the abstention rate. On the other hand, the variability of the abstention rate explained by the local models in municipalities such as Lisbon, Guarda, Viseu, Vila Real and Bragança is much smaller. Now, we will examine how each explanatory variables influences the abstention rate by mapping the regression coefficients for each variable: 29 Figure 6 – Coefficients for Ind_env Figure 7 – Standard Error of the Coefficients for Ind_env Figure 8 – Coefficients for Superior Figure 9 - Standard Error of the Coefficients for Superior 30 First of all, looking at the coefficients for Ind_env (Age index: number of people over 65 for every 100 people under 15), we can see that in the northern districts of Portugal, excluding Coimbra, an older population will result in a higher abstention rate. On the other hand, in the rest of the country, and especially in the districts of Setúbal and Faro, an older population will result in a lower abstention rate. Adding to this, we can say that, according to the Standard error of the coefficients for Ind_env, parameter estimates for locations to the north of Lisbon will be more reliable than those for the south. Looking at the coefficients for Superior (Percentage of residents with a Higher Education), we can see that in a higher percentage of the population with a University degree will generally result in a higher abstention rate, particularly in the districts of Beja, Castelo Branco, Évora, Faro, Portalegre and Viseu. We can also say that, looking at the Standard error of the coefficients for Superior, parameter estimates for municipalities close to Lisbon and Porto will be more reliable when compared to other municipalities. Finally, we mapped the values for the abstention rate predicted by the model and compare them to the actual values to evaluate the goodness of fit of the model (Figures 6 and 7). 37 Paldam, M. (2008). Vote and popularity functions. 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