# RECORD 36eac2e74fd054fe7faa7ad4134c3aa524d5616d
# TITLE: Mass appraisal for property tax purposes: approach using global and local spatial regression
# YEAR: 2023.0  VENUE: Revista Valorem  DOI: 10.29327/2290393.2.2-2
# AUTHORS: C. Zilli; Norberto Hochheim
# PAGES: 15  CANDIDATE TABLE PAGES (1-based): [9, 10]
# ABSTRACT: O IPTU é uma fonte de receita essencial para os municípios brasileiros, mas o sistema atual, baseado em valores de avaliação desatualizados, resulta em um tratamento desigual dos contribuintes. Para promovera equidade, são necessárias avaliações regulares de imóveis, especialmente no mercado imobiliário brasileiro, que evolui rapidamente. Um desafio chave neste processo é desenvolver um modelo que trate eficazmente os efeitos espaciais. Este estudo explora aaplicação de modelos de regressão espacial global e local para enfrentar a dependência e heterogeneidade espaciais em avaliações em massa de imóveis, visando garantir uma tributação justado imóvel. Analisando imóveis urbanos, avaliou-seo desempenho do modelo autoregressivo condicional (CAR) como um modelo de regressão global e o modelo de regressãogeograficamente ponderada(GWR) como um modelo de regressão local. As previsões de ambos os modelos são combinadas com a interpolação por krigagem ordinária para gerar uma Planta de Valores Genéricos (PVG)abrangente, representando a área de estudo. Nossos achados demonstram que a modelagem GWR supera o modelo CAR em precisão e sua capacidade de contabilizar a dependência e heterogeneidade espacial. Isso destacao GWR como uma alternativa viável para estabelecer uma base justa e equitativa para o cálculo de impostos sobre imóveis. Esta pesquisa contribui significativamente para o avanço das técnicas de avaliação em massa de imóveis, assegurando que a coleta do IPTU esteja alinhada com a natureza dinâmica do mercado imobiliário brasileiro. As percepções do estudo fornecem orientações valiosas para formuladores de políticas e profissionais na melhoria dos sistemas de tributação de propriedades, beneficiando, em última análise, tanto municípios quanto contribuintes.


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VALOREM | REVISTA DA SOCIEDADE BRASILEIRA DE ENGENHARIA DE AVALIAÇÕES | E-ISSN 2764-8532 
 
 Revista Valorem | v. 2 n. 2 (2023) | Página 10 
 
 
Revista da Sociedade Brasileira de Engenharia de Avaliações 
 
 
MASS APPRAISAL FOR PROPERTY TAX PURPOSES: APPROACH 
USING GLOBAL AND LOCAL SPATIAL REGRESSION
 
Avaliação em massa de imóveis para fins de imposto predial: abordagem utilizando 
regressão espacial global e local
 
 
Carlos Augusto Zilli 
http://orcid.org/0000-0002-4926-1186 
Instituto Federal de Santa Catarina (IFSC), 
Florianópolis, Brasil 
carloszilli@gmail.com 
 
Norberto Hochheim 
http://orcid.org/0000-0003-4670-4831 
Universidade Federal de Santa Catarina (UFSC), 
Florianópolis, Brasil 
hochheim@gmail.com 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Preenchimento dos Editores 
 
INFORMAÇÕES SOBRE O ARTIGO 
Submetido em
18/06/2023
Aceito em
22/09/2023
Publicado  em
25/09/2023
 
Comitê Científico Interinstitucional 
Editor-Responsável: Carlos Augusto Zilli 
Avaliado pelo Sistema Double Blind Review 
 
ABSTRACT 
Property tax is a vital revenue source for Brazilian municipalities, but the 
current system based on outdated assessment values results in unequal 
treatment of taxpayers. To promote fairness, regular property appraisals are 
necessary, especially in the rapidly evolving Brazilian real estate market. A 
key challenge in this process is developing a model that effectively 
addresses spatial effects. This study explores the application of global and 
local spatial regression models to tackle spatial dependence and 
heterogeneity in mass property appraisals, aiming to ensure fair property 
taxation. Analyzing urban properties, we evaluate the performance of the 
conditional autoregressive model (CAR) as a global regression model and 
geographically weighted regression (GWR) as a local regression model. 
Predictions from both models are combined with ordinary kriging 
interpolation to generate a comprehensive Property Value Map (PVM) 
representing the study area. Our findings show that the GWR modeling 
surpasses the CAR model in accuracy and its ability to account for spatial 
dependence and heterogeneity. This highlights GWR as a viable alternative 
for establishing a fair and equitable basis for calculating real estate taxes. 
This research significantly contributes to advancing mass appraisal 
techniques, ensuring property tax collection aligns with the dynamic nature 
of the Brazilian real estate market. The study's insights provide valuable 
guidance for policymakers and practitioners in improving property tax 
systems, ultimately benefiting municipalities and taxpayers alike.
Keywords: Mass appraisal of properties; Property tax; Property values 
map; Geographically weighted regression; Spatial econometrics.
RESUMO
O IPTU é uma fonte de receita essencial para os municípios brasileiros, 
mas o sistema atual, baseado em valores de avaliação desatualizados, 
resulta em um tratamento desigual dos contribuintes. Para promover a 
equidade, são necessárias avaliações regulares de imóveis, especialmente 
no mercado imobiliário brasileiro, que evolui rapidamente. Um desafio 
chave neste processo é desenvolver um modelo que trate eficazmente os 
efeitos espaciais. Este estudo explora a aplicação de modelos de regressão 
espacial global e local para enfrentar a dependência e heterogeneidade 
espaciais em avaliações em massa de imóveis, visando garantir uma 
tributação justa do imóvel. Analisando imóveis urbanos, avaliou-se o 
desempenho do modelo autoregressivo condicional (CAR) como um modelo 
de regressão global e o modelo de regressão geograficamente ponderada 
(GWR) como um modelo de regressão local. As previsões de ambos os 
modelos são combinadas com a interpolação por krigagem ordinária para 
gerar uma Planta de Valores Genéricos (PVG) abrangente, representando a 
área de estudo. Nossos achados demonstram que a modelagem GWR 
supera o modelo CAR em precisão e sua capacidade de contabilizar a 
dependência e heterogeneidade espacial. Isso destaca o GWR como uma 
alternativa viável para estabelecer uma base justa e equitativa para o 
cálculo 
de 
impostos 
sobre 
imóveis. 
Esta 
pesquisa 
contribui 
significativamente para o avanço das técnicas de avaliação em massa de 
imóveis, assegurando que a coleta do IPTU esteja alinhada com a natureza 
dinâmica do mercado imobiliário brasileiro. As percepções do estudo 
fornecem 
orientações 
valiosas 
para 
formuladores 
de 
políticas 
e 
profissionais na melhoria dos sistemas de tributação de propriedades, 
beneficiando, em última análise, tanto municípios quanto contribuintes.
Palavras-Chave: Avaliação em massa de imóveis; Imposto predial, Planta 
de valores genéricos; Regressão geograficamente ponderada; Econometria. 
 
 


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VALOREM | REVISTA DA SOCIEDADE BRASILEIRA DE ENGENHARIA DE AVALIAÇÕES | E-ISSN 2764-8532 
 
 Revista Valorem | v. 2 n. 2 (2023) | Página 18 
In the spatial regression error model, the lambda () represents the term of spatial error autocorrelation 
and is significant at the level of 5.0%. It is also possible to observe that all variables in the spatial regression 
error model were significant at the level of 5.0%, as shown in table 4. 
 
Table 4: Statistics related to the parameters of the CAR model. 
Variable1
Coefficients 
zcalc 
Significance 
Intercepto 
10.6470 
38.1924 
0.00000 
ln (FA) 
– 0.43255 
7.81141 
0.00000 
ln (DA) 
– 0.11770 
5.35113 
0.00000 
NB 
0.05794 
2.76309 
0.00573 
BT 
0.03583 
2.12263 
0.03379 
NG 
0.18520 
8.92755 
0.00000 
SP 
0.09129 
3.47056 
0.00052 
CQ 
0.20171 
12.3279 
0.00000 
 (lâmbda) 
0.56916 
4.57072 
0.00000 
1 Note: FA: floor area; DA: distance to Beira Mar Avenue; NB: number of bedrooms; BT: number of bathrooms; NG: number of 
parking spaces; SP: swimming pool; CQ: construction quality; : term of spatial autocorrelation. 
 
It is possible to observe that all regressors are significant and important for the formation of the model, 
confirmed by the low and close to zero significance. The signs of the regressors of the spatial regression 
error model confirm the expectation of the local real estate market, and are therefore consistent. 
Finally, the Breusch-Pagan test for the residuals of the spatial regression error model indicated that 
there are no problems of heteroscedasticity. With a BP value of 6.57 and a p-value of 0.4740, it can be said, 
at the 5.0% level of significance, that the autoregressive spatial error model is homocedastic. 
 
3.3. LOCAL SPATIAL REGRESSION RESULTS (GWR) 
 
The GWR model was adjusted with the same transformations and the same 190 market data used in 
the spatial regression of the error. To choose the best weighting function, whether fixed or variable, and the 
optimal bandwidth, the software GWR4 by Fotheringham et al. (2005) was employed, as described in the 
Methodology. Multiple simulations were conducted, which were separated according to selection criteria 
defined by the user in the GWR4 software interface. 
The type of weighting function was evaluated between fixed or variable Gaussian and fixed or variable 
bi-squared. The software also allows you to choose the stopping criterion and check the automatic checkbox 
for the best bandwidth. It then generates several models for different bandwidths, based on the selected 
weighting function and finally presents the one with the lowest value for the selected criterion (AIC, AICc and 
CV). 
Twelve models were generated and the chosen one would be the one that provided the lowest value in 
the Akaike information criterion (AIC), a parameter that will be used to make the comparison between the 
models generated in this study. The model that generated the lowest value in the AIC criterion was the fixed 
Gaussian weighting function with bandwidth  = 278.42 m. 
As seen, there is still the possibility that, among the seven explanatory variables considered in this 
study, one of them presents spatial stationarity, that is, it does not have a relevant spatial variability along the 
geographic plane. 
In general, the geographic variability for each coefficient can be tested by comparing models. To test 
the geographical variability of the k-th variable coefficient, a model comparison is performed between the 
adjusted GWR and a test model in which only the k-th coefficient is fixed, while other coefficients are 
maintained as in the geographically weighted adjusted model. 
In this case, if the adjusted GWR is better than the test GWR model, using the AICc as a comparison 
criterion, we can conclude that the k-th coefficient certainly varies in space. This test routine repeats this 
comparison between models for each geographically variable coefficient. 
For this test, as described by Leung et al. (2000), a positive DIFF Criterion, especially greater than 2.0, 
suggests that the local variable can be assumed as global. In the test, only the pool variable (PS) showed a 
positive result and greater than 2.0. In this study, this variable will be considered as global. The model, 
containing average coefficients for the local variables, and fixed coefficient for the pool variable, is in (9): 
 
ln (UV) =  0 + 1  ln (FA) + 2  ln (DA) + 3  (NB) + 4  (BT) + 5  (NG) + 6  (SP) + 7  (CQ) 
(9) 


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VALOREM | REVISTA DA SOCIEDADE BRASILEIRA DE ENGENHARIA DE AVALIAÇÕES | E-ISSN 2764-8532 
 
 Revista Valorem | v. 2 n. 2 (2023) | Página 19 
In the GWR model, the transformations adopted in the CAR model remained. It is noteworthy that the 
average coefficients were used because the GWR modeling, being a local regression, generates a 
regression equation for each sample point, weighted by the distance from its neighbors. The GWR model 
generated a report containing 190 regressions with the parameter estimates, the calculation of the t statistics 
and the standard error for each variable, as well as the predicted value, the local determination coefficient, 
the Cook distance and the influence, for each observation of the sample. Table 5 shows the statistics of the 
GWR model: 
 
Table 5: Statistics related to the parameters of the GWR model. 
Variable
Average Coefficients 
Median Coefficients 
DIFF Criterion 
Type 
Intercepto 
10.73166 
10.73542 
– 681.00 
Local 
ln  (FA) 
– 0.51700 
– 0.50740 
– 490.75 
Local 
ln (DA) 
– 0.08873 
– 0.10434 
– 685.68 
Local 
NB 
0.06497 
0.06628 
– 6.0723 
Local 
BT 
0.06173 
0.06011 
– 8.8288 
Local 
NG 
0.19071 
0.19394 
– 8.5496 
Local 
SP 
0.09732 
0.09732 
2.2167 
Global 
CQ 
0.20735 
0.22325 
– 13.272 
Local 
 
The signs of the regressors of the geographically weighted regression model (GWR) were the same as 
the signs of the spatial error model (CAR) and confirm the expectation of the local real estate market, being, 
therefore, coherent. 
 
3.4. GOODNESS OF FIT AND PERFORMANCE 
 
In this study, the quality of the fit of the models was analyzed using the Akaike (AIC), Bayesian (BIC) 
and Loglikelihood (LIK) information criteria, the determination coefficient (R²) and the standard error of the 
regression. Table 6 shows the indicators. 
 
Table 6: Quality indicators of the modeling performed. 
Statistics 
CAR Model 
GWR Model 
Determination Coefficient (R2) 
0.8169 
0.9239 
Log Likelihood (LIK) 
97.348 
182.005 
Schwarz Criterion  (BIC) 
– 152.80 
– 187.12 
Akaike Criterion  (AIC) 
– 178.70 
– 209.78 
Standard Error of Regression (Se) 
0.1425 
0.1303 
 
It is possible to notice that the GWR model proved to be superior to the CAR model in all the fit quality 
indicators in table 6. 
It was also verified how the absolute relative errors of the values predicted by the models were 
dispersed. Then, linear interpolation was performed to generate the surface of gradients, containing the 
absolute relative errors estimated for each modeling. These interpolations can be seen in figures 4a and 4b. 
 
 
Figure 4: Surface of relative absolute errors for 4a) CAR model and 4b) GWR model. 
 
Figures 4a and 4b show that the surface of relative errors for the GWR modeling was more uniform, 
with errors of lesser magnitude, evidenced by the dominance of the blue color. 


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VALOREM | REVISTA DA SOCIEDADE BRASILEIRA DE ENGENHARIA DE AVALIAÇÕES | E-ISSN 2764-8532 
 
 Revista Valorem | v. 2 n. 2 (2023) | Página 20 
As a way of visualizing the predictive power of the models, a dispersion graph of the observed versus 
adjusted values was constructed for each of the models, according to figures 5a and 5b. 
 
              
 
Figure 5: Dispersion diagram for 8a) CAR model and 8b) GWR model. 
 
It is possible to notice that the GWR model was more adjusted, with less dispersed points, indicating 
that it is an adjustment with greater predictive power of unit values than the CAR model. To analyze the 
performance of the modeling, we used the metrics established by the IAAO (2013). The limits established by 
the standard for the properties under study are shown in table 2. 
The data from the validation partition contained 17 properties in the Centro neighborhood, 6 properties 
in the Agronômica neighborhood and 12 properties in the Trindade neighborhood, as shown in table 1. The 
validation partition was analyzed to verify the presence of some discrepant data that could impair the results. 
It was found that the data AV_14, whose unit value of the property is R$ 17,500.00, is an outlier and, for this 
reason, was not used in the calculations. 
To calculate the performance of the geographically weighted regression, the prediction of the unit 
values of the validation partition was performed directly in the GWR4 software. The GWR4 software 
provides, for each of the validation partitions, based on the selected kernel and bandwidth, a prediction 
equation, with the respective t-statistic, standard error and local R² values. 
The results of the three indicators recommended by the IAAO (International Association of Assessing 
Officers) are shown in table 7. 
 
Table 7: Performance indicators of the modeling performed. 
Statistics 
CAR Model 
GWR Model 
Median of the Assessment Ratios (RATIO) 
0.968 
0.957 
Coefficient of Dispersion of the Median (COD) 
12.22% 
10.57% 
Price Related Differential (PRD) 
1.017 
1.015 
 
It can be observed that the two models presented values within the limits recommended by the 
standard. However, the GWR modeling proved to be superior, indicating greater uniformity in the 
assessment, as evidenced by the lower value in the COD. As for the PRD, the results indicate that there was 
no progressivity or regressivity in the predictions. Nevertheless, the GWR model exhibited slightly superior 
performance in this indicator. 
In addition, the square root of the mean square error (RMSE) was also calculated for the unit values 
predicted by the validation partition. The results are shown in table 8. 
 
Table 8: Root mean squared error (RMSE) for the validation partition. 
Statistics 
CAR Model 
GWR Model 
RMSE VALIDATION PARTITION 
1,412.27 
1,321.25 
Notes: The currency exchange rate in 05/2023 is US$ 1.00 = R$ 5.05 
 
In this case, the GWR modeling, which is a local spatial regression, again presented lower values for 
the RMSE. This phenomenon corroborates the idea that the value of a given property is strongly affected by 
the value of the properties in its surroundings.