# RECORD a7a83b631f85316bae926a6c965b325fb26c01c4 # TITLE: A gamma generalised linear model as an alternative to log linear real estate price functions # YEAR: 2019.0 VENUE: nan DOI: 10.4102/jef.v12i1.476 # AUTHORS: Dane Bax; T. Zewotir; D. North # PAGES: 11 CANDIDATE TABLE PAGES (1-based): [9, 10] # ABSTRACT: Orientation: Residential property markets play an important role in economies, informing policy development and decision-making. However, measuring quality-adjusted growth is difficult because of the heterogeneity of properties. Hedonic regression is frequently used in real estate econometric studies as a quality-adjusted technique to estimate residential property prices for the development of price indices. Log linear models are typically used to derive these hedonic price functions. Research purpose: This article develops hedonic pricing functions using generalised linear models for South African residential property listings over a 5-year period. Motivation for the study: A parametric alternative to the log linear model is investigated to address the limited studies conducted in South Africa. An important feature of this study is the inclusion of different property types and the geographic scope. Research approach/design and method: The data set consisted of 415 200 residential properties from all over South Africa. The data spanned a period from January 2013 to August 2017. Several generalised linear models were developed and compared. Main findings: The gamma generalised linear model provided the best overall fit, generalising well to the unseen validation data. An added benefit of this model is that the estimates were kept on the original scale, avoiding the need for back transformation which is an appealing feature of any model. A dummy locational variable was shown to account for the spatial dependency in the data. Practical/managerial implications: This framework provides property market participants with the ability to quantify the utility derived over the marginal distribution of the physical characteristics of properties. This research presents the groundwork to create a property price index where index number theory could be applied to the counterfactual predicted values obtained from hedonic price models to measure price inflation over time Contribution/value-add: This study analysed the South African residential property market based on an online company’s data, purportedly covering the entire market. No real estate hedonic price studies have been identified in South Africa with this level of scope. The gamma generalised linear model is a novel candidate to develop parametric real estate hedonic price functions. ===== PAGE 1 (score 4, numeric-paren 0, table-words 0, var-words 4) ===== https://www.jefjournal.org.za Open Access Journal of Economic and Financial Sciences ISSN: (Online) 2312-2803, (Print) 1995-7076 Page 1 of 11 Original Research Read online: Scan this QR code with your smart phone or mobile device to read online. Authors: Dane Bax1 Temesgen Zewotir1 Delia North1 Affiliations: 1School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa Corresponding author: Dane Bax, danebax@gmail.com Dates: Received: 28 Apr. 2019 Accepted: 02 Aug. 2019 Published: 05 Dec. 2019 How to cite this article: Bax, D., Zewotir, T. & North, D., 2019, ‘A gamma generalised linear model as an alternative to log linear real estate price functions’, Journal of Economic and Financial Sciences 12(1), a476. https://doi.org/​ 10.4102/jef.v12i1.476 Copyright: © 2019. The Authors. Licensee: AOSIS. This work is licensed under the Creative Commons Attribution License. Introduction The importance of measuring residential property price inflation is paramount to households and economies; however, the heterogeneity of properties makes it difficult (De Haan & Diewert 2011). Log linear models have been used extensively in real estate economics to estimate property prices and inflation. This study investigates generalised linear models as an alternative to the typical log linear approach. Cross-sectional hedonic price functions are developed and compared for the South African residential property market over a 5-year period. Background and objective Residential property is an important component of individual and national wealth where it is capitalised on household balance sheets, informing economic policy formulation (Hill 2013). Goodhart and Hoffman (2008) conducted a study providing evidence of relationships between house prices, credit and broad money. Using vector auto-regression fitted with ordinary Orientation: Residential property markets play an important role in economies, informing policy development and decision-making. However, measuring quality-adjusted growth is difficult because of the heterogeneity of properties. Hedonic regression is frequently used in real estate econometric studies as a quality-adjusted technique to estimate residential property prices for the development of price indices. Log linear models are typically used to derive these hedonic price functions. Research purpose: This article develops hedonic pricing functions using generalised linear models for South African residential property listings over a 5-year period. Motivation for the study: A parametric alternative to the log linear model is investigated to address the limited studies conducted in South Africa. An important feature of this study is the inclusion of different property types and the geographic scope. Research approach/design and method: The data set consisted of 415 200 residential properties from all over South Africa. The data spanned a period from January 2013 to August 2017. Several generalised linear models were developed and compared. Main findings: The gamma generalised linear model provided the best overall fit, generalising well to the unseen validation data. An added benefit of this model is that the estimates were kept on the original scale, avoiding the need for back transformation which is an appealing feature of any model. A dummy locational variable was shown to account for the spatial dependency in the data. Practical/managerial implications: This framework provides property market participants with the ability to quantify the utility derived over the marginal distribution of the physical characteristics of properties. This research presents the groundwork to create a property price index where index number theory could be applied to the counterfactual predicted values obtained from hedonic price models to measure price inflation over time Contribution/value-add: This study analysed the South African residential property market based on an online company’s data, purportedly covering the entire market. No real estate hedonic price studies have been identified in South Africa with this level of scope. The gamma generalised linear model is a novel candidate to develop parametric real estate hedonic price functions. Keywords: generalised linear models; real estate economics; model comparison and evaluation; spatial modelling; hedonic price functions. A gamma generalised linear model as an alternative to log linear real estate price functions Read online: Scan this QR code with your smart phone or mobile device to read online. ===== PAGE 9 (score 22, numeric-paren 0, table-words 4, var-words 8) ===== Page 9 of 11 Original Research https://www.jefjournal.org.za Open Access The analysis of deviance presented in Table 5 indicates that the residual deviance for each yearly gamma model was consistently lower than null deviance. This means that the covariates accounted for greater deviance explained than intercept only models and as such indicates a good fit. The modelling of spatial data in this study required the assessment of the assumption of independence, which was investigated using several plots and by performing hypothesis tests. Variograms quantify the spatial dependence in data by describing the spatial variance. The yearly gamma models’ residuals were plotted using spherical variograms and are presented in Figure 4 where similarities were found between all models. The ranges, distances beyond which the data are no longer correlated, are quite long, which suggests spatial autocorrelation is not an issue in the modelling results. The nugget effects as a percentage of the total sills are quite large which could indicate some variation at a small scale. A permutation test for Morans I was applied to formally test for the presence of spatial autocorrelation where, under the  null hypothesis, the data are randomly dispersed. The  Morans I statistic or correlation coefficient ranges between -1 and 1, where -1 shows perfect negative spatial a b d e c 4 2013 2 0 -2 Deviance residuals Log (size) 4 5 6 7 Log (size) 4 2 0 -2 Deviance residuals 4 5 6 7 2014 Log (size) 4 2 0 -2 Deviance residuals 4 5 6 7 2015 Log (size) 4 2 0 -2 Deviance residuals 4 5 6 7 2016 Log (size) 4 2 0 -2 Deviance residuals 4 5 6 7 2017 FIGURE 3: Gamma model residuals against transformed size covariate. TABLE 4: Gamma model results summary. Year 2013 2014 2015 2016 2017 βˆ p βˆ p βˆ p βˆ p βˆ p Intercept 9.913 2e-9 11.013 2e-9 10.932 2e-9 11.305 2e-9 11.148 2e-9 Log (size) 0.664 2e-9 0.626 2e-9 0.558 2e-9 0.479 2e-9 0.512 2e-9 Bedrooms 0.003 0.313 0.017 2e-9 0.021 2e-9 0.034 2e-9 0.025 2e-9 Bathrooms 0.111 2e-9 0.096 2e-9 0.112 2e-9 0.117 2e-9 0.112 2e-9 Cluster 0.090 2e-9 0.136 2e-9 0.146 2e-9 0.187 2e-9 0.187 2e-9 Duplex 0.003 0.874 0.025 0.104 0.035 0.024 0.086 2e-9 0.079 2e-9 House 0.027 3e-4 0.063 2e-9 0.103 2e-9 0.158 2e-9 0.141 2e-9 Simplex 0.061 4-e5 0.068 5e-7 0.078 2e-9 0.117 2e-9 0.087 2e-9 Townhouse 0.050 0.064 0.063 2e-9 0.077 2e-9 0.090 2e-9 0.099 2e-9 Note: Numbers were rounded to three decimal places and scientific notation was adopted for brevity. TABLE 5: Analysis of deviance. Year Residual deviance Null deviance 2013 1470.18 13 394.60 2014 4056.69 31 487.73 2015 4448.07 33 272.34 2016 6030.46 45 648.77 2017 4008.25 32 260.69 ===== PAGE 10 (score 11, numeric-paren 0, table-words 2, var-words 5) ===== Page 10 of 11 Original Research https://www.jefjournal.org.za Open Access autocorrelation and 1 shows perfect positive spatial autocorrelation. Hundreds of permutations were run, 999 in total, for each yearly gamma model. The results of the tests are presented in Table 6 which indicate a weak negative correlation. Formally, at an alpha of 0.05, there is not enough evidence to reject the null hypothesis of no spatial autocorrelation for each yearly gamma model. This coincides with the findings of Bourassa et al. (2007) where the addition of a location dummy variable accounted for spatial dependence adequately. Conclusion Residential property is a barometer of individual and collective wealth and acts as measure of financial stability in  an economy. Measuring residential property prices is difficult because of the heterogeneity thereof. The estimation of residential property prices using hedonic modelling is pervasive in real estate economic literature where log linear models are typically employed. This article investigated generalised linear models as an alternative to log linear models to develop hedonic price functions to estimate residential property listing prices in South Africa over a 5-year period. The gamma generalised linear model provided the best fit and good generalisability whilst keeping the expected values on the original scale, which is an appealing alterative to log linear models. The spatial  dependence of residential properties was effectively accounted for by including an area factor variable, supported by variograms and Morans I  permutation tests, showing no evidence to reject the null hypothesis of no spatial  autocorrelation. This  framework provides property market  participants with  the  ability to quantify the utility derived over the  marginal  distribution of the physical characteristics of residential properties. This research presents the groundwork to create a property price index where index number theory could be applied to the hedonic price models to measure price inflation over time. Acknowledgements Competing interests The authors have declared that no competing interests exist. Authors’ contributions D.B. contributed to the conceptual design, research methodology, cleaning and analysing the data, developing the models and visualisations using R, a statistical programming language. This article forms part of his PhD degree which he is currently pursuing. T.Z. contributed towards the conceptual design and research methodology as the supervisor of D.B.  D.N. contributed towards the conceptual design and research methodology as the supervisor of D.B. Funding information This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors. Data availability statement Data sharing is not applicable to this article as no new data were created or analysed in this study. Disclaimer The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of any affiliated agency of the authors. TABLE 6: Permutation test for Morans I. Year Statistic p 2013 -0.0312 0.999 2014 -0.0267 0.999 2015 -0.0129 0.999 2016 -0.0207 0.999 2017 -0.0345 0.999 FIGURE 4: Gamma model variogram plots. (a) variogram 2013 model, (b) variogram 2014 model, (c) variogram 2015 model, (d) variogram 2016 model, (e) variogram 2017 model. 0.06 0.02 1 2 3 4 5 6 Distance Semivariance 0.08 0.06 0.02 1 2 3 4 5 6 Distance Semivariance 0.04 0.06 0.02 1 2 3 4 5 6 Distance Semivariance 1 2 3 4 5 6 Distance Semivariance 0.08 0.06 0.02 0.04 1 2 3 4 5 6 Distance Semivariance 0.10 0.06 0.02 a b d e c ===== PAGE 11 (score 4, numeric-paren 0, table-words 0, var-words 3) ===== Page 11 of 11 Original Research https://www.jefjournal.org.za Open Access References Anselin, L., 2006, ‘Spatial econometrics’, in T.C. Mills & K. Patterson (eds.), Palgrave handbook of econometrics Vol 1, econometric theory, Palgrave Macmillan, New York, pp. 901–941. Blum, A., Kalai, A. & Langford, J., 1999, ‘Beating the hold-out: Bounds for K-fold and progressive cross-validation’, COLT ‘99 Proceedings of the twelfth annual conference on Computational learning theory, July 07–09, Santa Cruz, CA, pp. 202–208. 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