1# RECORD a7a83b631f85316bae926a6c965b325fb26c01c42# TITLE: A gamma generalised linear model as an alternative to log linear real estate price functions3# YEAR: 2019.0 VENUE: nan DOI: 10.4102/jef.v12i1.4764# AUTHORS: Dane Bax; T. Zewotir; D. North5# PAGES: 11 CANDIDATE TABLE PAGES (1-based): [9, 10]6# 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.789===== PAGE 1 (score 4, numeric-paren 0, table-words 0, var-words 4) =====1011https://www.jefjournal.org.za12Open Access13Journal of Economic and Financial Sciences 14ISSN: (Online) 2312-2803, (Print) 1995-707615Page 1 of 1116Original Research17Read online:18Scan this QR 19code with your 20smart phone or 21mobile device 22to read online.23Authors:24Dane Bax1 25Temesgen Zewotir1 26Delia North1 27Affiliations:281School of Mathematics, 29Statistics and Computer 30Science, University of 31KwaZulu-Natal, Durban, 32South Africa 33Corresponding author:34Dane Bax,35danebax@gmail.com 36Dates:37Received: 28 Apr. 201938Accepted: 02 Aug. 201939Published: 05 Dec. 201940How to cite this article:41Bax, D., Zewotir, T. & 42North, D., 2019, ‘A gamma 43generalised linear model as 44an alternative to log linear 45real estate price functions’, 46Journal of Economic and 47Financial Sciences 12(1), 48a476. https://doi.org/4910.4102/jef.v12i1.47650Copyright:51© 2019. The Authors. 52Licensee: AOSIS. This work 53is licensed under the 54Creative Commons 55Attribution License.56Introduction57The importance of measuring residential property price inflation is paramount to households 58and economies; however, the heterogeneity of properties makes it difficult (De Haan & Diewert 592011). Log linear models have been used extensively in real estate economics to estimate property 60prices and inflation. This study investigates generalised linear models as an alternative to the 61typical log linear approach. Cross-sectional hedonic price functions are developed and compared 62for the South African residential property market over a 5-year period.63Background and objective64Residential property is an important component of individual and national wealth where it is 65capitalised on household balance sheets, informing economic policy formulation (Hill 2013). 66Goodhart and Hoffman (2008) conducted a study providing evidence of relationships between 67house prices, credit and broad money. Using vector auto-regression fitted with ordinary 68Orientation: Residential property markets play an important role in economies, informing 69policy development and decision-making. However, measuring quality-adjusted growth is 70difficult because of the heterogeneity of properties. Hedonic regression is frequently used in 71real estate econometric studies as a quality-adjusted technique to estimate residential property 72prices for the development of price indices. Log linear models are typically used to derive 73these hedonic price functions.74Research purpose: This article develops hedonic pricing functions using generalised linear 75models for South African residential property listings over a 5-year period.76Motivation for the study: A parametric alternative to the log linear model is investigated to 77address the limited studies conducted in South Africa. An important feature of this study is the 78inclusion of different property types and the geographic scope.79Research approach/design and method: The data set consisted of 415 200 residential properties 80from all over South Africa. The data spanned a period from January 2013 to August 2017. 81Several generalised linear models were developed and compared.82Main findings: The gamma generalised linear model provided the best overall fit, generalising 83well to the unseen validation data. An added benefit of this model is that the estimates were 84kept on the original scale, avoiding the need for back transformation which is an appealing 85feature of any model. A dummy locational variable was shown to account for the spatial 86dependency in the data.87Practical/managerial implications: This framework provides property market participants 88with the ability to quantify the utility derived over the marginal distribution of the physical 89characteristics of properties. This research presents the groundwork to create a property price 90index where index number theory could be applied to the counterfactual predicted values 91obtained from hedonic price models to measure price inflation over time92Contribution/value-add: This study analysed the South African residential property market 93based on an online company’s data, purportedly covering the entire market. No real estate 94hedonic price studies have been identified in South Africa with this level of scope. The gamma 95generalised linear model is a novel candidate to develop parametric real estate hedonic price 96functions.97Keywords: generalised linear models; real estate economics; model comparison and evaluation; 98spatial modelling; hedonic price functions.99A gamma generalised linear model as an alternative 100to log linear real estate price functions101Read online:102Scan this QR 103code with your 104smart phone or 105mobile device 106to read online.107108109===== PAGE 9 (score 22, numeric-paren 0, table-words 4, var-words 8) =====110111Page 9 of 11112Original Research113https://www.jefjournal.org.za114Open Access115The analysis of deviance presented in Table 5 indicates that 116the residual deviance for each yearly gamma model was 117consistently lower than null deviance. This means that the 118covariates accounted for greater deviance explained than 119intercept only models and as such indicates a good fit.120The modelling of spatial data in this study required the 121assessment of the assumption of independence, which was 122investigated using several plots and by performing 123hypothesis tests. Variograms quantify the spatial dependence 124in data by describing the spatial variance. The yearly gamma 125models’ residuals were plotted using spherical variograms 126and are presented in Figure 4 where similarities were found 127between all models. The ranges, distances beyond which the 128data are no longer correlated, are quite long, which suggests 129spatial autocorrelation is not an issue in the modelling results. 130The nugget effects as a percentage of the total sills are quite 131large which could indicate some variation at a small scale.132A permutation test for Morans I was applied to formally 133test for the presence of spatial autocorrelation where, under 134the null hypothesis, the data are randomly dispersed. 135The Morans I statistic or correlation coefficient ranges 136between -1 and 1, where -1 shows perfect negative spatial 137a138b139d140e141c1424143201314421450146-2147Deviance residuals148Log (size)1494150515161527153Log (size)154415521560157-2158Deviance residuals15941605161616271632014164Log (size)165416621670168-2169Deviance residuals17041715172617371742015175Log (size)176417721780179-2180Deviance residuals18141825183618471852016186Log (size)187418821890190-2191Deviance residuals19241935194619571962017197FIGURE 3: Gamma model residuals against transformed size covariate.198TABLE 4: Gamma model results summary.199Year20020132012014202201520320162042017205βˆ206p207βˆ208p209βˆ210p211βˆ212p213βˆ214p215Intercept2169.9132172e-921811.0132192e-922010.9322212e-922211.3052232e-922411.1482252e-9226Log (size)2270.6642282e-92290.6262302e-92310.5582322e-92330.4792342e-92350.5122362e-9237Bedrooms2380.0032390.3132400.0172412e-92420.0212432e-92440.0342452e-92460.0252472e-9248Bathrooms2490.1112502e-92510.0962522e-92530.1122542e-92550.1172562e-92570.1122582e-9259Cluster2600.0902612e-92620.1362632e-92640.1462652e-92660.1872672e-92680.1872692e-9270Duplex2710.0032720.8742730.0252740.1042750.0352760.0242770.0862782e-92790.0792802e-9281House2820.0272833e-42840.0632852e-92860.1032872e-92880.1582892e-92900.1412912e-9292Simplex2930.0612944-e52950.0682965e-72970.0782982e-92990.1173002e-93010.0873022e-9303Townhouse3040.0503050.0643060.0633072e-93080.0773092e-93100.0903112e-93120.0993132e-9314Note: Numbers were rounded to three decimal places and scientific notation was adopted for brevity.315TABLE 5: Analysis of deviance.316Year317Residual deviance318Null deviance31920133201470.1832113 394.6032220143234056.6932431 487.7332520153264448.0732733 272.3432820163296030.4633045 648.7733120173324008.2533332 260.69334335336===== PAGE 10 (score 11, numeric-paren 0, table-words 2, var-words 5) =====337338Page 10 of 11339Original Research340https://www.jefjournal.org.za341Open Access342autocorrelation and 1 shows perfect positive spatial 343autocorrelation. Hundreds of permutations were run, 999 in 344total, for each yearly gamma model. The results of the tests 345are presented in Table 6 which indicate a weak negative 346correlation. Formally, at an alpha of 0.05, there is not enough 347evidence to reject the null hypothesis of no spatial 348autocorrelation for each yearly gamma model. This coincides 349with the findings of Bourassa et al. (2007) where the addition 350of a location dummy variable accounted for spatial 351dependence adequately.352Conclusion353Residential property is a barometer of individual and 354collective wealth and acts as measure of financial stability 355in an economy. Measuring residential property prices is 356difficult because of the heterogeneity thereof. The estimation 357of residential property prices using hedonic modelling is 358pervasive in real estate economic literature where log linear 359models are typically employed. This article investigated 360generalised linear models as an alternative to log linear 361models to develop hedonic price functions to estimate 362residential property listing prices in South Africa over a 3635-year period. The gamma generalised linear model provided 364the best fit and good generalisability whilst keeping the 365expected values on the original scale, which is an appealing 366alterative to log linear models. The spatial dependence of 367residential properties was effectively accounted for by 368including an area factor variable, supported by variograms 369and Morans I permutation tests, showing no evidence to 370reject the null hypothesis of no spatial autocorrelation. 371This framework provides property market participants 372with the ability to quantify the utility derived over 373the marginal distribution of the physical characteristics of 374residential properties. This research presents the groundwork 375to create a property price index where index number theory 376could be applied to the hedonic price models to measure 377price inflation over time.378Acknowledgements379Competing interests380The authors have declared that no competing interests exist.381Authors’ contributions382D.B. contributed to the conceptual design, research 383methodology, cleaning and analysing the data, developing the 384models and visualisations using R, a statistical programming 385language. This article forms part of his PhD degree which he is 386currently pursuing. T.Z. contributed towards the conceptual 387design and research methodology as the supervisor of 388D.B. D.N. contributed towards the conceptual design and 389research methodology as the supervisor of D.B.390Funding information391This research received no specific grant from any funding 392agency in the public, commercial or not-for-profit sectors.393Data availability statement394Data sharing is not applicable to this article as no new data 395were created or analysed in this study.396Disclaimer397The views and opinions expressed in this article are those of 398the authors and do not necessarily reflect the official policy or 399position of any affiliated agency of the authors.400TABLE 6: Permutation test for Morans I.401Year402Statistic403p4042013405-0.03124060.9994072014408-0.02674090.9994102015411-0.01294120.9994132016414-0.02074150.9994162017417-0.03454180.999419FIGURE 4: Gamma model variogram plots. (a) variogram 2013 model, (b) variogram 4202014 model, (c) variogram 2015 model, (d) variogram 2016 model, (e) variogram 4212017 model.4220.064230.02424142524263427442854296430Distance431Semivariance4320.084330.064340.02435143624373438443954406441Distance442Semivariance4430.044440.064450.02446144724483449445054516452Distance453Semivariance454145524563457445854596460Distance461Semivariance4620.084630.064640.024650.04466146724683469447054716472Distance473Semivariance4740.104750.064760.02477a478b479d480e481c482483484===== PAGE 11 (score 4, numeric-paren 0, table-words 0, var-words 3) =====485486Page 11 of 11487Original Research488https://www.jefjournal.org.za489Open Access490References491Anselin, L., 2006, ‘Spatial econometrics’, in T.C. Mills & K. 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