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Space and Residential Values Modelling Interactions Between Geographical Patterns and Property Attributes Marius Thériault, François Des Rosiers, Paul Villeneuve & Yan Kestens First Congress of the International Real Estate Society Anchorage, Alaska, July 25-28, 2001

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Problem Statement (1) n Handling Interaction Between Spatial Factors and Values of Specific Housing Attributes – In hedonic modelling, values are usually specified using fixed coefficients – The assumption is that values of attributes are invariant across the city – However, various types of households have different needs and tastes – These households form the market of home buyers (demand) – They are not distributed evenly over space (spatial heterogeneity) – This could locally distort the demand for specific structural attributes – There is a need to test, the interaction between the structural features of physical and urban space and those housing attributes which are putatively influenced by these structural features – spatially adjusted coefficients n This paper presents a methodology to handle interactions between geographical factors and the marginal contribution of each property attribute (model its variation over space)

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Problem Statement (2) n Spatial Autocorrelation Among Residuals of the Hedonic Model – Spatial autocorrelation is highly detrimental to the efficiency of standard statistical tests used to assess the statistical significance of OLS regression coefficients – A city-wide hedonic modelling approach must (1) test for the presence of spatial heterogeneity (2) implement alternative estimation techniques to handle spatial autocorrelation – In 1990, Casetti’s expansion method was used by Can for investigating spatial drifts in housing markets n We extend the procedure using several neighbourhood quality indexes, and combining them with comparisons of each property with its immediate neighbours – thus allowing for a locally sensitive assessment of spatial dependence and for a significant step towards an efficient control of spatial autocorrelation

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Conceptual Framework (1) n Increasing Social Diversity – During the last 40 years, rapid social and economic changes have been restructuring housing demand in North American cities – Three tendencies capable of profoundly modifying residential markets n Rapid increase in female labour force during the seventies has diversified household profiles and needs (dual earners, domestic outsourcing) n Rising income inequalities between households now counting on varying numbers of working members produce social polarisation into urban space n Development of the service sector of metropolitan areas and decentralisation of manufacturing activities has affected the social profile of many neighbourhoods, especially by redistributing services locations – All these changes have an effect on residential relocation behaviour, which is highly related to household structure and cycles n Social diversity and decentralisation of services increases the complexity of rent gradients influencing housing markets

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Conceptual Framework (2) n Tree types of gradients – City-wide gradients related to structural factors (E.g. urban form, socio- economic status of neighbourhoods, location rent) – Local gradients linked to externalities (E.g. noise caused by the proximity of a motorway) – Local gradients reflecting local market internal dynamics (E.g. diffusion of home renovation among neighbours) and /or emulation among neighbours (E.g. adding a swimming pool in the backyard) – First two are “true” gradients (exogenous effects), last one is “false” gradient (endogenous effect) – True gradients violate the stationarity assumption (fixed coefficient) and produce spatial autocorrelation among residuals of the hedonic model if their effect is not appropriately handled – False gradients are reflecting evolution of market trends and do not yield significant spatial autocorrelation among model residuals

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Conceptual Framework (3) n Socio-demographic profiles and housing market – Socio-demographic profiles of the population are certainly major determinants of housing market differentiation and form city-wide trends (exogenous effect) – However, the distribution of population is also constrained by the housing market (endogenous effect) – Those effects are intermingled as demand and supply are driven by availability and affordability of housing – Therefore, these interactions with housing attributes should be modelled explicitly and translated into space-varying coefficients using Casetti’s expansion method – Conceptually, household formation and household income are major determinants of housing demand. The rate of household formation has a quantitative effect on the demand level, while income has a qualitative effect on the type of demand and selection of needed attributes

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Conceptual Framework (4) n Endogenous Market Effects – As pointed out by Can (1990), most people prefer to live in neighbourhoods where they think the return for their housing investment will be highest – People are then willing to invest in maintaining dwelling where the return on such expenditures is sufficiently high – This suggest that home owners are observing their immediate neighbours and will be more prone to improve their property if the neighbourhood itself is upgrading – Conversely, home buyers generally try to find homes in neighbourhoods having socio-economic status similar to their own, implying that similar people, with the same needs, tend to agglomerate at specific locations – Hence the intrinsic nature of spatial dependence which governs real estate markets, each social group tending to value specific sets of housing attributes, while specific attributes may be valued differently by various segments of the population

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Objectives n Main Objectives – Measuring the effect of social differentiation and residential segregation on the valuation of specific amenities – Assess the impact of the conformity/difference of a house with its neighbours when put on the market n Example of research issues – A cosy house surrounded by poorly-maintained properties will loose a large part of its value, eventually more if it belongs to the higher segment of the local market (buyers with high socio- economic status) – Provide a procedure to integrate those ecological effects in hedonic models; methods to isolate the marginal impact of this spatial mismatch on the sale price and means to identify which specific attributes will loose their value

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Test Case n Market, Geographical Location and Social Status – 4040 bungalows (one-story single family house) sold within the Quebec Urban Community from January 1990 et December 1991 – Each property is described using a large set (nearly 80) of property- specific attributes (Table 3) and neighbourhood-related attributes (more than 100) computed using GIS functions (Tables 1, 2 and 3) – Among these neighbourhood characteristics, travel times to services (schools and shopping centres) and census socio-economic data are linked to each property through selecting the nearest street corner (accessibility) or using point-in-polygon algorithm (census data) – Neighbourhood-related attributes are replaced by factor scores derived from principal component analysis and orthogonal Varimax rotations (Tables 1 and 2, Maps 1 and 2) – Principal components avoid multicollinearity problems among independent variables and they summarize structural factors behind accessibility to services and heterogeneity of households’ needs

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Modelling Approach n Sub-Samples – Models are built using 3633 cases (90% of total) selected at random with proportional stratification by municipality (13), keeping the 407 remaining observations from an independent model effectiveness assessment n Tests to Avoid – Multicollinearity – Autocorrelation – Heteroskedasticity – Problem solving strategies

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Spatial Autocorrelation (1) n Spatial autocorrelation – Significance tests and measures of fit that ignore spatial autocorrelation may be misleading – The presence of spatial error autocorrelation can make the indication tests for heteroskedasticity highly unreliable – Ecological fallacy effects often arise strictly from the precence of spatial dependency – Spatial autocorrelation is based on positional information of geo- referenced data which is not captured by classical statistics, ncluding OLS multiple regression – Spatial autocorrelation is a measure of true but masked information needed to understand mechanisms of urban dynamics; its is an artifact of specification error in spatial modelling – If an important variable is missing from a regression equation, the spatial distribution of this variable constitutes a communality across regression residuals, causing them to appear to be autocorrelated

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Spatial Autocorrelation (2) n Measuring spatial autocorrelation – Moran’s I formula to assess spatial autocorrelation (Table 4) – Implemented within a GIS software (MapInfo) providing a network of 15 nearest neighbours n Measuring similarity/difference with the 15 nearest neighbours – Average status of the 15 nearest neighbours – Departure form the neighbours

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Models and Results (1) n Five Steps (Table 5) – 1) Standard Hedonic model using property specifics and principal components of geographical factors (Model A, Table 6) – 2) Try to replace each property specific by a combination of the 15 nearest neighbours weighted average and specific departure from the neighbourhood trend (Model B – not shown) – 3) Compute interactions between each principal component and each property attribute and externality index using Casetti’s expansion method ( ); try to add interactions to property specifics (model C – not shown) – 4) Integrate models A, B and C : specific attributes, neighbourhood trends/specific departures and spatial interactions (Model D, Table 7) – 5) Model the perceived tax burden/opportunity for over/under-assessed property (by the municipality) using a two-stage approach - Using estimates of Model D as proxy to house values (Model E, Table 8)

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Models and Results (2) n Adjusted R-square is increasing from 0.768 to 0.822 n Adding 23 spatial variables has a rather marginal effect on explanatory power (R-square increase) n Standard deviation of residuals are falling very slowly, from 0.118 to 0.103 (0.125 to 0.106 – control sample) n F ratios are decreasing until Model D (366 to 277) but show a net improvement for Model E (56 variables) n Moran’s I for Model E are closer to that of actual sale price (Graph 1) and shows the same distance range (1125 m) n The most important improvement concerns the reduction of spatial autocorrelation among residuals – Model A (0.16) to Model E (0.08) n The inclusion of spatial interactions (expansion method) and the relative tax differential is generating this improvement, reducing the range of spatial autocorrelation to about 375 metres (Graph 3)

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Discussion (1) n Space-dependent values of attributes – They generate significant interactions (exogenous socio-economic or externality effects) using Casetti’s expansion method – Their effect can be split in two parts, that of the property itself, that of the weighted average of its 15 nearest neighbours (endogenous effects) n Isolating functional form of the spatial relationships – Annex 1 shows the mathematical specification of Model E (multiplicative form) – Each space-dependent attribute may be isolated to define a spatially adjusted coefficient – 15 were found, most of them being very important attribute of the house (living area, lot size, age), attributes that could be seen outdoor (Skylight, Veranda, Shed), taxation rates, or internal attributes that are highly representative of the general maintenance of the property

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Discussion (2) n Computing spatially adjusted coefficients – Since interactions and neighbours adjustments involve geographical factors that can be mapped, it is possible to compute local values of space-Varying coefficients and to build map of their distribution controlling for both location and size-related (or presence) attributes n Map 4 shows an interpolated view of the location rent effect of a bungalow situated on a larger than average lot (1000 sq. m. versus a geometric mean of 622 sq. m. for all sold properties in 1990-91) – The absolute effect is very strong, from 44% to 60% of house value – The relative value of land increases with socio-economic status and is also related to life cycles, favouring mature suburbs inhabited by empty-nesters n Map 5 and 6 show depreciation-related spatial trends for a 10 years-old and a 40 years-old house – These functions integrate endogenous and exogenous gradients – The household maintenance/repair decisions are determined by income levels, but also by households’ perception concerning the future value of their asset, in turn determined by the signals they receive from their immediate neighbours, but mediated by their own age

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Discussion (3) n Map 7 shows the marginal effect of adding a second washroom – The spatial trend estimator is regional-level accessibility to services – Per se, a washroom adds 5,4% to the house value – There was an average of 1.25 washroom per house in that market – Young families with middle income far away from regional services are forced to go to new suburbs to access home ownership and often trade off the second washrooms – In older neighbourhoods, the second washroom is prevalent and there is a penalty for house having only one n Map 8 shows the marginal effect of having a shed on the lot – The value of a shed is highly variable in space and is increasing home value in sectors that are showing less than average household income and where the proportion of blue collar is higher than average

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Conclusion n These worked examples clearly show the benefits of including interactions with urban dynamics within hedonic models n The purpose is to enhance our understanding on the complex linkages between housing prices and the socio-economic evolution of North American cities n An important side benefit – it helps reduce spatial autocorrelation among model’s residuals n We can further improve handling of spatial dependence – 1) by considering a wider range of spatial attributes (E.g. environment) – 2) by defining more flexible ways of measuring spatial dependence (E.g. non linear functions, multivariate interactions) – 3) by considering information about home buyers (E.g. socio-economic status, revealed perceptions) – 4) by splitting principal components scores to distinguish city-wide trends (E.g. trend surface) and peculiarities of specific areas (departure from the trend)

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Acknowledgements n This project is funded by – the Quebec Province’s FCAR program – the Canadian Social Sciences and Humanities Research Council – the Canadian Network of Centres of excellence in Geomatics (GEOIDE) – the Canadian Natural Sciences and Engineering Research Council n The research was realised in co-operation with – the Quebec Urban Community Appraisal Division (CUQ) – the Quebec Ministry of Transport (MTQ) – the Quebec Urban Community Transit Society (STCUQ)

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