Spatial Effects in Hedonic Price Estimation: A Case …...Spatial Effects in Hedonic Price Estimation: A Case Study in the City of Toronto F. Long, A. Páez, S. Farber Centre for Spatial

Spatial Effects in Hedonic Price Estimation: A Case Study in the

City of Toronto

F. Long, A. Páez, S. Farber

020 June 2007

Spatial Effects in Hedonic Price Estimation: A Case Study in the City of Toronto F. Long, A. Páez, S. Farber

Centre for Spatial Analysis ‐ Working Paper Series

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Abstract Recognition of the limitations of traditional hedonic models to account for spatial effects has led in recent years to the development and use of spatial econometric and statistical techniques in real estate applications. It seems appropriate, as the number of applications grows, to evaluate the relative ability of some newer approaches in terms of producing accurate spatial predictions. This article compares a selection of techniques to assess their performance. The focus is on moving window approaches that can be conceptualized as sliding neighborhoods (i.e. soft market segmentations), and that can incorporate spatial dependency effects. Comparison of moving windows regression (MWR), geographically weighted regression (GWR), and moving windows Kriging (MWK) sheds light into the relevance of different spatial effects. Results using Toronto as a case study indicate that market segmentation may be more important than spatial dependencies. The findings suggest practical guidelines with regards to the use of the models investigated. Keywords: Hedonic prices, moving windows, geographically weighted regression, Kriging, spatial effects, goodness‐of‐fit

1. Introduction Hedonic analysis, a method for decomposing a commodity into its constituent characteristics and estimating their implicit prices (Lancaster, 1966; Rosen, 1974), is an indispensable tool in the study of real estate markets. Its extensive use in this field includes: estimating demand for housing and neighborhood attributes; making general improvements in house price indices; analyzing the impact of neighborhood externalities; measuring housing demand in residential mobility studies; estimating the benefits to accrue from public investment programs; appraising individual housing units; and examining the capitalization of a wide range of amenities (e.g. Can, 1992b; Malpezzi, 2003). In the history of hedonic price analysis, use of linear regression was the dominant approach until the early 1990’s. Since then, with the emerging technology of geographic information systems as well as progress in the fields of spatial econometrics and statistics, studies such as Can (1992a; 1992b), Dubin (1992), Pace and Gilley (1997), and Basu and Thibodeau (1998), have helped to bring hedonic analysis into a new domain ‐ exploiting the spatial nature of residential datasets. As expounded in the studies of Can (1992b) and Dubin (1998) among others, the existence of spatial autocorrelation and spatial heterogeneity precludes the straightforward use of the traditional hedonic model, an approach that is non‐spatial in character, and therefore fails the sufficiency criterion described by Griffith (1988, pp 9‐11) when used in the analysis and modeling of housing datasets. The awareness of many researchers to a major limitation of traditional hedonic analysis, namely its insensitivity to the spatiality of housing datasets, and of the potential



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consequences of omitting spatial effects on the statistical validity and reliability of models, has led to a different understanding and emphasis of price determination processes. Indeed, the last decade has seen a proliferation of hedonic studies that control for spatial dependence and/or spatial heterogeneity, including Chica‐Olmo (1995), Páez (2001), Tse (2002), Farber and Yates (2007), Bitter et al. (2007), and other papers in the specialized literature (e.g. Pace et al, 1998; Pace and Lesage, 2004). With the increase in the number of applications, it seems appropriate to assess the relative merits of different modeling techniques, in particular with respect to their ability to produce accurate spatial predictions useful in the evaluation of real estate markets. While recognizing the many different ways available to incorporate spatial effects in hedonic analysis, four newer techniques are deliberately selected such that the model formulations allow for the isolation and combination of two spatial effects, namely dependency and heterogeneity. These effects can be thought of as locational/adjacency effects and market segmentation respectively. In particular, the focus is set on moving window approaches that can be conceptualized as “sliding neighborhoods” (Dubin, 1992), or soft market segments, and that are amenable to the incorporation of spatial dependency effects. Through comparative analysis of prediction accuracy between moving windows regression (MWR), geographically (or locally) weighted regression (GWR), Kriging, and moving windows Kriging (MWK), we address the question of the impact of spatial dependence and spatial heterogeneity upon predictive accuracy. Using as a case study the city of Toronto, Canada, and a dataset with over 30,000 recorded transactions, we conduct out‐of‐sample validation experiments for the models selected. The empirical example shows that modeling moving window‐based market segmentations leads to superior results relative to single market (or global) approaches, and also to modeling spatial dependencies. Indeed, after implementation of the moving window approaches, there appears to be limited support for spatial autocorrelation analysis at the sub‐market level in this case study. The results suggest some practical guidelines for the application of the models studied. The paper is organized into 4 sections. Section 2 provides a review of the traditional hedonic and spatial hedonic modeling literature. Section 3 introduces the data set under study and the methods used to investigate the impacts of spatial effects. Section 4 contains a discussion of model results, and section 5 concludes the study by summarizing the results.

2. Background 2.1 Traditional Model Specifications Hedonic regression analysis is an essential component in real estate studies. It allows for the decomposition of housing expenditure into multiple characteristics internal or external to housing units. Therefore hedonic prices, i.e. the implicit prices of attributes, can be obtained from multivariate regression analysis between observed dwelling prices and the



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quantity and quality of characteristics associated with them. For a review of the economic foundations in hedonic modeling see Court (1939), Lancaster (1966), and Rosen (1974). As with any other econometric approach, the specification of hedonic models plays a crucial role in determining the accuracy and precision of modeling results (Can and Megbolugbe, 1997). Although there is no consensus in the literature regarding the variables to be included into the hedonic price function (and indeed, it is questionable whether precise hedonic specifications can be derived from economic theory; see Mason and Quigley, 1996), characteristics in three basic categories are generally considered: structural characteristics such as dwelling size and age; social and environmental attributes of the neighborhood in which the dwelling is located; and other location characteristics such as accessibility to employment, services and recreational facilities (Basu and Thibodeau, 1998; Bowen et al, 2001). House price is typically expressed as a function of the above characteristics such as:

i i ki ki pi pi ji ji ik p j

Y S N Lα β γ λ ε= + + + +∑ ∑ ∑ . (1)

As stated in the traditional regression model, ( 1, 2,..., )iY i n= , where n is the number of observations in the dataset, represents property values. S is a vector of structural attributes; N is a vector representing attributes of the surrounding neighborhood; L is a vector of variables capturing locational or proximity characteristics; and ε is a vector composed of random error terms which represents all those factors that affect sale prices but are omitted from the modeling process (Bowen et al, 2001). A compact version of the above function using matrix notation can be stated as: Y X β ε= + , (2) where Y is a ( *1)n vector of observed sale prices of n housing units on the market; X is a (n*k) vector of K structural, neighborhood and locational characteristics for housing units; β is a (k*1) vector of unknown coefficients; and ε is an (n*1) vector of stochastic disturbance terms. The traditional hedonic price function is a typical econometric regression model and ordinary least squares (OLS) is by far the standard technique to estimate the unknown coefficients computed as:

1ˆ ( )T TX X X Yβ −= . (3) Ordinary least squares provides a single set of coefficient estimates for all observations within the dataset, which means the traditional hedonic model assumes a set of spatially invariant or “global” coefficients. The appropriateness of this simplifying assumption (i.e. coefficient stability) has generated considerable debate by posing a simple question: what is the effect of adding one bedroom to a house located in the city centre compared to a house in a suburban area? Will this improvement add an equivalent monetary value to the market prices of these two houses? In many cases, the answer turns out to be no (Farber



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and Yeates, 2007; Goodman and Thibodeau, 1998; Goodman and Thibodeau, 2003; Hess and Almeida, 2007; Pavlov, 2000; see Schnare and Struyk, 1976). This variation in the behavior of a given process across space is often recognized in urban housing market studies and referred to as spatial heterogeneity, which is believed to be caused by the different spatial processes operating in local urban housing markets (Can, 1990) or linked to varying household preferences (Kestens et al, 2006). As is well‐known, in OLS the regression coefficients β are estimated by minimizing the sum of squared errors Tε ε , and accordingly the market value of a property with characteristics 0X is estimated as 0X β . To ensure that OLS is the best linear unbiased estimator and predictor, there is a set of ideal conditions that must be satisfied: X must be independent of the errors; and the errors themselves must be independent, homoskedastic and normally distributed. In the analysis of spatial data, the plausibility of these assumptions is suspect not only from a theoretical perspective, but has also been challenged by many empirical studies which often report interdependence among observations (i.e. spatial autocorrelation), and spatial variations of regression coefficients (i.e. spatial heterogeneity). It is theorized that locational effects and market segmentation are at the root of spatial dependency and spatial heterogeneity respectively. 2.2 Locational effects Can (1992b) distinguished two levels of locational effects: (1) neighborhood effects, which is the array of locational characteristics; and (2) adjacency effects, which are externalities associated with the location of properties relative to each other. The traditional hedonic model capitalizes the neighborhood effects by including a set of characteristics which account for the socioeconomic and physical make‐up of the neighborhood and the accessibility of properties to urban amenities. Interestingly, Can’s use of the spatial expansion model (Casetti, 1972), i.e., of a trend surface of spatial drift, did not improve model accuracy in a hedonic house price context. It has been argued that this may be a consequence of complex spatial variation of parameters that cannot be captured by simple expansion expressions (Brunsdon et al, 1996; Fotheringham et al, 1998). In addition to locational characteristics, adjacency effects also contribute to similarities among neighboring properties ‐ possible due the difficulties involved in measuring proxy variables for locational externalities (Basu and Thibodeau, 1998; Bowen et al, 2001), or because benefits (e.g. renovations, landscaping) or disbenefits (e.g. state of disrepair) tend to spillover to other properties. While it is natural to anticipate that adjacency effects (i.e. spatial dependencies) will diminish after introducing neighborhood effects into the hedonic model by making use of spatial variables, a number of studies (Basu and Thibodeau, 1998; Bowen et al, 2001; Farber, 2004; e.g. Pace and Gilley, 1997) have proved the superiority of spatial approaches over traditional hedonic regression in terms of improving the explanatory power and the statistical properties of the models. Recalling that residuals are assumed to be the stochastic disturbance terms which represent all those factors that have an impact upon



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house prices but are not explicitly incorporated into the modeling procedure, Dubin et al. (1999) note that if any remaining systematic spatial pattern is observed, such as the clustering of similar or dissimilar values, then it is likely that potentially valuable information has not been retrieved from the process. The use of techniques that use this residual patern (e.g. Kriging, spatial error autocorrelation models) is then recommended. 2.3 Market Segmentation Schnare and Struyk (1976) posit that urban housing markets should be organized into a series of submarkets each represented by a unique functional relationship between prices and property attributes. In order to capture the dynamics operating in the local housing market and provide more accurate marginal price estimates for attributes, one needs to stratify the data into different sectors along a segmentation line and fit a separate model to each independent sector. However, the segmentation scheme to define the boundaries of sub‐markets requires excellent knowledge of local markets and is not always available. In addition, a number of approaches devised so far to define sub‐markets or independent sectors have been disputed as being arbitrary and lacking explicit estimates of market elasticity. If these approaches cannot ensure that sub‐markets delineated under their stratifiers are homogeneous zones, the results will be misleading. Also, the disaggregation into discrete areas may impose unrealistic discontinuities in their effects, for example, if effects of certain neighborhood characteristics go beyond arbitrarily defined boundaries (Can, 1992b). In this situation, no set of fixed neighborhood boundaries can accurately describe the urban market structure. Out of such concern, moving window modeling frameworks are proposed as a way to incorporate spatial heterogeneity into hedonic price structure. Moving windows regression (MWR), and geographically (or locally) weighted regression (GWR: Brunsdon et al, 1996; LWR: McMillen, 1996) adopt the concept of a ‘sliding neighborhood’ previously mentioned (Can and Megbolugbe, 1997; Dubin, 1992), in which no rigid predefined neighborhood boundaries are required, and thus can be seen as soft market segmentation methods. As recent studies demonstrate, under this framework, variations in marginal attribute price can be measured in a continuous rather than a discrete manner across space (e.g. Kestens et al, 2006). 3. Data and Methods 3.1 Data This study investigates spatial autocorrelation and spatial heterogeneity in a number of spatial hedonic price functions using data from 33,494 transactions of single‐family detached houses sold between January 2001 and December 2003 in the City of Toronto. In order to avoid potential bias brought by distorted market forces like clearance sales, only open market sale records are included in the dataset. The primary source of information is from the Municipal Property Assessment Corporation (MPAC) of Ontario. Every year, MPAC prepares an assessment roll for every Ontario municipality that provides the assessed value of all the properties in a municipality, or in the jurisdiction of a school board with taxing authority



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(www.mpac.ca). The computer file provided by MPAC contains each property’s address as well as information of each property’s structural characteristics. With ArcView, each transaction is assigned geographic coordinates and is geocoded into the study area. Since it is desirable to test the predictive power of models using a different sample than the one used for estimation, in this study the original dataset, which consists of 33,449 observations, is divided into two groups by a random sampling procedure (q.v. Case et al, 2004). This gives an estimation sample, i.e., in‐sample observations, and a validation sample, i.e., out‐of‐sample observations. The estimation sample contains 30,145 observations, which is 90% of the observations available, whereas the validation sample is comprised of 3,349 observations, which constitutes 10% of the recorded transactions. With regard to the explanatory variables used in traditional hedonic analysis, the selection of variables is partially informed by the study of Farber (2004), combined with results from exploratory data analysis (e.g. correlation analysis, multicollinearity tests and stepwise regression). Seven variables were finally selected to be applied in various spatial hedonic model specifications. As mentioned in section 2, these determinants can be classified into three categories.

[FIGURE 1]

The first category contains the structural features of properties. Information on these features comes from MPAC’s file. Variables include: Area‐‐‐‐‐‐ Effective site area in square feet Front‐‐‐‐‐‐Effective site frontage in feet HouseAge‐‐‐‐‐‐The age of dwelling in decades Saledate‐‐‐‐‐‐Since transactions occurred during a period of 36 months from January 2001 to December 2003, a variable ranging from 1 to 36 is defined to represent the temporal component, seasonality and inflation. The second category includes the characteristics of the immediately surrounding natural and social environment. Information is obtained at the census tract level, coming from the 2001 Census of Population. Two important dimensions of neighborhood were identified: income level and ethnic composition, represented by the following variables: MeanIncome‐‐‐‐‐‐The mean household incomein the census tract level. PctImm‐‐‐‐‐‐The percentage of immigrants in the census tract. 1

1 Due to privacy concerns, information for census tracts which contain few households is suppressed by Statistic Canada. 4173 observations have their mean household income and percentage of immigrants set to value of 0. This will influence the modeling results for traditional hedonic model, but doesn’t have an impact on other spatial hedonic models since neighborhood attributes are not included in their modeling specification.



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The third category is the locational characteristics of properties. In correspondence with the polycentric nature of modern cities, locational characteristics are measured by proximity to social amenities rather than being measured simply by distance to the CBD. The variable identified under this category was derived using ArcView, based on data from DMTI’s street network file: DistTransit‐‐‐‐‐‐The straight line distance of a given property to the nearest Light Rapid Transit (LRT), subway or train station. It is important, to ensure that the findings are not unduly affected by a poorly selected validation sample, that the hold‐out observations be representative of the estimation sample. As the summary statistics for sale prices and explanatory variables in Table 1 indicate, the two samples are equivalent in terms of their means and distributions.

[TABLE 1] Regarding the functional form of the model, a linear model or a semi‐log functional form are commonly used. In this study, a linear functional form is favored for several reasons. Firstly, the major interest of this study lies in comparing the predictive ability of a variety of spatial hedonic models. Conditioned by this research objective, the logarithmic functional form can be problematic in the sense that unbiased estimation of regression coefficients in units of log price could be biased when transformed back to original prices (Goldberger, 1968). Secondly, several of the models studied in this paper involve the measurement of spatial autocorrelation among residuals or the calculation of variance‐covariance structures of residuals. Transformed residuals obtained in semi‐log functional form may have potential undesirable effects since this transformation may obscure the original spatial pattern and subsequently affect the measurements of spatial autocorrelation or the construction of variance‐covariance structures. Lastly, in a linear specification, the estimated coefficients can be easily interpreted as the implicit marginal prices of attributes (Bowen et al, 2001). 3.2 Methods In this section, the spatial models selected for this study are described. 3.2.1 Moving Windows Regression and Geographically Weighted Regression An intuitively appealing method for easing the restriction of spatially invariant relationships between variables implied by the traditional hedonic model is to estimate coefficients of attributes for each site by only using its neighboring observations for model calibration. More specifically, the idea is to impose a window centered at the point of interest, and only observations situated within the window are regarded as “neighbors” and consequently included into the regression for estimating coefficients at the given point. The window then moves to the next point, where a new parameters are estimated using only its neighboring observations. This procedure is repeated until the moving



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window visits all the estimation points in the study area. The above method is called moving windows regression. Essentially, it comprises a series of locally linear regressions and generates unique parameter estimates for every regression point. Thus MWR allows the model to alter over space to reflect the varying structure within the data. Geographically (or locally) weighted regression (Brunsdon et al, 1996; McMillen, 1996) refines the idea of moving windows regression by introducing a weighting scheme that decreases the influence of observations according to their location with respect to the regression point. Thus, instead of treating all points within the window equally, the scheme de‐emphasizes distant points according to the geographical principle of distance decay. The spatial weighting function employed in this study is the nearest neighbor (adaptive) bi‐square function which is defined as:

[ ]222 /1 ddw ijij −= if j is one of the n’th nearest neighbors of i; 0=ijw otherwise; (6)

where d is the distance from i to its n’th nearest neighbor. The value of n (the number of nearest neighbors used for local estimation) is determined by a cross‐validation procedure to achieve the best model performance. This adaptive bi‐square function ensures that the density of transactions does not affect estimation, since every moving window contains the same number of observations, and thus every local regression works with an equal sample size. In addition, this scheme takes into account potential anisotropy, since with increasing window size the window expands in the direction of the next nearest observation regardless of direction. It is worthwhile to note that moving windows regression can be regarded as a form of GWR when a simpler weighting scheme defined as follows is used:

1=ijw if j is one of the n’th nearest neighbors of i; 0=ijw .

In ordinary least squares, the coefficient estimates are calculated by minimizing the sum of the squared differences between predicted and observed values of the dependent variable. In MWR and GWR in contrast weighted least squares is used for coefficient estimation, where a weighting factor ijw is applied to each squared difference before minimizing. If Wi is a diagonal matrix of weights ijw , then local coefficients can be estimated as:

1( )t ti i iX W X X WYβ −= (8)

Based on the definitions described above, moving windows regression and geographically weighted regression are in essence similar approaches: they both endeavor to model spatial heterogeneity by performing regressions for each site of interest at a local level through the use of a moving window. 3.2.2 Kriging In order to capitalize on information contained in autocorrelated residuals when it exists,



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Kriging can be used to improve prediction accuracy. Its mathematical expression is as follows:

1

( ) ( ) ( )

ˆ ( ) ( ) ( )

T

n

ii

Y s x s U s

U s s U s

β

λ=

= +

=∑ (9)

where ( )U s is a zero mean process at location s with covariance function ()C . The generalized least squares approach is employed to estimate β . However, the values of

( )U s are not entirely unpredictable; an estimate ( ˆ ( )U s ) is computed as a weighted linear combination (with weights λ ) of estimated residuals to approximate ( )U s . As Goldberger (Lancaster, 1966) demonstrates, Kriging is an optimal interpolation technique, which aims to improve the accuracy of prediction in two ways: first, it considers spatial correlation in residuals and specifies it into parameter estimates; second, it adjusts the prediction values by adding a local component (or predicted error term denoted by U ) to the contextual drift, which is obtained from nearby properties as a weighted average of estimable residuals ( )U s (Bailey and Gatrell, 1995). Dubin has been a major proponent of Kriging in the analysis of real estate prices (Case et al, 2004; Dubin et al, 1999; Dubin, 1992), and other studies have confirmed the usefulness of this technique (e.g. Chica‐Olmo, 1995). 3.2.3 Moving Windows Kriging (MWK) A key assumption in Kriging is the spatial stationarity of attributes, that is the mean and variance of each distribution is the same at all locations and the correlation between observations can be represented as a function of the distance separating them. In reality, many spatial phenomena display both mean nonstationarity (i.e. trends) and location‐dependent variance. In addition, it is implausible that the spatial covariance of any pair of observations can be expressed by a universal function. A modification to Kriging, namely moving windows Kriging, was advanced by Haas (1990a) as a way to adapt to a non‐uniform spatial structure, and to improve prediction accuracy. This is accomplished by estimating a new set of parameters for each prediction site, using only observations within the immediate neighborhood. Moving windows Kriging makes use of a local spatial covariance structure that varies from site to site, a property believed to lead to a more accurate portrayal of the observed spatial phenomena (Haas, 1990a; Haas, 1995). Moving windows Kriging was conducted by Dubin in a real estate context (Case et al, 2004). 4. Results and Discussion The techniques described in the preceding section are designed to emphasize or focus on different spatial processes. MWR and GWR attempt to model a spatially heterogeneous process by carrying out a regression for each prediction site using neighboring observations. In addition, the two approaches may partially eliminate spatial autocorrelation by calibrating models within a local, more homogeneous context. Compared to MWR and GWR, Kriging concentrates on the use of the spatial dependency



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structure of residuals. Through the construction of a variance‐covariance matrix, Kriging improves prediction accuracy by adjusting parameter estimates and utilizing the systematic component of the error term. With respect to MWK, it accounts for both spatial autocorrelation and spatial heterogeneity. The former part of MWK is the same as MWR, that is, regression is performed within each moving window surrounding the prediction site. The latter part can be regarded as localized Kriging, i.e. estimating a location‐specific spatial dependency structure, therefore providing adjusted parameter estimates and predicted error term for each prediction site. In so doing, MWK models a spatially varying process with flexible relationships between some sets of variables, while remaining capable of exploiting the spatial information remaining in the residuals via the prediction of error terms. In this section, the above spatial models as well as the traditional benchmark model are applied. Their modeling performance in terms of out‐of‐sample prediction accuracy is evaluated and discussed. 4.1 Benchmark Model: Traditional Hedonic Regression Results The following set of variables is common to all models: Area, Front, SaleDate, HouseAge, Age^2 and DistTransit. A second set available only to global modeling techniques (the benchmark model and Kriging) adds neighborhood attributes, i.e. MeanIncome and PctImm. These variables are absent from the first specification due to concerns over indeterminate matrices during local estimation. Parameter estimates and other statistics of the traditional hedonic model using the two different sets of regressors are presented in Tables 2 and 3.

[TABLE 2] [TABLE 3]

Note that the use of neighborhood attributes substantially improves the fit of the model. Use of these variables, however, is not without problems, due to censoring (i.e. variables set to zero in sparsely populated enumeration units), and the fact that neighborhood attributes obtained at the census level may not vary enough in the relatively small geographical areas used for local model estimation, thus leading to matrix invertibility problems in estimation. Clearly, excluding the variables has a deleterious effect on the traditional model. On the other hand, a model that incorporates a spatially autocorrelated structure may not suffer much in terms of its predictive ability (Dubin, 1992). This is further explored below. 4.2 Model Comparisons 4.2.1 Cross Validation All moving window approaches require that a window size be calibrated. For GWR, MWR, and MWK, Fotheringham et al. (2002) and Haas (1995) suggest using cross‐validation to determine an appropriate size for the moving window. Cross‐validation is a



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technique in which the optimal window size or number of nearest neighbors is the one that minimizes the following function:

2

1

ˆ( )n

i ii

CV y y=

= −∑ , (10)

where n is the number of data points in the study area, iy is the observed value for the i ’th data point and iy is the predicted value of the i ’th point obtained when i itself is omitted from the computation. Subtracting the predicted value from the actual value generates prediction errors. The sum of the squared errors provides an overall measurement of performance of models at different window sizes. For MWK, optimum window size represents a trade‐off between being small enough to predict a locally stationary process but large enough to construct meaningful local covariance structures. For the purpose of comparing models, all window sizes in this study are defined in terms of cross‐validation minimization over the set of in‐sample observations. See Haas (1990a; 1990b), and Webster and Oliver (1992) for alternatives to cross‐validation for determining MWK window size. Dubin (in Case et al, 2004) uses a window of 200‐300 nearest neighbors. Hess and Almeida (2007), in estimating local regressions, select a distance buffer of ½ mile around each regression point. Neither study validates the window size. For MWR and GWR, 24 different window sizes ranging from 50 to 700 nearest neighbors are applied and the cross‐validation score (CV) is computed for each window size. By plotting the value of the CV score versus the bandwidth tested in Figure 2, it is observed that the lowest CV score is obtained at 190 and 250 neighbors for MWR and GWR respectively.

[FIGURE 2] Recalling the definition of MWR and GWR, these models originate from the same notion of modeling spatially varying relationships facilitated by local regressions at each point (Fotheringham et al, 1996). The difference between GWR and MWR is the use of a distance weighting scheme in GWR. But how effective this weighting scheme is or to what extent will GWR differ from MWR still remains in question. Inspection of their cross‐validation provides some relevant findings. First, their graphs display the usual trend: with increasing window size, the CV score decreases sharply at the beginning until it reaches a lower bound, then it stays relatively stable for a certain range, after which it increases towards the sum of squared errors obtained by the global model. However, as shown in Figure 2, MWR goes up more rapidly than GWR which implies that by employing the distance weighting scheme, GWR has a steadier performance that continues to produce better results for a wider range of window size selections. With respect to the contribution of the distance weighing scheme, according to the figure, beyond the bandwidth of 200 nearest neighbors, a substantial and rapidly increasing gap is observed between MWR and GWR from which the efficacy of the weighting scheme can be deduced. However, if the



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comparison is made at their lowest points (i.e. at their respective optimal window sizes), the difference is less marked which demonstrates the efficacy of MWR when an optimal window size is used. In addition, based on the same graph, it is clear that the optimum window for MWR ranges from 100 to 200 nearest neighbors, which is quite different from GWR whose best performance is achieved when 200 to 300 nearest neighbors are used. Therefore, the use of identical window size for both models is not justified, as it may have a relatively large impact on model estimation, especially when the interest is on assessing performance (contrast this with the results reported by Farber, 2004). Due to the computational burden of the MWK estimation procedure a more limited range of moving widow sizes are included in its cross validation, ranging from 50 to 450 nearest neighbors. The experiment sufficiently captures the general trend and the lowest CV score is found at 150 neighbors. Since MWK incorporates a variance‐covariance matrix of residuals, in addition to modeling spatial heterogeneity as MWR does, the model also makes use of residual spatial autocorrelation. Therefore, by comparing the cross‐validation results of MWR and MWK one may detect the isolated and combined effects of spatial autocorrelation and spatial heterogeneity upon model performance. It is observed in Figure 3 that MWK and MWR share similar CV‐Bandwidth trends in terms of direction and magnitude. It was anticipated that MWK would surpass MWR since spatial heterogeneity and spatial dependency are both accounted for in its model formulation. However, the results indicate that this is not necessarily the case. In fact, as will be seen in more detail below, the incorporation of spatial autocorrelation into model specification in addition to localized regressions may at times bring down the performance of the model.

[FIGURE 3] 4.2.2 Comparative Analysis of Predictive Accuracy of Various Hedonic Price

Functions Once an ideal window size has been determined for each modeling strategy, this information can be used to estimate models at the out‐of‐sample locations, to predict all property values not used in the calibration of the window size. Predicted property values can then be compared to the actual observed transactions, and a number of indices describing the prediction error of out‐of‐sample observations can be computed as quantitative measures of predictive performance. Table 4 presents a set of statistics comparing the performance of the various models investigated.

[TABLE 4] According to Table 4, the best predictive power is achieved by GWR in terms of the mean absolute error and the root mean squared error; MWR also attains satisfactory results with key statistics only marginally different from GWR; Kriging2 using extra neighborhood



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attributes performs only slightly better than Kriging1; as expected, the performance of OLS1 is not comparable to the performance of OLS2 due to the omission of neighborhood variables. Surprisingly, MWK has the lowest predictive power. Table 5 presents some additional statistics of model performance. The column termed ‘correlation’ contains Pearson correlation coefficients between predicted and observed values of the validation sample. The column entitled ‘% better than OLS’ is a percentage of observations for which the model in question obtains a better estimate than the traditional hedonic price function.

[TABLE 5] The contribution of two types of spatial effects ‐ spatial dependency and spatial heterogeneity ‐ can be detected and measured respectively by comparing the traditional hedonic price function OLS1 and other spatial hedonic models since they are using the same set of explanatory variables. Firstly, the improvement of Kriging1 over OLS1 can be attributed to the use of spatial error autocorrelation at the global level that is ignored by the traditional model. Secondly, MWR and GWR achieve the best prediction results in which 85% of prediction errors are within a 25% range of the original sale prices. The corresponding figure for Kriging1 is 52%. Thus it can be said that in this case capturing spatial heterogeneity by use of a moving windows approach had a greater impact on model performance than capturing spatial dependence. Also, in support of the cross‐validation results, the marginal impact of using a weighting scheme in GWR is almost negligible in the presence of a CV calibrated optimum bandwidth in MWR. The comparisons made thus far are useful in determining the individual impacts of spatial dependency and spatial heterogeneity. In addition to this, by comparing MWK to MWR one can identify the effect of spatial autocorrelation after spatial heterogeneity is built into the modeling process. As noted earlier, the difference between MWK and MWR in terms of cross validation is relatively small. In the out‐of‐sample prediction, on the other hand, it is observed that the performance of MWK is rather disappointing. While surprising, it is important to note that Basu and Thibodeau’s (1998) found that in submarkets where residuals are uncorrelated, Kriging will either have negative or no influence on prediction accuracy. The above result, together with visual inspection of MWR local residuals, strongly suggests that error autocorrelation is not present at the submarket level when an optimal window size is used, perhaps on account of the inclusion of structural and accessibility attributes and the fact that within each moving window a relatively homogenous area is obtained. In addition, comparison between OLS2 and the spatial hedonic price functions reveals another interesting finding. As observed in Tables 7 and 8, unlike Kriging1 over OLS1, Kriging2 improves on the performance of OLS2 only marginally by making use of residual



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spatial dependency. The similarities in results for OLS2 and Kriging1 support Dubin’s (1992) argument that in the presence of possible measurement error associated with the inclusion of neighborhood effects, it is preferred to model spatial dependency in the error term through a Kriging procedure. On the other hand, if the researcher is confident about the quality of available variables, improving the trend established by linear regression would have the additional benefit of discriminating between the mean and covariance structure of the process. Finally, consistent with cross validation results, MWR and GWR achieve more accurate predictions than OLS2 despite their lack of neighborhood characteristics. It can thus be interpreted that neighborhood attributes are not indispensable to hedonic price analysis with respect to a local modeling approach. One explanation could be that in the relatively homogeneous area within a moving window, neighborhood attributes contribute little to price explanation. Similarly, the improvement of Kriging1 over OLS2 suggests that the spatial variation in housing prices due to neighborhood characteristics can be successfully modeled as the predicted component of the spatially autocorrelated residuals. 4.3 Experiment 2: Another Combination of Variables The preceding results cast a shadow of doubt on the ability of MWK to obtain accurate predictions, despite the fact that this modeling approach combines both a moving windows strategy with the use of a covariance structure to capture spatial residual pattern. This suggests that the MWR model used above controls for most spatial autocorrelation at the moving window level, thus leaving little valuable information to permit MWK to perform a better job. In the same way, with a relatively complex drift composed of six determinants, the traditional hedonic model captures the primary features of the modeling process. In this situation, the location dependent variance‐covariance matrix generated by MWK may be redundant and may have the undesirable effect of obscuring the process. In order to explore the extent to which this might be the case, a second set of models is specified that make use of a simplified variable set, therefore increasing the potential for residual pattern at both the global and local scale. The questions motivating this experiment can be formulated as, how much effort should go into specifying a more complete model? Conversely, to what extent is it practical to use a simple model and transfer more of the burden of prediction to spatial residual pattern? In this modeling round, a simpler model is formulated with only two variables used by all models (Area and DistTransit), In addition, OLS2 and Kriging2 include the neighborhood variables previously used (MeanIncome and PctImm). Cross validation results are displayed in Figures 4 and 5; statistics for the comparison of models are reported in Tables 6 and 7.

[FIGURE 4] [FIGURE 5] [TABLE 6]



‐ 15 ‐

[TABLE 7] The figures and tables confirm previous findings that GWR and MWR perform best, with Kriging as a third. Note however, in comparison to Tables 4 and 5, that Kriging was not overly affected by the omission of variables. Most indicators for the Kriging models are only marginally different from previous results, and in the case of Kriging1 there is even a slight improvement in terms of prediction errors (see Table 6). Compared to the first modeling round, prediction performance of MWK improves in a minor fashion, with cross validation results somewhat more favorable than MWR and some key statistics for out‐of‐sample predictions slightly better than OLS1. This experiment confirms that MWK does show improved model performance when considerable spatial autocorrelation exists among residuals. However, its performance appears to be inconsistent as revealed by its varying predictive power exhibited in cross validation and out‐of‐sample estimation. This weakness is also verified by the descriptive analysis of its prediction errors which has the highest standard deviation and broadest range among all candidate models, as shown in Tables 6 and 7. A possible cause underlying the predictive weakness of any given local model, currently under investigation, is whether cross‐validation is not unduly affected by a small number of influential observations, leading to cross‐validation‐calibrated window sizes that are not sufficiently representative of the points in the out‐of‐sample used for validation (Farber and Páez, 2006). 5. Conclusions and Suggestions Traditional hedonic price functions make a series of postulates about the residuals and parameters in the modeling process. Due to the housing market’s characteristic features of locational effects and market segmentation, these assumptions are not tenable in many empirical applications. As the potential implications of violating the assumptions of independence and homogeneity dawn on ever more researchers, increasing numbers of applications of spatial econometric and statistical techniques continue to appear in the literature. With the rising popularity of these techniques, it seems important to assess the relative merits of various models, and to develop guidelines to support the use of different modeling strategies. Observed spatial dependency and non‐stationarity prompted the use in this study of several spatial modeling techniques. Kriging was adopted for its ability to incorporate systematic residual information (i.e. error autocorrelation) to obtain an improved predictive model. Moving windows regression and its enhanced counterpart, geographically weighted regression, are methods designed to model spatially heterogeneous processes. Lastly, moving windows Kriging was used since its design simultaneously incorporates locational effects and spatial heterogeneity.



‐ 16 ‐

The results of the spatial models were compared to each other and to the traditional hedonic modeling approach. The comparison was primarily focused on cross‐validation trends and goodness‐of‐fit criteria. The results indicate that traditional hedonic models, even in the presence of structural, neighborhood, and accessibility variables, did not adequately address spatial dependency or spatial heterogeneity issues. The superior results of Kriging underscore that advantages of giving explicit treatment to spatial dependency effects, in particular when the use of spatially varying attributes is unreliable (due to censoring, measurement errors, etc.), or when additional neighborhood attributes are unavailable. More interestingly, two moving window approaches (MWR and GWR) obtained the most accurate results. Finally, moving windows Kriging did not perform as well as anticipated. In terms of modeling guidelines, our empirical comparison suggests the following: 1. Examination of round 1 and round 2 model results implies that improving the benchmark model (OLS, traditional hedonic equation) tends to lead to improvements in the rest of the alternatives. The additional cost of obtaining a good benchmark model seems warranted, whenever variables of reasonable quality are available. 2. Omission of zonal‐based neighborhood attributes seems to negatively affect only the traditional hedonic equation. 3. Further to the topic of missing variables, the results also suggest that, in terms of predictive power, the ability of Kriging to make use of residual spatial pattern makes it the most robust alternative in the face of omitted variables. Kriging is thus, among the models studied in this article, the best alternative within a global modeling paradigm. 4. Two moving window approaches, MWR and GWR, produced the most accurate results. Although the cross‐validation procedure, and the estimation of local models is more time consuming and expensive, this additional cost is not as huge as it may appear. The gains in prediction accuracy may well offset a few computer‐hours of validation time. A conceptually appealing property of the moving window approaches is that they can be seen as a form of soft market segmentation, where market boundaries need not be rigidly predefined. 5. The predictive difference between MWR and GWR is relatively small. MWR can be easily implemented using standard software in combination with GIS technology. The advantage of GWR, in addition to slight gains in performance, is that the results appear to be substantially better for a wider range of window size selections. 6. Moving windows Kriging produced surprisingly poor results, in particular with respect to out‐of‐sample predictions. There are two possible reasons for this:



‐ 17 ‐

a) The result may be an indication of the ability of MWR to significantly reduce the degree of spatial error autocorrelation within the local windows, hence rendering the Kriging component of MWK ineffectual. The evidence would thus suggest that the impact of one spatial effect (spatial dependency) can be in some cases mitigated by the incorporation of the other spatial effect (spatial heterogeneity) into the modeling process. b) The cross‐validation procedure may result in a window size that is not sufficiently representative of out‐of‐sample observations. The general sensitivity of cross‐validation to influential observations is a topic currently under research (Farber and Páez, 2006). In terms of future research, further comparisons using a variety of datasets should help to verify the generality of the results reported in this article, and the robustness of the guidelines suggested. On a different but related topic, in this article Kriging was favored for its flexibility and explicitly predictive design. An alternative approach would be to consider the popular spatial autoregressive model (SAR) formulation (e.g.Anselin, 1988), for which good results have been reported in the past (e.g. Can, 1992b; Pace and Gilley, 1997; Páez et al, 2001). The only comparison available (to our knowledge) between the autoregressive model and the moving window approaches considered in the present article, indicates the advantage of local, moving windows models (Farber and Yeates, 2007). Comparison of the models selected for this paper and other models (SAR, splines, etc.) would be a worthwhile endeavor.



‐ 18 ‐

Figure 1: Locations of Estimation and Prediction Samples

Number of Nearest Neighbors Figure 2: Cross Validation Results for MWR(*) and GWR(∆)

Cross Validation Score



‐ 19 ‐

Number of Nearest Neighbors Figure 3: Cross Validation Results for MWR(*) and MWK(o)

Number of Nearest Neighbors Figure 4: Cross Validation Results for MWR(*), GWR(∆) and MWK(o) (2nd round)





‐ 20 ‐

Number of Nearest Neighbors Figure 5: Cross Validation Results for MWR(*), GWR(∆), MWK(o), OLS1(−) and OLS2(‐‐‐)

(2nd round)




‐ 21 ‐

Table 1: Summary Descriptive Statistics of Variables

Variable Mean Standard Deviation SalePrice ($1000)

384.406 293.1

Area (feet^2/100)

54.098 43.895

Front (feet)

42.198 16.323

HouseAge (decades)

4.808 2.447

SaleDate (indicator variable)

19.613 10.219

DistTransit (km)

1.872 1.269

MeanIncome ($1000)

74.166 58.098

Estimation Sample (n=30,145)

PctImm (%)

37.67 19.369

SalePrice ($1000)

389.603 269.132

Area (feet^2/100)

53.753 32.694

Front (feet)

41.964 15.414

HouseAge (decade)

4.805 2.491

SaleDate (indicator variable)

19.798 10.183

DistTransit (km)

1.856 1.27

MeanIncome ($1000)

75.811 60.118

Validation Sample (n=3,349)

PctImm (%)

36.969 19.665



‐ 22 ‐

Table 2: Traditional Hedonic Model Regression Results (OLS1)

Observations 30145 Standard Error 246.59

Degrees of Freedom 30138 F 2074.7

R Square 0.29 Significance of F 0

Adjusted R Square 0.29

Residual SSE 1.83E+09

Variable Estimate P‐value

Constant 350.0537 0.0000 Area 1.5602 0.0000 Front 5.2236 0.0000 Saledate ‐118.4245 0.0000 HouseAge 11.4954 0.0000 Age^2 2.5853 0.0000 DistTransit ‐46.1160 0.0000

Table 3: Traditional Hedonic Model Regression Results (OLS2)

Observations 30145 Standard Error 191.58

Degrees of Freedom 30136 F 5052.8

R Square 0.57 Significance of F 0

Adjusted R Square 0.57

Residual SSE 1.11E+09

Variable Estimate P‐value

Constant 279.8418 0.0000 Area 1.3206 0.0000 Front 3.3344 0.0000 Saledate ‐97.1914 0.0000 HouseAge 8.8965 0.0000 Age^2 2.8268 0.0000 DistTransit ‐19.4413 0.0000 MeanIncome 2.6656 0.0000 PctImm ‐3.0777 0.0000



‐ 23 ‐

Table 4: Summary Statistics for Prediction Errors

Model Mean

Absolute

Root MSE Minimum Maximum Standard

Deviation

OLS1*

OLS2**

MWR

GWR

Kriging1*

Kriging2**

MWK

139.37

102.22

66.08

64.03

109.76

99.3882

199.49

227.58

174.14

143.52

139.97

169.29

169.17

376.59

‐1767.26

‐1171.27

‐1845.57

‐1630.32

‐1079.57

‐1015.86

‐3475.57

2626.91

2464.75

2525.50

2526.55

2468.58

2595.54

3006.38

227.59

174.15

143.47

139.91

167.98

169.13

376.38

* without neighborhood attributes: MeanIncome, PctImm ** with neighborhood attributes: MeanIncome, PctImm

Table 5: Other Statistics for Model Prediction Performance

Case Correlation Pseudo R2 % Better than OLS1 % Better than OLS2

OLS1*

OLS2**

MWR

GWR

Kriging1*

Kriging2**

MWK

0.5406

0.7568

0.8594

0.8669

0.7946

0.7779

0.4058

0.2923

0.5729

0.7386

0.7515

0.6314

0.6051

0.1647

66.59%

77.66%

78.29%

58.17%

65.33%

47.51%

33.41%

70.50%

71.25%

45.98%

49.66%

37.47%




‐ 24 ‐

Table 6: Summary Statistics of Prediction Errors for Second Round Comparison

Case Mean

Absolute

Root MSE Minimum Maximum Standard

Deviation

OLS1*

OLS2**

MWR

GWR

Kriging1*

Kriging2**

MWK

150.0953

110.3593

77.6374

76.3025

106.3207

101.3898

146.5522

241.2678

186.1266

156.7824

154.2950

173.5379

176.2319

287.7564

‐742.56

‐1013.84

‐2385.62

‐2064.23

‐1290.82

‐1155.66

‐3938.73

2835.93

2451.56

2486.02

2487.15

2421.28

2549.85

2934.15

241.24

186.15

156.79

154.31

173.09

176.25

287.47


Table 7: Other Statistics for Second Round Model Comparison

Case Correlation Pseudo R2 % Better than OLS1 % Better than OLS2

OLS1*

OLS2**

MWR

GWR

Kriging1*

Kriging2**

MWK

0.4513

0.7230

0.8256

0.8300

0.7789

0.7557

0.5233

0.2037

0.5227

0.6816

0.6889

0.6067

0.5710

0.2738

68.26%

75.72%

76.53%

64.23%

71.42%

58.67%

31.74%

67.99%

68.32%

52.05%

53.96%

47.51%



‐ 25 ‐

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Spatial Effects in Hedonic Price Estimation: A Case …...Spatial Effects in Hedonic Price Estimation: A Case Study in the City of Toronto F. Long, A. Páez, S. Farber Centre for Spatial