document de travail 2001-008 - FSA ULaval

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DOCUMENT DE TRAVAIL 2001-008 ISOLATING SPATIAL FROM A-SPATIAL COMPONENTS OF HOUSING ATTRIBUTES USING KRIGING TECHNIQUES Francois Des Rosiers, Marius Thériault, Paul -Y Villeneuve and Yan Kestens

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09-2001

Isolating Spatial from A-spatial Components of Housing Attributes Using Kriging Techniques by François Des Rosiers1, Marius Thériault2, Paul-Y Villeneuve3 and Yan Kestens4 Paper presented at the European Real Estate Society 8th Annual Conference Alicante, Spain, June 26-29, 2001 (Final version) 1. Faculty of Business Administration, Planning and Development Research Centre, Laval University, Québec, Canada, G1K 7P4 Phone: 418-656-2131 ext. 5012, e-mail: [email protected] 2. Department of Planning, Director, Planning and Development Research Centre, Laval University, Québec, Canada, G1K 7P4 Phone: 418-656-2131 ext. 5899, Fax: 418-524-6701 e-mail: [email protected] 3. Department of Planning, Planning and Development Research Centre, Laval University, Québec, Canada, G1K 7P4 Phone: 418-656-2131 ext. 3791, e-mail: [email protected] 4. Ph.D candidate, Planning and Development Research Centre, Laval University, Québec, Canada, G1K 7P4 Phone: 418-656-2131 ext. 8878 e-mail: [email protected]

Key Words: GIS, Hedonic modelling, Accessibility, Neighbourhood factors, Urban externalities, Housing markets, Spatial analysis, Factor analysis, Kriging.

SUMMARY OF PAPER

The current piece of research is an attempt to isolate spatial from a-spatial components of housing attributes, using kriging techniques. The hedonic approach, on which is based this investigation, is applied to the bungalow (single-story, detached house) segment of the Quebec Urban Community (QUC) residential market. Some 3 638 properties transacted over the 199091 period (24 months) are considered in the analysis. Sale prices range from a low of $50 500 to a high of $336 000 while mean price stands at roughly $87 700. Combining the standard and stepwise regression procedures, a series of 33 descriptors are first selected which describe building and site characteristics (23), local taxation features (2), proximity attributes (3) as well as socio-demographic (3) and accessibility-to-services (2) factors obtained through principal component analyses (PCA). Eight building and site descriptors are then selected and their degree of spatial dependence tested using the Moran’s I index. The latter are applied a kriging procedure which results in two sets of transformed variables accounting for both the neighbourhood-specific and property-specific dimensions of the price setting process; these reflect local averages and departure from local averages, respectively. Findings clearly suggest that while kriging techniques do not substantially reduce the level of spatial autocorrelation affecting model residuals, they nevertheless provide a most useful insight into the house price determination process by sorting out the spatial and a-spatial components of residential values. In particular, this study shows that the marginal contribution of building attributes varies according to the neighbourhood profile of the housing stock and that, by and large, a high standard neighbourhood impacts positively on a single property’s market price, and vice-versa. -2-

1. INTRODUCTION : CONTEXT

AND

OBJECTIVE

OF

RESEARCH

The current piece of research is an attempt to isolate spatial from a-spatial components of housing attributes, using kriging techniques. The hedonic approach, on which is based this investigation, is applied to the bungalow (single-story, detached house) segment of the Quebec Urban Community (QUC) residential market for the 1990-91 period. As put by several authors, property values mirror complex overlapping effects which combine externalities and location factors (Krantz et al. 1982; Hickman et al. 1984; Shefer 1986; Yinger et al. 1987; Strange 1992; Can 1993; Dubin 1998; Hoch and Waddell 1993; Des Rosiers et al. 1996). While hedonic models have long proved their usefulness as an analytical device, previous research has shown that substantial portion of price variability remains unexplained (Anselin and Can 1986; Dubin and Sung 1987; Can 1993; Dubin 1998). The great deal of neighbourhood factors needed to adequately account for submarket specifics (Adair, Berry and McGreal 1996 & 1998) raises methodological issues linked to the presence of excessive multicollinearity between model attributes, as well as to structural heteroskedasticity and spatial autocorrelation among residuals; all of these are detrimental to the stability of regression coefficients (Dubin 1988; Anselin and Rey 1991; Can and Megbolugbe 1997; Basu and Thibodeau 1998; Pace, Barry and Sirmans 1998; Des Rosiers and Thériault 1999). Among the issues that need be addressed, spatial dependence - a current feature of property markets - deserves substantial research efforts. Indeed, several situations involve the presence of spatial autocorrelation, thereby violating the basic assumptions underlying hedonic models. For instance, while liveable area will partly mirror size features that are exclusive to a given piece of property, it also reflects neighbourhood characteristics which are common to all houses in the same market segment. Consequently, implicit prices of housing attributes may prove to be both biased and unstable (Can 1990 & 1993, Dubin 1998). In this paper, kriging techniques are used to address the problem. Once spatial dependence has been identified for each housing descriptor, relevant variables are then split into their spatial (neighbourhood-specific) and a-spatial (property-specific) components. This results in two sets of unbiased coefficients being calibrated. In so doing, space-related influences on prices can be isolated

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for each major attribute and their consistency assessed over space. Moreover, applying kriging to individual variables rather than to model residuals provides a “methodologically correct” device to tackle spatial dependence. Resorting to kriging requires an extensive use of Geographic Information Systems (GIS) and, in particular, of spatial statistics methods (Anselin and Getis 1992; Griffith 1993a; Zhang and Griffith 1993; Thériault and Des Rosiers 1995; Levine 1996) such as variography and spatial interpolation techniques (Dubin 1992; Panatier 1996). 2. PREVIOUS RESEARCH FINDINGS: CONTROLLING

FOR

COLLINEARITY

Previous research papers have addressed both the multicollinearity and spatial autocorrelation issues. Following other examples in the recent hedonic literature (Bourassa et al. 1999), Des Rosiers, Thériault and Villeneuve (2000a) resort to factor analysis (Thurstone 1947; Rummel 1970; Can 1992) in order to generate independent complex variables used as substitutes for initial attributes, thereby reducing collinearity to a minimum. The study is based on a sample of some 2 400 town cottages sold in the Quebec region from January 1993 to January 1997. Accessibility is computed at the level of individual properties (Nijkamp, Van Wissen and Rima 1993) and based on travel times, using some 19 250 street intersections (Thériault, Des Rosiers and Vandersmissen 1999). Census data used to define neighbourhood attributes are aggregated at the level of 604 enumeration areas. Using the principal component method (PCA) with a Varimax rotation, some 49 initial variables are thus being reduced to only six significant factors – four neighbourhood and two access principal components – which are in turn integrated in the hedonic equation as complex independent variables. Findings clearly suggest that factor analysis is highly efficient at sorting out access and neighbourhood dimensions. In particular, the educational profile of local residents as well as access to regional and local services emerge as powerful price determinants. As an extension to the above study, Des Rosiers, Thériault and Villeneuve (2000b) look at house price determinants from a dynamic perspective by

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investigating the influence of socio-demographic, as well as socio-economic, changes over the 1986-1996 period upon property values in the Quebec Urban Community (QUC – roughly 675 000 in population). The general data base includes some 14 400 single-family properties transacted between January 1993 and January 1997. Two market segments are analyzed, namely town cottages (1 708 sales) and condominium units (940 sales), with sales in the $100 000 - $250 000 price range. Here, enumeration areas are recomputed using spatial smoothing techniques in order to produce some 416 homogeneous delimitations that allow for comparison over time. Calculation of standardized, or time de-trended, socio-demographic changes between censuses (1986-91; 1991-96; 1986-96) are then performed using linear regression adjustments, while a series of five PCAs are computed on socio-demographic attributes: while state components are derived from each census (1986, 1991 and 1996), change components are computed on the 1986-91 and 1991-96 periods. A sixth PCA is also performed on access variables (1988 road network, extended to the whole Quebec Metropolitan Area). Finally, significant principal components are substituted for individual variables in the hedonic equations. Findings indicate that both socio-demographic structures (state variables) and their evolution (change variables) over time do affect house prices significantly, with most principal components retained in the analysis displaying strong t values. As expected then, neighbourhood profiles as well as changes are part of the price setting process and should not be overlooked in hedonic modelling. 3. DEALING WITH SPATIAL AUTOCORRELATION USING KRIGING Spatial autocorrelation, or dependence, refers to the relationships between neighbouring values in space. It measures the degree of resemblance (positive) or dissimilarity (negative) between places or individuals as a function of the distance which separates them. In real estate, it stems from the influence that geographical structures exert upon urban rent through differentials in access to services, socio-demographic structures and environmental quality. It also mirrors the space-related similarities among building attributes within homogeneous neighbourhoods. There are a number of statistical indexes for measuring the degree of spatial autocorrelation

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(Anselin 1995; Cressie 1993; Getis and Ord 1992; Griffith 1993a; Odland 1988; Ord and Getis 1995; Ripley 1981). The one most commonly used – also considered the most robust from a mathematical standpoint - was developed by the statistician Moran (1948 and 1950) after whom it was named (Moran's I). The Moran's I expresses the intensity of the relationship between any value located at a specific place (census tract or property) and the similar for its neighbours. It actually measures autocorrelation between values of a z vector, considering some weight (wij) that is a function of spatial proximity between any (i and j) pair of observations. For the purpose of this paper, the interaction between each property is weighted only by the squared inverse distance between them, thereby reflecting the decrease in influence under gravitational effect. The role the Moran’s I plays with respect to the analysis of spatial structures is similar to that of the correlation coefficient in conventional statistics. While there is no bounds for the index, in practice, its values range between +1 and -1. A positive value expresses an association by resemblance between neighbours; a negative value corresponds to a dissimilarity; and a zero value indicates the absence of any defined spatial structure. The Moran’s I thus provides a parametric test for assessing whether, and to what degree, observed spatial structures are random. Since multiple regression analysis requires that observations be independent from one another in order for hypothesis testing to be reliable (Anselin and Rey 1991; Griffith 1993b), testing for the presence of spatial autocorrelation is a prerequisite for good hedonic modelling. Solving for the problem leads us to consider kriging as a modelling tool. First developed by Krige (1951) to model geological structures, kriging is now considered as one of the most powerful spatial interpolation method (Panatier 1996). It is based on an analysis of the spatial variance of spacedistributed values. The latter are used to build experimental variograms whereby the variation in the observed means differentials between values – in this case, property sale prices – is expressed in terms of the distance which separates them in space (Cressie 1993). Variograms are then applied theoretical functions so as to obtain the best adjustment for value variations resulting from proximity, which is accounted for in the spatial interpolation of the phenomenon.

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Kriging is to the study of spatial distributions what regression analysis is to conventional statistics. While independently developed, spatial autocorrelation and kriging techniques rest upon similar spatial variance and covariance concepts (Griffith and Csillag 1993). Kriging makes it possible to model proximity relationships between the values of socio-demographic and economic attributes of adjoining neighbourhood units in order to compute space de-trended values which are representative of a local variation, and therefore independent from the regional trend as well as from propertyspecific patterns. In so doing, kriging can contribute to properly identify areas with space-specific, structural profiles that differ from those of adjacent areas while bringing out flaws in the urban fabric. Among other things, it should help understand the detrimental effect of an inadequate division of space for census purposes (Wong and Amrhein 1996). Finally, kriging adjusts continuous functions to principal components, thereby smoothing expressions of spatial structures for individual properties included in hedonic models. In a recent paper, Des Rosiers et al. (2001) resort to kriging in order to measure the effect of micro-level, neighbourhood profiles on house values and market differentiation in the presence of spatial autocorrelation. While principal component analysis (PCA) derived from former research serves as a sorting device to identify space-structuring factors, kriging is used to deal with spatial dependence among model residuals. A global sample of 2 405 town cottages sold in the QUC from January 1993 to January 1997 is used for the modelling process. Two randomly chosen, equal-size sub-samples are also defined for cross-validation purposes. Isotropic exponential and spherical variogram functions are calibrated for interpolation of model residuals. Once reinserted in the hedonic equations as independent vectors, kriged variables are shown to substantially improve both explanatory and predictive model performances. Thus, findings suggest that most spatially structured variations are being captured by the adjusted model, leaving the remaining spatial autocorrelation statistically non significant while regressors’ “t” values are greatly enhanced. Although kriging emerges as a promising device to complement the traditional modelling approach, some theoretical issues remain to be addressed. Firstly, a stable spatial structure over time must be identified in order for kriging to be adequately resorted to. Secondly,

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applying the procedure to a model’s initial set of residuals may result in the ensuing, adjusted residuals being highly correlated with the kriged variable, thereby violating the OLS error-randomness condition. To address the problem, kriging should therefore be performed individually on every relevant housing attribute so as to sort out in each case the spatial and aspatial components of price variations. This is precisely the object of the current piece of research. 4. SPATIAL

VS.

A-SPATIAL COMPONENTS

OF

HOUSE PRICE VARIATIONS

As mentioned earlier, space-related housing attributes will cause spatial dependence to alter the reliability of hedonic prices. While applying kriging on model residuals allows for an overall adjustment to be made on house values, it does not help isolate the spatial from the a-spatial components of implicit prices. These may differ substantially among housing attributes and will vary according to neighbourhoods as well as over time. The object of this research is to identify such components. In order to achieve that, kriging is performed on the QUC bungalow (single-family, one story, detached houses) segment of the residential market. Some 3 638 properties transacted over the 1990-91 period (24 months) are considered in the analysis. Sale prices range from a low of $50 500 to a high of $336 000 while mean price stands at roughly $87 700 (see Table 1). 4.1

Analytical Approach: the Basic Model

The initial modelling stage (Model A) is based on previous research findings. Combining the standard and stepwise regression procedures, a series of 26 descriptors are first selected which describe building and site characteristics (17), time and local taxation features (2), proximity attributes (1) as well as socio-demographic (4) and accessibility-to-services (2) dimensions. The two latter sets of descriptors consist of complex variables derived from applying principal component analysis (PCA) to an array of census and travel time attributes (Des Rosiers, Thériault and Villeneuve 2000a). The operational definition of all variables used in this study can be found in Table 1, together with main descriptive statistics. Finally, a semi-log functional form is resorted to here, in line with most hedonic studies found in the relevant literature. Two variables (apparent age and lot size) are applied a logarithmic transformation on the grounds of their

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distribution. Finally, extreme residuals (above 3 SEE) are discarded from the computation, thus leaving 3 599 cases in the analysis. Findings obtained with Model A are displayed in Table 2. Both explanatory (Adj. R-Square: 0,779; F Value: 489) and predictive (SEE: 0,1124) performances are quite good considering the model covers the whole QUC territory. All regression coefficients do conform in sign and magnitude with theoretical expectations and are statistically significant at the 0,001 level or below. As can be seen from both “t” values and standardized Beta coefficients, living area (LIVAREA), apparent age (LNAPPAGE), lot size (LNLOTSIZ), local tax rate (LTAXRATE), proximity to a highway exit (HIGHWEXT) as well as CSF3 (level-of-education component) and ACCF1 (access to regional services) emerge as prominent determinants of residential values. Finally, multicollinearity remains well within acceptable limits (maximum VIF at 5,1). 4.2

Measuring Spatial Autocorrelation of Building Attributes

While several variables (fiscal and proximity descriptors, census and access factors) are explicitly space-related attributes, it is not so for building and site characteristics whose property-specific content is generally taken for granted. Yet, it can be assumed that some of these also bear a significant neighbourhood-specific dimension, whose magnitude can only be brought out through a systematic analysis of the spatial dependence their inherent structure involves. In this paper, building and site attributes are selected on the grounds of their degree of spatial autocorrelation (Moran’s I > 0,10) and of their “t” value (Sig. < 0,001). The Surfer v.7 software is used for that purpose. Eight descriptors (LIVAREA, LNAPPAGE, QUALITY, WASHROOM, INFFOUND, FIREPLCE, FINBSMT and LNLOTSIZ) are thus identified; related correlograms are displayed in Figure 1. The Moran’s I test is also applied on sale price (LNSPRICE) for comparative purposes while the correlogram pertaining to QUALTY could not be computed for technical reasons. In most cases, spatial dependence remains significant within the 1000 – 1200 m. distance range while its influence is particularly strong with respect to LNAPPAGE. It extends well beyond 1 500 m. in the case of LNLOTSIZ whereas it is limited to roughly 400 m. for INFFOUND. Finally, spatial autocorrelation derived from QUALITY (not shown here) reaches 500 m. from any property. It is therefore relevant to recalibrate the initial

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model so as to sort out the spatial and a-spatial components of these building and site attributes. 4.3

Kriging Building and Site Attributes

Table 3 presents the regression results obtained with the “kriged” hedonic model (Model B) which includes all variables and factors previously defined

(33 descriptors). While model performances indicate a slight improvement when compared with Model A (Adjusted R-Square: 0,782; F value: 392; SEE: 0,1116), Model B multicollinearity remains well under control – most VIF standing below the 3 threshold -, in spite of the greater number of descriptors. To validate that assertion, a factor analysis (PCA) was performed on all kriged variables, but the three ensuing components did not perform as well as individual variables did; the latter were therefore maintained in the equation. Finally, all regression coefficients display high “t” values with related probabilities of 0,001 or below, safe for two variables (BSMT_KGD and CSF2). It is worth noting that kriging building attributes induces only minor changes in the magnitude of other model parameters. Only census-related coefficients (CSF1 to CSF4) do experience some modifications, which suggests that these could possibly interfere with kriged components. While this issue is not addressed in the current paper, it will be in a forthcoming paper focusing on the use of interactive variables (IRES Annual Meeting, Anchorage, July 2001). A visual representation of kriging outputs (kriged values, variograms and standard deviations) is displayed in Map 1. Findings clearly suggest that both kriged (KGD, neighbourhood-specific) and property-specific (DEP, departure from local average) components of initial variables significantly contribute to house price determination. While LNLOT_DEP was also found to exert a strong and significant impact on sale price, LNLOT_KGD did not emerge as statistically significant. Since interpreting the property-specific component in the absence of its neighbourhood-specific counterpart is rather meaningless, the original LNLOTSIZ attribute is used instead. Living area and apparent age – the two most prominent house price determinants -, both display highly significant neighbourhood (spatial) and property-specific (a-spatial) coefficients. In the former case, the first component adds some 0,7% to value for each square

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metre of living area while any departure from the local mean adds (removes) another 0,4%.

Table 4 may help interpreting the marginal contributions of spatial and a-

spatial components derived from kriging. It provides, for each kriged descriptor, the marginal contribution to value for different levels of housing attributes depending on the neighbourhood profile. Findings can be summarized as follows: ! The higher the average size of houses in the neighbourhood, the higher the contribution of a given property size to its market value; ! The age-related depreciation of a house is higher if this house is younger than the neighbourhood it is located in, and vice-versa; ! The higher the positive (negative) discrepancy in the quality level between a property and its neighbourhood, the higher the positive (negative) impact on its price; ! The marginal contribution of a washroom increases with the average number of washrooms in the surrounding properties; ! The detrimental effect of inferior foundations increases with the proportion of surrounding houses with concrete foundations; ! The added value of a fireplace rises with the average number of fireplaces in neighbouring houses; ! Similarly, the contribution of a finished basement is greater in neighbourhoods where a larger proportion of houses also have one. By and large then, it seems that, caeteris paribus, the market value of a property is driven upwards or downwards by the local housing stock standards: high overall building standards will tend to raise a given property’s sale price whereas low neighbourhood standards will exert a downward effect on its value. This “demand pull” adequately mirrors -11-

households’ preferences for housing attributes, resulting in relatively homogeneous urban neighbourhoods. This trend is reversed though with respect to overall quality features (QUALITY and INFFOUND), where positive or negative property-specific attributes are valued in contrast to the immediate neighbourhood housing profile. 4.4

After Kriging Spatial autocorrelation

Model B is validated using an exogenous data set (some 400 cases, or 10% of the original data bank). As shown in Table 5, ex post performances outmatch theoretical expectations. Most of all though, the Moran’s I test applied to remaining departure values (Figure 2) clearly indicates that most of the spatial autocorrelation present in the initial building and site variables is actually captured via the kriged attributes, thereby generating more stable estimates for property-specific price determinants. This is an important feature of this paper since it clearly suggests that spatial influences present in building and site attributes ought to be sorted out before any reliable interpretation of hedonic prices is brought forward. This comment applies even though spatial autocorrelation is still found in the kriged model residuals, as shown by Figure 3. 5.

SUMMARY

OF

FINDINGS

AND

SUGGESTIONS

FOR

FURTHER

RESEARCH

The current piece of research is an attempt to isolate spatial from a-spatial components of housing attributes, using kriging techniques. The hedonic approach, on which is based this investigation, is applied to the bungalow (single-story, detached house) segment of the Quebec Urban Community (QUC) residential market. Some 3 638 properties transacted over the 199091 period (24 months) are considered in the analysis. Sale prices range from a low of $50 500 to a high of $336 000 while mean price stands at roughly $87 700. Combining the standard and stepwise regression procedures, a series of 26 descriptors are first selected which describe building and site characteristics (17), time and local taxation features (2), proximity attributes (1) as well as socio-demographic (4) and accessibility-to-services (2) factors obtained through principal component analyses (PCA). Eight building and site descriptors are then selected and their degree of spatial

-12-

dependence tested using the Moran’s I index. The latter are applied a kriging procedure which results in two sets of transformed variables accounting for both the neighbourhood-specific and property-specific dimensions of the price setting process; these reflect local averages and departure from local averages, respectively. Findings clearly suggest that while kriging techniques do not substantially reduce the level of spatial autocorrelation affecting model residuals, they nevertheless provide a most useful insight into the house price determination process by sorting out the micro-level spatial and a-spatial components of residential values. In particular, this study shows that the marginal contribution of building attributes varies according to the neighbourhood profile of the housing stock and that, caeteris paribus, the market value of a property is driven upwards or downwards by the local housing stock standards. Such findings are consistent with the relative homogeneity of urban neighbourhood which mirrors households’ preference patterns. Where overall quality features are concerned though, positive or negative property-specific attributes are valued in contrast to the immediate neighbourhood housing profile. While kriging techniques already offers an interesting analytical device to improve hedonic modelling and study real estate markets, other approaches deserve further investigation. For instance, it is possible to address the spatial dependence issue using an approach which is directly derived from the Moran’s I procedure rather than through spatial interpolation. Jointly resorting to interactive variables may lead to an even more detailed interpretation of how residential values are designed.

REFERENCES

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Pace, R. K., R. Barry and C. F. Sirmans, 1998. Spatial statistics and real estate. The Journal of Real Estate Finance and Economics, 17(1): 5-14. Panatier, Y., 1996. Variowin: Software for Spatial Data Analysis in 2D. Statistics and Computing, Springer-Verlag, New York, 91 p. Ripley, B. D., 1981. Spatial Statistics. New York, Wiley. Rummel, R.J., 1970. Applied Factor Analysis. Evanston, Northwestern University Press. Shefer, D., 1986. Utility changes in housing and neighbourhood services for households moving into and out of distressed neighbourhoods, Journal of Urban Economics, 19: 107-124. Strange, W., 1992. Overlapping neighbourhoods and housing externalities. Journal of Urban Economics, 32: 17-39. Thériault, M., and F. Des Rosiers, 1995. Combining hedonic modelling, GIS and spatial statistics to analyze residential markets in the Quebec Urban Community. In Proceedings of the Joints European Conference on Geographical Information, EGIS Foundation, March 26-31, The Hague, The Netherlands, 2: 131-136. Thérault, M., F. Des Rosiers and H. Vandermissen, 1999. GIS-Based Simulation of Accessibility to Enhance Hedonic Modelling and Property Value Appraisal. Joint IAAO-URISA 1999 Conference, New Orleans, April 11-14, 10 pages. Thurstone, L.L., 1947. Multiple Factor Analysis. Chicago, University of Chicago Press. Wong, D. and Amrhein, C., 1996. The Modifiable Areal Unit Problem. Special issue of Geographical Systems, vol. 3, no. 2-3. Yinger, J. H., H. S. Bloom, A. Borsch-Supan and H. F. Ladd, 1987. Property Taxes and House Values: The Theory and Estimation of Intrajurisdictional Property Tax Capitalization. Studies in Urban Economics, Academic Press, Princeton Univ., 218 p. Zhang, Z., and D. Griffith, 1993. Developing user-friendly spatial statistical analysis modules for GIS: An example using ArcView. Computer, Environment and Urban Systems, 21(1): 5-29.

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Table 1: Operational Definition of Variables and Descriptive Statistics Sample of 3638 bungalows sold from January 1990 to December 1991 ($50 500 to $336 000) Variable

Type

Mean

Std. Dev

Min.

Max.

Definition

SPRICE LNSPRICE LIVAREA LIVAR_KGD LIVAR_DEP LNAPPAGE LNAPP_KGD LNAPP_DEP QUALITY QUAL_KGD QUAL_DEP WASHROOM WASH_KGD WASH_DEP INFFOUND INFFO_KGD INFFO_DEP FIREPLCE FIRE_KGD FIRE_DEP FINBSMT BSMT_KGD BSMT_DEP EXCAPOOL STAIR DISHWASH DETDGAR SKYLIGHT INFCEILQ OVEN VACUUM LNLOTSIZ LNLOT_KGD LNLOT_DEP WATRSEWR LTAXRATE NBMONTH

HIGHWEXT

CSF1 CSF2 CSF3 CSF4 ACF1 ACF2

Sale price of the property ($)

M

87 694

25 844

50 500

336 000

Natural logarithm of sale price ($) Building & Site Attributes Living area of the property (square metres) Kriged variable on LIVAREA (Nbhd-specific) Departure from LIVAR_KGD (Prop.-specific) Natural log, of apparent age of property (years) Kriged variable on LNAPPAGE (Nbhd-specific) Departure from LNAPP_KGD (Prop.-specific) House quality index (– below/ +above average) Kriged variable on QUALITY (Nbhd-specific) Departure from QUAL_KGD (Prop.-specific) Total number of washrooms (bath=1; toilet=0,5) Kriged variable on WASHROOM (Nbhd-specific) Departure from WASH_KGD (Prop.-specific) Presence of an inferior foundation quality Kriged variable on INFFOUND (Nbhd-specific) Departure from INFFO_KGD (Prop.-specific) Number of fireplaces Kriged variable on FIREPLCE (Nbhd-specific) Departure from FIRE_KGD (Prop.-specific) Presence of a finished basement Kriged variable on FIREPLCE (Nbhd-specific) Departure from FIRE_KGD (Prop.-specific) Presence of an excavated pool Presence of an indoor staircase made of hard wood Presence of built-in dishwasher (synthetic var.) Presence of a detached garage Presence of skylights Presence of an inferior ceiling quality Presence of a built-in oven and/or cooking top Presence of a central vacuum unit Natural logarithm of the lot size (square metres) Kriged variable on LNLOTSIZ (Nbhd-specific) Departure from LNLOT_KGD (Prop.-specific) House linked to municipal water & sewer networks Local Taxation & Time Attributes Global tax rate ($ per 100$ of appraisal value) Monthly inflation index : number of months elapsed between Jan. 1986 (reference) and date of sale Proximity Attributes (Euclidean distances and Buffer zones) Distance to nearest highway exit, in kilometers

M

11,3483

,246

10,830

12,725

M

96,357 96,413 -,045 2,483 2,492 -,009 -,034 -,034 ,000 1,254 1,253 ,002 ,038 ,039 ,000 ,188 ,188 ,001 ,568 ,568 ,001 ,039 ,077 ,553 ,106 ,013 ,009 ,106 ,059 6,433 6,441 -,008 ,990

17,887 10,924 12,388 ,896 ,705 ,436 ,220 ,140 ,163 ,330 ,184 ,264 ,192 ,060 ,169 ,418 ,238 ,328 ,495 ,175 ,453 ,194 ,267 ,497 ,307 ,112 ,092 ,308 ,235 ,323 ,196 ,214 ,099

16,360 69,514 -73,075 -,693 ,191 -3,365 -2,000 -1,975 -1,471 ,500 ,724 -1,246 ,000 ,000 -,342 ,000 -,070 -1,213 ,000 ,018 -,894 ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 5,273 5,706 -1,052 ,000

269,090 166,087 155,736 3,912 3,681 2,465 2,000 ,827 2,000 5,500 2,611 4,197 1,000 ,681 1,000 4,000 1,695 3,031 1,000 ,949 ,844 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 9,416 8,106 1,749 1,000

M

2,385 60,197

,428 6,732

1,253 49,000

3,084 72,000

D

3,086

1,652

,000

9,570

M

-,648 -,015 ,092 ,259 ,424 ,011

,530 1,195 ,869 ,775 ,915 ,780

-1,612 -2,290 -1,997 -2,993 -1,926 -1,448

2,554 2,668 2,776 3,291 2,509 3,832

Factor Scores – Census (1991) & Access (1988 network) Attributes Center-vs-Periphery component Family cycle component Level-of-Education component “X” generation location pattern Access-to-Regional-Services component Access-to-Local-Services component

M M M M M R R R M M M D M M M M M B M M D D D D D D D D M M M B M

M M M M M

Table 2: Model A - Basic Model Model Summary

Model 1

R ,884

Adjusted R Square ,779

R Square ,781

Std. Error of the Estimate ,1124

ANOVA Sum of Squares 160,740

Regression Residual Total

df 26

Mean Square 6,182

45,130

3572

,013

205,870

3598

F 489,319

Sig. ,000

Coefficients Unstandardized Coefficients B 1

Std. Error

Stand. Coeff. Beta

Collinearity Statistics t

Sig.

VIF

(Constant)

10,51

,054

194,46

,000

LIVAREA

,0050

,000

,363

35,53

,000

1,70

-,1097

,003

-,411

-36,72

,000

2,04

QUALITY

,0622

,009

,057

6,82

,000

1,13

WASHROOM

,0575

,007

,077

8,37

,000

1,39

INFFOUND

-,0686

,010

-,055

-6,71

,000

1,09

FIREPLCE

,0348

,005

,061

7,14

,000

1,17

FINBSMT

,0284

,004

,059

6,50

,000

1,33

EXCAPOOL

,0952

,010

,077

9,45

,000

1,08

STAIR

,0435

,007

,048

5,92

,000

1,09

DISHWASH

,0309

,004

,064

7,81

,000

1,10

DETGARAG

,0397

,006

,050

6,19

,000

1,08

SKYLIGHT

,0702

,017

,032

4,06

,000

1,03

INFCEILQ

-,1081

,021

-,042

-5,21

,000

1,05

OVEN

,0299

,006

,038

4,71

,000

1,07

VACUUM

,0368

,008

,036

4,49

,000

1,05

LNLOTSIZ

,0718

,007

,097

10,49

,000

1,38

WATRSEWR

,1076

,020

,045

5,43

,000

1,11

NBMONTH

,0014

,000

,040

5,13

,000

1,01

LTAXRATE

-,0598

,006

-,107

-9,30

,000

2,15

HIGHWEXT

,0154

,002

,107

9,19

,000

2,19

CSF1

,0321

,004

,071

7,22

,000

1,56

CSF2

-,0144

,003

-,072

-5,43

,000

2,85

CSF3

,0729

,004

,264

19,06

,000

3,12

CSF4

-,0220

,003

-,071

-7,28

,000

1,56

ACCF1

-,0771

,005

-,294

-16,63

,000

5,10

ACCF2

-,0307

,003

-,100

-9,20

,000

1,91

LNAPPAGE

Figure 1 : Moran’s I Correlograms Before Kriging Moran's I on Livarea

Moran's I on Lnsprice

0,6

Moran's I

0,2 0,0 12-199 m

200-397 m

400-598 m

600-794 m

800-999 m

1001-1175 m 1200-1391 m

1400-1574 m

1,2

0,4

1,0

0,3

0,8

0,2

0,6

0,1 0,0 12-199 m

200-397 m

400-598 m

600-794 m

1001-1175 m 1200-1391 m 1400-1574 m

-0,4

0,0

-0,2

-0,2

-0,3

-0,4

Moran's I

Moran's I on Inffound 0,2

0,1

0,1

0,1

600-794 m

800-999 m

1001-1175 m 1200-1391 m 1400-1574 m

Moran's I

0,3

0,2

400-598 m

0,0 12-199 m

200-397 m

400-598 m

600-794 m

-0,2

-0,3

-0,3

-0,4

-0,4

1001-1175 m 1200-1391 m 1400-1574 m

Moran's I

Confidence Limit (p=0,05)

Moran's I on Lnlotsiz 0,6

0,2

0,4

200-397 m

400-598 m

600-794 m

800-999 m

1001-1175 m 1200-1391 m 1400-1574 m

Moran's I

0,1 12-199 m

0,2 0,0 12-199 m

-0,2

200-397 m

400-598 m

600-794 m

800-999 m

1001-1175 m 1200-1391 m 1400-1574 m

-0,2

-0,3 -0,4

-0,4

Distance Lag

Distance Lag

Moran's I


200-397 m

400-598 m

600-794 m

800-999 m

1001-1175 m 1200-1391 m 1400-1574 m

-0,4

Moran's I on Finbsmt

-0,1

12-199 m

-0,3


0,0

0,0 -0,1

Distance Lag

Distance Lag

0,3


-0,2

Distance Lag

Moran's I

800-999 m

-0,1

-0,2

1001-1175 m 1200-1391 m 1400-1574 m

Moran's I on Fireplce

0,3

200-397 m

800-999 m

Moran's I


0,2

12-199 m

600-794 m

Distance Lag

0,3

-0,1

400-598 m

Distance Lag


0,0

200-397 m

-0,6

-0,4

Moran's I on Washroom

Moran's I

0,2

-0,1

Distance Lag

Moran's I

0,4

12-199 m

-0,2

Moran's I

800-999 m

Moran's I

Moran's I

0,4

0,5

Moran's I

0,8

Moran's I on Lnappage

Moran's I


Moran's I


Table 3: Model B - Kriging Building & Site Attributes Model Summary

Mode l

R

1

Adjusted R Square

R Square ,885

,784

Std. Error of the Estimate

,782

,1116

ANOVA Sum of Squares 161,095

Model Regression Residual Total

df 33

Mean Square 4,882

44,366

3560

1,246E-02

205,461

3593

F 391,715

Sig. ,000

Coefficients Unstandardized Coefficients B

Stand. Coeff.

Std. Error

(Constant)

10,37

,060

LIVAR_KGD

,0065

,000

LIVAR_DEP

,0044

,000

LNAPP_KGD

-,1198

LNAPP_DEP

Beta

Collinearity Statistics t

Sig.

VIF

171,41

,000

,294

20,90

,000

3,26

,217

25,05

,000

1,24

,005

-,354

-23,48

,000

3,75

-,1061

,005

-,191

-22,32

,000

1,20

QUAL_KGD

,0521

,015

,030

3,50

,000

1,24

QUAL_DEP

,0656

,012

,044

5,51

,000

1,05

WASH_KGD

,0911

,013

,070

7,09

,000

1,60

WASH_DEP

,0448

,008

,047

5,58

,000

1,19

INFFO_KGD

-,1663

,035

-,042

-4,74

,000

1,27

INFFO_DEP

-,0533

,012

-,037

-4,53

,000

1,13

FIRE_KGD

,0378

,009

,037

4,09

,000

1,37

FIRE_DEP

,0331

,006

,045

5,55

,000

1,08

BSMT_KGD

,0504

,018

,037

2,84

,005

2,79

BSMT_DEP

,0263

,005

,050

5,82

,000

1,21

EXCAPOOL

,0931

,010

,075

9,30

,000

1,08

STAIR

,0435

,007

,048

5,93

,000

1,10

DISHWASH

,0312

,004

,065

7,92

,000

1,10

DETGARAG

,0387

,006

,049

6,04

,000

1,09

SKYLIGHT

,0706

,017

,032

4,10

,000

1,03

INFCEILQ

-,1044

,021

-,040

-5,06

,000

1,05

OVEN

,0294

,006

,038

4,66

,000

1,08

VACUUM

,0377

,008

,037

4,64

,000

1,05

LNLOTSIZ

,0721

,007

,096

10,48

,000

1,39

WATRSEWR

,0911

,021

,036

4,37

,000

1,11

NBMONTH

,0014

,000

,039

5,00

,000

1,02

LTAXRATE

-,0652

,006

-,116

-10,05

,000

2,21

HIGHWEXT

,0147

,002

,102

8,75

,000

2,23

CSF1

,0249

,005

,055

5,23

,000

1,83

CSF2

-,0079

,003

-,040

-2,43

,015

4,41

CSF3

,0613

,004

,222

14,59

,000

3,80

CSF4

-,0184

,003

-,060

-5,65

,000

1,84

ACCF1

-,0748

,005

-,285

-15,92

,000

5,30

ACCF2

-,0304

,003

-,098

-8,99

,000

1,97

Map 1 : Kriged Values, Variograms and Kriged Standard Deviation Living Area

Ln of Apparent Age

3.6

180

3.4

170

3.2 3

160

2.8

150

2.6 2.4

140

2.2

130

2 1.8

120

1.6

110

1.4 1.2

100

1

90

0.8 0.6

80

0.4 0.2

70

0

60

1.05

Exponential Model Direction: 0.0 Tolerance: 90.0

Spherical Model Direction: 0.0 Tolerance: 90.0

1

0.7

0.95 0.9 0.6

200

0.85 0.8 0.75

0.5

0.7 Variogram

Variogram

150

0.65

0.4

0.6 0.55 0.3

100

0.5 0.45 0.2

0.4

50

0.35 0.3

0.1

0.25 0 0

500

1000

1500 Lag Distance

2000

2500

3000

0 0

500

1000

1500 Lag Distance

2000

2500

3000

Table 4: Interpreting Kriged Variables

80

90

100

110

120

130

140

150

80 90 100 110 120 130 140 150

1.682 1.758

1.718 1.795

1.754 1.833

1.791 1.872

1.829 1.912

1.868 1.952

1.908 1.994

1.948 2.036

1.837 1.919 2.006 2.096 2.190 2.289

1.876 1.960 2.048 2.140 2.237 2.337

1.916 2.002 2.092 2.186 2.284 2.387

1.956 2.044 2.136 2.232 2.333 2.438

1.998 2.088 2.181 2.280 2.382 2.489

2.040 2.132 2.228 2.328 2.433 2.542

2.083 2.177 2.275 2.377 2.484 2.596

2.128 2.223 2.323 2.428 2.537 2.651

Nhbd Average

5

10

15

20

25

30

35

40

0.825 0.766 0.734

0.817 0.759 0.727

0.812 0.755 0.723

0.809 0.752 0.720

0.807 0.749 0.718

0.805 0.748 0.716

0.803 0.746 0.715

0.801 0.745 0.713

0.712 0.695 0.682 0.671 0.661

0.705 0.689 0.675 0.664 0.655

0.701 0.685 0.672 0.661 0.651

0.698 0.682 0.669 0.658 0.649

0.696 0.680 0.667 0.656 0.647

0.695 0.678 0.665 0.655 0.645

0.693 0.677 0.664 0.653 0.644

0.692 0.676 0.663 0.652 0.643

Living Area Nhbd Average (Sq. meters) Property

Appar. Age (Years)

Property 5 10 15 20 25 30 35 40

Quality (Rank)

-2

-1

0

1

2

-2 -1 0 1 2

0.901 0.962

0.889 0.949

0.877 0.937

0.865 0.924

0.854 0.912

1.027 1.097 1.171

1.014 1.082 1.156

1.000 1.068 1.140

0.987 1.053 1.125

0.973 1.039 1.110

Nhbd Average

1

2

3

4

5

1 2 3 4 5

1.095

1.147

1.202

1.259

1.318

1.146 1.198 1.253 1.310

1.200 1.255 1.312 1.372

1.257 1.314 1.375 1.437

1.316 1.377 1.440 1.506

1.379 1.442 1.508 1.577

Nhbd Average

0

0.20

0.40

0.60

0.80

1

1.000 0.948

0.978 0.927

0.956 0.906

0.934 0.886

0.914 0.866

0.893 0.847

Nhbd Average Property

Washroom (#)

Property

Inf. Found. (0/1)

Property 0 1

Fireplace (#)

0

1

2

3

4

0 1 2 3 4

1.000

1.005

1.009

1.014

1.019

1.034 1.068 1.104 1.142

1.039 1.073 1.110 1.147

1.043 1.079 1.115 1.152

1.048 1.084 1.120 1.158

1.053 1.089 1.125 1.163

Nhbd Average

0

0.20

0.40

0.60

0.80

1

1.000

1.005

1.010

1.015

1.019

1.024

1.027

1.032

1.037

1.042

1.047

1.052

Nhbd Average Property

Fin. Bsmt (0/1)

Property 0 1

Table 5: Validating Models A and B Using Exogenous Data

Model A

Model B

Model Summary

Model Summary

R

Model 1

SAMP10PC = 1.00 (Selected) ,897

R

R Square ,805



SAMP10PC = 1,00 (Selected) ,903

Model 1

ANOVA Sum of Squares Regression Residual Total

df


R Square ,816


ANOVA

Mean Square

17,424

26

,670

4,208

374

1,1252E-02

21,632

400

F 59.561

Sig. .000

Regression Residual Total

Sum of Squares 17,570

33

Mean Square ,532

3,956

364

1,087E-02

21,526

397

df

F 48,989

Mean Prediction Error:

0,0784

Mean Prediction Error:

0,0770

Standard Deviation:

0,0660

Standard Deviation:

0,0634

Sig. ,000

Figure 2 : Moran’s I Correlograms After Kriging Moran's I on Livarea Departure Values 0,30 0,20 0,10 0,00 -0,10 -0,20 -0,30 -0,40

0,60 0,40 0,20 12-199 m 200-397 m

400-598 m

600-794 m

800-999 m

10011175 m

12001391 m

14001574 m

16101752 m

18011990 m

0,00 -0,20

12-199 m 200-397 m 400-598 m 600-794 m 800-999 m 1001-1175 1200-1391 1400-1574 1610-1752 1801-1990 m m m m m

-0,40

Moran's I

Confidence Limit p=(0.05)

Moran's I

Moran's I on Washroom Depart. Values

400-598 m

0,30

0,20

0,20 12-199 200- 400- 600- 800- 1001- 1200- 1400- 1610- 1801m 397 m 598 m 794 m 999 m 1175 1391 1574 1752 1990 m m m m m

12001391 m

14001574 m

16101752 m

18011990 m


0,00 -0,10

-0,80

-0,20

-1,00

12-199 200- 400- 600- 800- 1001- 1200- 1400- 1610- 1801m 397 m 598 m 794 m 999 m 1175 1391 1574 1752 1990 m m m m m

-0,30

-1,20

-0,40

Distance Lag

Distance Lag

Moran's I


Moran's I on Lnlotsiz Departure Values 0.40 0.20 0.00 12-199 200-

-0.20

10011175 m

0,10

-0,60


800-999 m

Moran's I on Finbsmt Departure Values

0,40

-0,40

600-794 m

Distance Lag

Moran's I on Inffound Departure Values

-0,20

Moran's I

12-199 m 200-397 m

Moran's I


0,00 12200- 400- 600- 800- 1001- 1200- 1400- 1610- 1801199 m 397 m 598 m 794 m 999 m 1175 1391 1574 1752 1990 m m m m m

0,30 0,20 0,10 0,00 -0,10 -0,20 -0,30 -0,40

Distance Lag

Distance Lag

0,30 0,20 0,10 0,00 -0,10 -0,20 -0,30 -0,40

Moran's I on Livarea Departure Values

Moran's I on Lnappage Depart. Values

m

400-

600-

800-

1001-

1200- 1400-

1610-

1801-

397 m 598 m 794 m 999 m 1175 m 1391m 1574 m 1752 m 1990 m

-0.40 Distance Lag

Moran's I


Distance Lag

Moran's I


Figure 3 : Moran’s I Correlograms on Models A and B Residuals

Moran's I on Model B Residuals

Moran's I

Moran's I on Model A Residuals 0,30

0,30

0,20

0,20

0,10

0,10

0,00 -0,10

0,00 12-199 200m 397 m

400598 m

600794 m

8001001- 1200- 1400999 m 1175 m 1391 m 1574 m

-0,10

-0,20

-0,20

-0,30

-0,30

-0,40

12-199 200-397 400-598 600-794 800-999 10011200140016101801m m m m m 1175 m 1391 m 1574 m 1752 m 1990 m

-0,40 Distance Lag

Moran's I


Distance Lag

Moran's I
