DOCUMENT DE TRAVAIL 2001-015 L S - FSA ULaval · 2001. 11. 13. · it is combining GIS technology and spatial statistics to define principal components of accessibility and socio-

Publié par : Published by : Publicación de la :

Faculté des sciences de l’administration Université Laval Québec (Québec) Canada G1K 7P4 Tél. Ph. Tel. : (418) 656-3644 Fax : (418) 656-7047

Édition électronique : Electronic publishing : Edición electrónica :

Aline Guimont Vice-décanat - Recherche et partenariats Faculté des sciences de l’administration

Disponible sur Internet : Available on Internet Disponible por Internet :

http ://www.fsa.ulaval.ca/rd [email protected]

DOCUMENT DE TRAVAIL 2001-015

MODELLING INTERACTIONS OF LOCATION WITH SPECIFIC

VALUE OF HOUSING ATTRIBUTES Marius THÉRIAULT, François DES ROSIERS, Paul VILLENEUVE and Yan KESTENS

Version originale : Original manuscript : Version original :

ISBN – 2-89524-136-8 ISBN - ISBN -

Série électronique mise à jour : One-line publication updated : Seria electrónica, puesta al dia

11-2001

Modelling Interactions of Location with Specific Value of Housing Attributes

Marius Thériault, Ph.D.

Director – Planning and Development Research Centre (CRAD), Félix-Antoine-Savard Building, 16th floor, Laval University, Canada G1K 7P4

Phone (418) 656-2131 ext. 5899; Fax (418) 656-2018 e-mail: [email protected]

François Des Rosiers, Ph.D.

Professor – Faculty of Business Administration, Laval University, Canada G1K 7P4 e-mail: [email protected]

Paul Villeneuve, Ph.D.

Professor – Department of Planning, Laval University, Canada G1K 7P4 e-mail: [email protected]

Yan Kestens

Ph.D. Candidate - Department of Planning, Laval University, Canada G1K 7P4 e-mail: [email protected]

1

Modelling Interactions of Location with Specific Value of Housing Attributes

Abstract

This paper presents a procedure for considering interactions of neighbourhood quality and property specifics

within hedonic models of housing price. It handles interactions between geographical factors and the marginal

contribution of each property attribute for enhancing values assessment. Making use of simulation procedures,

it is combining GIS technology and spatial statistics to define principal components of accessibility and socio-

economic census related to transaction prices of single-family homes. An application to the housing market of

the Quebec Urban Community (more than 3,600 bungalows transacted in 1990 and 1991) illustrates its

usefulness for building spatial hedonic models, while controlling for multicollinearity, spatial autocorrelation

and heteroskedasticity. Distance weighted averages of each property attribute in the neighbourhood and

interactions of property attributes with each principal component are used to detect any spatial effect on sale

price variations. This first-stage spatial hedonic model approximates market prices which are then used in

order to compare "expected" and actual property tax amounts, which are added to obtain a second-stage model

incorporating fiscal effects on house values. Interactions between geographical factors and property specifics

are computed using formulas avoiding multicollinearity problems, while considering several processes

responsible for spatial variability. For each property attribute, they define sub-models which can be used to

map variations, across the city, of its marginal value, assessing the cross effect of geographical location (in

terms of neighbourhood profiles and accessibility to services) and its own valuation parameters. Moreover,

this procedure distinguishes property attributes exerting a stable contribution to value (constant over the entire

region) from those whose implicit price significantly varies over space.

Keywords Hedonic modelling, Housing markets, Neighbourhood effects, Location, Spatial

interaction, Spatial gradients.

2

Problem statement

This piece of research presents a procedure aimed at handling interactions between spatial factors and housing

attributes. In traditional hedonic modelling, the contextual variations over space are usually specified using

“fixed” coefficients to assess their direct effect on housing values. This is based on the assumption that the

marginal prices of structural housing attributes are invariant across the city. This stable price assumption does

not hold if various types of households have different needs and are not, at the same time, distributed evenly

within the city. These households form the market of home buyers and sellers. The heterogeneity of their

distribution and tastes could locally distort the demand for specific structural attributes and amenities of

homes, thus creating significant geographical trends that should be reflected in the coefficients of hedonic

models.

Moreover, any spatial heterogeneity characterising the valuation process will produce spatial autocorrelation

among model residuals, if it is not appropriately handled in the model specification. Following Can (1990), we

believe that the influence of property specifics on house prices is, itself, influenced by the spatial variability of

demand, which is linked to heterogeneity in the distribution of household types and services within the city.

Therefore, hedonic modelling procedures cannot rely entirely on fixed space coefficients. There is a clear need

to test, and eventually incorporate, interactions between the structural characteristics of urban space and those

housing specifics which are putatively linked to them.

According to Griffith (1992, p. 278), “spatial autocorrelation may be defined as an average correlation

between observations based upon replicated realisations of the geographic distribution of some attribute.” It is

often linked to processes of diffusion occurring in geographical space. Anselin and Can (1986) have linked it

to the spatial distribution of externalities which produces urban density gradients. Autocorrelation is highly

detrimental to the efficiency of statistical tests used to assess the statistical significance of OLS (ordinary least

square) regression coefficients (Anselin, 1990b). In applied regression analysis, two methodological concerns

should be addressed: (1) testing for the presence of spatial heterogeneity and (2) implementing alternative

estimation techniques. Anselin (1990b, p. 186) suggests that: “With respect to estimation, the best-known

3

approach is probably the spatial expansion method of Casetti (1972, 1986), which introduces models of spatial

drift in the regression coefficients.” The expansion method has been used in numerous geographical

applications for investigating spatial and temporal drifts, especially in the study of migration and labour

market transformation. Can (1990) was first to use it in the context of hedonic modelling. This paper extends

its procedure, using several neighbourhood quality indexes, and combining them with comparisons of each

property with its nearest immediate neighbours. Thus, it is allowing for a much more sensitive assessment of

spatial dependence.

Conceptual framework

During the last twenty years, rapid social and economic changes have been restructuring the distribution of

activities in general, and housing demand in particular, in North American cities. These changes are related to

the pervasive processes associated with so-called “post-fordist” and “post-modern” emergent forms. The

social and demographic correlates of these emergent forms include three interrelated tendencies capable of

profoundly modifying the structure of residential markets. First, rapid increase in female labour force

participation rates, especially during the seventies, has diversified household profiles. Now, many working

couples, dual earner families and non-family households tend to rely on market solutions and practise

domestic outsourcing, because of their small size, the time constraints they operate under, or their high

disposable income (Rose and Villeneuve, 1993 and 1998). Second, rising income inequalities in the

occupational sphere, as well as among households now counting on varying numbers of working members,

produce forms of social polarisation that are inscribed into urban social space (Bourne, 1993). Third, the

development of the service sector of most cities, together with the decentralisation of manufacturing activities,

has also affected the social profile of many neighbourhoods, especially by redistributing services' locations.

All these changes have an effect on residential relocation behaviour, which is highly related to household

cycles (Nijkamp et al., 1993). As social diversity is rapidly increasing within cities, it is less likely to translate

into spatial variation in the price of housing structured around monocentric rent gradients. Distance from the

CBD as the sole measure of access to employment and consumption opportunities becomes less relevant

4

because households value access to places distributed at various locations. Therefore, assessing accessibility

requires the measurement of distance and travel time to a larger set of amenities. For Baltimore, Dubin and

Sung (1987) have found that the influence of suburban employment centres on housing prices is restricted to a

two-kilometre radius. Considering the existence of several non-CBD peaks in the rent gradient, estimation of

an area-wide function may give misleading results concerning the influence of the CBD on housing prices.

In the field of ecology, Legendre (1993) makes a useful distinction between “true” and “false” gradients. For a

trend (true gradient), values observed at specific locations can be modelled using a polynomial function of

their geographic coordinates (trend surface), plus an error term that is specific to each location. Error terms

around the trend surface are not spatially autocorrelated. A true gradient violates the stationarity assumptions

of most spatial-analysis methods and should be removed from the data. Otherwise, it should be explicitly

integrated in the model. Such trends are the results of exogenous processes, and they invalidate the

assumption of independence among the observations. In a “false” gradient, the trend-like structure is caused

by spatial autocorrelation. There is no change in expected value throughout the surface, although the value

observed at each locality is partly determined by neighbouring values: ( )∑ ++= ijjiij zfbz ε''0 . In this

model, the second term represents the sum of the effects of the points located within some distance from the

observation point. Here, autocorrelation is an outcome of endogenous processes (E.g. diffusion, competition,

association) while the phenomenon remains stationary. This raises the central issue of distinguishing true

gradients from false ones, a distinction which partly depends on the overall extension of the study area and on

the significance scale of the phenomenon variation over space.

For hedonic modelling purpose, we must distinguish three types of gradients: city-wide gradients related to

structural factors (E.g. urban form, socio-economic status of neighbourhoods, location rent), local gradients

linked to externalities (E.g. noise caused by a motorway), and local gradients reflecting local market internal

dynamics (E.g. diffusion of home renovation among neighbours). The first two types are true gradients

because they are mostly related to exogenous effects (not entirely driven by the housing market), the last one

is a “false” gradient related to an endogenous emulation effect. If an independent variable is spatially

structured and is assumed to be driving price formation, then, a true gradient, where present in both the

5

controlling and the dependent variables, is seen as a trend and should be explicitly handled (or removed using

trend surfaces and/or autoregressive techniques). If, however, the investigator has good reasons to believe that

the spatial structure (yielding autocorrelation on the independent variable) is the result of local market internal

dynamics (endogenous processes), then the situation should be assimilated to a “false” gradient which should

not be removed before modelling because this effect, being caused by the market itself, will not generate

additional spatial autocorrelation among the residuals of the hedonic model. These differences between types

of gradients are important when tests of statistical hypotheses are involved, and not important when the

purpose is only to describe spatial structures.

Socio-demographic profiles of the population are certainly major determinants of housing market

differentiation and form city-wide trends (exogenous effects). However, the distribution of population is also

constrained by the housing market (endogenous effect). Those effects are intermingled as demand and supply

are driven by the availability and affordability of housing. Therefore, their interactions with housing attributes

should be modelled explicitly and translated into space-varying coefficients using Casetti’s expansion method

to take into account the interactions of the housing market with urban socio-economic and accessibility

gradients.

How should we conceptualise these interactions? Let us first note that macro-economic analyses of the

housing sector pay great attention to household formation and household income as determinant of demand.

The rate of household formation is probably, with household size, the main quantitative component of housing

demand, determining as it does the number of units demanded, while household income certainly constitutes a

major qualitative component of housing demand (Miron, 1987). As put by Muth and Goodman (1988), both

determinants are closely related since household formation is as dependent upon economic conditions as it is

upon mere demographic factors. It so happens that household status and socio-economic status form basic

dimensions of residential differentiation in urban space, as identified through factorial ecologies (Le Bourdais

and Beaudry, 1988; Wyly, 1999). The others dimensions refer to ethnicity, a component that can be discarded

in the Quebec urban region because the population is highly homogenous in terms of language, origin and

religion. In general, for very similar houses within specific segments of the housing market, we would expect

6

a positive impact on demand, and hence, on prices, in neighbourhoods showing large households and/or high

rates of household formation. Similarly, with respect to socio-economic status, we should expect increasing

demand at the upper end of the market segment.

Furthermore, we should also expect the presence of local endogenous effects among housing attributes of

neighbouring homes. As pointed out by Can (1990, p. 256), most people prefer to live in neighbourhoods

where they think the return for their housing investment will be highest. For the same reason, people are

willing to invest in maintaining dwellings where the return on such expenditures are sufficiently high. This

suggests that home owners are observing their immediate neighbours and will be more prone to improve their

property if the neighbourhood itself is upgrading. Conversely, home buyers generally try to find homes in

neighbourhoods having socio-economic status similar to their own, implying that similar people, with the

same needs, tend to agglomerate at specific locations. Hence the intrinsic nature of spatial dependence which

governs real estate markets, each social group tending to value specific sets of housing attributes. Conversely,

a specific attribute may be valued differently by various segments of the population. If similar people are

concentrating at certain locations, thereby increasing residential segregation between social groups, then this

could seriously distort the valuation of specific amenities, hence the need to explicitly measure the conformity

of a house with its neighbours when put on the market. Obviously, a cosy house surrounded by poorly-

maintained properties will loose a large part of its value, eventually more if it belongs to the higher segment of

the local market.

Previous work

The hedonic approach, on which is based this investigation, is applied to the residential market of the Quebec

City region. As is known, this approach aims at explaining property prices on the basis of their physical and

neighbourhood-related characteristics. Its purpose is to evaluate the respective contribution of each attribute of

the residential bundle to market value (Can, 1990 and 1993; Dubin, 1998), using multiple regression analysis.

From a conceptual point of view, land – and property - prices are a combination of externality effects and

location rents (Krantz et al., 1982; Hickman et al., 1984; Shefer, 1986; Yinger et al., 1987; Strange, 1992;

7

Can, 1993; Dubin, 1998). Hoch and Waddell (1993) point out that the overlapping of access and

neighbourhood characteristics leads to highly complex influences on rent levels and values. As shown by Des

Rosiers et al. (1996 and 2001), specific transformation may be applied to the hedonic equation to account for

the non-monotonicity of some of the distance functions.

While hedonic models have long proved their usefulness as an analytical device, previous research has shown

that substantial portions of price variability remain unexplained (Anselin and Can, 1986; Dubin and Sung,

1987; Can, 1993; Dubin, 1998). Moreover, the appropriate neighbourhood factors needed to improve hedonic

models will vary among locations and market segments (Adair et al., 1996), making it difficult to integrate all

significant factors. These may include socio-demographic, household mobility and local housing stock

structural attributes, as well as macroeconomic determinants operating at a micro level (Adair et al., 1998).

Finally, multicollinearity of model attributes, as well as structural heteroskedasticity and spatial

autocorrelation among residuals are detrimental to the stability of regression coefficients, and even more to

the accuracy of their standard errors (Dubin, 1988; Anselin and Rey, 1991; Can and Megbolugbe, 1997; Basu

and Thibodeau, 1998; Pace et al., 1998; Des Rosiers and Thériault, 1999). These are issues deserving

substantial research efforts for improving robustness of hedonic models.

Previous research (Des Rosiers et al., 2000) demonstrates how principal component analysis can be

introduced into hedonic models of housing markets. Highly significant additive models of house prices were

built, while simultaneously controlling for multicollinearity and heteroskedasticity. The usefulness of

principal components to summarize geographical factors was illustrated using two sets of neighbourhood

attributes and factoring them using Hotelling’s (1933) method.

Take in Table I

The first set is formed of fifteen accessibility variables at various points of the road network computed

through simulation of travelling times in a transportation GIS (Thériault et al., 1998 and 1999). This set of

variables is decomposed into a regional component and a local component (Table I). The regional component

refers to locations with poor (positive loadings and factor scores) and good (negative loadings and factor

8

scores) access to regional-level services (universities and colleges, regional shopping centres, CBD). The local

component refers to locations with poor and good access to community-level services (primary schools, high

schools, neighbourhood shopping centres). These externalities have been previously identified for their

specific contribution to house values (Colwell, 1990, 1998; Dubin, 1992, 1998; Guntermann and Colwell,

1983; Shefer, 1986; Sirpal, 1994; Des Rosiers et al., 1996, 1999, 2001).

Another principal component analysis was performed on thirty socio-economic characteristics of

neighbourhoods taken from the 1991 population census, and known to be associated with spatial variations in

house values (Jackson, 1979; Anselin and Can, 1986; Can, 1990, 1992; Can and Megbolugbe, 1997; Dubin

and Sung, 1987; Dubin, 1992, 1998; Rodriguez et al., 1995; Des Rosiers and Thériault, 1992, 1999; Strange,

1992) or identifying socio-demographic and neighbourhood features having a potential qualitative or

quantitative effect on housing demand (E.g., children in families, age groups of population, single-person

households, age of houses in the neighbourhood, population density). Four components are found to extract

75% of the total variance and identify complementary dimensions of socio-economic urban structure (Table

II).

Take in Table II

The first component measures household structure, especially household size, larger households tending to be

family households and more frequently homeowners (positive loadings and factor scores), while the smaller

ones being more frequently non-family households and tenants (negative loadings). This component shows a

centre-periphery spatial gradient. We expect the scores on this component to be positively related to house

prices (families needing larger homes). The second component is more specific, opposing young families with

children living in newer suburbs and in gentrified neighbourhoods (positive loadings) to empty-nesters and

retirees living in older suburbs (negative loadings). With the aging of the population that pushes demand up in

older suburbs, scores on this component are negatively related to house values, especially since the first

component has already been hypothesised to account for the effect of larger households on house prices. The

third component clearly expresses the spatial structure of socio-economic status, with the upper town and its

9

westward extension showing high positive scores. As found by Can (1990) in Columbus (OH), we expect here

a strong positive influence of this component on prices. The fourth factor is much more specific. It opposes

young adults (positive loadings) still living with their parents in low density suburbs to retirees living in the

old city core (negative loadings). Similarly, because of a well-off aging population, a negative relationship

between this component and house values is expected.

With these two principal component analyses, Des Rosiers et al. (2000) were able to replace, in the hedonic

model, 45 highly interrelated variables by six independent (or nearly independent) factors obtained after

Varimax rotation with Kaiser’s normalisation is applied to the initial set of components in order to optimise

their contrasts. The principal components are mutually orthogonal within each set, a feature which provides an

efficient mean of controlling multicollinearity problems while including highly correlated neighbourhood

effects. Indeed, accessibility and socio-economic features of neighbourhoods are well known for their role in

the household location choice process (Landau et al., 1981; Shefer, 1986) through the complex interplay of

local positive and negative externalities (Colwell, 1990; Grieson, 1989; Hickman, 1984; Krantz et al., 1982;

Yinger et al., 1987). However, adding all these neighbourhood effects in hedonic models often raises

heteroskedasticity problems and hides the mechanisms by which they influence house values. Is it because

people are selecting their home location based on their neighbours’ socio-economic status (social attraction

process) or because the property attributes they are searching for appear only at some specific places, that is in

neighbourhoods offering the appropriate socio-economic status (enabling factor)? Is it because some major

externalities are concentrated in space (e.g. quality of view, noise related to motorways, power lines) (site

effect) or because amenities are not evenly distributed as a consequence of transportation-induced

disturbances (communication effect)? Obviously, simply adding geographical factors in hedonic models

cannot distinguish among all these effects.

This paper aims at illustrating the efficiency of interaction indexes to help answer these questions. Previous

work by Pavlov (2000) shows that space-varying regression coefficients can reveal geographical components

behind housing markets and detect the spatial dependence affecting value coefficients. Is the value of a square

meter of living area valid for the entire metropolitan region or is it changing over space? What could explain

10

these variations? It is the kind of issue Pavlov (2000) addresses using a semi-parametric approach. The same

issue is dealt with here, although using an approach based on OLS. Spatial interactions are formed using

Casetti’s expansion method (spatial trends or true gradients) and consideration is given to local endogenous

market features (false gradients) using distance-weighted averages of neighbouring property attributes.

Hedonic Modelling Procedure

Our case study develops hedonic models using as units of observation a sample of 4040 bungalows (one-story

single family house) sold within the Quebec Urban Community (QUC) from January 1990 to December 1991

(Figure 1). Each property is described using a large set of property-specific attributes (Table III) and

neighbourhood-related attributes. Among these neighbourhood characteristics, accessibility components are

linked with the home through selecting the nearest street corner and socio-economic components are

attributed using a point-in-polygon algorithm. Other externalities are measured using buffers around, or

distance statistics from, features that are known to produce externalities (E.g. motorways, power lines). Table

III shows the attributes found to have a significant relationship (Models A to E below) with the sale price of

bungalows sold in the QUC in 1990-91. It is a mix of property specifics, externality indicators and principal

components of neighbourhood-related features.

Take in Figure 1

Take in Table III

In order to provide a way to counter-validate results, the hedonic models are built using 90% of transactions,

selected at random with a proportional stratification by municipality. This provides 3633 cases for model

building, the 407 remaining observations being kept apart for an independent model effectiveness assessment.

Table III shows averages for each set of transactions and clearly illustrates the similitude of both model and

control samples on almost every attribute.

Although quite useful for explanatory purposes, integrating large sets of variables into a single regression

model may prove problematic. Multicollinearity, temporal or spatial autocorrelation and heteroskedasticity

11

isues should find solutions in order to allow reliable hypothesis testing (Anselin and Can, 1986; Anselin and

Rey, 1991; Goodman and Thibodeau, 1995). In particular, sorting out accessibility and neighbourhood

attributes can prove quite tricky considering the cross-influences between these two sets of factors. The

general procedure to be followed involves a succession of checks (Figure 2) designed to achieve optimal

model performances, subject to coefficient stability in the regression model. The order of checks is relevant.

While heteroskedasticity can exist without autocorrelation, temporal or spatial autocorrelation generates

heteroskedasticity. Therefore, two strategies could be used to avoid heteroskedasticity. The first one is to

define housing markets controlling simultaneously for home types, macro economic trends, homogeneity of

geographical areas and similar price segments (E.g. lower/middle/higher segments). This leads to very local

market segmentation valid for a specific period of time (economic cycle). Such models are useful for

assessment purpose and are using only property specifics. However, this approach cannot be used to study

urban dynamics at the city-wide level. Hence the need for the second strategy that specifically handles

variation of price effects over time and space, leading to autocorrelation problems.

Take in Figure 2

In the worked example, there is no temporal trend in the data and our attention will be focussed on

reducing spatial autocorrelation among model residuals. According to Anselin (1990b, p. 186),

“Significance tests and measures of fit that ignore spatial autocorrelation may be misleading. Also,

the presence of spatial error autocorrelation can make the indication tests for error heteroskedasticity

highly unreliable.” In fact, ecological fallacy effects often arise strictly from the presence of spatial

dependency (Griffith, 1992). Therefore, causality could falsely be attributed to the environmental

variables, when in fact the correlation merely reflects a common spatial structure present in both the

dependent and the independent data sets as a result of either inappropriate spatial aggregation or

some exogenous common underlying factor. Spatial autocorrelation is based on positional

information of geo-referenced data which is not captured by classical statistics, including OLS.

However, in most geographical study, including urban economics, location conveys non-trivial information

12

needed to understand the mobility of persons, land use patterns and socio-economic structures. According to

Griffith (1992, p. 273), spatial autocorrelation “[…] may be defined as a measure of true but masked

information content in geo-referenced data.” A complementary view point sees autocorrelation as an

artifact of specification error in spatial modelling. “[…] If a single variable is missing from a

regression equation, the spatial distribution of this variable constitutes a communality across

regression residuals, causing them to appear to be spatially autocorrelated. This problem is

exacerbated by multiple missing variables.” This is clearly the case for housing market modelling,

mainly because it is very difficult to quantitatively specify every factor influencing price formation.

Including all relevant attributes implies measuring such features as the quality of view, the effect of

vegetation, the attractiveness of historical monuments, the perception of crime rates by the

population, traffic noise, etc.

It is possible to work around this problem by adding autoregressive terms in the hedonic function to model

functional interdependence between the transaction price of a given house at time t and the prior sale prices

within its immediate neighbourhood (Ord and Getis, 1995). This approach was used by Can and Megbolugbe

(1997) in their hedonic model of 944 house transactions in Miami. They used various time and space lags to

build distance-weighted lagged price variables, combining these through the expansion method to assess the

spatial drift. Another way, used here, is to search for spatial dependence among housing attributes by trying to

measure the extent to which the departure of a given attribute from the surrounding values (each attribute of

each house being compared to its neighbours) can influence the spatial variation of its implicit price. Are the

“false” gradients on some attributes (neighbourhood specifics) influencing house values?

Take in Table IV

To answer this question, it is first useful to assess spatial autocorrelation (Odland, 1988). Table IV presents

autocorrelation measured among the 4040 bungalows. Their computation is based on the Moran’s I (Moran,

13

1950) formula:

( )( )

( )∑

∑∑

∑∑=

=

≠

=

=

≠

=

−

−−= n

ii

j

n

ii

ijn

j ij

n

i

ijn

j ijzz

zzzzd

d

nI

1

2

1

,

12

1

,

12

1

1.

Where, zi and zj are values at locations i and j; z is the average of all n values; 2ijd is the squared Euclidean

distance between location i and j. The use of squared-inverse distance is based on the assumption that mutual

influence among perceptions follows a gravity-like process (Niedercorn and Ammari, 1987). These

autocorrelation coefficients were computed using, for each home, the 15 nearest neighbours transacted in a

circular radius of less than two kilometres. This yielded 60340 pairs of neighbours (some have less than 15

neighbours within the distance threshold). Cases (8) having less than five neighbours in the two kilometre

radius were discarded from the analysis. Here, due to the high density of transactions in urban areas, we can

use the 15 nearest transactions as a “proxy” to compute the average of the property’s neighbourhood on each

housing attribute. Results indicate that most of the housing attributes are significantly autocorrelated. Only

three very scarce attributes did not show autocorrelation of their values (presence/absence) because less than

7% of the properties had these attributes: InfCeilQual (0.8%), CentVacuum (6%) and Balcony (2.9%). As

expected, sale prices are highly autocorrelated. The same holds true for accessibility and socio-economic

factors. We can hypothesise that part of this autocorrelation is reflected in the spatial dependence among sale

prices, and that explicitly integrating it into the hedonic model should help reducing spatial autocorrelation

among residuals.

For each attribute showing significant autocorrelation two new vectors of values were computed, one

conveying the weighted average of the 15 nearest neighbours (expressed as NhbdiAttribute in the tables),

while the other is the specific difference between the value of each house and the neighbourhood average

(PdifiAttribute). This formulation is similar to that used for computing Moran’s I:

ki

d

dAttribute

AttributeNhbd

k ik

k ik

k

i ≠=

∑

∑

=

= ;1

15

12

15

12 and AttributeNhbdAttributeAttributePdif iii −= .

Take in Table V

14

The hedonic analytical procedure rests upon five steps (models A to E, Table V), gradually improving spatial

attributes specification:

1. Build an hedonic model using only property specifics and geographical components in order to

identify attributes significantly contributing to the price (Model A).

2. Try to replace each property specific attribute by a combination of the 15 nearest neighbours

weighted average and specific departure from the neighbourhood trend; retain only those showing

significant relationships (t test) on both indicators, otherwise keep the original variable because there

is no measurable local endogenous gradient (Model B).

3. Compute interactions between each principal component and each initially defined property attribute

and externality index using the Casetti’s expansion method. In order to avoid multicollinearity

problems, both indicators should be centred before making the product:

))(( eFactorScoreFactorScorAttributeAttribute ii −− . Repeat step 2, using both interactions

and initial variable in a combined stepwise regression after inclusion of property specifics identified

at step 1 (Model C). Eventually, remove variables that fall below significance threshold.

4. Integrate specific attributes, neighbourhood trends/specific departures from the local mean and

spatial interactions, choosing, for each attribute the combination yielding higher t tests for all

coefficients. It is preferable to include or discard simultaneously both NhbdiAttribute and

PdifiAttribute. Indeed, if one of the two fails the significance test, it is better to keep only the

original attribute since it suggests that the local (false) gradient is not significant. Notwithstanding, a

significant interaction of this attribute with a geographical component may indicate the presence of a

spatial trend (true gradient). This integrated model is called Model D.

5. From our previous research, one specific variable, the " relative tax differential" was found to have a

highly significant influence on sale prices. When people try to find housing, they generally visit

many properties on the market and make their own estimation of their value. They also consider the

tax burden and the appraisal value. In Model D, the local tax rate is already included. It assesses the

effect of taxation at the regional level (13 municipalities) using global rates at time of transaction.

However, a property may be over/under-assessed by the municipality, thereby giving rise to

15

opportunities/externalities that could influence the offer price. If the property is an opportunity for

the buyer (lower taxes than expected from the price he is going to pay), or in contrast “over-taxed”,

the price will probably be adjusted somewhat (Timmermans and Golledge, 1990). The problem is

that the only indication we have to compute this differential between buyer’s and municipalities’

assessments is the sale price. Obviously, it should not be used to compute any independent variable

because this will result in heteroskedasticity. Therefore, a two-stage process is resorted to here, with

estimates from model D being used as a “proxy” for the estimated value. A relative tax differential

ratio is then computed and reintroduced in the equation at a second stage (using spatial expansion) to

build Model E (Annex 1):

{ } iiiiii EstModelDlueTaxationVaLTaxRateEstModelDLTaxRatelTaxDif )01.0()01.0(100Re −=

Summary of Results

Table V shows summary results from the five models. At each step, the adjusted R-square is increasing, from

0.768 to 0.822. Standard errors of estimates are decreasing, from 0.118 to 0.104, yielding fairly good

estimates of housing prices. Thus, adding 23 spatial variables has a rather marginal effect on explanatory

power. Standard deviations of residuals are falling very slowly from 0.1177 to 0.1027 for the model sample,

which is consistent with the trend for the control sample (0.1247 to 0.1055). F ratios decrease until model D

(366 to 247) but show a net improvement for model E (324), which has 56 significant independent variables.

All model estimates show spatial autocorrelation higher than that of the dependent variable (natural logarithm

of the sale price). However, Moran’s I for estimates of Model E (0.549) seems closer to that of the actual sale

prices and shows the same distance range (1125 metres).

The most important improvement concerns the reduction of spatial autocorrelation among models’ residuals.

It is dropping by nearly one half from Model A (0.16) to Model E (0.08). This is a clear indication that well

specified interaction variables can help reducing the spatial autocorrelation among residuals, thereby

improving in the long term, the reliability of OLS hypothesis testing. More importantly, despite the inclusion

of “false” gradients which did not lower the spatial range of autocorrelation among residuals (see correlogram

of Model B in Figure 3; Table V, section b), the inclusion of spatial interaction terms and of the relative tax

16

differential did a fairly good job, reducing the range of significant autocorrelation to about 375 metres (Table

V, sections c and e; Figure 3).

Take in Figure 3

A crude test for heteroskedasticity was carried out (Table V). It is using the Goldfeld and Quandt (1965)

approach dividing the total sample of transactions in three parts based on sale price: the lower end (40% of

transactions), the mid price (20%) and the higher segment of the market (40%). Table V shows sums of

squares of each model’s residuals for lower and higher market segment. Tests (F) indicate that Models “A”

and “E” seem to display a homoskedastic distribution of errors. However, Models "B", "C" and "D" have a

heteroskedastic distribution of error relative to values of the dependent variable. More sophisticated

approaches suggested by Anselin (1990a) will be needed to account for relationships of spatial variation,

Casetti's expansion and normality of residuals. However, this is far beyond the scope of this paper.

Take in Table VI

Table VI (Model A) shows a standard hedonic model based on the assumption that every coefficient is stable

in space. Multicollinearity (VIF) is low (lower than 5) reaching a maximum which is showing unsurprising

relationships between accessibility and socio-economic factors. Each coefficient expresses the relative

marginal contribution of the pertaining and all values are consistent with expectations.

Take in Table VII

Table VII (Model D) shows the results of step 4. Spatial factors and externalities are simply added to Model A

in order to test the existence of measurable spatial effects in the market under analysis. All the six components

do contribute significantly to explain sale prices and stay in the model as interactions and local endogenous

gradients are included. Slowly decreasing significance of their coefficients (Table VI versus Table VII)

suggest that a small part of their effect is transferred on the other spatial trend indicators. Twenty-three spatial

interaction and neighbourhood effects entered the model, all of these emerging as statistically significant. Lot

size, local tax rate, sheds and apparent age are showing multidimensional spatial effects. Apparent age of the

17

property (age adjusted for improvements) has both interaction and local gradient components. The interaction

component expresses a structural effect linked to the historic development of the city: in mature suburbs, non-

renovated bungalows are depreciating faster than elsewhere. Conversely, the local gradient is showing a

premium for renovating homes in older central neighbourhoods. Here, it is important to note that the

functional form of the relationship (Sc means spatially adjusted coefficient):

[ ] ( ) ( ) ( )( )( )01373.0248170.2ln01848.0ln12083.0ln11291.0ln +−+−+−= iiii CSFAppAgeAppAgePdifAppAgeNhbdi eAppAgeSc could also be

written [ ] ( ) ( ) ( )( )( )01373.0248170.2ln01848.0ln00079.0ln12083.0ln +−++−= iiii CSFAppAgeAppAgeNhbdAppAgei eAppAgeSc .

This second form is more convenient and will be used to specify equations of Model E (Annex 1). It is also

noteworthy that the most important determinants of sale price show spatial trends (AppAge, LotSize, LivArea,

LtaxRate).

Take in Table VIII

Table VIII (Model E) is very similar to the previous, except for the inclusion of RelTaxDif and its spatial

trend. Property specific attributes were retained to give a clearer view of the sub-models. Annex 1 presents the

mathematical specifications of the model with all its coefficients and constants. There is an increasing

collinearity in the AppAge spatial relationship but it is still acceptable (less than 5). Local tax effects become

highly multidimensional, suggesting strong spatial variations in the way taxation influences housing prices.

Analysing this complex relationship requires more than the mapping of coefficients and will be the subject of

a forthcoming paper. It is also important to point out that most of the internal attributes of the property did not

define spatial gradients. There are three exceptions (inferior ceiling quality, kitchen cabinets made of hard

wood and number of washrooms) which can be used to distinguish cheaper and more expensive properties.

This improved form of the model clearly identifies those attributes that are space-independent (i.e. they do not

generate interaction) and those needing space-dependent adjustments. More importantly, this procedure

specifically identifies which socio-economic and/or accessibility factors are mostly responsible for (or related

to) the spatial variation of the relative value. Finally, since the interactions involve geographical factors that

18

can be mapped, it is possible to compute local values of space-varying coefficients using equations of Annex

1 to build maps of their distributions.

Discussion

Take in Figure 4

Figure 4 shows an interpolated view of the location rent effect for a bungalow situated on a larger than

average lot (1000 sq. m. versus a geometric mean of 622 sq. m. for all sold properties in 1990-91). Building

this map implies calculating the following expression for every house in the market, replacing its lot size by

1000 sq. m. (fixed parameter) in order to obtain a location-dependent size-related spatially adjusted

coefficient: [ ] ( )( )( ) ( )( )( )09166.0343297.61000ln02342.001373.0243297.61000ln03047.0)1000ln05931.0(4 −−++−−+= ii CSFCSFi eMapSc . Then

local values are interpolated within a short radius (say 1000 meters) around the property, using IWD (inverse

weighted distance) with exponent 2 in order to estimate coefficients at every location. These maps were easily

realised using MapInfo GIS software. It takes about ten minutes to build such a map, allowing for easy

exploration of spatial dependence on many housing attributes. As the above expression suggests, the relative

value of land is related to socio-economic status (CSF3, positive influence) and family cycle A (CSF2,

negative scores), favouring mature suburbs inhabited by empty-nesters. The absolute effect is very strong,

from 44% to 60% of house value. The spatial pattern is closely associated with the major axis of the historic

development of the city, defining two sectors originating from Old Quebec (in white on the map because there

is no bungalow in that part of the city).

Take in Figures 5 and 6

Figures 5 and 6 show depreciation-related spatial trends for a 10 year-old and a 40 year-old house,

respectively, computed using the following functional forms:

[ ] ( ) ( ) ( )( )( )01373.0248170.210ln01855.0ln01234.010ln12424.05 +−++−= ii CSFAppAgeNhbdi eMapSc and

[ ] ( ) ( ) ( )( )( )01373.0248170.240ln01855.0ln01234.040ln12424.06 +−++−= ii CSFAppAgeNhbdi eMapSc . The absolute ranges of spatial

variability are smaller for new houses (79-75% versus 68-62%). These functions integrate two effects: a local

19

gradient that compare the property to its immediate neighbourhood (less depreciated if younger than average)

and a regional trend that favours older neighbourhoods. This is consistent with Can's (1997, p. 213) findings

that an older house in a transitional neighbourhood would be less valuable than an older house in a

neighbourhood that is undergoing gentrification. The effect of aging cannot be isolated from household

maintenance/repair decisions, which are not solely determined by households income levels but also by

households’ perception concerning the future value of their residential asset, in turn primarily determined by

the “signals” they receive from their immediate neighbourhood. Maps 5 and 6 clearly support this statement:

life cycle is more related to depreciation than socio-economic status and there is more incentive for

maintenance (higher depreciation rate) in older than in newer suburbs. This could be a spatial expression of

the growing market of housing renovation carried out by aging baby boomers.

Take in Figure 7

Figure 7 shows the marginal effect on house value of adding a second washroom. It is computed using the

ratio of two coefficients (2 versus 1 washroom) : [ ]( ) ( )( )( )

( )( )( )42520.0125406.1102594.005440.0

42520.0125406.1202594.005440.02

7 −−−+

−−−+

=i

i

ACF

ACF

i eeMapSc . The

spatial trend estimator is accessibility to regional-level services. Per se, a washroom adds 5.4% to the house

value. There was an average of 1.25 washroom per house in that market and the average accessibility

component score is 0.42520. Therefore being far away from regional-level services (positive scores) nearly

counterbalances the advantage of having a second washroom, although it is needed near the centre of the

agglomeration. This is probably related to young families with middle income being forced to go to new

suburbs in order to access home ownership. These new homes are generally equipped with only one

washroom. In older neighbourhoods, the second washroom is prevalent and there is a price penalty for houses

having only one.

Take in Figure 8

The last example concerns the marginal effect of having a shed on the lot. It is linked to both socio-economic

status and family-cycle B (young adults still at home versus retirees). Its spatial trend is shown on Figure 8:

20

[ ] ( )( )( ) ( )( )( )25839.044572.0102090.009166.034572.0101686.002050.08 −−+−−−+= ii CSFCSFi eMapSc . In general, a shed adds about 2%

to house value. However, people living in neighbourhoods with higher socio-economic status do not

appreciate this amenity. These areas are mostly inhabited by professionals and retirees. A large proportion of

them outsource maintenance of their lawn and garden. Conversely other people give a higher than average

value to the presence of a shed. These are mostly sectors showing less than average household income, where

the proportion of manual workers is higher than average.

Conclusion

All in all, these worked examples clearly show the benefits of including spatial interactions with urban

dynamics within hedonic models. The purpose is not primarily to improve explanatory power, but rather to

enhance our understanding on the complex linkages between housing prices and the socio-demographic

evolution of North American cities. An important side benefit of this approach is that it clearly helps reduce

spatial autocorrelation among residuals. At the present stage, autocorrelation is not completely removed and

autoregressive methods could be helpful for handling the remaining space-related effects. We can identify at

least four ways to improve the handling of spatial dependence in hedonic models: (1) by considering a wider

range of spatial attributes, especially those related to environment; (2) by defining more flexible ways of

measuring spatial dependence (non linear functions, multivariate interactions, may be some interaction among

property specifics); (3) by considering information on the socio-economic status and perceptions of home

buyers to complement information on socio-economic profiles of their new neighbourhood; (4) by improving

measurement of interactions adjusting trend surfaces over principal components thus dividing their meaning in

two parts: firstly the city-wide trend, the departure from the trend being secondly related to peculiarities of

specific areas.

Moreover, similar models must be designed for different market segments. It is also necessary to compare

temporal trends in the housing demand itself, and in the transformation of spatial interactions linked as they

are to population aging and its long-term replacement. In this research, we were advantaged by an

insignificant temporal trend and by the absence of interaction between time and space. In 1990-91, the

21

bungalow market in the Quebec urban region was experiencing stable, or slowly rising values. Comparison

with models built using data for the period of 1993 to 1996 (depressed market), and, eventually, 1997 to 2000

(progressive recovery) will allow testing for persistence of the spatial trends identified here. Finally,

additional work must be carried out to assess variations in coefficients’ error-terms over space.

Acknowledgement The authors gratefully acknowledge Martin Lee-Gosselin, Corinne Thomas, Josée Bouchard, Pierre Lemieux, Raynald Sirois, Marie-Hélène Vandersmissen and Maxime Pètre for their valuable help at various stages of this research. This project was funded by the Quebec Province’s FCAR program, the Canadian Social Sciences and Humanities Research Council, the Canadian Central Mortgage and Housing Corporation, the Canadian Network of Centres of Excellence in Geomatics (GEOIDE) and the Canadian Natural Sciences and Engineering Research Council. It was realised in close co-operation with the Quebec Urban Community Appraisal Division, the Quebec Ministry of Transport and the Quebec Urban Community Transit Society (STCUQ).

References Adair, A. S., Berry, J. N. and McGreal, S. W. (1998) “Assessing influences upon the housing market in

Northern Ireland”, Journal of Property Research, Vol. 15 No 2, pp. 121-134. Adair, A. S., Berry, J. N. and McGreal, S. W. (1996) “Hedonic modelling, housing submarkets and residential

valuation”, Journal of Property Research, Vol. 13 No 1, pp. 67-83. Anselin, L. (1990a) “Some robust approaches to testing and estimation in spatial econometrics”, Regional

Science and Urban Economics, Vol. 20 pp. 141-163. Anselin, L. (1990b) “Spatial dependence and spatial structural instability in applied regression analysis”,

Journal of Regional Science, Vol. 30 No 2, pp. 185-207. Anselin, L., and Can, A. (1986) “Model comparison and model validation issues in empirical work on urban

density functions”, Geographical Analysis, Vol. 18, pp. 179-197. Anselin, L., and Rey, S. (1991) “Properties of tests for spatial dependence in linear regression models”,

Geographical Analysis, Vol. 23 No 2, pp. 112-131. Basu, S., and Thibodeau, T. G. (1998) “Analysis of spatial autocorrelation in house prices”, The Journal of

Real Estate Finance and Economics, Vol. 17 No 1, pp. 61-86. Bourne, L. S. (1993) “Close together and worlds apart: An analysis of changes in the ecology of income in

Canadian cities”, Urban Studies, Vol. 30, pp. 1293-1317. Can, A. (1990) “The measurement of neighbourhood dynamics in urban house prices”, Economic Geography,

Vol. 66, pp. 254-272. Can, A. (1992) “Residential quality assessment: Alternative approaches using GIS”, The Annals of Regional

Science, Vol. 26, pp. 97-110. Can, A. (1993) “Specification and estimation of hedonic housing price models”, Regional Science and Urban

Economics, Vol. 22, pp. 453-474. Can, A., and Megbolugbe, I. (1997) “Spatial dependence and house price index construction”, Journal of Real

Estate Finance and Economics, Vol. 14, pp. 203-222. Casetti, E. (1972) “Generating models by the expansion method: Applications to geographical research”,

Geographical Analysis, Vol. 4, pp. 81-91.

22

Casetti., E. (1986) “The dual expansion method: An application for evaluating the effects of population growth on development”, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 16, pp. 29-39.

Colwell, P. F., Gujral, S. S. and Coley, C. (1985) “The impact of shopping centers on the value of surrounding properties”, Real Estate Issues, Vol. 10 No 1, pp. 35-39.

Colwell, P. F. (1990) “Power lines and land values”, Journal of Real Estate Research, Vol. 5 No 1, pp. 117-127.

Colwell, P. F. (1998) “A primer on piecewise parabolic multiple regression analysis via estimations of Chicago CBD land prices”, The Journal of Real Estate Finance and Economics, Vol. 17 No 1, pp. 87-98.

Des Rosiers, F., Thériault, M. and Villeneuve, P. Y. (2000) “Sorting out access and neighbourhood factors in hedonic price modelling”, Journal of Property Investment and Finance, Vol. 18 No 3, pp. 291-315.

Des Rosiers, F., and Thériault, M. (1992) “Integrating geographic information systems to hedonic price modelling: An application to the Quebec region”, Property Tax Journal, Vol. 11 No 1, pp. 29-57.

Des Rosiers, F. and Thériault, M. (1999) House Prices and Spatial Dependence: Towards an Integrated Procedure to Model Neighborhood Dynamics. Laval University, Faculty of Business Admin., Québec, Publ. 1999-002.

Des Rosiers, F., Lagana, A., Thériault, M. and Beaudoin, M. (1996) “Shopping centers and house values: An empirical investigation”, Journal of Property Valuation and Investment, Vol. 14 No 4, pp. 41-63.

Des Rosiers, F., Lagana, A. and Thériault, M. (2001) “Size and proximity effect of primary schools on surrounding house values”, Journal of Property Research, Vol. 18 No 2, pp. 1-20.

Dubin, R. A. (1988) “Estimation of regression coefficients in the presence of spatially autocorrelated error terms”, Review of Economics and Statistics, Vol. 70, pp. 466-474.

Dubin, R. A. (1992) “Spatial autocorrelation and neighbourhood quality”, Regional Science and Urban Economics, Vol. 22, pp. 433-452.

Dubin, R. A. (1998) “Predicting house prices using multiple listings data”, The Journal of Real Estate Finance and Economics, Vol. 17 No 1, pp. 35-60.

Dubin, R. A., and Sung, C. H. (1987) “Spatial variation in the price of housing: Rent gradients in non-monocentric cities”, Urban Studies, Vol. 24, pp. 193-204.

Goldfeld, S. M. and Quandt, R. E. (1965) “Some tests for homoskedasticity“, Journal of the American Statistical Association, Vol. 60, pp. 539-547.

Goodman, A. C., and Thibodeau, T. G. (1995) “Age-related heteroskedasticity in hedonic house price equations”, Journal of Housing Research, Vol. 6, No 25-42.

Grieson, R. E., and White, J. R. (1989) “The existence and capitalization of neighborhood externalities: A reassessment”, Journal of Urban Economics, Vol. 25 No 1, pp. 68-76.

Griffith, D. A. (1992) „What is spatial autocorrelation? Reflections on the past 25 years of spatial statistics”, L’Espace géographique, Vol. 21 No 3, pp. 265-280.

Guntermann, K. L., and Colwell, P. F. (1983) “Property values and accessibility to primary schools”, Real Estate Appraiser and Analyst, Vol. 49 No 1, pp. 62-68.

Hickman, E. P., Gaines, J. P. and Ingram, F. J. (1984) “The influence of neighbourhood quality on residential values”, The Real Estate Appraiser and Analyst, Vol. 50 No 2, pp. 36-42.

Hoch, I., and Waddell, P. (1993) “Apartment rents: Another challenge to the monocentric model”, Geographical Analysis, Vol. 25 No 1, pp. 20-34.

Hotelling, H. (1933) “Analysis of a Complex of Statistical Variables into Principal Components”, Journal of Educational Psychology, Vol. 24, pp 417-441 and 498-520.

Jackson, J. R. (1979) “Intraurban variation in the price of housing”, Journal of Urban Economics, Vol. 6, pp. 464-479.

23

Krantz, D. P., Waever, R. D. and Alter, T. R. (1982) “Residential tax capitalization: Consistent estimates using micro-level data”, Land Economics, Vol. 58 No 4, pp. 488-496.

Landau, U., Prashker, J. N. and Hirsh, M. (1981) “The effect of temporal constraints in household travel behaviour”, Environment and Planning A, Vol. 13, pp. 435-444.

Le Bourdais, C. and Beaudry, M. (1988) “The changing residential structure of Montreal 1971-81”, The Canadian Geographer, Vol. 32 No 2, pp. 98-113.

Legendre, P. (1993) “Spatial autocorrelation: trouble or new paradigm?”, Ecology, Vol. 74 No 6, pp. 1659-1673.

Miron, J. R. (2000) Housing in Postwar Canada, McGill-Queen’s University Press, Montreal. Moran, P. A. P. (1950) “Notes on continuous stochastic phenomena”, Biometrika, Vol. 37, pp. 17-23. Muth, R. F. and Goodman, A. C. (1988) The Economics of Housing Markets. Fundamentals of Pure and

Applied Economics, No. 31, Harwood Academic Publishers, 148 p. Niedercorn, J. H., and Ammari, N. S. (1987) “New evidence on the specification and performance of

neoclassical gravity models in the study of urban transportation”, The Annals of Regional Sciences, Vol. 21 No 1, pp. 56-64.

Nijkamp, P., Van Wissen, L. and Rima, A. M. (1993) “A household life cycle model for residential relocation behaviour”, Socio-Economic Planning Science, Vol. 27 No 1, pp. 35-53.

Odland, J. (1988) Spatial Autocorrelation. Scientific Geography, Vol. 9, Sage, Newbury Park. Ord, J. K., and Getis, A. (1995) “Spatial autocorrelation statistics: Distribution issues and an application”,

Geographical Analysis, Vol. 27 No 4, pp. 286-306. Pace, R. K., Barry, R. and Sirmans, C. F. (1998) “Spatial statistics and real estate”, The Journal of Real Estate

Finance and Economics, Vol. 17 No 1, pp. 5-14. Pavlov, A. D. (2000) “Space-varying regression coefficients: A semi-parametric approach applied to real

estate markets”, Real Estate Economics, Vol. 28 No 2, pp. 249-283. Rodriguez, M., Sirmans, C. F. and Marks, A. P. (1995) “Using geographic information systems to improve

real estate analysis”, The Journal of Real Estate, Vol. 10 No 2, pp. 163-173. Rose, D. and Villeneuve, P. Y. (1983) “Work, labour markets and households in transition”, in L. Bourne and

D. Ley (eds.), The Social Geography of Canadian Cities, McGill-Queens Press, Montreal, pp. 153-174. Rose, D. and Villeneuve, P. Y. (1998) “Engendering class in the metropolitan city: occupational pairing and

income disparities among two-earner couples”, Urban Geography, Vol. 19 No 2, pp. 123-159. Shefer, D. (1986) “Utility changes in housing and neighbourhood services for households moving into and out

of distressed neighbourhoods”, Journal of Urban Economics, Vol. 19, pp. 107-124. Sirpal, R. (1994) “Empirical modelling of the relative impacts of various sizes of shopping centres on the

values of surrounding residential properties”, Journal of Real Estate Research, Vol. 9 No 4, pp. 487-505. Strange, W. (1992) “Overlapping neighbourhoods and housing externalities”, Journal of Urban Economics,

Vol. 32, pp. 17-39. Thériault, M., Leroux, D. and Vandersmissen, M. H. (1998) “Modelling travel route and time within GIS: Its

use for planning”, in A. Bargiela and E. Kerckhoffs (eds.) Simulation Technology: Science and Art. Proceedings of the 10th European Simulation Symposium, The Nottingham-Trent University and Society for Computer Simulation International, pp. 402-407.

Thériault, M., Des Rosiers, F. and Vandersmissen, M. H. (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.

Timmermans, H., and Golledge, R. G. (1990) “Applications of behavioural research on spatial problems: preferences and choice”, Progress in Human Geography, Vol. 14, pp. 311-354.

24

Wyly, E. K. (1999) “Continuity and change in the restless urban landscape”, Economic Geography, Vol. 75 No 4, pp. 309-338.

Yinger, J. H., Bloom, H. S. Borsch-Supan, A. and Ladd, H. F. (1987) “Property Taxes and House Values: The Theory and Estimation of Intra-jurisdictional Property Tax Capitalization”, Studies in Urban Economics, Academic Press, Princeton Univ.

Travel time computed using transportation GIS to evaluate accessibility at 17,871 local street corners in Quebec Urban Community Total variance explained

Extraction sums of squared loadings Principal component analysis

Rotation sums of squared loadings Varimax with Kaiser normalisation

Components Total % of variance

Cumulative %

Total % of variance

Cumulative %

ACF1 9.683 64.556 64.556 6.313 42.086 42.086 ACF2 1.668 11.122 75.678 5.039 33.592 75.678

Rotated component matrix Components

Travel times simulated with TransCAD using the road network with impedance and turn penalties (minutes)

ACF1 ACF2

Travel time to nearest highway entrance by car 0.647 0.352 Travel time to nearest regional shopping centre by car 0.758 0.555 Travel time to nearest local shopping centre by car 0.557 0.727 Travel time to nearest neighbourhood shopping centre by car 0.546 0.671 Travel time to nearest high school by car 0.351 0.797 Travel time to nearest college or university by car 0.911 Travel time to Laval University by car 0.916 Travel time to downtown Quebec City by car 0.610 0.534 Travel time to downtown Sainte-Foy by car 0.893 Travel time to “La Capitale” shopping centre by car 0.320 0.701 Walking time to nearest neighbourhood shopping centre 0.407 0.776 Walking time to nearest primary school 0.724 Walking time to nearest high school 0.847 Walking time to nearest college or university 0.801 Walking time to Laval University 0.875 0.361 Accessibility to

regional services Accessibility to neighbourhood

services Interpretation of positive values

Far away from regional-level

services

Far away from local-level services

Interpretation of negative values

Close to regional-level

services

Close to local-level services

Adapted from Des Rosiers, F., Thériault M. and Villeneuve P. (2000) Sorting out access and neighbourhood factors in hedonic price modelling. Journal of Property Investment and Finance, vol. 18, no 3, p. 291-315.

Table I. Principal component analysis of distance to services

25

Principal component analysis on 1991 census attributes of 416 neighbourhoods in Quebec Metropolitan Area Total variance explained

Extraction sums of squared loadings Principal component analysis

Rotation sums of squared loadings Varimax with Kaiser normalisation

Components Total % of variance

Cumulative % Total % of variance

Cumulative %

CSF1 11.798 39.326 39.326 10.931 36.438 36.438 CSF2 4.724 15.748 55.074 4.789 15.964 52.402 CSF3 4.076 13.588 68.661 4.774 15.914 68.317 CSF4 1.791 5.971 74.632 1.895 6.315 74.632

Rotated component matrix Components

1991 Census Attributes CSF1 CSF2 CSF3 CSF4 % of persons 0-14 years -0.811 0.505 % of persons 15-24 years 0.808 % of persons 25-44 years 0.933 % of persons 45-64 years -0.883 % of persons 65+ years 0.648 -0.548 -0.370 % of women 0.545 -0.339 Persons per household -0.959 % of non-family households 0.945 % of single person households 0.933 Children per family -.0574 % of lone-parent families 0.759 -0.304 % of families with children -0.820 % of families with children 0-6 years 0.850 % of families with children 6-14 years -0.585 0.557 % of detached dwellings -0.932 % of dwellings in large buildings 0.473 Persons per room 0.425 -0.668 % of dwellings built before 1946 0.525 -0.494 % of dwellings built 1946-1960 0.309 -0.504 % of dwellings built 1961-1970 -0.575 0.636 % of tenants 0.910 % of households with housing costs >30% of income 0.635 -0.306 % of adults with highschool degree 0.903 % of adults with university degree 0.944 % of men with college degree 0.935 % of women with college degree 0.919 Average household income ($) -0.699 0.629 % households mowing during last 5 years 0.605 0.608 Population density (persons/hectare) 0.760 Dwelling density (density/hectare) 0.804 Centrality Family cycle

A Socio-economic

status Family cycle

B Interpretation of positive values

Small and non-family households tenants in the city

centre

Young families with children living in new

suburbs

Well educated persons with high

income in the upper town

Young adults living with their parents in low

density suburbs Interpretation of negative values

Family households homeowners in

suburbs

Empty-nesters and retirees living in

older suburbs

Low educated poor persons living mainly

in the lower town

Retirees living in the old city core

Table II. Principal component analysis of socio-economic census attributes

26

3633 bungalows used in the model Control

sample N=407

Variable Definition Type Minimum Maximum Average Std. Dev AverageSalePrice Sale price of the property ($) N 50500 336000 87779.23 25911.57 88382.69LnSalePrice Natural logarithm of sale price ($) N 10.82973 12.72487 11.34922 0.245897 11.35537LnAppAge Natural logarithm of apparent age (years) N -.69315 3.91202 2.48170 .89609 2.53634LnLotSize Natural logarithm of the lot size (sq. m.) N 5.27300 9.41638 6.43298 .32269 6.45284LivArea Living area of the property (sq. m.) N 16.36000 264.26000 96.28459 17.61973 96.56872WaterSewer Property is (1) / is not (0) linked to the municipal

aqueduct and sewer networks B 0 1 .99009 .09906 .98526

InfFound Property has(1) / does not have (0) inferior quality foundations

B 0 1 .03799 .19119 .02703

BaseFinh Basement is (1) / is not (0) finished B 0 1 .56813 .49541 .56757CathCeil Property has (1) / does not have (0) cathedral

ceiling B 0 1 .17066 .37626 .18182

InfCeilQual Property has (1) / does not have (0) inferior quality ceiling

B 0 1 .00853 .09199 .01229

Skylight Property has (1) / does not have (0) skylights B 0 1 .01239 .11062 .01229Quality House quality index (number of attributes

– below / + above average) N -2 2 -.03413 .21999 -.01966

Washrooms Total number of washrooms (bath = 1; toilet = 0.5)

N .5 5.5 1.25406 .32957 1.26536

Fireplace Number of fireplaces N 0 4 .18827 .41821 .18182HardwStair Presence (1) / absence (0) of an indoor staircase

made of hard wood B 0 1 .07707 .26674 .08845

Oven Presence (1) / absence (0) of a built-in oven and/or cooking top

B 0 1 .10570 .30749 .12285

KitchCab Presence (1) / absence (0) of kitchen cabinets made of hard wood

B 0 1 .11781 .32243 .12531

Dishwasher Presence (1) / absence (0) of a permanent dishwasher

B 0 1 .55216 .49734 .62162

CentVacuum Presence (1) / absence (0) of a central vacuum system

B 0 1 .05863 .23496 .07125

Veranda Presence (1) / absence (0) of a veranda B 0 1 .47702 .49954 .48403Balcony Presence (1) / absence (0) of a balcony B 0 1 .02945 .16909 .02703ExcavPool Presence (1) / absence (0) of an excavated pool B 0 1 .03881 .19317 .04668AttGarage Presence (1) / absence (0) of an attached garage B 0 1 .02753 .16363 .01966DetGarage Presence (1) / absence (0) of a detached garage B 0 1 .10515 .30679 .12776Shed80cf Presence (1) / absence (0) of a shed with more

than 80 cubic feet B 0 1 .45720 .49823 .46437

LTaxRate Local tax rate ($/$100 of assessed value) N 1.25310 3.08350 2.38474 .42820 2.37913TaxationValue Property value for taxation purpose ($) N 38000 307000 81946.42 23015.95 84044.23RelTaxDif Relative tax differential; adjusted tax using

estimated price – effective tax (% of estprice) N -2.19988 1.27402 .14690 .25444 .11640

NbMonthJ90 Number of month elapsed since January 1990 N 1.000 24.000 12.196 6.735 12.13022Motorway150 Property is (1) / is not (0) located at less than 150

metres from a motorway B 0 1 .01927 .13748 .00983

DistMotorExit Euclidean distance from the nearest motorway (km)

N .034 5.661 1.50665 .92099 1.52957

ACF1 Accessibility comp. #1 – Regional-level services N -1.92565 2.50911 .42520 .91465 .42634ACF2 Accessibility comp. #2 – Local-level services N -1.44784 3.83155 .01101 .78073 .00252CSF1 Census comp. #1 – Centrality N -1.61231 2.55393 -.64900 .52924 -.65465CSF2 Census comp. #2 – Family cycle A N -2.29043 2.66786 -.01373 1.19485 -.06834CSF3 Census comp. #3 – Socio-economic status N -1.99671 2.77619 .09166 .86869 .08234CSF4 Census comp. #4 – Family cycle B N -2.99326 3.29053 .25839 .77453 .25282

Table III. Operational definition of variables and descriptive statistics

27

4040 houses; 60340 valid pairs of neighbouring properties within a distance of 2000 metres.

Variable Distance Range of Significant

Autocorrelation (metres)

Moran’s I 0-2000 m

Probability (Ho I = 0)

Correlogram Shape

SalePrice 0-1125 0.39332 < 0.0001 Increase 0-625; fast decrease 625-1125 LnSalePrice 0-1125 0.44847 < 0.0001 Stable 0-1000; fast decrease 1000-1125 LnAppAge 0-1250 1.07452 < 0.0001 Fast decrease 0-375; decrease 375-1250 LnLotSize 0-1500 0.45483 < 0.0001 Fast decrease 0-250; stable 250-1500 LivArea 0-1125 0.42225 < 0.0001 Decrease 0-1125 WaterSewer 0-750 0.04132 0.0320 Stable 0-750 InfFound 0-750 0.10212 < 0.0001 Decrease 0-750 BaseFinh 0-875 0.19180 < 0.0001 Decrease 0-250; stable 250-875 CathCeil 0-500 0.46591 < 0.0001 Very fast decrease 0-500 InfCeilQual none 0.00928 0.3352 Non significant Skylight 0-500 0.03823 0.0428 Decrease 0-500 Quality 0-500 0.31956 < 0.0001 Very fast decrease 0-500 Washrooms 0-1125 0.13714 < 0.0001 Stable 0-1125 Fireplace 0-750 0.14418 < 0.0001 Decrease 0-750 HardwStair 0-750 0.08716 < 0.0001 Low 0-750 Oven 0-250 0.06379 0.0021 Decrease 0-250 KitchCab 0-500 0.15552 < 0.0001 Fast decrease 0-500 Dishwasher 0-500 0.07691 0.0003 Slow decrease 0-500 CentVacuum none 0.00215 0.4573 Non significant Veranda 0-750 0.10982 < 0.0001 Decrease 0-750 Balcony none 0.00630 0.3846 Non significant ExcavPool 0-750 0.06855 0.0010 Low 0-750 AttGarage 0-1000 0.11599 < 0.0001 Increase 0-750; fast decrease 750-1000 DetGarage 0-750 0.04978 0.0127 Stable 0-500; decrease 500-750 Shed80cf 0-1000 0.15184 < 0.0001 Decrease 0-1000 LTaxRate 0->2000 0.85783 < 0.0001 Stable 0-750; slow decrease 750-2000 TaxationValue 0-1150 0.45611 < 0.0001 Stable 0-1000; fast decrease 1000-1150 RelTaxDif 0-1125 0.33982 < 0.0001 Decrease 0-1125 NbMonthJ90 none 0.03363 0.0651 Non significant Motorway150 0-1000 0.51851 < 0.0001 Fast decrease 0-1000 DistMotorExit 0->2000 0.86620 < 0.0001 Stable 0-2000 ACF1 0->2000 0.89417 < 0.0001 Stable 0-2000 ACF2 0->2000 0.83594 < 0.0001 Stable 0-1850 CSF1 0->2000 0.62426 < 0.0001 Stable 0-1500; increase 1500+ CSF2 0->2000 1.18043 < 0.0001 Regular decrease 0-2000 CSF3 0->2000 0.80025 < 0.0001 Stable 0-1500; decrease 1500+ CSF4 0-1250 0.82272 < 0.0001 Fast decrease 0-1250

Table IV. Spatial autocorrelation of property and geographical attributes

28

a) Standard hedonic model of property attributes and geographical components

N of Cases : 3633 Indep. Var : 33 Max. VIF : 4.566 Adj. R-sq: 0.768 SEE : 0.1182 Constant : 10.60501 F ratio : 366.1 Df : 33 / 3599 Prob. : 0.000

Model Minimum Maximum Mean Std. Dev. N of cases Predicted 10.707112 12.749734 11.348533 0.215678 3633 Residual -0.460806 0.452916 0.0 0.117705 3633

Control sample Minimum Maximum Mean Std. Dev. N of cases Predicted 10.773187 12.218870 11.352782 0.222641 407 Residual -0.610692 0.621067 0.002583 0.124783 407

Autocorrelation Moran’s I Probability N of cases Pairs Range (m) Dependent 0.44847 < 0.0001 4040 60340 0-1125 Predicted 0.60066 < 0.0001 4040 60340 0-1375 Residuals 0.15971 < 0.0001 4040 60340 0-875 Heteroskedasticity SS Res. Low SS Res. High F D. of freedom Prob. Goldfeld-Quandt 24.0253 25.6554 1.0678 1616, 1616 0.0936

b) Weighted average of the 15 nearest neighbours and property difference on independent variables





c) Interactions between property attributes and geographical factors





Table V. Summary of hedonic model (Natural logarithm of sale price)

29

d) First stage integration of neighbourhood and geographical interaction relationships





e) Second stage integration of neighbourhood and geographical interaction relationships including relative tax differential



Control sample Minimum Maximum Mean Std. Dev. N of cases Predicted 10.794561 12.421995 11.361402 0.2364209 407 Residual -0.6270931 0.3649263 -0.00604 0.1056451 407

Autocorrelation Moran’s I Probability N of cases Pairs Range (m) Dependent 0.44847 < 0.0001 4040 60340 0-1125 Predicted 0.54906 < 0.0001 4040 60340 0-1125 Residuals 0.08338 0.0001 4040 60340 0-375 Heteroskedasticity SS Res. Low SS Res. High F D. of freedom Prob. Goldfeld-Quandt 18.2043 18.9623 1.0416 1616, 1616 0.2062

Table V (Continued). Summary of hedonic model (Natural logarithm of sale price)

30

Coefficients Significance Collinearity B Std Error Beta t Sig. VIF

(Constant) 10.60501 .05428 195.363 .000 LnAppAge -.11075 .00349 -.404 -31.719 .000 2.543 LnLotSize .06631 .00720 .087 9.209 .000 1.402 LivArea .00494 .00015 .354 33.042 .000 1.803 WaterSewer .10587 .02087 .043 5.073 .000 1.110 InfFound -.06440 .01079 -.050 -5.969 .000 1.105 BaseFinh .02209 .00457 .045 4.835 .000 1.331 CathCeil .01888 .00610 .029 3.094 .002 1.368 InfCeilQual -.10531 .02182 -.039 -4.826 .000 1.047 Skylight .07768 .01805 .035 4.304 .000 1.035 Quality .06296 .00948 .056 6.644 .000 1.129 Washrooms .06312 .00698 .085 9.037 .000 1.376 Fireplace .03825 .00510 .065 7.498 .000 1.183 HardwStair .03710 .00772 .040 4.807 .000 1.101 Oven .02539 .00665 .032 3.819 .000 1.085 KitchCab .02396 .00619 .031 3.871 .000 1.034 Dishwasher .02482 .00416 .050 5.961 .000 1.114 CentVacuum .03817 .00858 .037 4.451 .000 1.055 Veranda .01124 .00407 .023 2.763 .006 1.073 Balcony .03250 .01184 .022 2.744 .006 1.042 ExcavPool .08543 .01057 .067 8.083 .000 1.083 AttGarage .04333 .01271 .029 3.410 .001 1.124 DetGarage .04585 .00684 .057 6.707 .000 1.143 Shed80cf .02077 .00435 .042 4.773 .000 1.221 LTaxRate -.05500 .00672 -.096 -8.185 .000 2.151 NbMonthJ90 .00129 .00029 .035 4.404 .000 1.012 Motorway150 -.05084 .01466 -.028 -3.468 .001 1.055 DistMotorExit .01703 .00305 .064 5.584 .000 2.048 ACF1 Regional-level services -.06657 .00458 -.248 -14.524 .000 4.566 ACF2 Local-level services -.02736 .00348 -.087 -7.855 .000 1.920 CSF1 Centrality .03077 .00469 .066 6.563 .000 1.600 CSF2 Family cycle A -.01914 .00278 -.093 -6.890 .000 2.861 CSF3 Socio-economic status .08140 .00392 .288 20.757 .000 3.015 CSF4 Family cycle B -.01600 .00321 -.050 -4.991 .000 1.602 Location-related attributes are greyed.

Table VI. Model A : Standard hedonic model (Natural logarithm of sale price)

31


(Constant) 10.59499 .05382 196.873 .000 Nhbd LnAppAge -.11291 .00469 -.353 -24.060 .000 3.707 Pdif LnAppAge -.12083 .00436 -.246 -27.713 .000 1.359 LnAppAge * CSF2 .01848 .00257 .080 7.182 .000 2.111 LnLotSize .08031 .00787 .105 10.205 .000 1.836 LnLotSize * ACF2 -.01612 .00733 -.020 -2.199 .028 1.476 LnLotSize * CSF2 -.01897 .00699 -.026 -2.715 .007 1.565 LnLotSize * CSF3 .02591 .00742 .031 3.490 .000 1.313 Nhbd LivArea .00516 .00025 .279 20.761 .000 3.108 Pdif LivArea .00437 .00015 .265 28.561 .000 1.483 WaterSewer * CSF4 -.12834 .02695 -.038 -4.763 .000 1.081 Nhbd InfFound -.11313 .02040 -.049 -5.545 .000 1.337 Pdif InfFound -.06305 .01043 -.053 -6.043 .000 1.328 BaseFinh .02180 .00442 .044 4.935 .000 1.363 CathCeil .01907 .00609 .029 3.133 .002 1.494 CathCeil * CSF2 .01431 .00439 .029 3.258 .001 1.392 InfCeilQual -.08952 .02141 -.034 -4.182 .000 1.104 InfCeilQual * CSF2 .04358 .01924 .018 2.265 .024 1.070 Nhbd Skylight .16807 .03573 .041 4.704 .000 1.331 Pdif Skylight .06998 .01732 .035 4.040 .000 1.279 Nhbd Quality .09040 .01614 .050 5.599 .000 1.348 Pdif Quality .07668 .00963 .065 7.961 .000 1.146 Nhbd Washrooms .05701 .01335 .045 4.270 .000 1.873 Pdif Washrooms .05779 .00694 .081 8.324 .000 1.618 Washrooms * ACF1 -.01594 .00620 -.022 -2.572 .010 1.264 Nhbd Fireplace .03971 .00983 .040 4.038 .000 1.717 Pdif Fireplace .03180 .00494 .056 6.440 .000 1.290 HardwStair .03608 .00741 .039 4.871 .000 1.112 Oven .02297 .00638 .029 3.602 .000 1.095 Nhbd KitchCab .03908 .01175 .029 3.325 .001 1.283 Pdif KitchCab .02583 .00601 .036 4.296 .000 1.199 Dishwasher .02757 .00400 .056 6.894 .000 1.126 CentVacuum .03990 .00824 .038 4.842 .000 1.067 Veranda .01090 .00391 .022 2.788 .005 1.085 Veranda * ACF2 -.01192 .00488 -.019 -2.441 .015 1.025 Balcony .03783 .01137 .026 3.329 .001 1.052 Nhbd ExcavPool .06983 .02134 .030 3.272 .001 1.482 Pdif ExcavPool .08768 .01016 .075 8.626 .000 1.301 AttGarage .03861 .01231 .026 3.135 .002 1.156 DetGarage .04574 .00659 .057 6.944 .000 1.163 Shed80cf .01789 .00421 .036 4.245 .000 1.255 Shed80cf * CSF3 -.01351 .00467 -.023 -2.889 .004 1.114 Shed80cf * CSF4 .01544 .00512 .024 3.018 .003 1.093 LTaxRate -.04818 .00654 -.084 -7.370 .000 2.231 LTaxRate * CSF2 .03075 .00436 .067 7.051 .000 1.541 LTaxRate * CSF3 -.05924 .00694 -.103 -8.531 .000 2.509 LTaxRate * CSF4 -.02322 .00577 -.038 -4.027 .000 1.546 NbMonthJ90 .00139 .00028 .038 4.927 .000 1.021 Motorway150 -.05425 .01415 -.030 -3.833 .000 1.078 DistMotorExit * CSF2 .00711 .00244 .027 2.918 .004 1.422 ACF1 Regional-level services -.04799 .00387 -.179 -12.406 .000 3.564 ACF2 Local-level services -.03364 .00354 -.107 -9.513 .000 2.170 CSF1 Centrality .02196 .00473 .047 4.646 .000 1.781 CSF2 Family cycle A -.00850 .00314 -.041 -2.706 .007 4.011 CSF3 Socio-economic status .07212 .00406 .255 17.752 .000 3.546 CSF4 Family cycle B -.01156 .00329 -.036 -3.517 .000 1.846 Location-related attributes are greyed.

Table VII. Model D : First stage integration of neighbourhood and geographical interactions

32


(Constant) 10.68007 .04410 242.202 .000 LnAppAge -.12424 .00399 -.453 -31.107 .000 4.340 Nhbd LnAppAge .01234 .00484 .039 2.547 .011 4.698 LnAppAge * CSF2 .01855 .00235 .080 7.882 .000 2.100 LnLotSize .05931 .00698 .078 8.503 .000 1.717 LnLotSize * CSF2 -.03047 .00599 -.042 -5.085 .000 1.369 LnLotSize * CSF3 .02342 .00686 .028 3.416 .001 1.333 LivArea .00430 .00014 .308 30.692 .000 2.065 Nhbd LivArea .00070 .00020 .038 3.463 .001 2.428 WaterSewer * CSF4 -.11031 .02458 -.032 -4.489 .000 1.070 InfFound -.05870 .00960 -.046 -6.117 .000 1.140 Nhbd InfFound -.05662 .01671 -.024 -3.389 .001 1.067 BaseFinh .01080 .00407 .022 2.653 .008 1.377 CathCeil .01826 .00558 .028 3.272 .001 1.493 CathCeil * CSF2 .01709 .00402 .035 4.251 .000 1.389 InfCeilQual -.10458 .01964 -.039 -5.326 .000 1.105 InfCeilQual * CSF2 .07995 .01775 .033 4.504 .000 1.083 Skylight .07677 .01588 .035 4.834 .000 1.046 Nhbd Skylight .14678 .02932 .036 5.005 .000 1.067 Quality .07136 .00838 .064 8.521 .000 1.150 Washrooms .05440 .00635 .073 8.567 .000 1.484 Washrooms * ACF1 -.02594 .00569 -.036 -4.561 .000 1.266 FirePlace .02942 .00449 .050 6.548 .000 1.196 HardwStair .04459 .00679 .048 6.567 .000 1.111 Oven .02414 .00584 .030 4.135 .000 1.092 KitchCab .02565 .00551 .034 4.655 .000 1.069 Nhbd KitchCab .02122 .01000 .016 2.122 .034 1.106 Dishwasher .03104 .00366 .063 8.474 .000 1.125 CentVacuum .05108 .00757 .049 6.751 .000 1.071 Veranda .00763 .00359 .016 2.127 .034 1.089 Veranda * ACF2 -.01660 .00451 -.026 -3.681 .000 1.040 Balcony .03603 .01043 .025 3.455 .001 1.053 ExcavPool .10037 .00933 .079 10.754 .000 1.101 AttGarage .04600 .01127 .031 4.081 .000 1.152 DetGarage .04759 .00603 .059 7.886 .000 1.161 Shed80cf .02050 .00386 .042 5.307 .000 1.254 Shed80cf * CSF3 -.01686 .00428 -.029 -3.934 .000 1.113 Shed80cf * CSF4 .02090 .00469 .033 4.455 .000 1.094 LocalTaxRate * CSF2 .02339 .00416 .051 5.620 .000 1.670 LocalTaxRate * CSF3 -.05571 .00643 -.097 -8.669 .000 2.557 LocalTaxRate * CSF4 -.01779 .00526 -.029 -3.380 .001 1.533 RelTaxDif -.20769 .00748 -.215 -27.784 .000 1.226 RelTaxDif * ACF2 -.03092 .01047 -.023 -2.953 .003 1.197 RelTaxDif * CSF2 .02660 .00701 .031 3.795 .000 1.384 RelTaxDif * CSF3 -.03928 .00891 -.034 -4.407 .000 1.227 Motorway150 -.07162 .01299 -.040 -5.513 .000 1.081 DistMotorExit * CSF2 .01176 .00223 .044 5.264 .000 1.423 ACF1 Regional-level services -.05966 .00353 -.222 -16.914 .000 3.527 ACF2 Local-level services -.03799 .00311 -.121 -12.198 .000 2.003 CSF1 Centrality .01753 .00432 .038 4.061 .000 1.770 CSF2 Family cycle A -.00855 .00283 -.042 -3.017 .003 3.884 CSF3 Socio-economic status .07540 .00342 .267 22.056 .000 2.988 CSF4 Family cycle B -.01385 .00300 -.044 -4.621 .000 1.827 Location-related attributes are greyed.

Table VIII. Model E : Second stage integration of relative tax differential

33

0 2.5

kilometres5

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain RoadsPairs of Nearest Neighbours

Bungalows

Pair of Bungalows

Transactions in 1990-91

Figure 1. Location of bungalows and network of 15 nearest neighbours

Starting point

Stepwise MultipleRegressionProcedure

Yes Test forMulticollinearity

Test for SpatialAutocorrelation

No

Yes

Test forHeteroskedasticity

Unbiased andstable Hedonic

Model

No

No

Add Geographicand NeighborhoodAttributes and/or

Interactions

Yes

Property Attributes

Factor Analysis ofInterrelatedIndependent

Variables

Test for TemporalAutocorrelation Yes

Add TemporalCovatiates and/or

Interactions

No

Improve MarketSegmentation or

Use BinaryInteractions with

Property Specifics

Figure 2. Hedonic modelling procedure

34

Figure 3. Correlograms of Models Residuals

0 2.5

kilometres5

Spatial Coefficients1000 Square metres Lot

1.601.561.521.481.44

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain Roads

Figure 4. Location rent effect of a 1000 sq.-m. lot on house value (Bungalows 1990-91)

Correlogram of Residuals from Models A to E

-0.15000

-0.10000

-0.05000

0.00000

0.05000

0.10000

0.15000

0.20000

1-12

5

125-

249

250-

374

375-

499

500-

624

625-

749

750-

874

875-

999

1000

-112

4

1125

-124

9

1250

-137

4

1375

-149

9

1500

-162

4

1625

-174

9

Distance Lag (metres)

Mor

an's

IModel A ResidualsModel B ResidualsModel C ResidualsModel D ResidualsModel E Residuals

35

0 2.5

kilometres5

Spatial Coefficients10 years-old (Apparent Age)

0.790.780.770.760.75

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain Roads

Figure 5. Spatial trend of depreciation for a 10 years-old house (Bungalows 1990-91)

0 2.5

kilometres5

Spatial Coefficients40 years-old (Apparent Age)

0.6850.6680.6520.6350.619

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain Roads

Figure 6. Spatial trend of depreciation for a 40 years-old house (Bungalows 1990-91)

36

0 2.5

kilometres5

Spatial CoefficientNet Effect of the Second Washroom

1.121.091.061.031

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain Roads

Figure 7. Marginal effect of a second washroom on house value (Bungalows 1990-91)

0 2.5

kilometres5

Spatial CoefficientsPresence of a Shed

1.0561.0341.0130.9910.969

Municipalities

Hydrography

Road Network

Boundaries

Lake / River

RiverStream

MotorwaysMain Roads

Figure 8. Marginal effect of the presence of a shed on house value (Bungalows 1990-91)

37

[ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ]

[ ] [ ]

[ ] [ ] [ ]iiiii

ii

iiii

iiiii

ii

CSFCSFCSFCSFACF

ACFi

Motorwayii

iDetGarageAttGarageExcavPoolBalcony

i

CentVacuumDishwasheri

OvenHardwStairFireplace

iQuality

iiiBaseFinh

iiiiii

eeeee

exitDistMotorEScelTaxDifScLTaxRateSc

cfShedSceeeeVerandaSc

eeKitchCabSceee

WashroomsSceSkylightSclInfCeilQuaScCathCeilSce

InfFoundScWaterSewerScLivAreaScLotSizeScAppAgeScEstModelE

401385.0307540.0200855.0101753.0203799.0

105966.015007162.0

04759.004600.010037.003603.0

05108.003104.002414.004459.002942.0

07136.001071.0

Re

80

lnln59.43480$

−−−

−−

=

Where [1] [ ] ( ) ( ) ( )( )( )01373.0248170.2ln01855.0ln01234.0ln12424.0ln +−++−= iiii CSFAppAgeAppAgeNhbdAppAge

i eAppAgeSc

[2] [ ] ( )( )( ) ( )( )( )09166.0343297.6ln02342.001373.0243297.6ln03047.0)ln05931.0(ln −−++−−+= iiiii CSFLotsizeCSFLotSizeLotSizei eLotSizeSc

[3] [ ] ( ) ( )LivAreaNhbdLivAreai

iieLivAreaSc 00070.000430.0 +=

[4] [ ] ( )( ))25839.0499009.011031.0( −−−= ii CSFWaterSeweri eWaterSewerSc

[5] [ ] ( ) ( )InfFoundNhbdInfFoundi

iieInfFoundSc 05662.005870.0 −+−=

[6] [ ] ( )( )( )01373.0217066.001709.0)01826.0( +−−+= iii CSFCathCeilCathCeili eCathCeilSc

[7] [ ] ( )( )( )01373.0200853.007995.0)10458.0( +−+−= iii CSFlInfCeilQualInfCeilQuai elInfCeilQuaSc

[8] [ ] ( ) ( )SkylightNhbdSkylighti

iieSkylightSc 14678.007677.0 +=

[9] [ ] ( ) ( )( )( )42520.0125406.102594.005440.0 −−−+= iii ACFWashroomsWashroomsi eWashroomsSc

[10] [ ] ( ) ( )KitchCabNhbdKitchCabi

iieKitchCabSc 02122.002565.0 +=

[11] [ ] ( )( )( )01101.0247702.001660.0)00763.0( −−−+= iii ACFVerandaVerandai eVerandaSc

[12] [ ] ( )( )( ) ( )( )( ))25839.044572.08002090.009166.034572.08001686.0)8002050.0(80 −−+−−−+= iiiii CSFcfShedCSFcfShedcfShedi ecfShedSc

[13] [ ]( )( )( ) ( )( )( )

( )( )( )25839.043847.201779.009166.033847.205571.001373.023847.202339.0

−−−+−−−++−

= iiiiii

CSFLTaxRateCSFLTaxRateCSFLTaxRate

i eLTaxRateSc

[14] [ ]( )( )( ) ( )( )( )

( )( )( )01101.021469.0Re03092.009166.031469.0Re03928.001373.021469.0Re02660.0)Re20769.0(

Re −−−+−−−++−−

= iiiiiii

ACFlTaxDifCSFlTaxDifCSFlTaxDiflTaxDif

i elTaxDifSc

[15] [ ] ( )( )( )01373.025066.101176.0 +−= ii CSFayExitDistMotorwi exitDistMotorESc

[16] ki

d

dAttribute

AttributeNhbd

k ik

k ik

k

i ≠=

∑

∑

=

= ;1

15

12

15

12 and AttributeNhbdAttributeAttributePdif iii −=

[17] { } iiiiii EstModelDlueTaxationVaLTaxRateEstModelDLTaxRatelTaxDif )01.0()01.0(100Re −=

Appendix 1. Mathematical specification of Model E

Documents

DOCUMENT DE TRAVAIL 2001-015 L S - FSA ULaval · 2001. 11. 13. · it is combining GIS technology and spatial statistics to define principal components of accessibility and socio-