International Journal of Housing Markets and AnalysisEmerald Article: Using a Fourier polynomial expansion to generate a spatial predictorJean Dubé, Marius Thériault, François Des Rosiers

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To cite this document: Jean Dubé, Marius Thériault, François Des Rosiers, (2012),"Using a Fourier polynomial expansion to generate a spatial predictor", International Journal of Housing Markets and Analysis, Vol. 5 Iss: 2 pp. 177 - 195

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Using a Fourier polynomialexpansion to generatea spatial predictor

Jean DubeUniversite du Quebec a Rimouski, Quebec City, Canada, and

Marius Theriault and Francois Des RosiersResearch Center in Regional Planning and Development, Universite Laval,

Quebec City, Canada

Abstract

Purpose – Spatial autocorrelation in regression residuals is a major issue for the modeller because itdisturbs parameter estimates and invalidates the reliability of conclusions drawn from models. Thepurpose of this paper is to develop an approach which generates new spatial predictors that can bemapped and qualitatively analysed while controlling for spatial autocorrelation among residuals.

Design/methodology/approach – This paper explores an alternate approach using a Fourierpolynomial function based on geographical coordinates to construct an additional spatial predictorthat allows to capture the latent spatial pattern hidden among residuals. An empirical validation basedon hedonic modelling of sale prices variation using a large dataset of house transactions is provided.

Findings – Results show that the spatial autocorrelation problem is under control as shown by lowMoran’s I indexes. Moreover, this geo-statistical approach provides coefficients on environmentalamenities that are still highly significant by capturing only the remaining spatial autocorrelation.

Originality/value – The originality of this paper relies on the development of a new model thatallows considering, simultaneously spatial and time dimension while measuring the marginal impactof environmental amenities on house prices avoiding competition with the weight matrix neededin most spatial econometric models.

Keywords Regression analysis, Mathematics, Polynomials, Fourier polynomial expansion method,Geo-statistical approach, Hedonic price model, Spatial autocorrelation, Environmental amenities,Spatio-temporal analysis

Paper type Research paper

1. IntroductionInterdependence of spatial error terms is a major issue in regression analysis withgeographically located data, and one which has motivated 30 years of research inspatial econometrics (Anselin, 2010). Whereas geographic information systems (GIS)can be used to locate observations and create spatial predictors for assessing spatiallydistributed externalities such as proximity to urban amenities and environmentaldisturbances, most of the time, these variables only account for part of the total spatialvariance in the dependent variable. The literature reports that spatial autocorrelation

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This research is granted by the Fond quebecois de recherche sur la societe et la culture (FQRSC)and by the Social Sciences and Humanities Research Council of Canada (SSHRC). The authorsare grateful to the Quebec City’s appraisal division and to the Greater Quebec Real Estate Board(Chambre immobiliere de Quebec – CIQ) for giving access to their assessment and transactiondatabases.

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International Journal of HousingMarkets and Analysis

Vol. 5 No. 2, 2012pp. 177-195

q Emerald Group Publishing Limited1753-8270

DOI 10.1108/17538271211225922

among residuals can result either from a bad specification of the functional form orfrom omitted variables.

The application of spatial econometrics models allows capturing the spatialcorrelation among residuals. However, it neglects the fact that data can have anotherdimension than space (cross-section over space), such as time (cross-section pooled overtime), by limiting the spatial weight matrix used to geographical considerations. Thissituation can lead to overestimating the spatial autocorrelation component. Moreover,these models are so efficient at catching the latent spatial structure that they could exertinfluence on the statistical significance of other meaningful spatial attributes.

The objective of this paper is to develop a simple way to generate a global spatialpredictor, fixed over time, to capture most of the latent space information responsible forspatial autocorrelation. The suggest alternative is based on a geo-statistical modellingapproach using projected geographical coordinates (X, Y) of observations in order toadjust a polynomial function on the residuals using Fourier expansion transforms. It isempirically tested for efficiency and compared to the econometric spatial error model(SEM – LeSage, 1999). The purpose is to set a predictor of the latent spatial structurethat complements measured spatial variables. The model is estimated according to athree-stage procedure that could be implemented with large spatial databases, especiallywhere temporal variation are considered concomitantly, using any commercialstatistical software that includes the ordinary least square (OLS) approach.

The paper is divided into six sections. The next section summarizes approachesusually retained in the housing market literature for dealing with spatialautocorrelation issues depending on the presumed source of spatial dependence. Themethodological details of the Fourier polynomial expansion are presented in Section 3.Section 4 presents the database used to design the empirical test, while results obtainedwith model specifications are compared to SEM and analyzed in Section 5. Theusefulness of the Fourier polynomials to model the latent spatial structure is thenbrought out. The paper ends with a brief discussion on the ability of the three-stageprocedure for modelling the spatial structures linked to both measured and latentvariables (Section 6), followed by a conclusion (Section 7).

2. Literature reviewHedonic theory (Rosen, 1974) suggests that the price of a complex good can beexpressed as a function of the various attributes forming the bundle, so that eachcoefficient related to a given characteristic represents its implicit (hedonic) price. In thecase of housing, many amenities, both intrinsic and extrinsic to the property, have asignificant influence on the determination of the market value of units. Even so, a largepart of the total variance of house prices remains unexplained (Anselin and Can, 1986;Dubin and Sung, 1987; Can, 1992; Dubin, 1998). This situation stems, for a large part,from location effects that are not included in the hedonic equation and, consequently,that are transferred to the residuals of the model. The spatial autocorrelation in theresiduals, formally introduced by Cliff and Ord (1972, 1973, 1981) is a complex issuedue to its multidirectional character and to the feed-back and spillover effects itinvolves (Hordijk and Nijkamp, 1978; Hordijk, 1979).

Spatial econometrics literature proposes different methods to overcome spatialautocorrelation problems (Anselin, 1988, 2003, 2010; LeSage, 1999; Le Gallo, 2002).Econometric models depict the latent spatial structure with a spatial weight matrix

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function using an a priori exogenous specification based on distance amongobservations. The spatial weight matrix is defined considering spatial specificationand captures most of the spatial structure through the distance function used to computethe weights.

Even if spatial econometric techniques allow controlling for spatial dependenceamong residuals, identification of the source of spatial autocorrelation in the residualsremains a major issue in modeling spatial data (McMillen, 2010). According toMcMillen, the appropriate approach should depend on the view the modeller has aboutthe possible generation process that yields spatial autocorrelation. He argues thatgeographical coordinates are the only things that matter when analyzing spatial data.

This suggests considering other methods to handle spatial autocorrelation amongresiduals, such as geo-statistical techniques (Pannatier, 1996; Cressie, 1993; Zimmermanand Zimmerman, 1991; Krige, 1966). These techniques calibrate a regression model in atwo-step procedure whereby the spatial interactions are modelled, notably, through avariance-covariance matrix, which allows controlling the spatial autocorrelation in theresiduals by using space-related information. As noted by Griffith and Peres-Neto(2006), geo-statistical techniques can be viewed as a generalized least square (GLS)approach where the main goal is to create new spatial predictors that can beincorporated into the regression model.

Since real estate modelling applications (e.g. hedonic price models) relies ontransactions that are not only distributed in space, but also ordered in time, one canargue that a strictly spatial model could potentially lead to spurious spatio-temporalautocorrelation issue (Dube et al., 2011a). Then, one may reconsider relying on an apriori spatial distance-decay function to adjust the hedonic prices, mainly because sucha choice assumes that observations are strictly simultaneous, thus introducing somecolinearity among spatial variables and the weight matrix.

These considerations militate in favour of new developments based on geo-statisticalapproaches when spatial autocorrelation is significant, time dimension is relevant andresearchers try to assess the impact of environmental amenities rather than strictlycontrolling for spatial autocorrelation among residuals. Thus, a general approachshould account for that possibility, even more so when researchers are interested inexplaining complex statistical spatial structures (Unwin and Hepple, 1974).

Such an approach is in line with another stream of literature – seldom referenced inregional science papers – developed by ecologists (Borcard et al., 1992; Legendre, 1993)for dealing with issues of spatial autocorrelation in regression models. Being interestedin modelling spatial structures embedded in natural processes, ecologists distinguishbetween “true” (trend) and “false” gradients. On the one hand, in a trend, any value canbe expressed as a function of its geographic coordinates, plus an error term that isindependent among locations. On the other hand, a “false” gradient is caused by spatialautocorrelation within the process, meaning that the variation in values at a location ispartly determined by neighbouring values. The problem is therefore to distinguish truegradients from false ones.

3. MethodologyIn this context, we suggest developing a geo-statistical approach based on the Fourierpolynomials expansion method (Gallant, 1981, 1982) performed on the projectedgeographic coordinates (X, Y) and using it as a spatial smoothing technique for assessing

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the latent spatial pattern of dependency among the residuals of a hedonic model. Fourierpolynomials coefficients then generate a latent spatial predictor that could have someinfluence on house price, but that is not measurable otherwise. While this approachaccounts for the remaining unobservable spatial characteristics that impact on houseprices, it allows for mapping the latent component remaining in modelled pricevariation.

This paper aims at empirically testing, for a case study, the efficiency of ageo-statistical modelling approach using projected geographical coordinates (X, Y) ofobservations in order to adjust a polynomial function on residuals using Fouriertransforms. The parameters are estimated in a three-stage procedure.

The first step searches for the optimal order of a Fourier polynomial functionapplied to models residuals and that is fitted on the unexplained part of an auxiliaryfunction using trigonometric terms based on X, Y coordinates. The second step yieldsthe latent spatial predictor given the optimal order and the estimated coefficients of thetrigonometric terms. At the last step, the latent spatial predictor is added to the hedonicprice model as a new, explicit, spatial variable that is expected to exert some latentinfluence on price determination. This procedure can be implemented using OLSregression method[1].

While no consensus exists on the choice of the functional form of the hedonic priceequation in the literature (Halvorsen and Pollakowski, 1981), the semi-log and log-logforms remain among the most popular. They are still appropriate in empirical analysisif the main goal is to identify the average effect of amenities in a given geographic area.The functional form used in this study is a mix between the log-log form and thesemi-log form equation, with the natural log of the i the house price at time t, pit, beingregressed on the C continuous characteristics and the D dummy descriptors(equipments, sub-markets, etc.). That is, the vector of the k characteristics is a partitionvector, Zkit, that includes both continuous variables and dummies, with Z ¼ (C, D).Finally, Ts is a dummy descriptor that take a value of one is house is sell at time periods while it takes the value zero otherwise (equation (1)):

pit ¼XS

s¼1

asTs þXK

k¼1

bkZkit þ eit; ð1Þ

The bk vector represents the mean implicit (hedonic) prices relative to the various kobservable characteristics estimated by the model, with the parameters beingassumed constant over time. The as vector accounts for the sample difference(Wooldridge, 2002) and serves for building house price index over time. It can beviewed as housing price inflation and reflects the evolution of nominal prices.

It is assumed that spatial autocorrelation is created by a latent space-relatedinformation, or spatial predictor, that can be expressed as a function of thegeographical coordinates of the i house sold, Xi and Yi, taking place of the usualautoregressive specification[2] (equation (2)):

eit ¼ l½ f ðXiÞ þ gðYiÞ� þ uit; ð2Þ

where l is an autoregressive parameter (to be estimated), f ( · ) and g ( · ) are generalspatial functions to be defined and uit is a random independent and identicallydistributed error term (white noise).

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Spatial functions can be approximated in various ways, including throughpolynomials and splines. Some examples in the real estate literature include localweighted regression (Cleveland and Devlin, 1988; Case et al., 2004) and krigingtechniques (Martinez and Lorenzo, 2000; McCluskey et al., 2000). This said, the Fourierexpansion (McMillen, 2003; McMillen and Dombrow, 2001) can efficiently account fornonlinearity without assuming regular distances between each point while being easilyestimated using any statistical software that handles OLS; hence our choice for thisapproach. The Fourier expansion polynomial approximates an unknown function in acontinuous way by smoothing variations in geographically close observed residuals. Infact, it is a trend surface adjusted over geo-located residuals while using atrigonometric transformation so as to insure that values range between 0 and 2p.

In order to do so, transformations on local geographic coordinates (Xi and Yi) areapplied (equations (3) and (4)):

X*i ¼ 2p½Xi 2 minðXÞ�

½maxðXÞ2 minðXÞ�ð3Þ

Y*i ¼ 2p

½Yi 2 minðY Þ�

½maxðY Þ2 minðY Þ�ð4Þ

Then, using transformations, the spatial functions follow (equations (5) and (6)):

f ðXiÞ ¼ p0x þ p1xX*i þ p2xX

*2i þ

XQ

q¼1

kqx sin qX*i

� �þ gqx cos qX*i

� �� �ð5Þ

gðYiÞ ¼ p0y þ p1yY*i þ p2yY

*2i þ

XQ

q¼1

kqy sin qY*i

� �þ gqy cos qY*

i

� �� �ð6Þ

where p0x;p1x;p1x; kqx;gqx are parameters to be estimated for the spatial functionover the X geographical coordinates while p0y;p1y;p1y; kqy; gqy are parameters to beestimated for the spatial function over the Y geographical coordinates and q is theorder selected. It is assumed that these new functions, of order q, could capturedifferences between observed and estimated house prices that are assigned to locationand are the results of hidden spatial processes or patterns that could not be measuredotherwise.

The latent predictor, based on geographical coordinates, needed to be calibrated toidentify its optimal order, Q. This step is done by applying Schwartz’s (1978)information criterion (equation (7)) to the original model (equation (1)), incorporatingthe transformations on geographical coordinates (equations (5) and (6)):

BICðqÞ ¼ logðs 2Þ þ ð4qþ 4ÞlogðnÞ

n; ð7Þ

where s 2 is the sum of squares of residuals from an auxiliary regression includingFourier parameters, n is the total number of observations and 4q þ 4 is the totalnumber of non-constant parameters to be estimated.

As a final step though, and in order to insure that the parameters associated to thelatent spatial variable can compare with the one derived from the SEM approach,

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the variable needs to be standardized in an index lying between zero and one(equation (8)):

h*i ¼

½ fðXiÞ þ gðYiÞ�2 min½ fðXÞ þ gðY Þ�

max½ fðXÞ þ gðY Þ�2 min½ fðXÞ þ gðY Þ�; ð8Þ

Once the optimal order is determined, the latent spatial predictor can be generated, usingthe estimated coefficients of the Fourier expansion (equations (5) and (6)) and thetransformation on the geographical coordinates (equations (3) and (4)), to estimate thefinal hedonic model (equation (9)) by substituting the definition of the latent spatialfunction (equation (2)) by its value (equation (8)) in the original hedonic price equation(equation (1)):

pit ¼ at þXK

k¼1

bkZkit þ lh*i þ uit; ð9Þ

The adjusted residuals obtained with equation (9), uit, can be tested for the presence ofspatial autocorrelation so as to make sure that the space-related information derivedfrom the Fourier expansion adequately handles the problem.

This procedure can thus be summarized in three steps:

(1) first, the auxiliary regression using the error term on the original price model isestimated using different value of q on the trigonometric variables to define theoptimal order, Q;

(2) using the coefficients associated with the trigonometric variables, the latentspatial variable is then built and rescaled between zero and one; and

(3) the resulting predictor is then used to re-estimate the hedonic price model whichis tested for the presence of spatial autocorrelation in the residuals.

The crucial issue is to establish whether or not this approach offers a reliablealternative to the usual SEM for handling spatial autocorrelation problems. Whether ornot this spatial predictor is able to capture the latent space-related information is theobject of the following empirical comparative analysis.

4. Empirical validationThe database used to build the hedonic model for validation consists of19,357 single-family homes (single-detached, one-story houses, town cottages and rowhouses) sold between 1986 and 1996 within Quebec Urban Community (QUC), Canada(Figure 1). Each transaction is geo-located using a GIS. These data have been used in aprevious study by Dube et al. (2011b) and all the details about the physical characteristicsof the houses are presented in this former paper[3]. The goal of the paper is not to developa new way to specify the price equation, but instead to develop a method capable ofcontrolling for space-related information responsible for spatial autocorrelation withoutusing a spatial weight matrix. The space-related information then relies on constructing anew spatial predictor accounting for a (constant) spatial structure over time.

Some location factors are accounted for in the hedonic price model using twoaccessibility indices that combine car and walking access times from home tosurrounding services (Des Rosiers et al., 2000). Moreover, three socio-economic status

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variables (proportion of lone-parent families, median household income and share ofpopulation holding a university degree) are also used to isolate household profile effectsin houses prices and are drawn from the 1991 Canadian census. Finally, seven fixedeffects dummies are introduced for capturing space-related peculiarities withinrelatively homogeneous sub-markets (Voisin et al., 2010).

5. ResultsSince previous research have shown that the marginal contribution of amenities is byand large constant over the whole area of Quebec City for the 1986-1996 period(Dube et al., 2011a, b), the coefficients are assumed to be invariant over the period ofanalysis in as much though as the price equation is specified using the log-log functionalform and that fixed location and temporal effects are incorporated in the specification.

While accounting only for the physical amenities in the model allows to explainslightly more than 70 per cent of the total variance in house prices, residuals showimportant problems related to spatial autocorrelation, as shown by the high value ofthe Moran’s I (Table I). When spatial amenities (sub-markets, accessibility indicesand socio-economic status) are added on to the model, the R 2 statistic rises by about10 per cent while both the value of the Moran’s I and of the Akaike (1974) informationcriterion (AIC) drop, thereby suggesting an improvement in the global modelperformance and its specification (Table I). Even if spatial autocorrelation decreaseswhen the spatial amenities are included in the original specification, it remainssignificant, which suggests using an appropriate technique in order to solve the issue.

The Fourier polynomial expansion optimal order (Q), is calibrated using the originalhedonic price function. In this particular case, the optimal order, Q, is set to six (6) forall specifications (Table II).

Figure 1.Spatial distribution of

houses sold in theQUC, 1986-1996

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The parameters assigned to trigonometric variables are used to build a single spatialtrend surface predictor, representing space-related information, that allows visualizingthe spatial drift unexplained by other variables included in the hedonic price model andreflected in each auxiliary (initial) model residuals. Such variations can generate a“latent location rent map” that may change given the variables inserted in the hedonicmodel (Figures 2-4). The interpretation of the map can be related to the error of theinitial hedonic model (auxiliary function): while positive residuals indicate that themodel underestimates the observed value of houses, negative residuals suggest thatthe model overestimates the actual sale price. Put otherwise, incorporatingspace-related information will tend to reduce the actual underestimation of the saleprice predicted by the model for values close to zero (grey and light red) while reducingits overestimation for values close to one (dark red). These maps enable a qualitativeinterpretation of the spatial structure that could not be captured with explicit variablesand could lead to hypotheses of additional indicators to include in the model(e.g. quality of view).

Usual OLS approachPhysical

amenities (1)Fixed effectdummies (2)

Accessibilityindices (3)

Socio-economicprofiles (4)

R 2 0.7076 0.7817 0.7889 0.8017n 19,357 19,357 19,357 19,357Log-likelihood 6,807.4 9,633.8 9,961.9 10,564.4Akaike informationcriteria 213,550.8 219,191.5 219,843.7 221,042.8Moran’s I 0.1676 * * * 0.0719 * * * 0.0478 * * * 0.0304 * * *

Notes: Significant at: *p , 0.05, * *p , 0.01 and * * *p , 0.001; (2) – specification (1) withphysical amenities, dummies variables for fixed location effects (sub-markets) and transaction dates(3) – specification (2) with accessibility indices (4) – specification (3) with socio-economic profiles

Table I.Synthesis of statisticsgiven the specification –OLS method

Determination of the optimal Fourier expansion order (Q)Order q Fixed effect dummies (2) Accessibility indices (3) Socio-economic profiles (4)

q ¼ 0 23.81439 23.84727 23.90799q ¼ 1 23.87596 23.87790 23.92710q ¼ 2 23.89699 23.89851 23.93814q ¼ 3 23.89904 23.90119 23.93893q ¼ 4 23.90436 23.90724 23.94227q ¼ 5 23.90346 23.90646 23.94313q ¼ 6 23.90532 23.90798 23.94474q ¼ 7 23.90443 23.90741 23.94393q ¼ 8 23.90274 23.90557 23.94199q ¼ 9 23.90100 23.90377 23.94046q ¼ 10 23.90063 23.90328 23.93996

Notes: (2) – specification with physical amenities, dummies for fixed location effects and transactiondates (3) – specification (2) with accessibility indices (4) – specification (3) with socio-economicprofiles

Table II.Determination of theoptimal order of theFourier spatial function –Schwartz criterion

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Figure 2.Latent spatial functioncalibrated for the QUC,

1986-1996Note: Using fixed effect dummies

Figure 3.Latent spatial functioncalibrated for the QUC,

1986-1996Note: Using fixed effect dummies and accessibility indices

Fourierpolynomialexpansion

185

Unsurprisingly, the variation in the spatial trend surface vary according to the spatialvariables include in the hedonic price model specification. While existing sub-marketsdo play an important role in the determination of house values, the space-relatedinformation function derived from the Fourier expansion helps filling a gap in theunderstanding of the residential dynamics that prevails in the market (Figure 2). Thefirst space-related variable reveals an important historical pattern in pricedetermination: the dark red zone is in fact a residential development that was firstassigned to industrial workers. The building was hastily constructed with the quality ofhousing being somewhat poor, which explains why the actual price is lower that whatthe model predicts.

Adding accessibility and socio-economic variables have progressively revealed astrong north-south divide in the residuals (steep gradient), clearly isolating thenorth-eastern part of the city from the south-western suburbs (Figures 3 and 4). Thisdivide in the market is well known to local real estate brokers, while it cannot beexplained through any of the variables included in the model. It comes from along-term, historical, social division between upper and lower town populations, goingfurther than mere differences in income and education. The residential development onthe south-west axis has been characterised by wealthier households with highereducation level (from Sillery to St-Augustin-de-Desmaures).

Results suggest that the space-related information component have severalimplications on price determination, as noted by the magnitude of the coefficientsassociated to the spatial predictor and its significance (Table III) which slowlydecreases as additional spatial variables are integrated in the model. Moreover,the estimated Moran’s I index is now pretty low and close to zero, even if it

Figure 4.Latent spatial functioncalibrated for the QUC,1986-1996

Note: Using fixed effect dummies, accessibility indices and socio-economic profile status

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HPM with physicalamenities and fixedeffects variables (2)

HPM with accessibilityindexes and fixed effects

variables (3)

HPM with socio-economic status,accessibility and

fixed effectsvariables (4)

Variables Parameters Parameters Parameters

Living area (m2) – ln 0.4179 * * * 0.4181 * * * 0.4047 * * *

Lot size (m2) – ln 0.1042 * * * 0.1049 * * * 0.0999 * * *

Age – (years) ln 20.1074 * * * 20.1086 * * * 20.1037 * * *

Bungalow Reference Reference ReferenceCottage 20.0296 * * * 20.0309 * * * 20.0300 * * *

Attached 20.1383 * * * 20.1398 * * * 20.1379 * * *

Building quality index 0.0884 * * * 0.0884 * * * 0.0856 * * *

Number of bathrooms 0.0398 * * * 0.0397 * * * 0.0403 * * *

Finished basement 0.0345 * * * 0.0344 * * * 0.0320 * * *

Brick facing (51 per cent or þ ) 0.0188 * * * 0.0176 * * * 0.0161 * * *

Number of fireplaces 0.0535 * * * 0.0539 * * * 0.0488 * * *

High quality floors 0.0135 * * * 0.0138 * * * 0.0152 * * *

Hard wood stairs 0.0492 * * * 0.0492 * * * 0.0473 * * *

Superior quality counter 0.0387 * 0.0367 * 0.0363 *

Inferior luminosity 20.0201 * * * 20.0200 * * * 20.0183 * * *

Simple attached garage 0.0884 * * * 0.0869 * * * 0.0856 * * *

Double attached garage 0.0868 * * * 0.0853 * * * 0.0879 * * *

Simple detached garage 0.0245 * * * 0.0237 * * * 0.0267 * * *

Double detached garage 0.0575 * * * 0.0569 * * * 0.0611 * * *

Terrace 0.0216 * * 0.0218 * * 0.0169 *

Excavated pool 0.0998 * * * 0.1001 * * * 0.0940 * * *

Water sewer connected 0.1152 * * * 0.1163 * * * 0.0947 * * *

Cathedral ceiling 0.0319 * * * 0.0316 * * * 0.0289 * * *

Central vacuum 0.0404 * * * 0.0406 * * * 0.0400 * * *

Local tax rate 20.0477 * * * 20.0482 * * * 20.0514 * * *

Sub-market 1 Reference Reference ReferenceSub-market 2 20.4609 * * * 20.4537 * * * 20.3642 * * *

Sub-market 3 20.0993 * * * 20.0956 * * * 20.0917 * * *

Sub-market 4 20.4312 * * * 20.4245 * * * 20.3539 * * *

Sub-market 5 20.3883 * * * 20.3965 * * * 20.3404 * * *

Sub-market 6 20.3398 * * * 20.3310 * * * 20.2867 * * *

Sub-market 7 20.4110 * * * 20.4095 * * * 20.3364 * * *

Transaction date 1986 Reference Reference ReferenceTransaction date 1987 0.0791 * * * 0.0784 * * * 0.0776 * * *

Transaction date 1990 0.2318 * * * 0.2321 * * * 0.2280 * * *

Transaction date 1991 0.2508 * * * 0.2511 * * * 0.2480 * * *

Transaction date 1993 0.2914 * * * 0.2922 * * * 0.2894 * * *

Transaction date 1994 0.2886 * * * 0.2898 * * * 0.2885 * * *

Transaction date 1995 0.2755 * * * 0.2767 * * * 0.2734 * * *

Transaction date 1996 0.2756 * * * 0.2768 * * * 0.2743 * * *

Regional accessibility index 0.0397 * * * 0.0375 * * *

Local accessibility index 0.0129 * * * 0.0247 * * *

Single-parent family index 20.0043 * * *

(continued )

Table III.Estimated coefficients –

Fourier polynomial model

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remains significant. One factor explaining why spatial autocorrelation indices are stillsignificant is that transactions are distributed over time, which is ignored in thecalculation of the indices, even though inflation is accounted for[4]. This indicates that,although still significant statistically, the presence of spatial autocorrelation amongresiduals is far less problematic as the approach do correct for the problem of omittedspatial variables in the specification of the price equation.

One interesting outcome is that coefficients related to physical amenities are quitesimilar in the Fourier model (Table III) and in the SEM (Table IV). However, coefficientsassociated with measured spatial variables are fairly higher (and more significant) withthe Fourier model than with the spatial econometric model. This suggests that thespatial effects of the spatial econometric model competes with spatial factors that areknown to impact on price. The parameter estimates pertaining to the spatial amenitiesreach a higher statistical significance when the Fourier expansion method is used.

Finally, the estimated Moran’s I indices are still significant for this specificationeven if the coefficients are pretty low, suggesting that our previous hypothesis(no consideration of time dimension) cannot be rejected. This said, the Fourierapproach can be viewed as a suitable alternative to the spatial econometric model sinceit efficiently corrects for spatial autocorrelation in the residuals by introducing a newspatial predictor. The new space-related variable thus provides measurable spatialdeterminants of price that can be included in the model.

6. DiscussionIn addition to producing a realistic representation of the latent structure of pricevariations over space, the Fourier expansion approach can adequately control forspace-related information generating spatial autocorrelation in the model residuals.While the Moran’s I statistic remains significantly different from zero, its magnitude isconsiderably lessened and is globally comparable to the one obtained with the SEM.

HPM with physicalamenities and fixedeffects variables (2)

HPM with accessibilityindexes and fixed effects

variables (3)

HPM with socio-economic status,accessibility and

fixed effectsvariables (4)

Variables Parameters Parameters Parameters

Median income index 0.0125 * * *

Percentage with universitydegree 0.0030 * * *

Spatial effect 0.8194 * * * 0.6841 * * * 0.5238 * * *

Constant 8.4076 * * * 8.5004 * * * 8.6044 * * *

R 2 0.8035 0.8042 0.8115n 19,357 19,357 19,357Log-likelihood 10,652.0 10,687.7 11,058.2Akaike information criteria 221,226.1 221,293.4 222,028.5Moran’s I 0.0108 * * * 0.0090 * * * 0.0040 * * *

Notes: Significant at: *p , 0.05, * *p , 0.01 and * * *p , 0.001; spatial radius of influence is set to3,000 m; to insure every observation have at least one neighbourTable III.

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HPM with physicalamenities and fixedeffects variables (2)

HPM withaccessibility indices

and fixed effectsvariables (3)

HPM with socio-economic status and

fixed effectsvariables (4)

Variables Parameters Parameters Parameters

Living area (m2) – ln 0.3967 * * * 0.3970 * * * 0.3959 * * *

Lot size (m2) – ln 0.0995 * * * 0.0994 * * * 0.0993 * * *

Age – ln 20.1044 * * * 20.1046 * * * 20.1038 * * *

Bungalow Reference Reference ReferenceCottage 20.0337 * * * 20.0337 * * * 20.0331 * * *

Attached 20.1335 * * * 20.1336 * * * 20.1321 * * *

Quality construction index 0.0824 * * * 0.0825 * * * 0.0825 * * *

Number of bathrooms 0.0387 * * * 0.0387 * * * 0.0389 * * *

Basement finished 0.0323 * * * 0.0323 * * * 0.0320 * * *

Brick facing (51 per cent or þ ) 0.0158 * * * 0.0158 * * * 0.0153 * * *

Number of fireplaces 0.0445 * * * 0.0446 * * * 0.0443 * * *

High quality floor 0.0167 * * * 0.0167 * * * 0.0167 * * *

Hard wood stairs 0.0417 * * * 0.0417 * * * 0.0415 * * *

Superior quality counter 0.0443 * * 0.0442 * * 0.0421 * *

Inferior luminosity 20.0193 * * * 20.0194 * * * 20.0190 * * *

Simple attached garage 0.0790 * * * 0.0790 * * * 0.0791 * * *

Double attached garage 0.0882 * * * 0.0882 * * * 0.0883 * * *

Simple detached garage 0.0279 * * * 0.0278 * * * 0.0283 * * *

Double detached garage 0.0634 * * * 0.0633 * * * 0.0640 * * *

Terrace 0.0143 0.0143 0.0133Excavated pool 0.0926 * * * 0.0927 * * * 0.0918 * * *

Water sewer connected 0.0526 * * * 0.0521 * * * 0.0510 * * *

Local tax rate 20.0417 * * * 20.0416 * * * 20.0410 * * *

Cathedral ceil 0.0285 * * * 0.0285 * * * 0.0282 * * *

Central vaccum 0.0405 * * * 0.0406 * * * 0.0403 * * *

Area 1 Reference Reference ReferenceArea 2 20.3372 * * * 20.3293 * * * 20.2749 * * *

Area 3 20.2202 * * * 20.2002 * * * 20.1854 * * *

Area 4 20.3037 * * * 20.2826 * * * 20.2375 * * *

Area 5 20.3743 * * * 20.3435 * * * 20.2785 * * *

Area 6 20.3140 * * * 20.2918 * * * 20.2442 * * *

Area 7 20.4230 * * * 20.3902 * * * 20.3229 * * *

Transaction date 1986 Reference Reference ReferenceTransaction date 1987 0.0815 * * * 0.0815 * * * 0.0807 * * *

Transaction date 1990 0.2343 * * * 0.2344 * * * 0.2342 * * *

Transaction date 1991 0.2548 * * * 0.2550 * * * 0.2548 * * *

Transaction date 1993 0.2970 * * * 0.2972 * * * 0.2971 * * *

Transaction date 1994 0.3017 * * * 0.3019 * * * 0.3018 * * *

Transaction date 1995 0.2845 * * * 0.2848 * * * 0.2844 * * *

Transaction date 1996 0.2816 * * * 0.2819 * * * 0.2819 * * *

Regional accessibility index 0.0143 0.0199 * *

Local accessibility index 0.0057 0.0115 *

One-parent family index 20.0026Median income index 0.0039 *

Percentage with university degree 0.0016 * * *

(continued )

Table IV.Estimated

coefficients – SEM

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However, the Fourier expansion seems most successful at capturing spatial effectsincorporated in the error terms since it shows only the “residual” latent spatialstructure and leaves more room for the observed spatial variables to highlight the“true” impact they exert on the dependent variable. This is quite important whenresearchers try to measure the effect of spatial variables on a given phenomena.

As mentioned, the impact of such approach is that it allows capturing statisticallysignificant impact of environmental amenities even when correcting for spatialautocorrelation among residuals. The accessibility indices analysis of several travel timesto the nearest facilities had identified two orthogonal PCA components (i.e. access toregional and local services) which are always significant in hedonic models. That is notoccurring by chance, since it relates to the marginal utility consumers draw fromagglomeration economies linked to the proximity of services. When aggregated, suchexternalities gives rise to a well-known spatial structure: the bid rent curve for land.However, the fundamental explanation is not with the structure, but with the urbanfunctions (including forms and networks) that generate the structure. Obviously, thisstructure is full of space-related information and it contributes to adding some in theresiduals of a hedonic model when not considered. Allowing a latent spatial variable(distance weighting) to compete with accessibility when modelling hedonic prices wouldeventually remove part of the functional relationship of interest through the spatialstructure which is closely related to a distance decay function, thus somewhat muting themeasured, and theoretically sound, variable to the benefit of an unknown latent predictor.

Because it is applied once other spatial variables have been inserted in the auxiliaryfunction, the Fourier predictor can compete marginally, and only a posteriori, with thoseobserved spatial variations. Conversely, spatial econometric models uses a distancedecay weight matrix for an optimal control over spatial autocorrelation in the residuals,eventually competing with other spatial factors embedding similar distance decay intheir definition. Therefore, during this simultaneous modelling process, spatial structureincluded in intrinsic and extrinsic variables is likely partitioned between measuredspatial variation and latent spatial variation, the outcome being dependent uponminimizing single criterion, the log-likelihood (best fit from a mathematical pointof view).

HPM with physicalamenities and fixedeffects variables (2)

HPM withaccessibility indices

and fixed effectsvariables (3)

HPM with socio-economic status and

fixed effectsvariables (4)

Variables Parameters Parameters Parameters

Spatial effect 0.8366 * * * 0.8283 * * * 0.7926 * * *

Constant 9.1687 * * * 9.1528 * * * 9.0746 * * *

R 2 0.8215 0.8214 0.8217n 19,357 19,357 19,357Log-likelihood 11,362.8 11,364.7 11,405.9Akaike information criteria 222,649.6 222,649.4 222,725.7Moran’s I 0.0026 * * * 0.0024 * * * 0.0026 * * *

Notes: Significant at: *p , 0.05, * *p , 0.01 and * * *p , 0.001; spatial radius of influence is set to3,000 m; to insure every observation have at least one spatial neighborTable IV.

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In the end, the concern these considerations lead us to remains: what is the bestsolution? This clearly depends on the modeller’s objectives and theoretical discussionon reasons to include such and such spatial variable in the analysis and to eventuallyallow for spatial autocorrelation control. This is part of the rationale for justifyingdevelopment of the three-step Fourier predictor: it lets spatial variable say their word“before” spatial autocorrelation control is enforced.

Moreover, spatial statistics are not designed to handle the temporal dimension;consequently, if there is a temporal trend in data, the null hypothesis (i.e. the absence ofany spatial autocorrelation) may be rejected more often than it should be. Thatstrengthens the interest of a geo-statistical approach for dealing with spatialautocorrelation while considering its temporal context. A geo-statistical approach isalso free from the arbitrary of choosing the form of the spatial function while it allowsfor a global, rather than local, adjustment on the whole study area withoutpredetermining a specific kernel influence. That clearly deserves further investigation.

Above all though, the usefulness of the Fourier spatial predictor stems from itsability to account for a series of issues unaddressed by the classical OLS approach andfrom the additional information it conveys about the spatial structure of the residentialmarket that prevails over a given area. For example, the steep gradient observed inFigure 4 highly suggests revisiting the delineation of sub-market 6 (Figure 1) used inthe analysis (for handling fixed effects). There is obviously a social-economic fault linethat could be integrated. Finally, the Fourier approach has the noticeable advantage ofdealing adequately and parsimoniously with spatial autocorrelation related problemsthat invalidate the classical assumptions of the OLS procedure. It can as well handleheterogeneity and large databases.

7. ConclusionIn this paper, a novel approach is developed for handling the latent spatial dimension inhedonic price models. Based on the Fourier expansion method, the spatial component ofprices is captured through the coefficients associated with trigonometrictransformations on the geographical coordinates. The mathematical transformationsperformed on the spatial coordinates and introduced as a latent spatial predictor in thehedonic price model are shown to adequately handle spatial autocorrelation amongresiduals.

The computation of the latent spatial predictor is quite straightforward and can bedone in three steps using the OLS or the GLS procedures. First, an auxiliary function isestimated using an OLS approach and a Fourier expansion on the geographiccoordinates. Second, the estimated coefficients on the transformed geographiccoordinates are used to construct a new single spatial predictor that captures thelatent spatial structure. This predictor is then used to re-estimate the hedonicmodel while capturing a large part of the unobservable spatial dependence amongresiduals.

An interesting point with this approach is that the new spatial predictor can bemapped, thereby allowing the exploration of the “latent location rent”. In the applicationexample developed in this paper, the smoothed predictor captures most of theunexplained spatial variation in prices. This novel approach stems from ideas developedby Legendre (1993) about the spatial autocorrelation issue and the usefulness of a spatialautocorrelation handling paradigm, in complement to the usual modelling problem.

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Another benefit comes from implicit handling of the temporal dimension in real estatedata. Since spatial components that influence sale price are deemed stable over a certainperiod of time, the latent spatial predictor discloses valuable information onunknown factors of price determination over space that could not be capture bymeasurable variables. Moreover, the predictor of the remaining spatial structure is notrelated to an a priori choice of kernel radius of influence, as it is the case for the definitionof the weight matrices used in spatial econometric methods.

As the results show, the approach is most promising, particularly where a largedatabase is used. While further research, both quantitative and qualitative, is still neededin order to investigate the usefulness of the “latent location rent” as an explanatoryvariable in hedonic price models, there is no a priori reason why it should not begeneralized to other geographical contexts. In that respect, this paper corroboratesPinske and Slade’s (2010) statement that alternative approaches for dealing with thespatial autocorrelation issue can be developed from an empirical perspective. However,there is still a need for further research about the best approaches to partition spatialvariance between measured variables and latent predictors in order to better knowledgethe interaction processes behind the price setting process.

Notes

1. The overall procedure can be seen as a GLS approach and can be estimated using any usualstatistical software. In the present case, Stata was used.

2. The usual spatial autoregressive function for the error term (SEM) is usually expressed byeit ¼ lWeit þ uit, where W is an exogenous spatial weight matrix and l is the spatialautoregressive parameter to be estimated.

3. More details on the original database can be found in Des Rosiers (1991), Des Rosiers et al.(2002, 2001, 1999, 1996), Kestens et al. (2004) and Theriault et al. (2005, 2003).

4. The significance of both indices can be related to the length of the time span (ten years)considered since their computation relies on the assumption that only the spatial dimensionmatters. However, if it is realistic to view residuals observed in 1993 as being connected tothose observed in 1990, the reverse is not true unless the market can be perfectly anticipated,which is not the case.

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Corresponding authorJean Dube can be contacted at: jean_dube@uqar.qc.ca

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