Do peer effects shape property values?

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JPIF 29,4/5

Do peer effects shape property values?

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Urban & Real Estate Management, Faculty of Business Administration, Laval University, Quebec City, Canada

Franc¸ois Des Rosiers

Received December 2010 Accepted March 2011

Jean Dube´ De´partement socie´te´s, territoires et de´veloppement, Universite´ du Que´bec a` Rimouski (UQAR), Rimouski, Canada, and

Marius The´riault Graduate School of Land Planning and Regional Development, Laval University, Quebec City, Canada Abstract Purpose – Both hedonics and the traditional sales comparison approach are derived from a similar paradigm with respect to how prices, hence market values, are determined. While the hedonic approach can provide reliable estimates of individual attributes’ marginal contribution, it may – unlike the sales comparison approach – underestimate the prominent influence that surrounding properties exert on any given nearby housing unit and sale price. This paper seeks to develop a simple method for reconciling the two approaches within a rigorous conceptual and methodological framework. Design/methodology/approach – Peer effect models, an analytical device developed, and mainly used, by labour economists, are adapted to the hedonic price equation so as to incorporate nearby properties’ influences, thereby controlling for non-observable neighbourhood effects. In addition to basic, intrinsic, building and land attributes, the ensuing model accounts for three types of effects, namely endogenous interactions effects (i.e. comparable sales influences, or peer effects), exogenous, or neighbourhood, effects and, finally, spatial autocorrelation effects. Findings – Preliminary findings suggest that integrating peer effects in the hedonic equation allows bringing out the combined impacts of endogenous, exogenous and spatially correlated effects in the house price determination process, with spatial autocorrelation of model residuals being significantly reduced, even without resorting to a spatial autoregressive procedure. Research limitations/implications – Further investigation is still needed in order to find out which submarket delineation should be used to obtain optimal model performances. Originality/value – The paper leads to the conclusion that the comparable sales approach, as used in traditional appraisal practice, is valid, although its application is typically flawed by the too small sample size generally used by appraisers. Further investigation is still needed, however, in order to find out which submarket delineation should be used to obtain optimal model performances. This raises the paramount question as to whether the peer effect variable is adequately measured and addresses the tricky issue of kernel determination in spatial statistics and related applications, such as GWR. Keywords Hedonic price modelling, Peer effect models, Property values, Asset valuation, Property Paper type Research paper

Journal of Property Investment & Finance Vol. 29 No. 4/5, 2011 pp. 510-528 q Emerald Group Publishing Limited 1463-578X DOI 10.1108/14635781111150376

This research was funded by the Canadian SSHRC (Social Sciences and Humanities Research Council) and by the Quebec Research Fund on Society and Culture (FQRSC). Authors are grateful to Quebec City’s Assessment Service for giving them access to the 1986-1996 home transactions database. All three authors are regular members of Laval University’s Centre de Recherche en Ame´nagement et De´veloppement (CRAD).

1. Context and objective of research Under the hedonic theory (Rosen, 1974), the market price of a complex good (car, housing, etc) may, under a series of assumptions, be broken down into its attributes’ marginal contributions, with the shadow, or implicit, prices of attributes also reflecting the willingness-to-pay (WTP) of buyers. In the case of housing, property attributes include building and land intrinsic, physical characteristics (size, age, quality, presence of a garage, etc) as well as a series of exogenous location, neighbourhood and environmental attributes. Under traditional appraisal standards, comparable sales[1] of nearby properties, assumed to bear greater resemblance to a subject property than sales located further away, are used in order to derive a reliable market value. Although the hedonic framework can be said to vary substantially from the traditional sales comparison approach used in real estate appraisal in that the former rests on much stronger conceptual grounds than the latter while benefiting from large transaction samples that enable statistical inference, both are derived from a similar paradigm with respect to how prices, hence market values, are determined. Consequently, the two approaches could be viewed as complementing each other, each one presenting advantages and drawbacks. Thus, while the hedonic approach is much more explicit about the determinants of property values and can provide reliable estimates of individual attributes’ marginal contribution, it may – unlike the sales comparison approach – underestimate the prominent influence that surrounding properties exert on any given nearby housing unit and sale price. Over the past 15 years, the hedonic framework has benefited from major methodological developments, such as Geographically Weighted Regression (GWR – Fotheringham et al., 1998, 2002), which allows local implicit prices to be derived. While the latter accounts for the influences that neighbouring properties exert on local implicit prices, it is not exempt from methodological flaws, which can eventually lead to major inconsistencies (Bitter et al., 2007). In this paper, a simple method is developed for reinserting the comparable sales concept within a rigorous theoretical and methodological framework. It is based on peer effect models, an analytical device developed, and mainly used, by labour economists, which we adapt to the hedonic price equation so as to incorporate nearby properties’ influences, thereby controlling for non observable neighbourhood effects. In addition to basic, intrinsic, building and land attributes, the ensuing model accounts for three types of effects, namely endogenous interactions effects (i.e. comparable sales influences, or peer effects), exogenous, or neighbourhood, effects and, finally, spatial autocorrelation (SA) effects. The paper is composed of six sections. Section 2 provides a literature review on peer effect models and their applications. The conceptual framework of peer effect models is developed in the third section, where it is shown how the concept may be adapted to the hedonic price equation. In the fourth section, the database used for model calibration is presented while regression results are reported and discussed in the fifth section. Finally, a summary of findings and concluding comments end the paper. 2. Literature review Peer effects can be defined as the influence that members of a group exert on a given individual in the group. While such effects seem obvious and have been mentioned in the literature (Leibenstein, 1950; Veblen, 1899), the modern economic theory has mainly focused on the mathematical formulation of the mechanisms underlying market

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equilibrium and price determination. It is only since the 1970s, on the grounds of the Game theory (Von Neumann and Morgenstern, 1944; Nash, 1950), that the foundations of economics were challenged and that the door was opened to the possibility of social interactions in the determination of market equilibrium. Pioneering work by Asch (1956) and by Becker and Becker (1998) emphasized such interactions with respect to the individual choice process. With regard to labour economics, it is the publication, in 1966, of the Coleman report (Coleman et al., 1966) on student performances that sparked off this new paradigm. Among the main statements of the report is the fact that social interactions emerge as a major determinant of students’ level of achievement and that peer characteristics remain the best predictor of school performances. While such statements are interesting from an applied perspective, some methodological weaknesses in the report result in peer effects being overestimated. In the literature on peer effects, the research by Manski (1993) remains a seminal piece of work. According to the author, three factors lead the member of a group to adopt a behaviour which is similar to that of the other members: (1) endogenous interactions; (2) exogenous interactions; and (3) correlated effects. With endogenous interactions, any individual in the group is affected by the average behaviour of the group. While such influences are currently recognized in real estate valuation through the principle of conformity and taken account of through the comparable sales approach, they are usually ignored in traditional hedonic price models although they might explain part of the variations in prices (Dubin, 1998; Can, 1992; Dubin and Sung, 1987; Anselin and Can, 1986). As for exogenous, or contextual, interactions, they namely refer to the influence exerted on individuals by the socio-economic profile of group members (average income, educational level, household composition, level of criminality, etc) as well as by neighbourhood and environmental attributes (individual mobility, access to services and infrastructures, presence of mature trees, etc). Such dimensions are already amply documented in the hedonic modelling literature. Finally, correlated, or latent, effects stem from non-observable environmental attributes that apply to all group members and should be distinguished from measurable effects. Technically speaking, such latent influences can be brought forward as an explanation for the presence of SA in the model residuals. While this phenomenon is often encountered in spatial analysis and may cause some bias in the estimated regression parameters (Pace et al., 1998a; Can and Megbolugbe, 1997; Anselin and Rey, 1991; Can, 1990; Anselin and Griffith, 1988), several methods have been designed for dealing with the issue, among which spatial autoregressive procedures (Dubin et al., 1999; Pace et al., 1998b) and geographically weighted regression, or GWR (Fotheringham et al., 2002, 1998). Peer effect models are mainly resorted to in microeconomics and labour economics. Some authors have investigated the impact of peers on individual productivity at work (Ichino and Maggi, 2000) or on entrepreneurship determinants (Nanda and Sørensen, 2008). Others have focused on how peer influence affects school performances (Hallinan and Sørensen, 1983; Sacerdote, 2001; Zimmerman, 2003). In that respect, some

studies have laid emphasis on the assessment of either exogenous (Aaronson, 1998) or endogenous (Gaviria and Raphae¨l, 2001; Fertig, 2002; Rivkin, 2001; Evans et al., 1992; Case and Katz, 1991) effects since a joint assessment of both impacts raises econometric problems related to the estimation of model parameters as well as to the identification of true social interaction effects, as opposed to unobservables, within a group (Lee, 2006). Yet, recent research by Lin (2005) and Lee (2006) has led to simultaneously estimating exogenous and endogenous peer effects using a spatial autoregressive procedure, while controlling for correlated effects stemming from unobservables. According to Lee (2006), it is possible to use a fixed-effects model while identifying endogenous influences, provided within-group variations are important enough. Durlauf and Young (2001) are among authors who can be credited with having influenced this novel methodological trend, referred to as the new social economy. Several similarities exist between empirical issues dealt with in labour economics and those encountered in urban and real estate economics. Already, social interaction models have been used as a basis for modelling urban housing markets (Meen and Meen, 2003). Moreover, the idea that some spatial, hard-to-measure, determinants do affect the price determination process is widely accepted in real estate modelling research. While both exogenous and correlated effects are well documented in the relevant literature, investigating the impact of endogenous effects on property prices has not triggered much attention among authors. This is the gap this research aims at filling. 3. Accounting for peer effects in hedonic price modelling The hedonic price model was first developed to evaluate the implicit price of the different physical characteristics forming a bundle of goods (Rosen, 1974). Under the hedonic theory, the sale price, y, of a complex good, such as real estate, is expressed as a function of all the observable physical characteristics, X, of the goods (equation 1): y ¼ X b þ u;

ð1Þ

where b is a vector of the implicit (or hedonic) price of the characteristics, while u is an error term, which is supposed to be independent and identically distributed (iid). However, since real estate goods have an important geographic dimension, some non-negligible part of the variance in sale price is usually related to environmental amenities, which must consequently be incorporated in the hedonic equation (equation 2): y ¼ X b þ Zu þ u;

ð2Þ

where u is a vector of the implicit price of the environmental amenities (Z). The latter hedonic price equation can be viewed as a special case of peer effect models. Following Manski (1993), the peer effect model is obtained by regressing the dependent variable for an individual i belonging to the group g, termed Yig, not only on the individual’s attributes, Xki, and on the contextual characteristics of the group (excluding individual i ), Xg, but also on the mean value of the endogenous variable for the group, Yg, with individual i being excluded from the calculation. Thus, we can write: Y ig ¼ gD Y g þ b1 X ki þ b2 X g þ 1ig ;

ð3Þ

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where the gD and bz parameters measure the endogenous and exogenous (or contextual) effects, respectively. In the context of the property price equation, Xki accounts for property attributes while the contextual term, Xg, may be viewed as the vector of neighbourhood characteristics, including homeowners’ socio-economic profile. The originality of this hedonic price equation lies with the addition of the endogenous effect variable, Yg, which stands for the mean sale price of comparable houses nearby property i. An issue may be raised here as to whether the proposed approach is a relevant one as applied to real estate appraisal, in light of the spatial autocorrelation (SA), or dependence, problem, which systematically affects space-related phenomena. Let us recall that spatial dependence may stem from various spatial autoregressive processes involving either the error term (induced spatial dependence) or the response variable (inherent spatial autocorrelation) (Kissling and Carl, 2008; Anselin, 2003). In the former case, which is the most widespread, a spatial error model (SEM) procedure is required to correct for spatial dependence influences and the hedonic equation can write as: Y ¼ X b þ l We þ u;

ð4Þ

where l is the spatial autoregression coefficient, W is the spatial weights matrix, b is a vector representing the slopes associated with the explanatory variables in the original predictor matrix X, and u is the usual error term independently and identically distributed (iid) while e is the spatially autocorrelated error term. In the latter case, it is assumed that the autoregressive process occurs only in the dependent variable Y, which calls for a SAR-Spatial Lag (SAR) correction procedure. The hedonic equation is then expressed as: Y ¼ rWY ¼ X b þ u;

ð5Þ

where r is the autoregression coefficient. Finally, spatial autocorrelation can affect both response and explanatory variables, which requires that a SAR mixed model (SARmix) be used instead. As shown by Kissling and Carl (2008), SEM models tend to outperform standard OLS and other SAR procedures in terms of reliability, that is, for minimizing unpredictable biases in parameter estimates. While the authors’ findings apply to ecological data, there is some empirical evidence that such a conclusion still holds in a hedonic price modelling context. In that sense, the peer effect approach may rightfully be considered as an adequate substitute for the SAR procedure while conveniently complementing the SEM model. Indeed, considering that some SA remains in the model even after the SEM procedure has been performed on the OLS residuals, including a peer effect variable within the latter should be expected to lessen spatial dependence almost completely. This said, the peer effect model specification adopted in equation 3 is somewhat similar to the usual SAR, lagged-variable, model (equation 5) used to correct for the presence of SA among residuals. However, the peer effect specification, used in this paper as a substitute for the SAR approach, differs from the latter in that the peer effect variable, Yg, is not built using a spatial weight matrix based on pure geographic distance but, instead, is computed using the mean sale price over a given sub-market. In order to avoid the estimation problem that may arise due to the fact that the latter is itself endogenous to the system and, consequently, ought to be instrumentalized[2],

yg is being expressed, for any given submarket, as the mean sale price of houses for the previous quarter; moreover, property i sale price is excluded from the computation. Such information is quite easily available, in as much as submarket delimitations are known a priori. Consequently, the usual OLS approach may be resorted to for estimating the parameters of the hedonic equation. While the suggested approach may solve most of the SA problems encountered, it is impossible to tell, a priori, whether it can eliminate them completely. For the residual spatial autocorrelation to be handled, a spatial error model (SEM – equation 4) may be adopted as a complement to the peer effect specification. Computing the Moran’s I allows to assess the importance of correlated effects in the model and to determine whether the use of a SEM procedure is needed for getting rid of sub-optimality in the coefficients. Since the Moran’s I statistics only deals with the spatial dimension though, it cannot account for time and should therefore be interpreted with caution[3]. 4. The database This research rests on a Canadian database provided by the former Quebec Urban Community (CUQ) Assessment Division[4] on some 15,729 sales of single-family detached[5] houses that took place on the former CUQ territory between January 1990 and December 1996, with prices ranging from $50,000 (Can.) to $250,000. The database contains reliable information on sale prices and major property attributes, namely: building type and age, living area and lot size, interior quality descriptors, presence of specific features (a fireplace, a finished basement, hardwood floors, a garage, etc.) and access to local water and sewerage systems. A time variable is included in the model in order to control for the trend in prices characterizing the study period. Socioeconomic and household structure dimensions are also accounted for, with descriptors being drawn from Canada’s 1996 census data that include the average household income, the number of single-parent families as well as the percentage of university degree in the neighbourhood. As for access to regional services, it is accounted for through a PCA-derived factor score obtained by applying factor analysis (principal component analysis – PCA) to a set of car travel times from home to selected regional services[6] (The´riault et al., 2003). Quebec City territory is divided into seven submarkets (Figure 1) derived from a discriminant analysis and capturing major features of historical development stages, social fabric and built environment (Voisin et al., 2010). While the type of market segmentation may affect the optimal functional form of the hedonic function (Dube´ et al., 2010) and that other approaches may be used for that purpose, the one used here proves quite satisfactory. Thus, based on physical and environmental housing attributes, it was possible to correctly classify some 88 per cent of properties as opposed to only 79 per cent when administrative boundaries are used. Finally, endogenous effects are estimated using mean submarket house price for the previous quarter. Descriptive statistics for main building and land attributes are reported for each submarket in Table I. As expected, mean sale prices are highest for the Upper-City and Cap-Rouge/St-Augustin/Laurentien submarkets, as a result of larger liveable areas and substantially higher household income indices[7]. Yet, it is on the grounds of educational level that these two submarkets most distinguish themselves from the rest

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Figure 1. Residential submarket delimitation for Quebec City

1990

1991

1993

Upper-City (1) No. sales 409 406 333 Sale price ($) 128,531 132,168 140,456 Liveable area (m2) 130 131 131 Lot size (m2) 582 583 585 Building age (years) 31 34 36 Lower-City (2) # Sales 204 179 183 Sale price ($) 73,286 71,567 79,032 Liveable area (m2) 109 111 109 Lot size (m2) 488 461 457 Building age (years) 31 38 40 Cap-Rouge/St-Augustin/Laurentien (3) No. sales 461 416 396 Sale price ($) 116,196 121,378 124,160 122 127 126 Liveable area (m2) Lot size (m2) 687 693 883 Building age (years) 9 10 12 South Charlesbourg (4) No. sales 461 408 429 Sale price ($) 87,710 90,744 91,513 Liveable area (m2) 105 108 107 Lot size (m2) 642 677 659 Building age (years) 20 21 23 South Beauport (5) No. sales 419 439 356 Sale price ($) 87,026 88,614 93,371 Liveable area (m2) 103 106 109 Lot size (m2) 605 582 604 Building age (years) 13 14 17 Ancienne-Lorette/Neufchaˆtel/Lorretteville/Lebourgneuf (6) No. sales 659 590 580 Sale price ($) 87,357 89,870 95,868 2 103 105 106 Liveable area (m ) Lot size (m2) 616 628 635 Building age (years) 14 15 16 Peri-urban area (7) No. sales 666 657 751 Sale price ($) 75,023 74,464 80,143 2 Liveable area (m ) 93 93 95 Lot size (m2) 813 755 784 Building age (years) 13 13 13

1994

1995

1996

315 137,861 132 593 38

222 136,018 133 587 39

259 125,847 129 603 37

210 80,770 114 398 33

156 76,549 109 447 42

159 78,849 113 518 42

348 119,852 126 752 14

277 122,468 127 691 14

148 124,427 130 662 12

376 93,719 111 660 23

340 90,280 108 674 25

78 91,027 106 681 24

338 92,157 107 751 19

283 89,911 103 590 17

85 92,090 110 648 21

643 97,115 108 640 17

441 93,684 108 646 19

324 98,945 112 641 17

736 80,686 94 829 14

561 79,504 96 713 14

486 79,433 96 731 15

Notes: aSafe for No. sales, statistics refer to mean values

of the city: indeed, well above 40 per cent of local residents hold a university degree, which is between twice to four times as much as what is found elsewhere. As shown in previous research on Quebec City, the correlation between house prices and educational level is even stronger than that obtained with income.

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Table I. Descriptive statistics for main building and land attributes, 1990-1996a

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Table II summarizes the descriptive statistics relative to access and socio-economic attributes. Accessibility indices are computed based on commute time from home, by car and on foot, to nearby services, with the information being drawn from the 1991 Origin-Destination (O-D) survey carried out at the region level. Such indices – which are expressed here as standardized values – are derived from a principal component analysis (PCA – Des Rosiers et al., 2000) and stand for two types of services operating at different hierarchical levels, namely regional services (major shopping centres, colleges and universities, etc) and local services (neighbourhood shopping centres, primary and secondary schools, etc). In so doing, all the information is preserved. Accessibility indices have been shown to emerge as significant determinants of value in hedonic price models without causing excessive colinearity among explanatory variables (VIFs not reported in this paper). Finally, three socio-economic indices have been computed based on the 1991 Canadian census, which stand for exogenous effects within submarkets. The first two are standardized values and refer to the number of lone-parent families and median household income in the neighbourhood, respectively, while the third index reports the percentage of individuals holding a university degree and is used as a proxy for the level of gentrification. 5. Main regression findings 5.1 Ordinary least squares method Hedonic price modelling is performed using a semi-log, log-linear functional form, with liveable area, building age and lot size also being applied a logarithmic transformation[8]. Four different specifications are used, whose results are reported in Table III. The first specification, referred to as the Base Model (Column 1), only includes property physical and land attributes as well as time dummies (reference ¼ 1990). In the second specification (Column 2), exogenous, or neighbourhood, influences are added to the Base Model. Finally, the third specification yields the Global Model (Column 3) which incorporates both exogenous and endogenous, or peer, effects, thereby allowing us to assess whether, and to what extent, these combined influences affect the magnitude and statistical significance of regression parameters as well as the presence of spatial dependence among residuals[9]. As can be seen from Table III, building, land and time attributes of the Base Model explain, as expected, as much as 69 per cent of house price variations. All property descriptors’ coefficients are most consistent in signs and magnitudes and corroborate previous findings. Time dummy coefficients also suggest that sale prices have experienced a rise until 1993, followed by a substantial drop thereafter. This is consistent with the property slump, which affected the Quebec metropolitan region (QMR) for most of the decade, with an inversion in the trend by 2000. Considering the detrimental effect of spatial dependence though, as shown by the high value of the Moran’s I statistics, it is premature to comment on the true value of parameter estimates at this point. Furthermore, comparing Table III models, it is obvious that some property attributes display greater instability in their magnitude than others, depending on the specification used. This is particularly the case with the local tax rate, which encompasses a large array of local amenities and is therefore prone to being

1990 Upper-City (1) Regional Acessibility Index Local Accessibility Index Nb. of Lone-parent Hslds. Hsld. Median Income ($) % of University degree in Nbhd. Lower-City (2) Regional Acessibility Index Local Accessibility Index Nb. of Lone-parent Hslds. Hsld. Median Income ($) % of University degree in Nbhd.

1991

1993

1994

1995

1996

1.308 1.282 1.296 1.286 1.293 1.262 20.105 20.131 2 0.087 2 0.089 20.094 20.094 l 2.260 2.247 2.285 2.198 2.194 2.257 u 3.921 3.905 3.802 3.738 4.004 3.645 44.205 43.168 44.114 43.437 44.139 42.632

l u

0.515 0.482 2.281 2.520 13.178

0.554 0.487 2.548 2.429 13.653

0.594 0.422 2.541 2.391 13.609

0.619 0.296 2.897 2.444 12.915

0.569 0.481 2.578 2.377 12.485

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0.557 0.491 2.452 2.442 13.401

Cap-Rouge/St-Augustin/Laurentien (3) Regional Acessibility Index 0.226 0.252 0.218 0.232 0.210 0.388 Local Accessibility Index 21.258 21.291 2 1.421 2 1.421 21.438 21.017 Nb. of Lone-parent Hslds. l 1.750 1.763 1.743 1.740 1.770 1.972 u 4.361 4.347 4.250 4.206 4.235 4.050 Hsld. Median Income ($) % of University degree in Nbhd. 41.950 41.132 40.773 39.457 40.313 37.959 South Charlesbourg (4) Regional Acessibility Index Local Accessibility Index Nb. of Lone-parent Hslds. Hsld. Median Income ($) % of University degree in Nbhd.

20.672 20.675 2 0.728 2 0.667 20.749 20.651 0.769 0.764 0.764 0.753 0.773 0.735 l 2.543 2.516 2.540 2.481 2.441 2.222 u 3.391 3.475 3.375 3.457 3.391 3.595 20.125 20.819 20.079 20.476 20.110 21.937

South Beauport (5) Regional Acessibility Index Local Accessibility Index Nb. of Lone-parent Hslds. Hsld. Median Income ($) % of University degree in Nbhd.

20.835 20.836 2 0.900 2 0.866 20.869 20.918 0.190 0.195 0.177 0.173 0.143 0.228 l 1.982 1.935 1.928 1.914 1.858 1.880 u 3.212 3.181 3.132 3.120 3.156 3.144 16.462 16.345 16.129 15.575 16.156 14.836

Ancienne-Lorette/Neufchaˆtel/Lorretteville/Lebourgneuf (6) Regional Acessibility Index 20.550 20.550 2 0.555 2 0.483 20.527 20.319 Local Accessibility Index 0.355 0.370 0.352 0.335 0.349 0.281 Nb. of Lone-parent Hslds. l 2.153 2.135 2.058 2.068 2.101 1.872 Hsld. Median Income ($) u 3.331 3.272 3.390 3.340 3.340 3.328 % of University degree in Nbhd. 19.469 18.656 19.626 19.011 19.691 20.368 Peri-urban Area (7) Regional Acessibility Index Local Accessibility Index Nb. of Lone-parent Hslds. Hsld. Median Income ($) % of University degree in Nbhd.

21.175 21.208 2 1.172 2 1.206 21.169 21.158 20.551 20.556 2 0.605 2 0.639 20.565 20.606 l 1.959 1.921 1.943 1.960 1.984 1.993 u 2.895 2.822 2.881 2.873 2.835 2.847 10.769 10.681 10.697 10.430 10.788 9.853

Notes: Legend: l, u: standardised indices

Table II. Descriptive statistics for access and socio-economic attributes, 1990-1996

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Table III. Regression results – ordinary least square (OLS) method

Base model (prop. and time attr.) Coefficient Sig. Ln_Liveable Area (m2) Ln_Lot Size (m2) Ln_Building age (years) Cottage Attached Quality Index Nb. of bathrooms Finished basement Brick facing (51% þ ) Nb. of fireplaces Superior quality floors Hard wood staircase Sup. quality kitchen cabinet Inferior luminosity Cathedral ceiling Central vacuum Simple attached garage Double attached garage Simple detached garage Double detached garage Presence of a terrace Excavated pool Access to water and sewage Local Tax Rate Year 1991 Year 1993 Year 1994 Year 1995 Year 1996 Regional accessibility index Local accessibility index Nb. of lone-parent families Median household income Percentage of university degree holders Endogenous effects Intercept n R-squared RMSE AIC Moran’s I

0.5479 0.0273 2 0.0788 2 0.0716 2 0.1299 0.1481 0.0550 0.0528 0.0438 0.0607 0.0386 0.0492 0.0532 2 0.0259 0.0471 0.0385 0.1345 0.0942 0.0366 0.0339 0.0500 0.0993 0.1661 2 0.2144 0.0305 0.0458 0.0258 0.0068 0.0088

Base þ NBHD attr. Global model (all var. (exog. effects) included) Coefficient Sig. Coefficient Sig.

*** *** *** *** *** *** *** *** *** *** *** *** * *** *** *** *** *** *** *** *** *** *** *** *** *** ***

0.4339 0.0752 2 0.1033 2 0.0549 2 0.1548 0.1148 0.0416 0.0435 0.0180 0.0433 0.0202 0.0401 0.0190 2 0.0193 0.0331 0.0431 0.1123 0.0920 0.0320 0.0586 0.0329 0.0908 0.1238 2 0.0756 0.0232 0.0632 0.0604 0.0430 0.0438 0.0535 0.0335 2 0.0060 0.0080

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ***

0.4293 0.0744 20.1053 20.0503 20.1487 0.1145 0.0397 0.0443 0.0183 0.0448 0.0188 0.0392 0.0239 -0.0186 0.0334 0.0438 0.1101 0.0933 0.0346 0.0620 0.0320 0.0934 0.1216 20.0421 0.0101 0.0420 0.0400 0.0320 0.0245 0.0455 0.0334 2 0.0088 0.0103

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ** *** *** *** * *** *** *** *** *** *** *** ***

0.0058 * * * 9.0482 * * * 8.9648 15,729 15,729 0.6918 0.7666 0.1797 0.1564 2 9,329.83 2 13,691.68 0.1486 * * * 0.0919

0.0040 * * * 0.2478 * * * *** 6.1428 * * * 15,729 0.7720 0.1546 214,055.03 *** 0.0446 * * *

Notes: Signif. level: * p , 0.05; * * p , 0.01; * * * p , 0.001

affected downward when other contextual attributes are brought in the equation. Time dummies also prove to be quite unstable when the endogenous effects alone are added to the Base Model, with related coefficients exhibiting counterintuitive signs and loosing their statistical significance.

Adding exogenous determinants results in a substantial improvement in the model explanatory power (R-squared raises to nearly 0.77) and, to a lesser extent, in its predictive performance (RMSE down to 0.156). All contextual parameters emerge as highly significant and with consistent signs. As expected, regional accessibility impacts more heavily on property prices than local accessibility does. Accounting for exogenous effects also reduces the marginal contribution of several building characteristics (living area, brick facing, superior quality kitchen cabinet, access to public services and local tax rate) while others have theirs raised (lot size and building age). Year dummy coefficients are also affected upward: not only do they yield higher price increases than those obtained with the Base Model, but the trend thus brought out is more consistent with the facts; indeed, while house values have actually dropped following 1993, they never got back to their 1990 level, as suggested with the first specification. Above all though, spatial dependence is greatly reduced when accessibility factors and socio-economic variables are accounted for, although it remains highly significant with a Moran’s I value of 0.092. Finally, regression findings suggest that that the Global Model, which includes both endogenous and exogenous effects, provides the best performances on all fronts. This is clearly brought out by the Akaike (1974) Information Criterion (AIC)[10] which makes it possible to compare the relative performance of models that differ as to their specification and/or functional form. Adding on accessibility and socio-economic attributes translates into a higher R-squared, a lower RMSE and a Moran’s I value which, although still significant, displays a slight improvement over the Peer Effect Model. As can be seen, the coefficient of the endogenous effect variable, which stands at 0.248, displays a high statistical significance. As for property, contextual and time variables, they display stabilized estimates that are consistent in both sign and magnitude, with significance levels of 0.05 or lower – safe for one descriptor. Considering that latent spatial effects are still significant in the residuals in spite of the fact that peer effects are being accounted for in the model, resorting to a SEM procedure[11] is advisable. 5.2 Applying an SEM procedure Regression results obtained with a SEM procedure are reported in Table IV. The model specifications applied here are the same as those used within the OLS framework. First, it can be seen that the SEM procedure clearly yields improved overall performances when compared with the OLS method – all the more so for the Base Model specification, which generates a 0.77 R-squared. As expected, and as shown by the magnitude of the Lambda parameter, in excess of 0.97 and significant at the 0.001 level, it is most efficient at solving SA problems, at least to a large extent. Thus, even with the first specification, the Moran’s I value drops from 0.1486 (OLS) to a mere 0.0092 (SEM); with the Global Model, it is down to 0.0006 and only significant at the 0.05 level. Second, controlling for spatial dependence results in regression parameters displaying a much greater consistency throughout the spectrum of housing characteristics, even for attributes whose coefficients tend to exhibit pronounced variability under the OLS procedure. This is particularly the case with the local tax rate attribute – as it is also with time dummies – whose stabilized impact estimate at, roughly, 2 0.06 seems quite sensible in light of the literature on tax capitalization issues (Yinger et al., 1987). A possible explanation for such variability rests with the

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Table IV. Regression results – spatial autoregressive error model procedure (SEM)

Base model (property attr.) Coefficient Sig. Ln_Liveable area (m2) Ln_Lot size (m2) Ln_Building age (years) Cottage Attached Quality index Nb. of bathrooms Finished basement Brick facing (51% þ ) Nb. of fireplaces Superior quality floors Hard wood staircase Sup. quality kitchen cabinet Inferior luminosity Cathedral ceiling Central vacuum Simple attached garage Double attached garage Simple detached garage Double detached garage Presence of a terrace Excavated pool Access to water and sewage Local tax rate Year 1991 Year 1993 Year 1994 Year 1995 Year 1996 Regional accessibility index Local accessibility index Nb. of lone-parent families Median household income Percentage of university degree holders Endogenous effects Intercept Lambda n R-squared RMSE AIC Moran’s I

0.4374 0.0983 20.1173 20.0494 20.1438 0.1157 0.0417 0.0473 0.0251 0.0507 0.0194 0.0454 0.0344 20.0207 0.0336 0.0463 0.0970 0.0845 0.0206 0.0461 0.0310 0.0999 0.1380 20.0618 0.0187 0.0594 0.0580 0.0435 0.0439

Base þ NBHD attr. (exog. effects) Coefficient Sig.

Global model (all var. included) Coefficient Sig.

0.4125 * * * 0.0970 * * * 2 0.1132 * * * 2 0.0459 * * * 2 0.1434 * * * 0.1136 * * * 0.0408 * * * 0.0445 * * * 0.0209 * * * 0.0440 * * * 0.0201 * * * 0.0434 * * * 0.0229 2 0.0200 * * * 0.0291 * * * 0.0455 * * * 0.0970 * * * 0.0890 * * * 0.0233 * * * 0.0528 * * * * 0.0226 * ** 0.0925 0.1184 * * * 2 0.0578 * * * 0.0202 * * * 0.0626 * * * 0.0631 * * * 0.0469 * * * 0.0492 * * * 0.0139 * * 0.0288 * * * 2 0.0049 * * * 0.0089 * * *

0.4072 * * * 0.0967 * * * 20.1133 * * * 20.0441 * * * 20.1441 * * * 0.1155 * * * 0.0401 * * * 0.0441 * * * 0.0202 * * * 0.0445 * * * 0.0199 * * * 0.0434 * * * 0.0244 20.0200 * * * 0.0297 * * * 0.0452 * * * 0.0952 * * * 0.0898 * * * 0.0237 * * * 0.0543 * * * * 0.0197 * ** 0.0942 0.1206 * * * 20.0572 * * * ** 0.0104 0.0412 * * * 0.0392 * * * 0.0321 * * * 0.0250 * * * * 0.0127 * * * 0.0214 2 0.0051 * * * 0.0102 * * *

0.0043 * * *

0.0036 * * * 0.2623 * * * 5.9887 * * * 0.9713 * * *

*** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** *** ** *** *** *** *** *** *** *** ***

8.4993 * * * 0.9953 * * * 15,729 0.7699 0.1551 2 13,343.80 0.0092

8.9106 * * * 0.9844 * * *

15,729 0.7824 0.1508 2 14,225.50 *** 0.0039

Notes: Signif. level: * p , 0.05; * * p , 0.01; * * * p , 0.001

15,729 0.7852 0.1499 214,429.40 *** 0.0006

*

fact that the local tax rate is the only “spatial” variable included in the first specification. Consequently, its parameter captures most of the spatial effect, as suggested by the drop in the coefficient obtained with the second specification of the OLS procedure. Last but not least, regression findings obtained with the SEM procedure corroborate the usefulness of the endogenous effect variable in the hedonic price function, even where spatial dependence is controlled for. Indeed, under the Global Model specification, its parameter reaches 0.2623 (p , 0:001) and remains highly significant. Accounting for peer effects also contributes to lowering SA in the residuals, as suggested by the sharp drop in the Moran’s I value which stands at 0.0006 in the Global Model, down from 0.0039 as obtained with the second model specification. Based on the AIC, the latter once again outperforms all other specifications: its explanatory performance stands at 78.5 per cent, its RMSE is down to 0.15 – which makes it suitable for predictive purposes – while it is quasi-exempt from any spatial autocorrelation influence. Moreover, all regression coefficients are in line, in both magnitude and sign, with theoretical expectations. 5.3 Discussion Findings suggest that peer, endogenous, effects do act as a significant determinant of property values. Furthermore, when used in combination with exogenous attributes in the hedonic price equation, they prove quite efficient at reducing the extent of spatial dependence in the model residuals. Finally, even where a spatial autoregressive procedure is applied so as to explicitly account for spatial autocorrelation influences, the peer effect variable parameter still emerges as being highly significant and contributes to lessen SA still further. A major issue, which should be addressed here refers to the optimal size of the submarkets which serve as a basis for peer effect computation. In that respect, one may question the relevance of using only seven submarkets as a basis for measuring peer effects, which could be expected to affect neighbouring properties only. Indeed, a priori, the larger the submarket, the greater the variance of the lagged, peer effect variable within that submarket. This argument is all the more appropriate considering that, in this paper, peer effects are measured based on sales that occurred over the last three months – and not over the whole six-year period – which provides the assurance that some variability will be found in the variable. As discussed above, such a procedure also takes care of the endogeneity problem. This said, in order to assess the sensitivity of peer effect model results with respect to submarket delimitation, the model has been re-estimated using 17, as opposed to seven, spatial sectors. Regression findings for selected variables are reported in Table V. While year dummies as well as socio-economic attributes display a somewhat greater sensitivity in their parameter estimate following a change in submarket delimitation, regression coefficients remain, by and large, quite stable whether seven or seventeen submarkets are used. This said, such a shift results in a sharp decrease in the endogenous effect regression coefficient, which drops from 0.2478 to 0.1066 while still remaining statistically significant at the 0.001 level. This suggests that increasing the number of submarkets does not, as expected, necessarily enhance the explanatory contribution of peer effects to house value; quite the opposite, it seems to lessen it, at least in this particular case. This may be due to the fact that increasing the number of

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Table V. OLS regression results for selected variables – Global Model – comparing 7 versus 17 submarkets

7 Submarkets Coefficient Sig. Ln_Liveable area (m2) Ln_Lot size (m2) Ln_Building age Cottage Attached Quality index Access to water and sewage Local tax rate Year 1991 Year 1993 Year 1994 Year 1995 Year 1996 Regional accessibility index Local accessibility index Nb. of lone-parent families Median household income Percentage of university degree holders Endogenous effects Intercept n R-squared RMSE AIC Moran’s I

0.4293 0.0744 2 0.1053 2 0.0503 2 0.1487 0.1145 0.1216 2 0.0421 0.0101 0.0420 0.0400 0.0320 0.0245 0.0455 0.0334 2 0.0088 0.0103 0.0040 0.2478 6.1428 15,729 0.7720 0.1546 214,055.03 0.0446

** ** ** ** ** ** ** ** * ** ** ** ** ** ** ** ** ** ** **

**

17 Submarkets Coefficient Sig. 0.4342 0.0749 20.1040 20.0555 20.1542 0.1141 0.1210 20.0573 0.0177 0.0541 0.0520 0.0396 0.0372 0.0472 0.0330 2 0.0066 0.0069 0.0053 0.1066 7.7301 15,729 0.7677 0.1561 2 13,761.44 0.0408

** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** ** **

**

Notes: Significant level: * p , 0.05; * * p , 0.001

spatial sectors reduces the number of available sales which peer effect computation is based on, thereby affecting downwards the overall influence of endogenous effects. Yet another possible explanation may lie with the fact that submarket homogeneity, rather than mere size, conditions the overall performance of the peer effect model. In summary, where peer effects are computed by including all sales that occurred over the last three months, the best model performances are obtained using seven homogeneous submarkets, as shown by most indicators reported in Table V, with the exception of the Moran’s I which is slightly lower with seventeen submarkets. 6. Conclusion – summary of findings and suggestions for further research This research aims at reconciling the hedonic framework with the traditional sales comparison approach used in real estate appraisal. For that purpose, a simple method is put forward based on peer effect models, an analytical device developed, and mainly used, by labour economists. In a house price setting context, the classic hedonic price equation may be adapted so as to incorporate nearby properties’ influences, thereby controlling for non observable neighbourhood effects. In addition to basic, intrinsic, building and land attributes, the ensuing model accounts for three types of effects, namely endogenous interactions effects (i.e. comparable sales influences, or peer effects), exogenous, or neighbourhood, effects and, finally, spatial autocorrelation effects.

The paper rests on a Canadian database provided by the former Quebec Urban Community (CUQ) Assessment Division, on some 15,700 sales of single-family detached houses that took place on the former CUQ territory between January 1990 and December 1996, with prices ranging from $50,000 (Cdn.) to $250,000. Seven spatial submarkets, derived from a discriminant analysis and capturing major features of historical development stages, social fabric and built environment, are used for the study. Hedonic price modelling is performed using a semi-log, log-linear functional form, with liveable area, building age and lot size also being applied a logarithmic transformation. Three different specifications are used, starting with the Base Model, which only includes property structural and land attributes as well as time dummies. Exogenous, or environmental, influences are then added in the second specification. Finally, endogenous, or peer, effects, expressed as the mean sale price of houses in a given submarket for the previous quarter, are added on in the third specification, referred to as the Global Model specification which includes all categories of attributes. Finally, both OLS and SEM methods are applied to the data, with the latter generating the best performances on all fronts, in as much though as peer effects are accounted for in the modelling process. To conclude, our findings lead to the conclusion that the comparable sales approach, as used in traditional appraisal practice, is a valid one, although its application is typically flawed by the too small sample size generally used by appraisers. The peer effect model allows revisiting the conformity principle which grounds the comparable sales approach, although within the more structured and rigorous framework of the hedonic price method. As discussed above, further investigation is still needed though in order to find out which submarket delineation should be used to obtain optimal model performances. This raises the paramount question as to whether the peer effect variable is adequately measured and addresses the tricky issue of kernel determination in spatial statistics and related applications, such as GWR. Notes 1. Comparable sales, or comparables, can be defined as “recently sold properties that are similar in important respects to a property being appraised. The sale price and the physical, functional, and location characteristics of each of the properties are compared to those of the property being appraised in order to arrive at an estimate of value” (AccuriZ, 2008). 2. In attempting to estimate the causal effect of some variable X on a dependent variable Y, an instrument is a third variable Z which affects Y only through the effect it exerts on X. In order to be used in a linear model, an instrumental variable (IV): should be correlated with both the dependent and the explanatory variables, conditional on the other covariates; and should not be correlated with the error term of the regression, that is, the instrument cannot suffer from the same problem as the original predicting variables. 3. A substitute to the Moran’s I statistics is actually needed if this limitation is to be overcome; this, however, is well beyond the scope of this paper. 4. Now, Quebec City Assessment Division. 5. Single-family detached units include “bungalows”, defined as single-storey units and used here as the reference category, and “town-cottages”, defined as multi-storey units. 6. Computations are performed on the basis of the 1994 regional road network. 7. This holds in spite of relatively old properties in the Upper-City.

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