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Abstract
Spatial heterogeneity, spatial dependence and spatial scale constitute key features of spatial analysis of housing markets. However, the common practice of modelling spatial dependence as being generated by spatial interactions through a known spatial weights matrix is often not satisfactory. While existing estimators of spatial weights matrices are based on repeat sales or panel data, this paper takes the approach to a cross-section setting. Specifically, based on an a priori definition of housing submarkets and the assumption of a multifactor model, we develop maximum likelihood methodology to estimate hedonic models that facilitate understanding of both spatial heterogeneity and spatial interactions. The methodology, based on statistical orthogonal factor analysis, applied to the urban housing market of Aveiro (Portugal) at two different spatial scales, provides exciting inferences on the spatial structure of the housing market.
RÉSUMÉ L'hétérogénéité spatiale, la dépendance spatiale et l’échelle spatiale sont des caractéristiques clé de l'analyse spatiale dans les marchés de l'immobilier. Toutefois, la pratique habituelle de la modélisation de la dépendance spatiale comme étant le résultat d'interactions spatiales par le biais d'une matrice de poids spatiaux n'est souvent pas satisfaisante. Alors que les estimateurs existants des matrices de poids spatiaux sont basés sur des données de panel ou des ventes répétées, la présente communication adopte le principe d'un cadre transversal. Plus spécifiquement, sur la base d'une définition à priori des sub-marchés de l'immobilier, et de l'hypothèse d'un modèle multifactoriel, nous créons une méthodologie de probabilité maximale pour estimer des modèles hédoniques qui facilitent les connaissances de l'hétérogénéité spatiale et des interactions spatiales. Cette méthodologie, basée sur une analyse des facteurs orthogonaux, appliquée au secteur de l'immobilier urbain à Aveiro (Portugal) à deux échelles spatiales différentes, fournit des inférences excitantes en ce qui concerne la structure spatiale du secteur de l'immobilier.
EXTRACTO La heterogeneidad, dependencia y escala espaciales constituyen características clave del análisis espacial de los mercados de la vivienda. No obstante, la práctica común de modelar la dependencia espacial como algo generado por interacciones espaciales a través de una matriz conocida de pesos espaciales, a menudo, no es satisfactoria. Aunque los estimadores existentes de matrices de pesos espaciales se basan en ventas repetidas o datos de panel, este estudio lleva el planteamiento a un marco de corte transversal. Específicamente, basados en una definición a priori de los submercados de la vivienda y en la presuposición de un modelo de múltiples factores, desarrollamos una metodología de probabilidad máxima para estimar modelos hedónicos, que facilita la comprensión de la heterogeneidad espacial y las interacciones espaciales. La metodología, basada en el análisis estadístico de factores ortogonales y aplicada al mercado de la vivienda urbana de Aveiro (Portugal) en dos escalas espaciales diferentes, proporciona interesantes inferencias sobre la estructura espacial del mercado de la vivienda.
Acknowledgements
The detailed review, comments and constructive criticism by two anonymous referees helped us revise and improve this article substantially. Their contribution is gratefully acknowledged. The authors are also grateful to Glen Bramley, Bernie Fingleton, George Galster, Lung-fei Lee, Duncan Maclennan, Steve Malpezzi, Jesús Mur and Gwilym Pryce, and to participants at the 9th Spatial Econometrics and Statistics Workshop (Orléans, 2010) for many helpful comments and suggestions. The usual disclaimer applies. The authors acknowledge financial support from the Portuguese Foundation for Science and Technology (FCT) on the project DONUTS(PTDC/AUR-URB/100592/2008), together with the Competitiveness Factors Thematic Operational Programme (COMPETE) of the Community Support Framework III (European Commission) and the European Community Fund FEDER. Casa Sapo and another anonymous real estate agency are thanked for permission to use their data for the empirical analyses.
Notes
1. There is considerable debate in the literature as to which of these alternatives constitute an appropriate criterion, and even whether submarkets are truly spatial entities (see Rothenberg et al., Citation1991). Here, we abstract from these issues somewhat and assume that our submarkets, at the given spatial scale, have a spatial context which we examine in terms of spatial heterogeneity and spatial dependence.
2. Inference on cross-submarket heteroscedasticity is a by-product of our methodology. However, we do not focus on this issue in the paper.
3. We are grateful to an anonymous referee for valuable suggestions that helped us understand better the role of the factor model in our setting.
4. For representative applications using hedonic models in a spatial econometric setting, see Can (1992), Pace & Gilley (1997), Basu & Thibodeau (Citation1998) and Anselin et al. (Citation2010).
5. For a setting with n spatial units under study, W is an n×n matrix with zero diagonal elements. The off-diagonal elements are typically either dummy variables for contiguity or inversely proportional to the distance between a pair of units, so that spillover between a pair of units that are farther apart is lower.
6. An analogy with well-researched urban areas in the USA is illuminative. A city like Chicago, where house prices have risen in the CBD but grown even faster in the suburbs, would imply positive spatial interaction between the centre and the periphery. By contrast, in Detroit, where prices in the centre have declined at the same time as suburban housing prices have continued to rise, suggests negative spatial interaction. We are grateful to Steve Malpezzi for pointing out this connection with urban structure.
7. See Bhattacharjee & Holly (Citation2011) for a review and discussion, as well as an application to network interactions within a committee.
8. Note that, since the spatial weights matrix is unknown in our setting, it is necessary to row-standardize W to enable identification of both W and the autoregressive parameter (λ) in Equation (Equation1). The assumption that the intra-submarket spatial weight is the same across all submarkets is not necessary, but retained here for computational simplicity.
9. That is, W 0 is symmetric, which holds if ω kl =ω lk for all l≠k. W is the row-standardized version of W 0 , and therefore will not be symmetric in general.
10. Here, A 1/2 denotes the symmetric square root of a positive definite matrix A, and A −1/2 denotes its inverse. In other words, A −1/2 has the same eigenvectors as A, but with the eigenvalues replaced by the reciprocal of the square root of the corresponding eigenvalues of A.
11. See Bhattacharjee & Holly (Citation2011) for further discussion on partial identification and structural constraints in this context.
12. We thank an anonymous referee whose comments encouraged us to examine the special features of the factor model in the cross-section context, thereby improving upon the methodology and its discussion substantially.
13. See Bhattacharjee & Holly (Citation2011) for further discussion of the conceptual distinction between strong and weak dependence and their connection to the spatial weights matrix.
14. In principle, one can allow the within-submarket spatial weights to vary across submarkets. In our empirical exercise, we abstract from this issue for the sake of computational simplicity.
15. The name of the agency is withheld because of a confidentiality agreement.
16. Flats tend to be located in areas with high residential density, which in turn generate scale economies for the provision of these infrastructure facilities.
17. Given the small sample sizes in each submarket, it is not surprising that many regression coefficients are not statistically significant. The estimates indicate that, despite small sample sizes, it is important to allow for spatial heterogeneity. Further, the limitation of sample size is counterbalanced by the benefits of estimating spatial econometric models (spatial error and spatial lag models) by maximum likelihood, which is computationally difficult on large datasets.
18. The value of the Moran's I statistic ranges from 1 (perfectly positive spatial autocorrelation) to −1 (perfectly negative spatial autocorrelation), a value near zero indicating no spatial autocorrelation.
19. Rothenberg et al. (1991) define submarkets in terms of bundle ‘quality’ (that is, close hedonic substitutability), and these sets of close substitute units may or may not have any spatial content.
20. The consolidation of a single urban area corresponding to the municipalities of Aveiro and Ílhavo was built on a territory previously organized as a set of small urban and rural clusters, each with its own provision of small scale services. Factor 2 reflects the proximity to such local centres.
21. Paradoxically, the main reason mitigating against more formal analysis of spatial structure using the estimated spatial weights matrix is large sample size. It is computationally very difficult to conduct ML-based inferences in spatial econometric models when sample size is large. Suitable ML computation for large sample applications, based perhaps on regularization or subsampling, is planned for the future.