Housing rental submarkets in hedonic regression: econometric arguments and practical application

Abstract

The paper deals with a hedonic model of housing rents focusing on geographical submarkets in the Belgian region of Wallonia. The question is what is the best way to use the delineated submarkets under the condition of a relatively small simple size. While the set of submodels for the submarkets provides an important improvement in the predictive capacity and a considerable reduction in the standard error as well as in the spatial autocorrelation in residuals, this approach has an important drawback unacceptable for the users of the Automated Valuation System “rent calculator”: in the submodels, some crucial structural variables, such as those for living area or building age, are often insignificant. This problem, caused, at least partly, by limited size of subsamples, manifests itself much less in the overall models (OLS and spatial error) with “location-quality values”. The practical advantages of these models are more important than the relative econometric superiority of the geographically weighted and multi-level alternatives.

Appendices

Appendix 1. Procedure of submarkets’ delineation

To group the municipalities into rental zones, we first select the municipalities with at least 10 observations. There are 99 such municipalities (of 262). In each of these municipalities, the median value of the “location-quality value” is calculated. Ten clusters of these municipalities are created using the Ward method. Each of the six biggest Walloon cities makes a single rental area, namely Charleroi, Liège, Namur, Mons, Tournai, and Verviers.

To create the other zones, we start with the “value influence centres”. These are municipalities with at least 30 observations, a very high (or a very low) “location-quality value” and suspected of significantly influencing their neighbouring municipalities. The examples of high value “centres” are Braine-l’Alleud, Wavre and Ottignies-Louvain-la-Neuve in the North of Walloon Brabant and Arlon in the South-East of the Luxembourg province. If adjacent municipalities belong to the same cluster, we group them to the same zone. This zone can extend further, to the zones of influence of other “centres”.

During this iterative zoning process, the following criteria are applied:

• 1) The number of observations in each zone must be more than 30.

• 2) The difference between the median values of the “location-quality values” of adjacent zones should not exceed 15%.

• 3) Administrative division (the elements of provincial borders) is considered, if possible.

• 4) The classification of municipalities by Belfius (2007) is considered (“central” municipalities, agglomerations, residential and rural areas).

Appendix 2. Model comparison

Model*	Number of insignificant variables	Adj. R²/ pseudo R²	Moran’s I in residuals and its significance	Predictions within 10%	Predictions within 20%
OLS regional initial R = X B struct. + ɛ	0	0.5999	0.137 (0.000)	46.4%	77.7%
OLS regional with submarkets’ dummies R = X (B struct., submarkets dummies) + ɛ	3 internal + 4 submarkets	0.6665	0.013 (0.000)	50.9%	81.8%
OLS regional with “location-quality” R = X (B struct., location-quality) + ɛ	3 internal	0.6672	0.015 (0.000)	51.3%	81.8%
OLS submodels For each submodel i: R i = X i B struct. i + ɛ i	10 internal to 23 internal	-	0.009 (0.004)	56.9%	86.6%
SEM regional with submarkets’ dummies R = X (B struct., submarkets dummies) + u u = λWu + ɛ	3 internal + 5 submarkets	0.6718	0.0022 (0.445)	50.6%	81.8%
SEM regional with “location-quality” R = X (B struct., location-quality) + u u = λWu + ɛ	3 internal	0.6712	0.0025 (0.396)	51.4%	81.6%
GWR regional For each observation k: r k = X B struct. + ɛ B struct. = (X T WX) −1 X T WR	1 internal*	0.6963	0.017 (0.000)	57.1%	87.1%
GWR regional, estimates aggregated by submarkets For each submodel i: ${b}_{i}=\sum _{k=1}^{n}\frac{{b}_{k}}{n}$	0 internal to 17 internal**	-	0.025 (0.000)	52.8%	84.2%
Multi-level model R = X B struct. + ɛ r1 + ɛ r2	5 internal (fixed effects) + 22 internal*** (random effects)	-	0.010 (0.002)	54.9%	86.4%

1. * The equations are presented in matrix form: R – vector of rents (r), B – vector of coefficients (structural and others, b), ɛ - error term, W – spatial weight matrix, ɛ_r1 - error term for regional level, ɛ_r2 –error term for submarket level.
2. ** In the GWR, the number of variables with no significant estimates is reported (the cases without a particular variable in a particular submarket are not counted).
3. *** In the multi-level model, the variables with zero variances are counted as well.