and ) as many quantiles have been defined on the conditional distribution of rents. In order to estimate the implicit rent for owners, researchers should first assign each home owner to a quantile, hmm,qq which is not straightforward. According which distribution should owners be ordered to? Gasparini and ββ̂ Escudero (2004) use income distribution, assuming that the monetary value of the demand for nonobservable characteristics is related monotonically to the distribution of income, so that the conditional quantiles of the distribution of non-observables coincides with the quantiles of the distribution of income. In other words, individuals in the th quantile with respect to the rent distribution are also found in the th quantile of the income distribution. This might not be the case if, for instance, the quality of rented and owners-occupied dwellings differs. qq qq

# **2.1.6 Spatial Models**

Since the most important determinant of a house value and its related rental value is location, house prices are most likely to be spatially dependent. Not only dwellings in the same location are likely to have similar characteristics because they were probably developed at the same time, but they also share the same neighborhood amenities (parks, school, hospitals), or suffer from the same unpleasant features, for instance pollution or crime. Although some neighborhood characteristics can be explicitly accounted for in the hedonic model (vector h in equation 1), a location omitted variable bias might still persists (Hill, 2013). This can occur either because of true unobservable characteristics, or because of limitations in the set of variables in the questionnaire. The growing availability in surveys of geospatial NN coordinates for each dwelling may allow overcoming this problem.

Spatial dependence can be captured either in the regressors or the error term. The first approach, i.e., including location as an explanatory variable using geospatial data, is the most straightforward one. Spatial autoregressive (SAR) models allow for both spatially lagged dependent variables and spatially lagged disturbance terms. Although the SAR models are the most commonly applied, there is a wide range of alternatives, particularly semi-parametric and non-parametric spatial approaches, for example: _kriging_ (Diggle and Ribeiro 2007, Montero and Larraz, 2010), _spatial smoothing_ (Wood 2006; Wood et al. 2008), approaches based on _spatial penalization_ (Fahrmeir et al. 2013), _geographically weighted least squares_ (Fotheringham et al. 2002), and _spatial scaling factor models_ (Brunauer et al., 2010). For an elaborate discussion and a review of the literature on spatial dependence, the use of geospatial data and also on nonparametric estimation, we refer the reader to Hill (2013). A more recent approach is the inclusion of explicit spatial (distance) variables obtained from _GIS data_ (Lozano-Gracia and Anselin, 2012).

In general, the methods to introduce the spatial information in a hedonic model can be classified in the following three families: first-order autoregressive spatial models (SAM); first-order autoregressive errors models (SEM); and first-order spatial Durbin error models (SDM). In all the three cases, spatial dependence is modeled through a matrix h of spatial weights (where each entry is represented by ), with zeros on the diagonal (so that the distance between a dwelling and itself is zero). ΛΛ hkk In the case of SAM models, location is included as an explanatory variable by using the weighted λλ independent variable (the rent h ) as regressor, where weights come from the space matrix (equation 6): RR h h 0 h (6) + �MM hkk mm hmm hmm gg(R ) = �λλ ff(R ) + αα mm=1ff (XX )ββ + εε kk 8