Hedonic pricing and the spatial structure of housing data – an application to Berlin

ABSTRACT

Housing prices are largely determined by physical location. By applying the outsample prediction accuracy of rental prices as evaluation criteria, we examine whether the choice of the hedonic model additionally depends on the spatial structure of housing data, i.e. accounting for locational effects by either district fixed effects or spatial econometric modelling. Our results show that a generalised spatio-temporal model outperforms a district fixed effects model only if the spatial density – the weighted mean distance to nearest neighbours – is relatively small. Moreover, we use the required density thresholds to deduce a pseudo indifference curve, thereby showing that the ratio of the weighted spatial distance-to-the mean district diameter increases with the mean sample size per district. This emphasises the role of data structure and district choices for model selection. Differences in data can thereby serve as an explanation for contradictory findings in literature, whether spatial econometric methods or simple district fixed effects are used.

KEYWORDS:

• Hedonic pricing
• spatial-temporal modelling
• rental prices

Acknowledgment

The authors thank the editor and three anonymous reviewers for their very valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. See also Figure 1, representing four examples of spatially structured point data with a detailed description.

2. Gelfand et al. (2004) examine property prices through the use of temporally indexed price processes by means of a more generalised class of spatio-temporal models. In order to control for heterogeneous effects endogenously, Nappi-Choulet and Maury (2011) expand the former model by means of the spatial autoregressive local estimation approach proposed by Pace and LeSage (2004). Liu (2013) expands the model proposed by Pace et al. (1998) by a yearly varying spatial dependence parameter.

3. Moreover, the use of sparse lower triangular weighting matrices speeds up the estimations, which is notably suitable in particular for repetitive computation.

4. All data in this paragraph are taken from the statistical office of Berlin-Brandenburg and the regional database from empirica AG (rental prices and vacancy rates).

5. See Rosen (1974) for a two-stage identification approach and a discussion of the assumptions under which implicit prices refer to the willingness-to-pay.

6. In the literature, no consensus has so far arisen as to how one should observe crucial locational information and on how to select and implement appropriate neighbourhood variables. However, in the last two decades the traditional hedonic pricing approach has been augmented by methodologies from spatial statistics and econometrics in order to address the mentioned limitations; see Basu and Thibodeau (1998) and Dubin et al. (1999), among others.

7. Bourassa et al. (2007) find that simple ordinary least squares models do well in terms of predictive ability. However, in consideration of the aim of this study, when comparing implicit rental prices over space and time as well as along the price distribution, there exists the need to control for locational determinants by means of a more generalised concept.

8. Various spatial weightings for W_r, such as k-nearest neighbours, contiguity neighbours and graph and distance-based neighbours, are described at full length in Bivand, Gmez-Rubio, and Pebesma (2008).

9. LeSage and Pace (2009) present and discuss various estimation techniques.

10. Form of graphical representation is inspired by Fuess and Koller (2016).

11. Thus, we measure the adequacy of our model specification, notably by controlling for locational influences, by examining the spatio-temporal correlation remaining in the residuals. As we have seen lately, the generalised spatio-temporal model is able to control for all existing correlation in the error term.

12. For a more detailed derivation and description of the approach, see Pace et al. (2000).

13. Several options for restrictions are presented in Pace et al. (1998).

14. The computation of the determinant of the Jacobian is a key issue in the estimation of spatial models and is discussed in length by LeSage and Pace (2009).

15. Locational determinants, such as green spaces, that are temporally invariant will be considered by the spatio-temporal matrices anyhow, provided the spatial density of housing observations is high enough.

16. For a detailed discussion of the conditions, see Pace et al. (2000).

17. For example, Pace et al. (2000) uses nn = 15 for the examination.

18. ^minρ,nn $\sum _{i=1}^N{\left[y_i-W(\rho ,nn)(Y+X)\right]}^2$ Contrary to conventional cross-sectional spatial econometric models, the configuration of spatial weighting improves the fit of the empirical model.

19. Since constant weightings perform well, Pace et al. (ibid.) do not assume more complicated specifications. Due to the lack of initial comparable housing, the first observations of p have to be discarded for estimation purposes.

20. The data are made available by empirica-systeme.

21. Naturally, our data do not include any rental prices of flats with off-record transactions. Our analyses should therefore be seen as an evaluation of all publicly offered rental flats without claiming to hold true for all newly offered rental contracts.

22. See Thomschke (2016) for details.

23. According to a survey of the Statistical office Berlin-Brandenburg, the residence duration (on average) is shortest in the central districts Berlin-Mitte (7.9 years) and Friedrichshain-Kreuzberg (8.1 years). The longest average residence duration amounts to 12.3 years in Treptow-K¨openick and 10.8 years in Marzahn-Hellersdorf – both districts are located at the edge of the city.

24. Decreasing the spatial density of observations, however, mitigates the reversion of those marginal effects (see Table A1). Grid-search (SSE minimization) results for two predefined numbers of neighbours, m_s, are presented in the bottom plots of Figure A2 as examples. For a random sample of 3000 observations, these results show that the optimal weighting parameters of neighbouring observations vary with the spatial closeness of those dwellings. The related weighting scheme for resulting ρ-values is presented in the top figure of Figure A2.

25. See Tables A2 and A3. We also computed the mean weighted distance to neighbours for each sample, but they do not differ significantly among in- and out-samples (Table A4). The out-sample size is smaller due to the truncation of the first ms observations. We use the RMSD and MAE for measuring prediction accuracy which are pretty standard in the out-sample forecasting literature: $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}=\sqrt{\sum _{i=1}^n{({\hat{y}}_i-y_i)}^2/n}$ and $\mathrm{M}\mathrm{A}\mathrm{E}=\sqrt{\sum _{i=1}^n\left|{\hat{y}}_i-y_i\right|/n}$ whereby $\hat{y}$ represents the vector of predicted values and y vector with the original values, respectively. Besides of accuracy measures (adjusted) R², Root-mean-square deviation (RMSD) and mean absolute error (MAE), the table also includes selected percentiles of the predicted errors terms. These distributions do not exhibit any anomalies.

26. See Table A2 and Table A3 for tests on differences in accuracy measures.

27. The weighting scheme is derived from the calculation of the S-matrices, and is thus also dependent upon the weighting parameter $\rho$ which is estimated beforehand via grid search minimization.

28. The number of observations range from 293 transactions in Militino et al. (2004) to 50,000 in Case et al. (2004).

29. Here this is done by increasing the predefined number of neighbours along the temporal dimension.

Additional information

Notes on contributors

Sören Gröbel

Sören Gröbel is a research assistant and PhD student at the Institute of Spatial and Housing Economics at the University of Münster.

Lorenz Thomschke

Lorenz Thomschke is a PhD student at the Institute of Spatial and Housing Economics at the University of Münster and is working as a research assistant at the research institute empirica ag, Berlin.