## Abstract

We use adaptive weights smoothing (AWS) of Polzehl and Spokoiny (J R Stat Soc Ser B 62:335–354, 2000; Ann Stat 31:30–57, 2003; Probab Theory Relat Fields 135:335–362, 2006) to estimate a map of land values for Berlin, Germany. Our data are prices of undeveloped land that was transacted between 1996 and 2009. Even though the observed land price is an indicator of the respective land value, it is influenced by transaction noise. The iterative AWS applies piecewise constant regression to reduce this noise and tests at each location for constancy at the margin. If not rejected, further observations are included in the local regression. The estimated land value map conforms overall well with expert-based land values. Our application suggests that the transparent AWS could prove a useful tool for researchers and real estate practitioners alike.

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## Notes

See Polzehl and Spokoiny (2006). For simplicity, we nonetheless refer to the approach as AWS throughout the paper.

Fitting a third degree polynomial for the distance to the CBD gives such a decreasing function with an \(R^2=0.2345\).

Scatterplots of the estimated AWS land values, \(\hat{\theta }(\mathbf {x}_i)\), against the land prices, \(y_i\), (not reported) as well as kernel density estimates of the estimated AWS residuals, \(\hat{\epsilon }_i\), (not reported) indicate that the assumption of normally distributed and homoscedastic (log) land prices is approximately satisified in our application below.

While the exact sampling distribution of \(T^k_{ij}\) can only be derived in iteration step \(k=1\) if the bandwidth \(h^0\) is very small, the \(\chi ^2_1\)-distribution may still be a good approximation in every iteration step \(k\) under the assumptions of normally distributed, homoscedastic errors.

The condition involves the probability that the Kullback Leibler divergence between the adaptive AWS estimate and the globally constant \(\theta \) is bounded and that the bound does not increase in the iteration process for a given propagation level \(\epsilon .\) Becker and Mathé (2013) also propose a method for estimating this probability from the simulated data.

We use the sequence \(\mathbf {h}=\left\{ 1, 2, 3, 4, 5, 7, 9, 11, 14, 18, 22, 28, 35, 44, 55, 69, 86, 108, 135\right\} \) to increase \(h^{k}\) in the \(k^{*}=19\) iteration steps.

\(\lambda =3.8415\) corresponds to the \(5\,\%\) percentile of the \(\chi ^2_1\)-distribution.

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## Acknowledgments

We have benefited from comments received at the Applicable Semiparametrics Conference in Berlin 2013 and from two anonymous referees. Financial support from the Deutsche Forschungsgemeinschaft, CRC 649 Economic Risk, is gratefully acknowledged. The usual disclaimer applies.

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## Appendix

### Appendix

We adjust the land prices as follows. First, we run the regression

where \(p_{j,t}\) is the log price of site \(j\) transacted in quarter \(t\). The column vector \(\mathbf {d}_{j}\) has \(T\) elements: the first element for the overall constant is one, the period \(t\) element is one if site \(j\) was transacted in this period, zero otherwise. 1996Q1 is the omitted reference period. The vector \(\mathbf {x}_{j}\) contains binary indicators for unusual features of the site and for unusual aspects of the business dealings. The vector \(\mathbf {z}_{j}\) contains binary indicators for Berlin’s 96 administrative sub-districts (Ortsteile). The site will be located in one of these sub-districts. The vector \(\mathbf {z}_j\) also contains a binary indicator for site location adjacent to a lake or the bank of a river. Finally, the vector contains binary indicators for site’s location rating. This rating comes from Berlin’s Senate Department for Urban Development and rates natural amenities, the quality of existing buildings, access to public transport and shopping facilities with in the neighborhood. The rating for a site takes one of four values: low, medium, high, very high. The variables in \(\mathbf {z}_j\) control crudely for location effects. Without the inclusion, the estimates of \(\varvec{\alpha }\) and \(\varvec{\beta }\) may suffer from omitted variable bias. Table 4 presents least squares estimates of the model in Eq. 15. The in-sample fit, as measured by the \(R^2\), is reasonably good. Except for the coefficient for ground monument, all coefficients are statistically significant at the 5 % significance level. The signs of the point estimates, as well as their magnitude, are plausible.

Second, given the coefficient estimates, we compute the adjusted log real land price as

The first entry of \(\mathbf {d}_{b}\) is one, the entries for the four quarters of the year 2009 are 0.25 each, the remaining entries are zero. The term in brackets in Eq. 16 converts prices to the base year 2009. The estimated value of \(\mathbf {z}_{j}\varvec{\gamma } \) is not considered for \(p_j\), because it enters Eq. 15 only to prevent bias. The resulting \(p_{j}\) is in real terms and adjusted for unusual circumstances of the site. Using it in our analysis puts us on an equal footing with the land price information used by local surveyors to produce the BRW. The summary statistics for prices in natural scale in Table 1 are computed using \(P_j=\exp \!\left\{ p_{j} + 0.5 \widehat{\sigma }^2_{\epsilon }\right\} \), where \(\widehat{\sigma }^2_{\epsilon }\) is the estimated variance of the error term in Eq. 15 (Kennedy 1983).

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Kolbe, J., Schulz, R., Wersing, M. *et al.* Identifying Berlin’s land value map using adaptive weights smoothing.
*Comput Stat* **30**, 767–790 (2015). https://doi.org/10.1007/s00180-015-0559-9

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DOI: https://doi.org/10.1007/s00180-015-0559-9