Hedonic House Price ModelingBased on Multilevel Structured Additive Regression - Computational Approaches for Urban Environments - page 113

individual effects for each and every quantile. GAMLSS regression also allows
for different marginal effects due to the two estimated parameters of the gamma
distribution. Mean regression, in contrast, only reveals one marginal effect for each
covariate, shifts it according to the estimated variance and converts it to natural units
using the transformation property of the lognormal distribution.
While the differences in the estimated effects between the mean regression and
the GAMLSS regression were rather small for the expected house price per sq. m.,
there are now substantial differences for the quantiles. Compared to the results of
the quantile regression and the GAMLSS regression, the mean regression seems to
slightly underestimate the prices for the 20- and 50 %-quantile, while the results
for the 80 %-quantile are more similar. This indicates a more skewed distribution of
house prices than can be captured by the mean regression.
Figure 5.4 illustrates this distribution by showing the posterior 10-, 20-, 50-, 80-,
90 %-quantile estimates of the three different methods for each structural covariate.
Additionally, the posterior mean estimates are displayed for the mean regression
and the GAMLSS regression. Particularly the effects of the age of the building
considerably differ between the individual quantiles both in the GAMLSS and the
quantile regression, revealing the limits of the mean regression where the marginal
effects are almost the same for all quantiles.

5.7.3 Spatial Effects

The total amount of spatial heterogeneity is composed of spatial effects on munici-
pal (level-2), district (level-3), and county level (level-4). Continuous neighborhood
effects (see Sect. 5.7.1 ) explain spatial heterogeneity explicitly to a certain extent
on two of these levels (municipal and district). We call this the explained spatial
heterogeneity. The remaining i.i.d. spatial random effects " 5 , " 5;6 and " 5;6;3 as well
as the correlated district-specific effect f 5;6;2 .d i st / in ( 5.5 ) account for unexplained
spatial heterogeneity. In the following, we analyze the distribution of spatial
heterogeneity over Austria. For the sake of clarity, we only show the estimation
results for the 50 %-quantiles.

5.7.3.1 Total Spatial Heterogeneity

Figure 5.5 visualizes the posterior 50 %-quantile estimates of the total spatial effects
(explained plus unexplained heterogeneity) for the mean regression, the GAMLSS
regression and the quantile regression evaluated at the average effect. We can
see a very pronounced spatial heterogeneity showing single-family homes in the
western counties to be considerably more expensive than in the eastern and southern
counties. Furthermore, we find a clearly positive effect in urban areas with a strong
peak in Vienna.