Hedonic House Price ModelingBased on Multilevel Structured Additive Regression - Computational Approaches for Urban Environments - page 99

effect controlling for spatial heterogeneity. The municipality effect is a rather
complex spatial effect and is further decomposed through the remaining levels
in the hierarchy of the model. The level-2 equation contains nonlinear effects
of continuous municipality-specific covariates and a spatial district effect, which
is decomposed in the level-3 equation. Two of the covariates on level-2 enter
the equation logarithmically (denoted by the prefix “ ln_ ”), namely, the share of
academics and the population density. The reason for this is that the distributions of
these covariates are strongly positively skewed, which results in volatile estimation
results on the natural scale. District-specific spatial heterogeneity is modeled
through the correlated spatial effect dist in the level-3 equation by Markov random
fields (see Sect. 5.4.3 ). We denote this by the superscript “ mrf ”. The level-3 equation
is additionally composed of a nonlinear effect of the district-specific covariate
wko _ ind and a spatial county effect. The level-4 equation constitutes a usual
county specific i.i.d. random effect and for technical reasons (improved mixing)
the intercept of the model.
On all levels, for continuous covariates, possibly nonlinear functions f 1 ; f 2 ;:::
modeled by P-splines (see Sect. 5.4.2 ) are assumed. The categorical covariates on
level-1, describing the quality and condition of the house, are encoded as dummy
variables and subsumed in the design matrix X with estimated parameters

.

5.3.2 Beyond Mean Modeling: Distributional Regression

The model implied by Eq. ( 5.3 ) can equivalently be written as

y N. ; 2 I / D N. Z 1 ˇ 1 C ::: C Z q ˇ q C X ; 2 I /; (5.6)

that is, given the covariates the response vector is multivariate normal with mean

and homoscedastic covariance matrix 2 I . Hence, so far only the mean of the
response distribution is modeled in dependence of covariates. As already pointed out
in Fahrmeir et al. ( 2004 ), not only the mean but also the variances of the response
may depend on covariates when modeling real estate data. For instance, the analysis
of data on the monthly rent of apartments in Munich in Fahrmeir et al. ( 2004 )
revealed that apartments built in the 1950s-1970s generally exhibit lower variability
than modern apartments that have been built recently. The reason is that the postwar
period in Germany is characterized by a quite homogeneous construction style based
on low-quality construction material.
To model the variance of the responses, we may assume

/ D exp. Z 1 ˇ 1 C ::: C

Z q ˇ q C X

2 D exp. Q /;

where the Z j and X are design matrices of further covariates z 1 ;:::; z q and
x 1 ;:::; x p modeling the variance. Of course, some (or all) of these covariates may