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A possible way to assure identifiability is to assume monotonicity for f (either
monotonically increasing or decreasing) and to restrict the spread of f ,for
example, by assuming
X
K
ˇ k
D c:
kD1
The constant c can, for example, be chosen such that the squared sum of the
coefficients is identical to that of the model without scaling factors. Imposing
monotonicity constraints is easily done using the methodology of Brezger and
Steiner ( 2008 ).
An application of generalized random slope modeling in the context of real
estate data is given in Brunauer et al. ( 2010 ) when modeling rents of apartments in
dependence of covariates. The nonlinear price gradients are assumed to be district
specific and modeled in the form of generalized random slopes as proposed above.
5.6
Software
The multilevel STAR models described above can be estimated with the open
source software package BayesX (Brezger et al. 2005 ). To facilitate exploration and
visualization of fitted models, the R ( R Development Core Team 2013 ) package
R2BayesX (Umlauf et al. 2012 ) has been developed, which provides a fully
interactive R interface to BayesX with the usual R modeling “ look & feel ”. In
the following, we exemplify the usage of the software estimating the four-level
hierarchical STAR model ( 5.5 ).
We first load the required packages and data sets.
R> library("R2BayesX")
R> library("spdep")
R> load("AustriaHouse.rda")
R> load("DistrictsBnd.rda")
The file AustriaHouse.rda contains four data sets, one for each spatial resolu-
tion as described in Sect. 5.2 . The data set with the highest resolution including the
house prices and housing attributes is called HousePrice , data on the municipal
level is provided in the data frame Municipal , on the district and county level
in objects District and County , respectively. The file DistrictsBnd.rda
contains a boundary map object DistrictsBnd that is used to compute the
necessary neighborhood structure for estimating the level-3 correlated spatial effect
of the districts in Austria. After transforming the class “ bnd ” object to an object of
class “ SpatialPolygons ” with
R> DistrictsSp <- bnd2sp(DistrictsBnd)
the final neighborhood object DistrictsNb , which is used for fitting the model,
can be generated by
R> DistrictsNb <- poly2nb(DistrictsSp)
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