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structure. On this level, an externally provided home price index indicating the
neighboring house price level,
wko_ind
, is available.
Level-4
is the county level (9 counties); we do not employ any further explana-
tory covariates on this level.
5.3
Structured Additive Distributional Regression
5.3.1
Hierarchical STAR Models
Suppose that observations .y
i
;
z
i
;
x
i
/, i
D
1;:::;n, are given, where y
i
is a
continuous response variable, and
z
i
D
.
z
i1
;:::;
z
iq
/
0
,and
x
i
D
.x
i1
;:::;x
ip
/
0
are
vectors of covariates. For the variables in
z
, possibly nonlinear effects are assumed,
whereas the variables in
x
are modeled in the usual linear way. The components of
z
are not necessarily continuous covariates. A component may also indicate a time
scale, a cluster, or a spatial index (e.g., municipality, district, or county) a certain
observation pertains to. We assume an additive decomposition of the effects of
z
ij
(and x
ij
) and obtain the model
D
f
1
.
z
i1
/
C
:::
C
f
q
.
z
iq
/
C
x
i
C
"
i
:
y
i
(5.1)
Here, f
1
;:::;f
q
are nonlinear functions of the covariates
z
i
and
x
i
is the usual
linear part of the model. The errors "
i
are assumed to be mutually independent
Gaussian with mean 0 and variance
2
,thatis,"
i
N.0;
2
/.
The nonlinear effects in (
5.1
) are modeled by a basis functions approach, that
is, a particular function f of covariate
z
is approximated by a linear combination of
basis or indicator functions
K
X
f.
z
/
D
ˇ
k
B
k
.
z
/:
(5.2)
kD1
ˇ D
.ˇ
1
;:::;ˇ
K
/
0
is a vector of unknown
regression coefficients to be estimated. Defining the n
K design matrix
Z
with elements
Z
Œi; k
D
B
k
.
z
i
/, the vector
f
D
.f .
z
1
/;:::;f.
z
n
//
0
of function
evaluations can be written in matrix notation as
f
D
Z
The B
k
's are known basis functions and
ˇ
. Accordingly, we obtain
y
D C " D
Z
1
ˇ
1
C
:::
C
Z
q
ˇ
q
C
X
C "
;
(5.3)
where
y
D
.y
1
;:::;y
n
/
0
,
"
N.
0
;
2
I
/.
Recently Lang et al. (
2014
) have proposed a multilevel version of STAR models
to cope with the hierarchical nature of the data on house prices. Suppose that
covariate
z
j
D
.
1
;:::;
n
/
0
and
2f
1;:::;K
g
is a unit or cluster index and
z
ij
indicates the cluster
observation i pertains to. In our case,
z
ij
denotes the municipality where the i th
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