Geoscience Reference
In-Depth Information
house is located. Then the design matrix
Z
j
is a n
K incidence matrix with
Z
j
Œi; k
D
1 if the i -th observation belongs to cluster k and zero else. The K
1
parameter vector
ˇ
j
is the vector of regression parameters, that is, the k-th element
in
corresponds to the regression coefficient of the k-th cluster. We now define the
second-level equation
ˇ
ˇ
j
D
j
C "
j
D
Z
j1
ˇ
j1
C
:::
C
Z
jq
j
ˇ
jq
j
C
X
j
j
C "
j
;
(5.4)
where the terms
Z
j1
ˇ
j1
;:::;
Z
jq
j
ˇ
jq
j
correspond to additional nonlinear functions
f
j1
;:::;f
jq
j
and
X
j
j
comprises additional linear effects of cluster level covari-
ates. For the house price data, these are the covariates on municipality level, that
is, the purchase power index, share of academics, etc. The “errors”
"
j
N.
0
;
j
I
/
comprise a vector of i.i.d. Gaussian random effects. Using the compound prior (
5.4
),
we obtain an additive decomposition of the cluster-specific effect. By allowing a
full STAR predictor (as in the level-1 equation), a rather complex decomposition
of the cluster effect
ˇ
j
including interactions is possible. A special case arises
if cluster-specific covariates are not available. Then the prior for
ˇ
j
collapses to
ˇ
j
D "
j
N.
0
;
j
I
/, and we obtain a simple i.i.d. Gaussian cluster-specific
random effect with variance parameter
j
.
A third or fourth level in the hierarchy is possible by assuming that the
second or third level regressions contain additional cluster-specific random effects
whose parameters are again modeled through STAR predictors of cluster-level
covariates.
In our model, we distinguish four levels: Single-family homes (level-1) belong to
municipalities (level-2), which are nested in districts (level-3), which are themselves
nested in counties (level-4). Then our model can be written as the following four-
level hierarchical STAR model:
level-1:
lnp
qm
D
f
1
.
area
/
C
f
2
.
areaplot
/
C
f
3
.
age
/
C
f
4
.
timeindex
/
C
f
5
.
muni
/
C
X
C "
D
Z
1
ˇ
1
C
Z
2
ˇ
2
C
Z
3
ˇ
3
C
Z
4
ˇ
4
C
Z
5
ˇ
5
C
X
C "
level-2:
ˇ
5
D
f
5;1
.
ppind
/
C
f
5;2
.
lneduc
/
C
f
5;3
.
ageind
/
C
f
5;4
.
comm
/
C
f
5;5
.
lnden
/
C
f
5;6
.
dist
/
C "
5
D
Z
5;1
ˇ
5;1
C
Z
5;2
ˇ
5;2
C
Z
5;3
ˇ
5;3
C
Z
5;4
ˇ
5;4
C
Z
5;5
ˇ
5;5
C
Z
5;6
ˇ
5;6
C "
5
D
f
5;6;1
.
wkoind
/
C
f
mrf
level-3:
ˇ
5;6
5;6;2
.
dist
/
C
f
5;6;3
.
county
/
C "
5;6
D
Z
5;6;1
ˇ
5;6;1
C
Z
5;6;2
ˇ
5;6;2
C
Z
5;6;3
ˇ
5;6;3
C "
5;6
;
level-4:
ˇ
5;6;3
D
1
0
C "
5;6;3
:
(5.5)
The level-1 equation contains the main predictor. Apart from usual linear effects,
the predictor is composed of possibly nonlinear effects of the continuous covari-
ates
area
,
areaplot
,
age
,and
time
_
index
as well as an uncorrelated municipality
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