Civil Engineering Reference
In-Depth Information
2.3
Benchmark Model
We take time factor into nonparametric part to construct benchmark model, consider
that coefficients are time varying, construct semiparametric time-varying model
based on model above:
π
(
t
i
)=
7
a
(
t
i
)
E
t
i
π
t
i
+
1
(
t
i
)+
b
(
t
i
)
π
t
i
−
1
(
t
i
)+
c
(
t
i
)
y
t
i
−
1
(
t
i
)+
G
(
t
i
)+
ε
(
t
i
)
(2)
n
j
=
1
z
j
(
t
i
)(
π
(
t
j
)
−
a
(
t
j
)
E
t
i
π
t
i
+
1
(
t
j
)
−
b
(
t
j
)
π
t
i
−
1
(
t
j
)
−
c
(
t
j
)
y
t
i
−
1
(
t
j
))
G
(
t
i
)=
(3)
t
j
−
t
i
t
l
−
t
i
h
1
n
n
where
z
j
,
Kand
h
1
denote kernel function and bandwidth, and
h
1
is figured out by the cross-
validation method.
In order to solve time-varying coefficients
a
(
t
i
)
denotes weight,
j
=
1
z
j
(
t
i
)=
1, select
z
j
(
t
i
)=
K
(
)
/
∑
l
=
1
K
(
)
∑
h
1
(
t
i
)
,
b
(
t
i
)
,
c
(
t
i
)
,and
G
(
t
t
i
)
,using
Taylor expansion,
a
(
t
i
)=
a
1
+
a
2
(
t
−
t
i
)
,
b
(
t
i
)=
b
1
+
b
2
(
t
−
t
i
)
,
c
(
t
i
)=
c
1
+
c
2
(
t
−
t
i
)
(4)
2.4
Path Model
Recentlymore and more scholars believe that the relation among money, house
prices, and interest rates inflation is nonlinear (
Duan
(
2007
);
Zheng and Liu
(
2008
);
Xiang et al.
(
2011
);
Zheng and Liu
(
2010
); etc.). So we take money growth, housing
price index, and interest rate, respectively, into G and then obtain three path models
based on previous benchmark model:
π
(
t
i
)=
a
(
t
i
)
E
t
i
π
t
i
+
1
(
t
i
)+
b
(
t
i
)
π
t
−
1
(
t
i
)+
c
(
t
i
)
y
t
−
1
(
t
i
)+
G
(
t
i
,
Δ
M
t
i
−
1
)+
ε
(
t
i
)
(5)
π
(
t
i
)=
a
(
t
i
)
E
t
i
π
t
i
+
1
(
t
i
)+
b
(
t
i
)
π
t
−
1
(
t
i
)+
c
(
t
i
)
y
t
−
1
(
t
i
)+
G
(
t
i
,
h
t
i
−
1
)+
ε
(
t
i
)
(6)
π
(
t
i
)=
a
(
t
i
)
E
t
i
π
t
i
+
1
(
t
i
)+
b
(
t
i
)
π
t
−
1
(
t
i
)+
c
(
t
i
)
y
t
−
1
(
t
i
)+
G
(
t
i
,
r
t
i
−
1
)+
ε
(
t
i
)
(7)
If the sample size is large enough the models (5), (6), and (7) will be convergent
in probability to model (2) (
Xu 2010
):
n
j
=
1
z
j
(
t
i
)(
π
(
t
j
)
−
a
(
t
i
)
E
t
i
π
t
i
+
1
(
t
i
)
−
b
(
t
j
)
π
t
i
−
1
(
t
j
)
−
c
(
t
j
)
y
t
i
−
1
(
t
j
))
(8)
G
(
t
i
,
Δ
M
t
i
−
1
)=
(
,
)
,
(
,
)
(
)
where
G
t
i
h
t
i
−
1
G
t
i
r
t
i
−
1
can be expressed by the same token,
z
j
t
i
denotes
Δ
M
t
j
−
Δ
M
t
i
Δ
M
t
l
−
Δ
M
t
i
t
j
−
t
i
h
1
n
j
n
l
t
l
−
t
i
h
1
weight,
1
z
j
(
t
i
)=
1
,
z
j
(
t
i
)=
K
(
)
K
(
)
/
∑
1
K
(
)
K
(
)
,
∑
=
−
h
2
h
2
K
h
1
,and
h
2
denote kernel function and bandwidth, and
h
2
is calculated by the
cross-validation method.
,
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