Civil Engineering Reference
In-Depth Information
the asymmetry kurtosis parameters and the degrees-of-freedom parameter, respec-
tively.
λ
is restricted within
(
−
1
,
1
)
.When
λ
<
0 that skewed to the left. When
λ
>
0, it implies skewed to the right.
η
is restricted within
(
2
,
+
∞
)
.If
λ
=
0, it
implies the skewed-
t
distribution will turn toward the Student-
t
distribution.
Here we employed two families of copula model to describe the dependence
structure between the total value of import and export with exchange rate. Gaussian,
Student-
t
, Gumbel, and Clayton copula were used to capture different dependence
structure.
2
The advantage of Gaussian and Student-
t
copulas is that one can specify
different levels of correlation between the marginals. In contrast to elliptical copulas,
Archimedean copula can capture the right tail dependence and left tail dependence.
These copula models are briefly discussed below.
The dynamic Gaussian copula is the density of the joint standard uniform
variables (
m
t
,
w
t
), with a time-varying correlation,
ρ
t
. The density of the time-
varying Gaussian copula is then
exp
2
x
t
−
y
t
x
t
+
y
t
1
ρ
t
x
t
y
t
−
C
Gau
t
(
m
t
,
w
t
|
ρ
t
)=
√
1
+
.
(6)
−
ρ
t
t
2
2
(
1
−
ρ
)
The density of the time-varying Student-
t
copula is
1
−
n
+
2
2
x
t
+
y
t
1
+
−
2
ρ
t
x
t
y
t
+
C
t
T
(
,
|
ρ
,
)=
√
1
n
1
t
.
m
t
w
t
n
(7)
t
−
ρ
−
ρ
t
The density of the time-varying Gumbel copula is
exp
τ
t
x
τ
t
−
1
y
τ
t
−
1
a
t
b
t
−
x
τ
t
−
1
y
τ
t
−
1
1
C
Gum
t
(
m
t
,
w
t
|
τ
t
)=
+
(8)
1
−
τ
t
τ
t
2
x
τ
t
−
1
)
τ
t
−
1
)
τ
t
−
1
1
−
2
τ
t
)
τ
t
−
1
−
+(
y
+(
δ
t
−
1
)
(
x
+(
y
.
(9)
The density of the time-varying Clayton copula is
2
τ
t
+
1
τ
t
C
Glay
a
−
τ
t
b
−
τ
t
)
−
a
−
τ
t
−
1
b
−
τ
t
−
1
(
m
t
,
v
t
|
τ
t
)=(
τ
t
+
1
)(
−
.
(10)
t
t
t
t
t
2
Gaussian and Student-
t
copula belong to family of elliptical copula. And Gumbel and Clayton
copula are Archimedeans copula.
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