where etr· stands for exptrace·. It follows from 2
that
− 1
t
=
− 1
t ′
, 5
where = Ŵ
− 1′
is an upper unit triangular matrix.
Denote B = b , B
1
and b = vecB. The likelihood func- tion of b, ,
1
, . . . ,
T
is then [
y|b, ,
1
, . . . ,
T
] ∝ |
|
T T
t= 1
|
t
|
− 12
etr −
1 2
T t=
1
S
t
b
− 1
t ′
, 6
where
S
t
b = y
t
− b
− B
1
x
t
y
t
− b
− B
1
x
t ′
. 7
Denote
X =
1 1
· · · 1
x
1
x
2
· · · x
T
, = diag
1
, . . . ,
T
= diagλ
11
, λ
21
, . . . , λ
p 1
, λ
12
, . . . , λ
pT
.
The likelihood of b, , is then Lb, , = [y|b, , ]
∝ | |
T
| |
− 12
etr −
1 2
[ y − X
′
⊗ I
p
b]
′
I
T
⊗ ×
− 1
I
T
⊗
′
[y − X
′
⊗ I
p
b]
= | |
T
| |
− 12
etr −
1 2
[I
T
⊗
′
y − X
′
⊗
′
b]
′
×
− 1
[I
T
⊗
′
y − X
′
⊗
′
b] .
8 For a full Bayesian analysis, one needs to assign a prior to
parameters. In the likelihood function 8
, b = vecb , B
1
and are parameters pertaining to the regression model
1 , in-
volves parameters a , β, A
1
, δ in SV equation
3 . We now
discuss the prior specification. 2.2
Priors of b , B
1
,
We assume the priors of components of b, are indepen-
dent. Here are the marginal priors.
i Priors of b =
b
10
, . . . , b
p ′
. We assume that the inter- cept b
i
is always included in the model. We also assume inde- pendent priors for b
i
: b
i indep
∼ Nb
i
, ξ
2 i
, i =
1, . . . , p. 9
ii Priors of B
1
={b
ij
}
p×q
. Each element b
ij
is associated with an indicator γ
b,ij
. If γ
b,ij
= 1, b
ij
is included; and if γ
b,ij
= 0,
b
ij
is excluded. Then b
ij
has a two-stage prior: for fixed p
b,ij
∈ 0, 1,
Pγ
b,ij
= 1 = 1 − Pγ
b,ij
= =
p
b,ij
, i =
1, . . . , p, j = 1, . . . , q. 10
For given γ
b
= γ
b, 11
, γ
b, 12
, . . . , γ
b,pq ′
, we let b
ij
|γ
b,ij indep
∼ 1 − γ
b,ij
N0, κ
2 b,ij
+ γ
b,ij
N0, c
2 b,ij
κ
2 b,ij
11 for i = 1, . . . , p and j = 1, . . . , q, where κ
b,ij
are small and c
b,ij
are large constants. If we write η
b,ij
= c
γ
b,ij
b,ij
= 1,
if γ
b,ij
= c
b,ij
, if γ
b,ij
= 1,
and D
b,j
= diagη
b, 1j
κ
b, 1j
2
, . . . , η
b,pj
κ
b,pj 2
, then
11 is
equivalent to
b
j
| γ
b,j indep
∼ N
p
0, D
b,j
for j = 1, . . . , q. 12
Combining the priors in i and ii, we can write the prior for b as
b|γ
b
∼ N ¯b, ¯
, 13
where ¯
b = b
10
, . . . , b
p
, 0, . . . , 0
′
, ¯
= diagξ
2 10
, . . . , ξ
2 p
, η
b, 11
κ
b, 11
2
, . . . , η
b,pq
κ
b,pq 2
. iii Priors of . For j = 2, . . . , p, let ψ
j
be a vector contain- ing the non-redundant elements of the jth column of , that is,
ψ
j
= ψ
1j
, . . . , ψ
j− 1,j
′
. Also, define a vector of indicators of
length j − 1, γ
ψ, j
= γ
ψ, 1j
, . . . , γ
ψ, j−
1,j ′
. We assume that ele-
ments of ψ
j
may be included in the model γ
ψ, ij
= 1 or not
γ
ψ, ij
= 0. Let the model index for ψ
ij
, γ
ψ, ij
, be independent Bernoullip
ψ, ij
random variables: for fixed p
ψ, ij
∈ 0, 1,
Pγ
ψ, ij
= 1 = 1 − Pγ
ψ, ij
= =
p
ψ, ij
, i =
1, . . . , j − 1, j = 1, . . . , p. 14
For given γ
ψ, j
= γ
ψ, 1j
, . . . , γ
ψ, j−
1,j ′
, we assume that ψ
ij
|γ
ψ, ij
indep
∼ 1 − γ
ψ, ij
N0, κ
2 ψ,
ij
+ γ
ψ, ij
N0, c
2 ψ,
ij
κ
2 ψ,
ij
15 for i = 1, . . . , j − 1 and j = 2, . . . , p, where κ
ψ, ij
are small and c
ψ, ij
are large constants. If we write η
ψ, ij
= c
γ
ψ, ij
ψ, ij
= 1,
if γ
ψ, ij
= c
ψ, ij
, if γ
ψ, ij
= 1,
and D
ψ, j
= diagη
ψ, 1j
κ
ψ, 1j
2
, . . . , η
ψ, j−
1,j
κ
ψ, j−
1,j 2
, then
15 is equivalent to
ψ
j
| γ
ψ, j
indep
∼ N
j− 1
0, D
ψ, j
16 for j = 2, . . . , p. Note that a slight modification of the setting
allows for modeling whether some elements of are equal. Instead of centering the prior of these elements at 0, we can
set them at a common mean in prior 15
. When the parameter index γ = 0 the corresponding element approximately equals
to the common mean and when γ = 1 it is not restricted. But to simplify notation, throughout the article we only consider priors
centered at 0.
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:11 11 January 2016
2.3 Priors of a
, β, A
1
, δ
We assume that a , β, A
1
, and δ have mutually independent priors.
i Priors of a =
a
10
, . . . , a
p ′
. For fixed ¯a
j
, σ
a
, we as- sume that
a
j indep
∼ N¯a
j
, σ
2 a
. 17
ii Priors of β = β
1
, . . . , β
p ′
. For fixed ¯
β
j
, σ
β
, we assume that
β
j indep
∼ N ¯
β
j
, σ
2 β
. 18
iii Priors of A
1
. Let the model index for a
jk
, γ
a,jk
be in- dependent Bernoullip
a,jk
random variables: for fixed p
a,jk
∈ 0, 1,
Pγ
a,jk
= 1 = 1 − Pγ
a,jk
= =
p
a,jk
for j = 1 . . . , p, k = 1, . . . , r. 19
For given γ
a,j
= γ
a,j 1
, γ
a,j 2
, . . . , γ
a,jr ′
, we assume that a
jk
|γ
a,jk indep
∼ 1 − γ
a,jk
N0, κ
2 a,jk
+ γ
a,jk
N0, c
2 a,jk
κ
2 a,jk
, 20
where κ
a,jk
would be small and c
a,jk
would be large con-
stants. Later, we also write A
1
in terms of its row vectors:
A
1
= a
′ 1
, . . . , a
′ p
′
. Here a
j
= a
j 1
, . . . , a
jr ′
, j = 1, . . . , p. De- note
