∗ † ‡
∗
†
‡
a
s
1
= a+e
1
π
s
2
= a + e
2
c = 1 c = 0 a ∼ nid(0, 1) e
1
∼ nid(0, σ
2
1
)
e
2
∼ nid(0, σ
2
2
) s
1
s
2
d = 1
d = 0
Ω
θ E(a|Ω) > θ h = 1 h = 0
π a
θ
d
E(a|s
1
, s
2
, d = 1) = κ
2
s
1
+ κ
1
s
2
E(a|s
1
, s
2
, d = 0) = −ψκ
1
ω + (
1
1 + σ
2
1
κ
1
+ κ
2
)s
1
κ
1
κ
2
s
1
s
2
ω
ψ
P (c = 1|d = 0)
d = 1
d = 0
ψκ
1
ω
E(a|s
1
, s
2
, d = 1) > E(a|s
1
, s
2
, d = 0)
d(s
1
, s
2
, ) = c.1
"
s
2
−
1
1 + σ
2
1
s
1
> −
0.8ψ
1 −0.64ψ
ω
#
δE(s
2
|a)
δa
> 0
δE(s
2
|a,s
1
)
δa
> 0
∂P (h|s
1
,s
2
,d=1)
∂s
2
> 0
E(a|h = 1, d = 1) > E(a|h = 1, d = 0)
κ
2
∂
2
P (h|s
1
,s
2
,d=1)
∂s
2
∂κ
2
< 0
P (d = 1) = πΦ
0.8ψ
1−0.64ψ
ω
˜κ
2
˜κ
2
∂
2
P (h|s
1
,s
2
,d=1)
∂σ
2
1
∂s
2
=
σ
2
2
(σ
2
2
+σ
2
1
σ
2
2
+σ
2
1
)
2
> 0
∂P (h|s
1
,s
2
,d=1,female)
∂s
2
>
∂P (h|s
1
,s
2
,d=1,male)
∂s
2
π
ψκ
1
ω
∂
2
P (h)
∂aδπ
> 0
•
•
•
y
is
= βRef
i
+ λ
s
+ µ
k
+ e
s
y
is
i
s Ref
i
i λ
s
µ
k
e
s
β
a
y
is
= βRef
i
+ γa
s
+ δRef
i
∗ a
s
+ µ
k
+ e
s
γ
γ + δ
γ
δ
y
ij
= βT
i
+ γX
i
+ δy
bs
ij
+ λ
j
+ e
i
y
ij
i j
y
bs
ij
λ
j
Ref
i
= γT + βX + e
i
T
i
y
is
= µ
k
+ λ
s
+ βRef
i
+ γscore
s
+ δRef
i
∗ score
s
+ e
s
γ score
δ
y
ij
= βT
i
+ γX
i
+ λ
j
+ e
i
y
ij
i j
X
i
λ
j
y
ij
= βT
i
+ γX
i
+ λ
j
+ e
i
i
j
T
i
∗∗ ∗∗ ∗∗ ∗ ∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗
∗∗ ∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
R
2
∗∗ ∗ ∗∗ ∗
R
2
R
2
β
fem
= β
male
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗∗ ∗∗ ∗∗
∗∗∗ ∗∗∗ ∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗ ∗
∗∗ ∗∗∗ ∗∗
∗∗
∗∗∗
∗∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗ ∗∗
∗∗ ∗∗
R
2
β
Inf
β
Mon
β
Inf
β
Mon+Inf
β
Mon
β
Mon+Inf
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗
∗ ∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗ ∗∗ ∗∗ ∗∗∗ ∗∗ ∗∗∗ ∗∗∗
∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗∗ ∗∗∗
∗∗ ∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗ ∗∗ ∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗
R
2
∗∗
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
∗∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗
∗
∗
∗∗ ∗∗
∗
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
R
2
R
2
R
2
∗
p < 0.10
∗∗
p < 0.05
∗∗∗
p < 0.01
E(a|s
1
, s
2
, d) = β
00
+ β
01
d + β
10
s
1
+ β
11
s
1
d + β
21
s
2
d(s
1
, s
2
, d) = c.1(E(a|s
1
, d = 0) < E(a|s
1
, s
2
, d = 1))
d(s
1
, s
2
, ) = c.1(β
01
+ β
11
s
1
+ β
21
s
2
> 0) = c.1(s
2
> −
β
01
β
21
−
β
11
β
21
s
1
)
s
1
s
2
E(a|s
1
, s
2
, d = 1)
E(a|s
1
, s
2
, d = 1) = (β
00
+ β
01
) + (β
10
+ β
11
)s
1
+ β
21
s
2
β
10
+ β
11
=
σ
2
2
σ
2
2
+ σ
2
1
σ
2
2
+ σ
2
1
β
21
=
σ
2
1
σ
2
2
+ σ
2
1
σ
2
2
+ σ
2
1
β
00
+ β
01
= 0
E(a|s
1
, s
2
, d = 1) =
σ
2
2
σ
2
2
+ σ
2
1
σ
2
2
+ σ
2
1
s
1
+
σ
2
1
σ
2
2
+ σ
2
1
σ
2
2
+ σ
2
1
s
2
s
2
E(a|s
1
, d) = E(E(a|s
1
, s
2
, d)|s
1
, d) = β
00
+ β
01
d + β
10
s
1
+ β
11
s
1
d + β
21
dE(s
2
|s
1
, d)
s
2
s
1
E(s
2
|s
1
, d = 0) = P (c = 0|d = 0)E(s
2
|s
1
, c = 0) + P (c = 1|d = 0)E(s
2
|s
1
, s
2
< −
β
01
β
21
−
β
11
β
21
s
1
)
ψ ≡ P (c = 1|d = 0) κ
1
≡
σ
2
1
σ
2
2
+σ
2
1
σ
2
2
+σ
2
1
κ
2
≡
σ
2
2
σ
2
2
+σ
2
1
σ
2
2
+σ
2
1
ω =
q
(1 + σ
2
2
)(1 − ρ
2
)
E(s
2
|s
1
, d = 0) =
1 − ψ
1 + σ
2
1
s
1
+
ψ
1 + σ
2
1
s
1
− ψ
q
(1 + σ
2
2
)(1 − ρ
2
)
φ
−
β
01
β
21
−
β
11
β
21
s
1
−
1
1+σ
2
1
s
1
√
(1+σ
2
2
)(1−ρ
2
)
Φ
−
β
01
β
21
−
β
11
β
21
s
1
−
1
1+σ
2
1
s
1
√
(1+σ
2
2
)(1−ρ
2
)
∼
=
−ψ(0.64
β
01
β
21
+ 0.8
q
(1 + σ
2
2
)(1 − ρ
2
) + (
1 − 0.64ψ
1 + σ
2
1
− 0.64ψ
β
11
β
21
)s
1
s
2
E(s
2
|s
1
, d = 0)
E(a|s
1
, d = 0) = (β
00
+ β
01
) + (β
10
+ β
11
)s
1
+ β
21
E(s
2
|s
1
, d = 0)
∼
=
−0.64β
01
ψ − 0.8β
21
ψ
q
(1 + σ
2
2
)(1 − ρ
2
) +
1 − 0.64ψ
1 + σ
2
1
β
21
− 0.64ψβ
11
+ (β
10
+ β
11
)
!
s
1
E(a|s
1
, d = 0) = β
00
+ β
10
s
1
β
00
= −
0.8ψ
1 − 0.64ψ
κ
1
β
10
=
1
1 + σ
2
1
κ
1
+ κ
2
β
01
=
0.8ψ
1 − 0.64ψ
κ
1
β
11
= −
1
1 + σ
2
1
κ
1
β
21
= κ
1
d(s
1
, s
2
, ) = c.1
"
s
2
− 1/(1 + σ
2
1
)s
1
> −
0.8ψ
1 − 0.64ψ
ω
#
E(a|s
1
, s
2
, d) = 1.
"
0.8ψ
1 − 0.64ψ
κ
1
ω + −
0.8ψ
1 − 0.64ψ
κ
1
ωd + (
1
1 + σ
2
1
κ
1
+ κ
2
)s
1
− (
1
1 + σ
2
1
κ
1
)s
1
d + κ
1
s
2
d > θ
#
π (P (c = 1))
P (d(s
1
, s
2
, ) = 1) = π.P (s
2
−
1
1 + σ
2
1
s
1
> −
0.8ψ
1 − 0.64ψ
ω)
c d
P (d = 1|c = 1)
P (h = 1)
E(a|h = 1)
P (c = 1) → 1