Bayesian Inference in the Seemingly Unrelated Regressions Model

8
Collecting all these results, substituting into equation (18), and letting | E(1) ′ E(1)
| be
absorbed into the proportionality constant, we can write
⎡ ( β1−β# 1) ′ X ′
1 Q(1) X1( β1−β#
1)
⎤
f( β1| y, β2, β3,..., βM
) ∝ ⎢v1+
⎥
2
⎢
s#
1
⎥
⎣
⎦
− ( K + v )/2
1 1
(19)
where v1 = T − K1
and s# 1 2 = ( y1− X1β # 1 )′
Q(1) ( y1− X1β
# 1)/
v1. Equation (19) is in the
form of a multivariate t-distribution with degrees of freedom v 1 , mean β # 1 , and
v v − s# X ′ Q X − . See, for example, Zellner (1971,
covariance matrix ( /( 2) ) 2 ( )
1
1 1 1 1 (1) 1
p.383). The conditional posterior pdf’s for other β i are similarly defined.
E. Conditional Posterior pdf for ( Σ | β
)
Viewing the joint posterior pdf in equation (12) as a function of only Σ yields the
conditional posterior pdf for Σ given β. It is the inverted Wishart pdf (see, for
example Zellner 1971, p.395)
− ( T+ M+ 1) 2 1 −1
2
f( Σ| β, y) ∝ Σ exp{ − tr( AΣ )}
(20)
It has T degrees of freedom, and parameter matrix A.
F. Marginal Posterior pdf for Σ
The marginal pdf for Σ , obtained by using the result in (13), and then using properties
of the multivariate normal pdf to integrate out β , is given by
∫
f( Σ | y) = f( β, Σ| y)
dβ
−1 −1/2 − ( T+ M+
1) 2 1 ˆ −1
T
2
∝ X′ ( Σ ⊗I ) X Σ exp{ − ( y− Xβ)( ′ Σ ⊗I )( y− Xβˆ)}
−1 −1/2 − ( T+ M+
1) 2 1 ˆ −1
T
2
= X′
( Σ ⊗I ) X Σ exp{ − tr( AΣ
)} (21)
T