Bayesian Inference in the Seemingly Unrelated Regressions Model
Bayesian Inference in the Seemingly Unrelated Regressions Model
Bayesian Inference in the Seemingly Unrelated Regressions Model
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8<br />
Collect<strong>in</strong>g all <strong>the</strong>se results, substitut<strong>in</strong>g <strong>in</strong>to equation (18), and lett<strong>in</strong>g | E(1) ′ E(1)<br />
| be<br />
absorbed <strong>in</strong>to <strong>the</strong> proportionality constant, we can write<br />
⎡ ( β1−β# 1) ′ X ′<br />
1 Q(1) X1( β1−β#<br />
1)<br />
⎤<br />
f( β1| y, β2, β3,..., βM<br />
) ∝ ⎢v1+<br />
⎥<br />
2<br />
⎢<br />
s#<br />
1<br />
⎥<br />
⎣<br />
⎦<br />
− ( K + v )/2<br />
1 1<br />
(19)<br />
where v1 = T − K1<br />
and s# 1 2 = ( y1− X1β # 1 )′<br />
Q(1) ( y1− X1β<br />
# 1)/<br />
v1. Equation (19) is <strong>in</strong> <strong>the</strong><br />
form of a multivariate t-distribution with degrees of freedom v 1 , mean β # 1 , and<br />
v v − s# X ′ Q X − . See, for example, Zellner (1971,<br />
covariance matrix ( /( 2) ) 2 ( )<br />
1<br />
1 1 1 1 (1) 1<br />
p.383). The conditional posterior pdf’s for o<strong>the</strong>r β i are similarly def<strong>in</strong>ed.<br />
E. Conditional Posterior pdf for ( Σ | β<br />
)<br />
View<strong>in</strong>g <strong>the</strong> jo<strong>in</strong>t posterior pdf <strong>in</strong> equation (12) as a function of only Σ yields <strong>the</strong><br />
conditional posterior pdf for Σ given β. It is <strong>the</strong> <strong>in</strong>verted Wishart pdf (see, for<br />
example Zellner 1971, p.395)<br />
− ( T+ M+ 1) 2 1 −1<br />
2<br />
f( Σ| β, y) ∝ Σ exp{ − tr( AΣ )}<br />
(20)<br />
It has T degrees of freedom, and parameter matrix A.<br />
F. Marg<strong>in</strong>al Posterior pdf for Σ<br />
The marg<strong>in</strong>al pdf for Σ , obta<strong>in</strong>ed by us<strong>in</strong>g <strong>the</strong> result <strong>in</strong> (13), and <strong>the</strong>n us<strong>in</strong>g properties<br />
of <strong>the</strong> multivariate normal pdf to <strong>in</strong>tegrate out β , is given by<br />
∫<br />
f( Σ | y) = f( β, Σ| y)<br />
dβ<br />
−1 −1/2 − ( T+ M+<br />
1) 2 1 ˆ −1<br />
T<br />
2<br />
∝ X′ ( Σ ⊗I ) X Σ exp{ − ( y− Xβ)( ′ Σ ⊗I )( y− Xβˆ)}<br />
−1 −1/2 − ( T+ M+<br />
1) 2 1 ˆ −1<br />
T<br />
2<br />
= X′<br />
( Σ ⊗I ) X Σ exp{ − tr( AΣ<br />
)} (21)<br />
T