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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6<br />
E( β| y, Σ ) =β= ˆ [ X′ ( Σ ⊗I ) X] X′<br />
( Σ ⊗ I ) y<br />
(15)<br />
−1 −1 −1<br />
T<br />
T<br />
and posterior covariance matrix equal to<br />
−1 −1<br />
T<br />
V( β| y, Σ ) = [ X′<br />
( Σ ⊗ I ) X]<br />
(16)<br />
The last two expressions are familiar ones <strong>in</strong> sampl<strong>in</strong>g <strong>the</strong>ory <strong>in</strong>ference for <strong>the</strong> SUR<br />
model. They show that <strong>the</strong> traditional SUR estimator, written as<br />
β= ˆ [ X′ ( Σˆ ⊗I ) X] X′<br />
( Σˆ<br />
⊗ I ) y<br />
(17)<br />
−1 −1 −1<br />
T<br />
T<br />
where ˆΣ is a 2-step estimator or a maximum likelihood estimator, can be viewed as<br />
<strong>the</strong> mean of <strong>the</strong> conditional posterior pdf for<br />
β<br />
given Σ ˆ.<br />
The traditional covariance<br />
matrix estimator<br />
[ X′ ( Σˆ<br />
⊗ I ) X]<br />
can be viewed as <strong>the</strong> conditional covariance<br />
−1 −1<br />
T<br />
matrix from <strong>the</strong> same pdf. S<strong>in</strong>ce this pdf does not take <strong>in</strong>to account uncerta<strong>in</strong>ty from<br />
not know<strong>in</strong>g Σ (<strong>the</strong> fact that ˆΣ is an estimate is not recognised), it overstates <strong>the</strong><br />
reliability of our <strong>in</strong>formation about β . This dilemma was noted by Fiebig and Kim<br />
(2000) <strong>in</strong> <strong>the</strong> context of an <strong>in</strong>creas<strong>in</strong>g number of equations.<br />
C. Marg<strong>in</strong>al Posterior pdf for β<br />
The more appropriate representation of our uncerta<strong>in</strong>ty about β is <strong>the</strong> marg<strong>in</strong>al<br />
posterior pdf f ( β | y)<br />
. It can be shown that this pdf is given by<br />
f( β | y) = f( β, Σ| y)<br />
dΣ<br />
∝<br />
∫<br />
T 2<br />
A −<br />
(18)<br />
The <strong>in</strong>tegral <strong>in</strong> (18) is performed by us<strong>in</strong>g properties of <strong>the</strong> <strong>in</strong>verted Wishart<br />
distribution (see, for example, Zellner 1971, p.395). For<br />
2<br />
f( β| y)<br />
∝ A −T<br />
to be