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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22<br />
The conditional posterior pdf for ( Σ | y)<br />
is <strong>the</strong> same <strong>in</strong>verted-Wishart distribution as<br />
was given <strong>in</strong> equation (20). The marg<strong>in</strong>al posterior pdf for β is<br />
−T<br />
2<br />
f( β | y) ∝ A I S ( β )<br />
(45)<br />
The posterior pdf for β 1 conditional on <strong>the</strong> rema<strong>in</strong><strong>in</strong>g β i is <strong>the</strong> truncated multivariate<br />
t-distribution<br />
− ( K + v )/2<br />
1 1<br />
⎡ ( β1−β# 1) ′ X ′<br />
1 Q(1) X1( β1−β#<br />
1)<br />
⎤<br />
f( β1| y, β2, β3,..., βM) ∝ ⎢v ⎥<br />
1+ I ( )<br />
2<br />
S β<br />
⎢<br />
s#<br />
1<br />
⎥<br />
⎣<br />
⎦<br />
(46)<br />
Of <strong>in</strong>terest is how to best use <strong>the</strong>se pdfs to draw observations on β , and<br />
possibly Σ , from <strong>the</strong>ir respective posterior pdf’s. The conditional posterior pdf’s for<br />
( β | Σ ) and ( Σ | β ) can be used with<strong>in</strong> a Gibbs sampler provid<strong>in</strong>g <strong>the</strong> <strong>in</strong>equality<br />
restrictions are sufficiently mild for a simple acceptance-rejection algorithm to be<br />
practical when sampl<strong>in</strong>g from <strong>the</strong> truncated multivariate normal distribution. By a<br />
“simple acceptance–rejection algorithm”, we mean that a draw is made from a<br />
nontruncated multivariate normal distribution and, if it lies outside <strong>the</strong> feasible region,<br />
it is discarded and replaced by ano<strong>the</strong>r draw. This procedure will not be practical if<br />
<strong>the</strong> probability of obta<strong>in</strong><strong>in</strong>g a draw with<strong>in</strong> <strong>the</strong> feasible region is small, which will<br />
almost always be <strong>the</strong> case if <strong>the</strong> number of <strong>in</strong>equality restrictions is moderate to large.<br />
Thus, we are us<strong>in</strong>g <strong>the</strong> term “mild” <strong>in</strong>equality restrictions to describe a situation<br />
where <strong>the</strong> maximum number of draws necessary before a feasible draw is obta<strong>in</strong>ed is<br />
not excessive.<br />
If <strong>the</strong> <strong>in</strong>equality restrictions are not mild, <strong>the</strong>n a Metropolis-Hast<strong>in</strong>gs<br />
algorithm can be employed. In <strong>the</strong> steps we described <strong>in</strong> Section III, if a candidate