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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20<br />
Suppose a set of J l<strong>in</strong>ear restrictions is written as<br />
⎛η⎞<br />
Rβ = ( R1 R2)<br />
⎜ ⎟=<br />
r<br />
⎝γ<br />
⎠<br />
(37)<br />
where R<br />
1<br />
is ( J × J ) and nons<strong>in</strong>gular, R<br />
2<br />
is ( J × ( K − J )) , and η and γ are J and<br />
( K − J ) dimensional sub-vectors of β , respectively. To make this partition, it may be<br />
necessary to reorder <strong>the</strong> elements <strong>in</strong> β . Correspond<strong>in</strong>gly, we can reorder <strong>the</strong> columns<br />
of X and partition it so that <strong>the</strong> l<strong>in</strong>ear SUR model can be written as<br />
⎛η⎞<br />
y = Xβ + e= ( X1 X2)<br />
⎜ ⎟+<br />
e<br />
⎝γ<br />
⎠<br />
(38)<br />
This reorder<strong>in</strong>g may destroy <strong>the</strong> block-diagonal properties of X . From (37), we can<br />
solve for η as<br />
−1<br />
1 2<br />
η= R ( r−R<br />
γ )<br />
(39)<br />
Substitut<strong>in</strong>g (39) <strong>in</strong>to (38) and rearrang<strong>in</strong>g yields<br />
− 1 −1<br />
1 1 2 1 1 2<br />
y− X R r = ( X − X R R ) γ + e<br />
or<br />
z = Zγ + e<br />
(40)<br />
where<br />
−1<br />
z = y − X R r , and<br />
1<br />
1<br />
−1<br />
2 1 1 2<br />
Z = X − X R R represent new sets of “observations”.<br />
In general, Z and γ can no longer be partitioned unambiguously <strong>in</strong>to M separate<br />
equations. However, <strong>the</strong> stochastic properties of e rema<strong>in</strong> <strong>the</strong> same. Thus, all <strong>the</strong><br />
results <strong>in</strong> Sections II and III that did not rely on a partition<strong>in</strong>g of X and β can still be<br />
applied to <strong>the</strong> model <strong>in</strong> (40). In particular, a Gibbs sampler can be used to draw γ and