Bayesian Inference in the Seemingly Unrelated Regressions Model

15
Although the above remarks on convergence and numerical standard errors
were made in the context of the Gibbs sampler for β and Σ , they also apply to other
MCMC algorithms including the Gibbs sampler for β and the Metropolis Hastings
algorithm described below.
F. Gibbs Sampling with β
If the number of equations is large, making Σ of high dimension, then it may be
preferable to use a Gibbs sampler based on the conditional posterior pdfs for the β i
from each equation. Note, however, that this alternative is not feasible if crossequation
restrictions on the β i , as discussed in Sections V and VII, are present.
To proceed with this Gibbs sampler, we begin with starting values for all
coefficients except the first, say
(0) (0) (0)
2 3
( β , β ,…, β M ) and then sample iteratively using
the following steps for the $ -th draw:
1. Draw
2. Draw
( )
β $ 1 from ( ( $ − 1) ( $ −
f β 1)
1| β2
, , βM
)
… .
( )
β $ 2 from ( ( $ ) ( $ − 1) ( $ −
f β 1)
2 | β1 , β3
, , βM
)
… .
!
i. Draw
( )
β $ i from ( ( $ ) ( $ ) ( $ − 1) ( $ −
f β | 1)
i β1 , , βi− 1, βi+
1 , , βM
)
… … .
!
M. Draw
( )
β $ M from ( ( $ ) ( $
f β | )
M β1 , , βM−1)
… .
The conditional posterior pdfs are multivariate t-distributions from which we can
readily draw values (see the Appendix). Ordinary least squares estimates are adequate
for starting values.