Bayesian Inference in the Seemingly Unrelated Regressions Model

29
*
The procedure for deriving the predictive pdf is to begin with the joint pdf
f( y , βΣ , | y)
and to then integrate out Σ and β , either analytically or via a
numerical sampling algorithm. Now,
−M
/2 −1/2
1
−1
* 2 * *
′
* *
f( y | β, Σ ) = (2 π) Σ exp{ − ( y − X β) Σ ( y − X β)}
−1/2 1 −1
A
2 *
∝Σ exp{ − tr( Σ )}
(51)
where A* = [ y− X* β][ y− X * β ]′
. Thus, using the posterior pdf in equation (12) (no
inequality restrictions), we have
f( y , β, Σ | y) = f( y | β, Σ) f( β, Σ | y)
* *
− ( T + M + 2) 2 1
1
A A
−
2
*
∝Σ exp{ − tr[( + ) Σ ]}
Using properties of the inverted Wishart distribution to integrate out Σ yields
∫
f( y , β | y) = f( y , β, Σ| y)
dΣ
* *
*
( T 1) 2
∝ A+
A − +
(52)
Because analytical integration of β out of equation (52) is not possible, we consider
the conditional predictive pdf f( y*
| β , y)
. It turns out that this pdf is a multivariate
student t. Thus, f( y*
| y ) and its moments can be estimated by averaging quantities
from f( y*
| β , y)
over draws of β obtained using one of the MCMC algorithms
described earlier.
To establish that f( y*
| β , y)
is a multivariate t-distribution, we first note that
(see, for example, Dhrymes 1978, p. 458)
−1
* * *
′
* *
A+ A = A (1 + ( y − X β) A ( y − X β ))
(53)
Thus,