Bayesian Inference in the Seemingly Unrelated Regressions Model

7
proper, we require T ≥ M + rank( X*
) (Griffiths et al 2001). Also, this pdf is not of a
standard recognisable form. Except for special cases, analytical expressions for its
normalising constant and moments are not available. Estimating these moments, and
marginal pdf’s for individual coefficients β ik
we describe some more pdf’s that will prove to be useful.
, is considered in the next section; first,
D. Conditional Posterior pdf for ( β 1| β 2,..., β
M )
It is possible to show that the posterior pdf for the coefficient vector from one
equation, conditional on those from other equations, is a multivariate t-distribution.
To derive this result, we will consider the posterior pdf for β 1 , conditional on
( β , β ,..., β ) . We write a partition of ( Y − X*
B)
into its first and remaining
2 3
M
( M − 1) columns as
( 1 1 1 (1))
Y − X*
B= y − X β E
The corresponding partition of A is
⎡( y1− X1β1)( ′ y1− X1β1) ( y1− X1β1 )′
E(1)
⎤
A = ⎢ ⎥
⎢ E ′
(1) ( y1 X1 1 ) E ′
⎣
− β
(1) E(1)
⎥⎦
Using a result on the determinant of a partitioned matrix, we have
−
(( )( ′ ) ( )′
( ′ ) )
1 ( )
A = E ′ E y − X β y − X β − y − X β E E E E ′ y − X β
(1) (1) 1 1 1 1 1 1 1 1 1 (1) (1) (1) (1) 1 1 1
Defining
Q = I − E E ′ E E ′, and
−1
(1) T (1) ( (1) (1) ) (1)
β # = ′ ′ the second
1
1 ( X1 Q(1) X1)
− X1 Q(1) y1
term in the above equation can be written as
( y − X β )′ Q ( y − X β ) = ( y − X β # )′ Q ( y − X β # ) + ( β − β # )′
X ′ Q X ( β − β # )
1 1 1 (1) 1 1 1 1 1 1 (1) 1 1 1 1 1 1 (1) 1 1 1