Bayesian Inference in the Seemingly Unrelated Regressions Model

11
equation (16)). Like in (22), the average in (23) is computed for, and plotted against,
a grid of values for β ik .
B. Estimating Posterior Means and Standard Deviations
Corresponding to the three ways given for estimating posterior pdf’s, there are three
ways of estimating their posterior means and variances. The first way is to use the
sample mean and covariance matrix of the draws. That is,
N
1
Eˆ( β | y ) = β = β
( )
∑ $
N $ = 1
(24)
and
N
N $ = 1
( ) ( )
( )( )
1
Vˆ( β | y ) = ∑ β $ − β β $ − β ′
−1
(25)
The second and third approaches use the results (1) an unconditional mean is
equal to the mean of the conditional means, and (2) the unconditional variance is
equal to the mean of the conditional variances plus the variance of the conditional
means. Applying these two results to the conditional posterior pdf in (19) yields
1 N
N
( $ ) 1
′
( $ ) −1 ′
( $ )
i ∑ i ∑ i () i i i () i i i
N $ = 1 N $ = 1
Eˆ( β | y ) = β # = ( X Q X ) X Q y =β# (26)
and
# # # #
( )( )
⎛
N
N
v 1 2( ) ( ) 1 1 ( ) ( )
ˆ( | )
i
⎞ $
V i y si ( X ′
$
i () i i )
−
$ $
β = Q X
′
⎜ ⎟ ∑#
+ ∑ βi − βi βi − βi
⎝vi
−2⎠N $ = 1 N −1$
= 1
(27)
yields
Applying the two results to the normal conditional posterior pdf’s in (14)