Bayesian Inference in the Seemingly Unrelated Regressions Model

17
Section V), then f( β * | y) = 0. When r > 1, β * is a more likely value than
( )
β $
in the sense that it is closer to the mode of the distribution. When
r < 1, β * is further into the tails of the distribution. If r > 1, β * is
accepted; if r < 1, β * is accepted with probability r . Thus, more draws
occur in regions of high probability and fewer draws occur in regions of
low probability. Details of the acceptance-rejection procedure follow in
step 3.
3. Draw a value u for a uniform random variable on the interval (0,1).
If
u ≤ r , set
( $ +1 )
β = β*
.
If
u > r , set
( )
β = β
$ +1 ( $ )
.
Return to step 1 with $ set to $ + 1.
Let
* ( )
( | )
q β β $ be the distribution used to generate the candidate value β * in step 1. In
our case it is a normal distribution. In more general Metropolis-Hastings algorithms,
where our choice of distribution is not necessarily utilized, r is defined as
r =
* ( $ ) *
f( β | y) q( β | β )
.
( $ ) * ( $ )
f( β | y) q( β | β )
In our case
( $ ) * * ( $ )
q( β | β ) = q( β | β )
. Various alternatives for q (.) have been
suggested in the literature.
IV.
NONLINEAR SUR
Many economic models are intrinsically nonlinear, or a nonlinear model may result
from substituting nonlinear restrictions on β into a linear model. The Gibbs sampling