Bayesian Inference in the Seemingly Unrelated Regressions Model

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value β
*
is infeasible, then r = 0 , and the retained draw is automatically the last
accepted feasible draw. That is,
β
( $ + 1) ( )
= β
$ .
If the inequality restrictions are not mild, but are linear, then using a Gibbs
sampler on subcomponents of β might prove successful. For example, using the
truncated multivariate t-distributions for each of the β i , as specified in equation (46),
could be useful. Also within different contexts, sampling from truncated multivariate t
and multivariate normal distributions has been broken down into sampling from
univariate conditional distributions by Geweke (1991) and Hajivassiliou and
McFadden (1990). Also, see the Appendix.
VII.
THREE APPLICATIONS
A. Wheat Yield
In Griffiths et al (2001) the following model was used for predicting wheat yield in
five Western Australian shires.
2 3 2 2 2
t 1 2 3 4 5 t 6 t 7 t 8 t 9 t 10 t t
Y = β + β t+ β t + β t + β G + β G + β D + β D + β F + β F + e (47)
Yield ( Y t ) depends on a cubic time trend to capture technological change and on
quadratic functions of rainfall during the germination period ( G t ) , the development
period ( D t ) , and the flowering period ( F t ). The rainfalls are measured relative to
their sample means. Inequality restrictions are imposed to ensure that the response of
yield to rainfall, at average rainfall, is positive. That is, for germination rainfall, for
example, ∂Y
/ ∂ G =β 5+ 2β 6 > 0. Thus, the feasible region for this example is
{ }
S( β ) = β | β + 2β > 0, β + 2β > 0, β + 2β > 0
(48)
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