Bayesian Inference in the Seemingly Unrelated Regressions Model

19
[ A] = [ y −h ( X, β)][ ′ y − h ( X, β )]
(34)
ij i i j j
The joint posterior pdf for ( β, Σ ) is
− ( T+ M+ 1) 2 1 −1
2
f( βΣ , | y) ∝f( β) Σ exp{ − tr( AΣ )}
(35)
and, integrating out Σ , the marginal posterior pdf for β is
2
f( β| y) ∝ f( β ) A −T
(36)
Thus, the posterior for β in the nonlinear SUR model involves the same determinant
of sums of squares and cross products of residuals as it does in the linear model. A
more general prior has been added. (Of course, it also could have been included in
the linear model.)
The Metropolis-Hastings algorithm described in Section III can be readily
applied to the posterior pdf in equation (36). Because the earlier results on conditional
posterior pdfs for β and the β i no longer hold, the draws need to be used directly to
estimate posterior pdfs and their moments.
V. IMPOSING LINEAR EQUALITY RESTRICTIONS
Economic applications of SUR models frequently involve linear restrictions on the
coefficients. For example, the same coefficient may appear in more than one equation,
the Slutsky symmetry conditions in demand models lead to cross-equation
restrictions, or one might want to hypothesize that all equations have the same
coefficient vector. Under the existence of cross-equation linear restrictions, the Gibbs
sampler using β and Σ , and the Metropolis-Hastings algorithm, can still be used.
However, the Gibbs sampler involving only β is no longer applicable. If the
restrictions are all within equation restrictions, all three algorithms are possible.