Bayesian Inference in the Seemingly Unrelated Regressions Model

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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 However, the Gibbs sampler involving only β is no longer applicable.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.
20 Suppose a set of J linear restrictions is written as ⎛η⎞ Rβ = ( R1 R2) ⎜ ⎟= r ⎝γ ⎠ (37) where R 1 is ( J × J ) and nonsingular, R 2 is ( J × ( K − J )) , and η and γ are J and ( K − J ) dimensional sub-vectors of β , respectively. To make this partition, it may be necessary to reorder the elements in β . Correspondingly, we can reorder the columns of X and partition it so that the linear SUR model can be written as ⎛η⎞ y = Xβ + e= ( X1 X2) ⎜ ⎟+ e ⎝γ ⎠ (38) This reordering may destroy the block-diagonal properties of X . From (37), we can solve for η as −1 1 2 η= R ( r−R γ ) (39) Substituting (39) into (38) and rearranging yields − 1 −1 1 1 2 1 1 2 y− X R r = ( X − X R R ) γ + e or z = Zγ + e (40) where −1 z = y − X R r , and 1 1 −1 2 1 1 2 Z = X − X R R represent new sets of “observations”. In general, Z and γ can no longer be partitioned unambiguously into M separate equations. However, the stochastic properties of e remain the same. Thus, all the results in Sections II and III that did not rely on a partitioning of X and β can still be applied to the model in (40). In particular, a Gibbs sampler can be used to draw γ and
Page 1 and 2: Bayesian Inference in the Seemingly
Page 3 and 4: 2 The objective of this chapter is
Page 5 and 6: 4 A Y X B ′ Y X B = ( − * )(
Page 7 and 8: 6 E( β| y, Σ ) =β= ˆ [ X′ (
Page 9 and 10: 8 Collecting all these results, sub
Page 11 and 12: 10 N ( $ ) ( $ ) ( $ ) ( $ ) ( 1
Page 13 and 14: 12 N N 1 ˆ( ) 1 1( ) 1 1( ) ˆ( |
Page 15 and 16: 14 1. Draw ( ) β $ from ( −1) f(
Page 17 and 18: 16 G. A Metropolis-Hastings Algorit
Page 19: 18 algorithms that we described are
Page 23 and 24: 22 The conditional posterior pdf fo
Page 25 and 26: 24 Although Griffiths et al (2001)
Page 27 and 28: 26 σ ij = β i ij S S j + 1 i ≠
Page 29 and 30: 28 m x ∑ m k k k 0 = x − ∑k
Page 31 and 32: 30 −1 f( y* | β, y) ∝ ⎡1 ( y
Page 33 and 34: 32 ⎛ U U U y ⎞ ⎛ () t X ⎞
Page 35 and 36: 34 B. Multivariate t Distribution C
Page 37 and 38: 36 REFERENCES Albert, J.H., Chib, S
Page 39 and 40: 38 Handbook of Applied Econometrics
Page 41: 40 Zellner, A., 1962. An efficient

Bayesian Inference in the Seemingly Unrelated Regressions Model

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