Bayesian Inference in the Seemingly Unrelated Regressions Model

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25 4 ∑β i = 1 i= 1 4 ∑ j= 1 β = 0 ij β ij = β ji 3. Inequality restrictions are required for the functions to satisfy concavity and monotonicity. These restrictions are • Monotonicity 0 < S i < 1 • Concavity B− S + ss′ is negative semidefinite where ⎡β ⎢ ⎢β B = ⎢β ⎢ ⎢⎣ β 11 21 31 41 β β β β 12 22 32 42 β β β β 13 23 33 43 β β β β 14 24 34 44 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦ ⎡S ⎢ S = ⎢ ⎢ ⎢ ⎢⎣ 1 S 2 S 3 ⎤ ⎥ ⎥ ⎥ ⎥ S ⎥ 4 ⎦ ⎡S ⎢ ⎢S s = ⎢S ⎢ ⎢⎣ S 1 2 3 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦ Note that B− S + ss′ is negative semidefinite if and only if its largest eigenvalue is nonpositive. Since S i depends on the input prices, a decision concerning the input prices at which S i is evaluated, and the inequality restrictions imposed, needs to be made. The inequality restrictions were imposed at average input prices for each of the 23 years. 4. Given the severe inequality restrictions that were imposed, the Metropolis- Hastings algorithm was used. 5. The quantities of interest are nonlinear functions of the parameters. They are the elasticities of substitution
26 σ ij = β i ij S S j + 1 i ≠ j and input demand elasticities ii ηii = + Si −1 Si η ij β β = S ij i + S j C. Expenditure Functions Our third example involves two expenditure functions estimated from a sample of 1,834 Bangkok households, and deflated by an “equivalence scale” measure of household size (Griffiths and Chotikapanich 1997). For the t-th observation, the functions are α m β m ( x − α m − α − α m 1 1t 1 1t t 1 1t 2 2t 3 3t w1 t = + e1 t xt xt ( β1m1 t + β2m2t + β3m3t ) m ) + α m β m ( x − α m − α − α m 2 2t 2 2t t 1 1t 2 2t 3 3t w2t = + + e2t xt xt ( β1m1 t + β2m2t + β3m3 t ) m ) m = 1+δ n +δ n 1t 21 2t 31 3t m = 1+δ n +δ n 2t 22 2t 32 3t m = 1+δ n +δ n 3t 23 2t 33 3t where w jt = expenditure proportion for commodity j, x t = total expenditure, m jt = equivalence scale for commodity j,
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 and 20: 18 algorithms that we described are
Page 21 and 22: 20 Suppose a set of J linear restri
Page 23 and 24: 22 The conditional posterior pdf fo
Page 25: 24 Although Griffiths et al (2001)
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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