Bayesian Inference in the Seemingly Unrelated Regressions Model

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27 n 2t = number of extra adults (each household has at least one adult), n 3t = number of children. The unknown parameters are (α 1 , α 2 , α 3 , β 1 , β 2 , β 3 , δ 21 , δ 31 , δ 22 , δ 32 , δ 23 , δ 33 ) This SUR model has the following characteristics: 1. The equations are nonlinear in the parameters. 2. A number of inequality restrictions were imposed, namely, 0 < β1, β 2 < 1 Additional expenditure from a one-unit increase in supernumerary income must lie between zero and one. 0≤δ ≤δ ≤1 Expenditure requirements for extra adults are less than 3j 2 j those for the first adult but greater than those for children. ⎧ eit ⎫ α i < min ⎨ ⎬ i = 1, 2 t ⎩1+δ 2in2t +δ3in3t ⎭ The smallest level of consumption in the sample must be greater than subsistence expenditure, a constraint from the utility function. 3. Given the nonlinear equations and the inequality constraints, the Metropolis-Hastings algorithm was used. 4. Two nonlinear functions of the parameters are of interest. They are the general scale or "household size":
28 m x ∑ m k k k 0 = x − ∑k α kmk + ∑k α k∑k β kmk β and the elasticities. Expressions for the latter can be found in Griffiths and Chotikapanich (1997). VIII. FORECASTING Suppose that we are interested in forecasting dependent variable values in the next period. The shire-level wheat yield application in the previous section is an example of where such a forecast would be of interest. In that case the objective is to forecast yield for each of the five shires. Since the yields are correlated via the stochastic assumptions of the SUR model, a joint forecast is appropriate. We can write next period’s observation as y* = X* β + e* (49) where y * is an M-dimensional vector, X * is an ( M × K) block diagonal matrix with the i-th block being a 1× K ) row vector containing next period’s explanatory ( i variables for the i-th equation, X * ⎡x1* ⎤ ⎢ x ⎥ 2* = ⎢ ⎥ ⎢ " ⎥ ⎢⎣ x3* ⎥⎦ (50) and e* ~ N (0, Σ ) is next periods ( M ×1 ) random error vector. The conventional Bayesian forecasting tool is the predictive pdf f( y* | y ). Graphing marginal predictive pdf’s from this density function, and computing its means, standard deviations and probabilities of interest are the standard ways of reporting results.
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 and 26: 24 Although Griffiths et al (2001)
Page 27: 26 σ ij = β i ij S S j + 1 i ≠
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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