By Alfred Taudes (auth.), Univ.-Prof. Mag. Dr. Alfred Taudes (eds.)
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Extra info for Adaptive Information Systems and Modelling in Economics and Management Science
2004) that such a full conditional Gibbs sampler is sensitive to the way model (1) is parameterized. There exist two ways to parameterize the model, depending on whether X i1 and Xi2 have common columns or not. The partly marginalized Gibbs sampler suggested in Fr¨uhwirth-Schnatter et al. (2004) where the random effects β N are integrated out 59 G ) turned out to be insensitive to the paramewhen sampling S N , α, and (β1G , . . , βK terization. In Fr¨uhwirth-Schnatter et al. (2004) the partly marginalized Gibbs sampler was introduced for homogeneous error variances.
The covariance matrices are sampled for each group separately. from the inQ Q N G verted Wishart distribution p(QG k |β , βk , Si = k) ∼ IW (νk , Sk ) with νkQ = ν0Q + Nk /2, N SkQ = S 0Q + 1/2 (k) Di (βi − βkG )(βi − βkG ) , i=1 (k) where Di = 1 iff Si = k and zero otherwise, Nk = #(Si = k) and ν0Q and S 0Q are the parameters of the prior inverted Wishart distribution of Q G k. (iv b) Sampling of the error variance parameter σε2 : Applying Bayes theorem yields the posterior inverted gamma distribution for σε2 .
2004b). Different heterogeneity models M1 , . . , ML are compared through their posterior probabilities: P (Ml |y N ) ∝ p(y N |Ml )P (Ml ), where the model likelihood p(y N |Ml ) is identical with the integrated likelihood function p(y N |φ) for model Ml : p(y N |Ml ) = p(y N |φ)p(φ)dφ. (7) G G 2 , η, QG φ = (α, β1G , . . , βK 1 , . . , QK , σε ) is the vector of unknown model parameN ters. Note that in (7) p(y |φ) is the marginal likelihood, where all latent variables like S N , β N and, for heterogeneous error variances, λN are integrated out.
Adaptive Information Systems and Modelling in Economics and Management Science by Alfred Taudes (auth.), Univ.-Prof. Mag. Dr. Alfred Taudes (eds.)