BayesianSAE using hierarchical bayes.pdf

U i σ 2 u n 0 σ 2 u and v i represents spatially

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u i | σ 2 u N (0 , σ 2 u ) and v i represents spatially correlated random effects. Initially, we consider the intrinsic Conditional Autoregressive (CAR) specification for v i (Besag et al., 1991). Under this specification the conditional distribution of v i given values v - i in all the remaining areas only involves the neighbouring areas v i | v - i , σ 2 v N ( X j δ i v j | δ i | , σ 2 v | δ i | ) (8) where δ i is the set of neighbours of area i and | δ i | the number of neighbours. In addition, we have added the constraint that the sum of the values of all the random effects v i is zero to make the intercept and the random effects identifiable (see, Banerjee et al., 2004, pages 163–164). As an alternative to this conditional specification, we can model the mean μ ij by including spatial random effects w i which are correlated according to the distance d kl between two areas k and l (Diggle et al., 1998): μ ij = α + x ij β + w i (9) with w distributed as a Multivariate Normal w | Σ MV N (0 , Σ); Σ kl = σ 2 w exp {- ( φd kl ) } (10) σ 2 w is the variance at any given point and φ is a smoothing parameter that controls the scale of the correlation between areas. Unlike model (7), we do not include a separate independent random effect u i in model (9). The motivation for doing so in model (7) lies with the fact that the spatial dependence of the intrinsic CAR random effects (8) is pre-determined by the neighbourhood structure. Hence unstructured effects are also included to allow for Bayesian learning about the strength of spatial dependence in the data, via the relative contribution of the u i and v i to the posterior (Besag et al., 1991; Eberley and Carlin, 2000). In the case of model (9), Bayesian learning about the strength of spatial dependence of the w i random effects takes place directly
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8 V. G´omez-Rubio et al. via the posterior estimation of the correlation parameter φ in (10) ( φ 0 implies no spatial correlation). It is technically possible to include a separate independent random effect term in (9), but in practice this can result in poorly identified posterior distributions (Diggle et al., 2002). For all these models, a sensible area level estimate is ˆ μ b,i = E ·| y [ α + X i β + z i ] = ˆ α + X i ˆ β + ˆ z i E ·| y [ . ] denotes posterior expectation and z i denotes the random effects, which are specified as either z i = u i + v i (as in (6)) or z i = w i (as in (9)). In this case, we compute the posterior means ˆ α , ˆ β and ˆ z i of α , β and z i , respectively, assuming that area level averages of the covariates X i are available. Model 1 is essentially the one proposed by Battese et al. (1988) modified to include different types of random effects. It assumes the same within-area variation ( σ 2 e ) for individuals in all areas, which is usually unrealistic because individual variation is likely to differ between areas. We therefore consider an extension of this model to the more general case in which we have a different variance σ 2 i in each area: Model 2 y ij | μ ij , σ 2 i N ( μ ij , σ 2 i ) σ 2 i vague prior (11) In this case, each area variance is estimated using the information only from
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  • Spring '16
  • Yessi
  • Regression Analysis, Mean squared error, Bayesian inference, Bayesian statistics

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