The log likelihood value for the model in (5) is calculated as
) ) , | ( * ) | 1 ( ) , | ( * ) | 0 ( l n ( * ) | ( l n 2 2 , 1 2 2 , 1 1 0 u v t t t u v t t t t y f Y S p y f Y S p Y f ∑ ,
2 v 1
] 2 5 v .
The prior log density can be expressed as * ) * , ( l n * ) ( l n * ) ( l n * ) l n 2 2 q p u v ,
2 u 0
2 u 1
The posterior density is calculated using the method described in Chib (1995) to simulate marginal conditional densities. The log posterior density can be expressed as
* ) * , , | * * , ( ~ l n * ) , | * ( ~ l n ) | * ( ~ l n ) | * ~ l n 2 2 2 2 2 u v t v t u t v t Y q p Y Y Y .
Derivation of the posterior density for the model in (5) is presented in Appendix B.
Marginal likelihood values for each of the economic sector portfolios for each of the four model specifications are presented in Table 4. For each model, the priors are non-informative and the Gibbs sampler is run 8,000 times after 2,000 initial Gibbs runs to realize some convergence for the parameters for each step of the simulation. To evaluate the relative strength of each model, we use the guidelines for model comparison in Kass and Raftery (1995). The guidelines for interpreting the log Bayes factors for model l and model j are
ln Bl, j
Evidence against model j
0 to 1 1 to 3 3 to 5 >5
Not worth more than a bare mention Positive Strong Very Strong