Estimation of each model is performed using Gibbs sampling Markov chain Monte Carlo (MCMC) integration. A brief outline of the Gibbs sampling sequence for the model in (5) is presented in Appendix A. Details of the output of the Gibbs sampler for the model in (5) are presented in greater detail in a later section. Each of the alternative models presented in this section are simplified versions of that in (5) with fewer components. We choose the Bayesian methodology for evaluating our model for two primary reasons. First, the Gibbs sampling methodology allows for more stable estimation of multi-parameter non-linear models than does typical maximum likelihood estimation. Secondly, model comparison using marginal likelihood values provides a way to compare alternative non-nested model specifications not available in the classical framework. From here forward in this section, for simplified notation, the portfolio subscript i is dropped.
Calculating Bayes Factors for Model Comparison
To assess which model explains the data best, marginal likelihood values are calculated for each of the models presented in the previous section. From these marginal likelihood values, Bayes factors comparing any of the two models can then be computed as the ratio of the marginal likelihood values.
) ( ) ( t BFl, j j t l Y m Y m
where BFl, j
is the Bayes factor of the posterior odds in favor of model l over model j.
The marginal likelihood values for each model are calculated from
) ( t Y m
* ) * ) | ( t Y f
) | * ~ t Y
Here the numerator on the right side of (7) is the product of the likelihood value of the data at
* and the prior density at *. Here, * indicates that the parameter space is evaluated at
the posterior mean from the initial Gibbs sampling runs as presented in Appendix A. The denominator is the simulated posterior density of *. For computational efficiency, the relation in (7) can be rewritten in log form as
) | * ~ l n * ) l n * ) | ( l n ) ( l n t t t Y Y f Y m .