worse than for the FLS data, as this model quite often leads to the worst behavior and is soundly beaten by the much more conservative null model. This seems in line with the fact that the number of regressors, _k_ , is now larger, so the full model is even more overparameterized than in the FLS case. Considering the LPS values for BMA, priors with _g_ = 1 _/n_ seem to have a slight edge, except for the case with a fixed _θ_ = 0 _._ 5. Finally, the choice of _m_ is again virtually immaterial for the random _θ_ cases, and affects those with fixed _θ_ .

Finally, Panel C in Table 5 collects prediction results for the SDM data, where the number of regressors is even larger with _k_ = 67. This is immediately felt in the behavior of the full model, which is now virtually always the worst model. The _{_ Min _,_ Mean _,_ Max _}_ values for LPS of the full model are _{_ 2 _._ 53 _,_ 5 _._ 16 _,_ 11 _._ 75 _}_ for _g_ = 1 _/n_ and _{_ 2 _._ 65 _,_ 6 _._ 95 _,_ 23 _._ 21 _}_ for _g_ = 1 _/k_<sup>2</sup> , whereas the null model leads to _{_ 1 _._ 72 _,_ 2 _._ 08 _,_ 3 _._ 09 _}_ .

As before, the case with _θ_ = 0 _._ 5 and _g_ = 1 _/n_ leads to the worst BMA performance, allowing the null model to beat it more than half of the time! This is due to those cases where prediction is relatively difficult for which the conservative null model performs better. Throughout, the choice of _g_ = 1 _/k_<sup>2</sup> does better in avoiding large values of LPS, reducing the cases for which BMA is beaten by the null model. On the other hand, it is more conservative and lowers the percentage of cases where BMA leads to the best predictive behavior. As we know from Subsection 3.3, the choice of _g_ = 1 _/k_<sup>2</sup> implies a larger size penalty, and thus stays closer to the rather conservative null model. As in the previous examples, the effect of _m_ is far larger for fixed _θ_ than for random _θ_ . For the combinations of random _θ_ with _g_ = 1 _/k_<sup>2</sup> BMA seems less dominant than for the other cases. This is simply a consequence of the fact that the best model then accounts for more than 60% of the probability mass (see Table 4), so that the best model also predicts very well here (albeit not quite as well as BMA). In fact, either BMA or the best model predict best in 90 of the 100 samples, while the null model does best for the remaining 10 samples.

If we measure robustness as in Subsection 5.1 by the standard deviation of the LPS values for BMA, Table 5 also shows us that cases with random _θ_ and _g_ = 1 _/k_<sup>2</sup> are the most robust. Note that differences in robustness can be quite large, and the case with fixed _θ_ = 0 _._ 5 and _g_ = 1 _/n_ stands out as the least robust combination by far.

To summarize the predictive behavior in terms of the choice of prior setting, we can state that a random _θ_ prior seems preferable in view of the lack of sensitivity to _m_ . Also, the combination of fixed _θ_ = 0 _._ 5 and _g_ = 1 _/n_ is to be avoided as it can lead to relatively bad forecasting behavior and BMA is not as dominant as it is under other priors. For these other priors there seems no clear guidance for the choice of _g_ on the basis of predictive behavior alone.

22