so this is an ideal candidate for BMA. We consider below two datasets which have been used in many studies on this topic.

# The FLS data

The first dataset we use contains _k_ = 41 potential regressors for modelling average per capita growth over the period 1960-1992 for a sample of _n_ = 72 countries. It was used in Fern ́andez _et al._ (2001b) (FLS), which presents more details.

First, we consider the analysis of the 50 subsamples which each consist of 62 observations. In line with the higher model size penalty implicit in the benchmark Beta prior with _c_ = 1 and the case with fixed _g_ = _k_<sup>2</sup> , we note that marginal posterior inclusion probabilities of the regressors tend to be (sometimes much) smaller for these cases. In keeping with the higher _g_ values induced by these priors, the posterior mean of _βj_ for model _Mj_ will be larger in absolute value (less shrinkage to the prior mean of zero), but this will typically be counteracted by more mass on smaller models. The resulting marginal posterior means of the regression coefficients can be smaller or larger than those for the other priors on _g_ . For the other priors posterior inclusion probabilities and posterior inference on the regression coefficients are quite similar, even though the posterior inference on _g_ can be somewhat different. Figure 6 presents boxplots over the 50 subsamples for the posterior medians of _g_ (on a log scale), the number of visited models and the posterior mean model size. As with the simulated data, posterior medians of _g_ are smallest for the hyper- _g_ priors and largest for the benchmark prior with _c_ = 1 and the case with fixed _g_ = _k_<sup>2</sup> . This leads to most models visited in the chain for the hyper- _g_ priors and least models in the chain for the benchmark with _c_ = 1 and _g_ = _k_<sup>2</sup> cases. Posterior model sizes are lower for the latter two cases, but not that different between the other priors. Note also that the prior assumptions on model space have almost no effect on posterior inference on _g_ , but prior mean model size _m_ does substantially affect the number of visited models and the posterior mean model size when we use a fixed _θ_ approach. In line with the results in Ley and Steel (2009), however, this dependence on _m_ virtually disappears when we use the random _θ_ prior in (5). Thus, for the rest of the section we will only present results for random _θ_ with _m_ = 7.

Prediction based on LPS (for 10 observations in each prediction subsample) is summarized in the top panel of Figure 7, where we present boxplots of the LPS values for BMA over the 50 prediction subsamples. As lower values of LPS are associated with better predictions, this suggests that the benchmark prior with _c_ = 1 and the fixed _g_ = _k_<sup>2</sup> case tend to predict somewhat worse than the rest. The other priors are quite close in predictive performance. This is consistent with these priors leading to quite similar posterior inclusion probabilities for the regression variables, and rather different from those obtained with the priors that result in worse predictions. BMA predicts best (compared with the highest posterior probability model, the null model and the full model) in most of the subsamples (in between 52% and 62%, depending on which prior we choose for _g_ ) and is never beaten by the null model, except for a few cases (up to 6%) with the benchmark priors with _c_ = 1 and _c_ = 0 _._ 1 and the fixed _g_ specifications.

The information contained in the data regarding _g_ is summarized by the marginal likeli23