# _5.2. Prediction_

In order to compare the predictive performance associated with the various prior choices, we turn now to the task of predicting the observable, growth, given the regressors. Of course, the predictive distribution is also derived through model averaging, as explained in, _e.g._ , FLS. As a measure of how well each model predicts the retained observations, we use the log predictive score, which is a strictly proper scoring rule, described in FLS. In the case of i.i.d. sampling, LPS can be given an interpretation in terms of the Kullback-Leibler divergence between the actual sampling density and the predictive density (see Fern ́andez _et al._ 2001a), and smaller values indicate better prediction performance.

We compare the predictions based on: _(i)_ BMA, _(ii)_ the best model (the model with the highest posterior probability), _(iii)_ the full model (with all _k_ regressors), and _(iv)_ the null model (with only the intercept).

Panel A in Table 5 summarizes our findings for the FLS data: the entries indicating “best” or “worst” model or how often a model is beaten by the null are expressed in percentages of the 100 samples for that particular prior setting.

- _(i)_ BMA—The predictive performance of BMA is much superior to that of the other procedures— which corroborates evidence in _e.g._ Raftery _et al._ , 1997, Fern ́andez _et al._ , 2001a and FLS. It is never the worst predictor and leads to the best predictions in more than half of the sampled cases (with the exception of the prior with fixed _θ_ = 0 _._ 5 and _g_ = 1 _/n_ ).

- _(ii)_ Best—Basing predictions solely on the model with highest posterior probability is clearly a lot worse: it almost never gives the best prediction and leads to the worst prediction in 18 to 46% of the cases; moreover, it is beaten by the simple null model in more than 35% of the cases.

- _(iii)_ Full—The use of the full model can lead to good forecasts, but is very risky, as it also has a substantial probability of delivering the worst performance: in fact, for _g_ = 1 _/k_<sup>2</sup> the proportion of the latter always exceeds the fraction of best performances. This behavior is also illustrated by the fact that _{_ Min _,_ Mean _,_ Max _}_ for LPS of the full model is _{_ 0 _._ 74 _,_ 1 _._ 77 _,_ 3 _._ 77 _}_ for _g_ = 1 _/n_ and _{_ 0 _._ 67 _,_ 2 _._ 15 _,_ 5 _._ 82 _}_ for _g_ = 1 _/k_<sup>2</sup> . For comparison, the null model leads to _{_ 1 _._ 67 _,_ 2 _._ 05 _,_ 2 _._ 72 _}_ .

Having established that BMA is the strategy to adopt for prediction, we can focus on the forecast performance of BMA to compare the predictive ability across the different prior settings. Mean values of LPS (over all 100 samples) are not that different, but the maximum values indicate that fixing _θ_ at 0.5 is the most risky strategy. This suggests using random _θ_ . In addition, BMA always performs better with respect to the other prediction strategies for _g_ = 1 _/k_<sup>2</sup> . Indeed, the worst case scenario for BMA appears to be _θ_ = 0 _._ 5 with _g_ = 1 _/n_ . Finally, the choice of _m_ almost leaves the random _θ_ results unaffected, but has a substantial effect on the cases with fixed _θ_ , in line with our expectations.

Panel B in Table 5 presents the same quantities for the MP data (where _k_ = 54), and the _{_ Min _,_ Mean _,_ Max _}_ values for LPS of the full model are _{_ 1 _._ 13 _,_ 2 _._ 55 _,_ 6 _._ 05 _}_ for _g_ = 1 _/n_ and _{_ 1 _._ 12 _,_ 2 _._ 79 _,_ 7 _._ 48 _}_ for _g_ = 1 _/k_<sup>2</sup> , whereas the null model leads to _{_ 1 _._ 70 _,_ 2 _._ 03 _,_ 2 _._ 63 _}_ .

The superiority of BMA is even more pronounced, with the prior setting _θ_ = 0 _._ 5 with _g_ = 1 _/n_ again being the least favorable for BMA. The performance of the full model is now considerably 20