Maruyama-George and Zellner-Siow priors (leading to a zero posterior density at _g_ 0), the F-Z with _a_ = 2 + 2 _/n_ , the hyper- _g/n_ and horseshoe- _n_ priors and the case with _g_ = _n_ , as a consequence of the very different value of _n_ for this application. In this application, the truncation induced by the Forte _et al._ prior means that the prior support does not include the chosen value of _g_ 0 (and is truncated quite a bit above that), which certainly makes this prior less appropriate for large values of _n/k_ . Thus, prior and posterior density values are both zero at _g_ 0 and the Bayes factor can not be computed for this case.<sup>17</sup> Let us summarize the information provided by these applications to real economic datasets. Predictive performance seems quite similar across priors, although possibly a bit worse for the benchmark prior with _c_ = 1 and the case with _g_ = _k_<sup>2</sup> . There is more variation in the Bayes factors, with most consistent data support provided for the Bottolo-Richardson, horseshoe, hyper- _g/n_ and horseshoe _/n_ priors, whereas the Maruyama-George, Forte _et al._ and Zellner-Siow priors (as well as the case with fixed _g_ = _n_ ) do quite well in two of the three applications, but rather badly in the one with large _n_ .

# **9. Concluding Remarks and Recommendations**

Combining the properties listed in Table 1 with the evidence from both simulated and real data, we can now come up with a recommendation for practitioners. We assume that users will want their priors to be consistent, to avoid the information paradox and to perform well in a wide variety of situations. The Bottolo-Richardson and horseshoe- _n_ priors are consistent (albeit at a slow rate) and do well on real data, but underperform on the simulated data. The truncation of the Forte _et al._ . prior makes it hard to recommend for situations where _n_ is appreciably larger than _k_ . The benchmark Beta prior with _c_ = 1 does not fare well in prediction for one application, and the benchmark priors with _c_ = 1 and _c_ = 0 _._ 1 do not do well in terms of Bayes factors. The Zellner-Siow and Maruyama-George priors perform quite well, except for the last application (with large _n_ ), where they get very little support from the data. Nevertheless, we feel they do deserve a place in the econometrician’s toolbox, especially if _n_ is relatively small (comparable to _k_ , say). In addition, if one wants to avoid numerical methods to compute Bayes factors between models with different sets of regressors, the Maruyama-George prior should be recommended, although this comes at the (small) cost of making the prior model-specific, which may make it slightly harder to interpret the prior on _g_ .

In our view, the two priors that stand out by not having displayed any truly bad behaviour in our experiments are the benchmark Beta prior with _c_ = 0 _._ 01 and the hyper- _g/n_ prior (with _a_ = 3).<sup>18</sup> Thus, these priors provide an interesting compromise and would be our general recommendations to practitioners.

> 17 Using a much larger value of _g_ 0 ( _g_ 0 _>_ 40) it would be possible, in principle, to compute a Bayes factor, but we can safely assume the Tobias-Li data do not support this prior at all, as it puts no mass whatsoever on areas with large values for the marginal likelihood _ly_ ( _g_ ).

> 18 For the hyper- _g_ prior we find (as do Feldkircher and Zeugner, 2009) that the choice between _a_ = 3 and _a_ = 4 makes little difference. As a consequence, we would also expect the hyper- _g/n_ prior with _a_ = 4 to perform very similarly.

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