The hierarchical Bayesian model explored in this paper has a hyperprior on the covariate inclusion probability _θ_ and a hyperprior on _g_ , which leads to an integral for the marginal likelihood that is solved by running the MCMC sampler over models and _g_ jointly. The advantage of these hyperpriors is to make the analysis more robust with respect to often arbitrary prior assumptions. We now allow the data to inform us on variable inclusion probabilities and the appropriate region for _g_ . This will affect the model size penalty (and, to a much lesser extent, the lack-of-fit penalty) for each given model. Putting a prior on both _θ_ and _g_ makes the analysis naturally adaptive and avoids the information paradox (Liang _et al._ , 2008), which affects analyses with fixed _g_ . It is important to stress that we are not claiming that using hyperpriors on _θ_ and _g_ performs better (according to some criterion) than a model with fixed values of _θ_ and _g_ that are chosen to optimize this same criterion. The point is rather that we don’t know what the “optimal” values are and getting them wrong can lead to very poor performance. We feel the model used here with the recommended priors on _g_ can be considered a safe “automatic” choice for use in Bayesian Model Averaging in the types of linear regression problems that typically arise in a variety of econometric settings.

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