Table 2 also presents the first two moments of the regression coefficients. The first set of moments is marginalized over all models visited, including the zeros for those models that exclude the regressor in question. These numbers reveal a strong positive correlation (equal to 0.89) between the absolute values of the normalized posterior mean and the inclusion probabilities. This ensures us that there are no variables with high inclusion probability but counteracting effects in different substantial sets of models. Any serious doubt as to the sign of the effect would be reflected in a posterior mean close to zero and a large posterior variance. This is reassuring as it implies that the direction of the influence of variables with high inclusion probabilities tends to be clear. Plots of the posterior densities of the _βi_ ’s can be found in FLS, which shows that the variable “Fraction Catholic” has modes on different sides of the origin. Indeed, this is the one variable where the value of the normalized posterior mean is a lot lower than could be expected on the basis of the (moderate) inclusion probability.

The second set of moments presented in Table 2 are computed conditionally upon inclusion— _i.e._ , by averaging over only those models which actually include the coefficient in question. These moments are important for assessing the effect of each regressor, given that it is included in the model. Clearly, differences with the unconditional moments can be very large if the regressor has a low or even moderate posterior inclusion probability.

# _3.2. Application to the Data Set of SDM and DW_

SDM, DW and DWa use a larger data set, and model annual GDP growth per capita between 1960 and 1996 for _n_ = 88 countries as a function of _k_ = 67 potential drivers. This leads to an even larger model space with 2<sup>67</sup> = 1 _._ 5 _×_ 10<sup>20</sup> models, and requires a slight update to the code used for FLS. In the Fortran code used in FLS, an unsigned double-precision variable was used to index models for all the MCMC accounting. This results in an upper limit of 2<sup>52</sup> models.<sup>7</sup> We have updated the code by using instead two model indices, making it suitable for any _k ≤_ 104. The revised code is available at http://www.warwick.ac.uk/go/msteel/steel ~~h~~ omepage/software/.

This very large model space is remarkably well explored by an MCMC chain with 2 million retained model visits, after discarding the first million as a burn-in, as evidenced by a correlation between model visit frequencies and actual posterior probabilities of the set of visited models equal to 0.995. The chain visits 126,844 models and is estimated to cover 47% of the total posterior mass. This is a very high coverage, given that we visit less than one model for every 10<sup>15</sup> models in the space. Any additional models (visited by running longer chains) will have virtually zero posterior probabilities, and will not affect the conclusions in any way.

Despite the much larger model space, Table 4 shows that posterior model probabilities are more concentrated than in the previous case and that the models with high posterior probability tend to be smaller, containing around 6–7 regressors.

> 7 Double-precision numbers cannot distinguish between 252 and 1+252 because they have the same internal representation. The 64-bit double format uses 1 bit for the sign and 11 for the exponent leaving 52 bits available for the fraction—thus the upper limit on ‘counting’ of 2<sup>52</sup> . An additional index will double the upper limit to 2<sup>104</sup> , which should be enough in most practical applications.

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