previous examples, and it appears that the case with convergence problems struggles to adequately describe the posterior distribution on the model space _M_ . While most of the mass is covered by a relatively small number of models, the prior assumptions induce a very fat tail of models with little but nonnegligible mass. However, inference on most things of interest, such as the regressor inclusion probabilities is not much affected by running longer chains. For all other prior settings, the very large model space is remarkably well explored by the MCMC chain.

# **5. Robustness and Predictive Analysis: Results from 100 Subsamples**

In the previous section we have illustrated that the choice of, perhaps seemingly innocuous, prior settings can have a dramatic impact on the posterior inference resulting from BMA. Posterior model probabilities and identification of the most important regressors can strongly depend on the prior settings we use in our analysis of growth regressions through BMA.

We now address the issue of whether small changes to the data set would result in large changes in inference— _i.e._ , data robustness. In addition, we want to assess the predictive performance of the model under various prior settings. We will use the same device to investigate both issues, namely the partition of the available data into an inference subsample and a prediction subsample. We will then use the various inference subsamples for the evaluation of robustness and assess prediction on the basis of how well the predictive distribution based on the inference subsamples captures the corresponding prediction subsamples.

We take random partitions of the sample, where the size of the prediction subsample is fixed at 15% of the total number of observations (rounded to an integer), leaving 85% of the sample to conduct inference with. We generate random prediction subsamples of a fixed size by using the algorithm of McLeod and Bellhouse (1983). We use 100 random partitions and compute the posterior results through an MCMC chain of 500,000 drawings with a burn-in of 100,000 for each partition. This led to excellent convergence, with the exception of the cases with _θ_ fixed at 0 _._ 5 and _g_ = 1 _/n_ . Thus, for this combination we have used a chain of length 1,000,000 after a burn-in of 500,000, which leads to reliable results.<sup>8</sup> To increase comparability, we use the same partitions for all prior settings.

# _5.1. Robustness_

Figure 6 indicates the distribution of the posterior mean model size across 100 inference subsamples of the FLS data (left panel). Values corresponding to the full sample are indicated by vertical lines. The right panel of the same Figure shows the posterior inclusion probabilities of the ten regressors with highest posterior inclusion probabilities in the full-sample analysis with the FLS prior setting— _i.e._ , _θ_ = 0 _._ 5 and _g_ = 1 _/_ max _{n, k_<sup>2</sup> _}_ .

A striking characteristic of both panels is the sensitivity of the results to the choice of _m_ for the fixed _θ_ priors, whereas the effect of _m_ is very small for the cases with random _θ_ . The choice of _g_ ,

> 8 In fact, the results are very close to those obtained with 100,000 burn-in and 500,000 retained drawings, with the only noticeable difference in the maximum LPS values.

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