In the extreme jointness case discussed above (when _i_ and _j_ appear always together), while _Jij_<sup>_DW_</sup> = _P_ ( _i ∩ j_ )<sup>_−_1</sup> , we have that _Jij_<sup>_DW a_</sup> = _∞_ , _Jij_<sup>_⋆_= 1 and</sup><sup>_Jij_=</sup><sup>_∞_, which are more intuitive and practical</sup> as extreme values. In case that _P_ ( _i_ ) _→_ 1, we obtain _Jij_<sup>_⋆→P_(</sup><sup>_j_)and</sup><sup>_Jij→P_(</sup><sup>_j_)</sup><sup>_/_(1</sup><sup>_−P_(</sup><sup>_j_)),</sup> which are well-behaved quantities in line with intuition.

Furthermore, note that _Jij_ does indeed correspond to the posterior odds ratio of the models including both _i_ and _j vs_ the models that include them only individually. Since this fact leads to a straightforward interpretation and to a statistically meaningful metric, _Jij_ is our preferred measure of jointness. In particular, there is no need for calibration, as this measure has a clear and direct interpretation in terms of model probabilities.

Another advantage of the measures proposed here is that they are easily extended to the case of more than two regressors. In particular, we define _multivariate jointness_ for general sets of regressors _S_ through two quantities: _P_ ( _S_ ), which is the total posterior probability assigned to those models having all regressors in _S_ , and _P_ ( _⊂S_ ), defined as the posterior mass assigned to all models including only proper subsets of _S_ . Then we can generalise the measures above as follows: Thus, the measure _JS_<sup>_⋆_istheposteriormassofthemodelscontainingallof</sup><sup>_S_asafractionof</sup> the posterior mass assigned to all models having any (or all) of the regressors in _S_ . As before, the measure _JS_ is the posterior odds ratio between those models having all the variables in the set _S_ and the models including only proper subsets of these variables.

# **3. Jointness of Growth Determinants**

# _3.1. The FLS Data_

We first illustrate the behaviour of our proposed measures of jointness using the growth data of FLS. The latter data set contains _k_ = 41 potential regressors to model the average per capita GDP growth over 1960-1992 for a sample of _n_ = 72 countries.<sup>3</sup> The results are based on an MCMC chain on model space with 2 million recorded draws, after a burn-in of 1 million. The correlation between model visit frequencies and probabilities computed on the basis of the exact posterior odds of the visited models is 0.992, indicating excellent convergence. An estimate of the total model probability captured by the chain, computed as in George and McCulloch (1997), is 70%.<sup>4</sup> Results are virtually identical to those obtained in

> 3 The dataset and the code used in FLS are available on the _Journal of Applied Econometrics_ code and data archive at http://qed.econ.queensu.ca/jae/. The updated code used here has been uplodaded to that website, and is also available at http://www.warwick.ac.uk/go/msteel/steel homepage/software/.

> 4 This is quite high, given that we only visit one in every 15 million models. Longer chains will capture marginally more of this posterior probability, but they will only add models with very small posterior probabilities and this will not affect any of the conclusions.

5