The empirical literature of growth has steadly improved the econometric methods used mainly to address the effect of cross-country heterogeneity in the estimated convergence rate. In this paper, we highlight an important implication of this process of econometric refinement that has so far received little attention. We show that the picture that emerges from models that allow for generalised heterogeneity changes our view of the process of convergence within the OECD. Estimation methods that allow for non or partial heterogeneity stress the importance of transitional dynamics in the process of convergence. Thus sigma-convergence is mostly accounted for by beta-convergence. On the contrary, when generalised parameter heterogeneity is (tested and) allowed for we find that the observed reduction in the dispersion of per capita income within the OECD has little bearing on transitional dynamics. sigma-convergence in this case happens because the long run features of these countries are becoming increasingly similar (convergence in steady states). There are also striking differences across estimated models as regards the evolution of the relative position of the average country with respect to its steady state income per capita level.