Results, theories, and metatheories in convergence estimation

I read Phillips and Sul's "Transition modeling and econometric convergence tests" today. The authors' recent work has been vaunted as state-of-the-art for analysing convergence of quantities like countries' incomes. Their paper is available online through Google Scholar.

My interest in the paper was threefold. Firstly, the paper's tests are useful in themselves. Secondly, I wanted to see the theoretical methods used, with a view to adopting them elsewhere. Thirdly, I wanted to see whether the methods could be considered as part of larger theories with more general application.

The paper tests the convergence of quantities y(i,t) which can be expressed as the product of a time specific function and a country specific function. The country function can also vary over time. The idea is to take ratios across countries to get rid of the time specific component, then look at variances across groups. If the ratio's variance is going to zero over time, we have convergence. A tidy equation follows from the expansion of the ratio which allows for regression testing.

Here is the general methodology used:
1. Transform the base equation to get an expression dependent on only one key quantity. Expand the expression into a familiar form.
2. Determine asymptotics.
3. Apply moment conditions.

Here are candidate metatheories corresponding to each stage. The list is by no means exclusive or exhaustive, and other taxonomies could be made.

1. A theory classifying all transformations of base equations into expressions dependent only on quantities of interest. Perhaps group theory from pure mathematics may help here.
2. Weiner process theory.
3. GMM analysis.

I would like to bring recent work on estimation into a similar classification and so combine the theories in a systematic way. Have to see how it goes.