They argue that Lucas, who modeled an effect of improving health on growth and Nelson and Phelps who modeled an effect of the level of health on growth are both correct.
Their evidence comes mainly from a 96 country cross sectional average growth regression where both the initial level of life expectancy and the growth of life expectancy over the sample have positive and significant coefficients, both in LS and IV models.
Of the two results, they claim the effect of initial life expectancy is more robust.
I like the piece because they take a very reduced form approach. It's health and health improvements on growth, with basically nothing else in the model.
I dislike the piece because they, as do so many others, abuse the Hansen test of over-identifying restrictions to justify their instruments.
First, failing to reject the null, or "passing" the Hansen test, does not validate your identification, the test is on over-identifying instruments. Consider that in an exactly identified equation the test cannot be performed.
Second, failing to reject the null doesn't mean you don't have an instrument problem. A p level of .13 on a Hansen test means you don't reject the null at conventional levels, but it also means (more or less, I am speaking imprecisely here), there is an 87% chance that the null is false and your instruments are suspect. Another way to say this is we are rarely given any information about the power of the test, which is crucial when failing to reject the null is what guides our modeling choices.