> "generalized method of moments" approaches to cross-country analysis (of e.g. the effectiveness of aid)
Which is an entirely reasonable criticism. GMM is a complex mathematical process, wiki suggests [0] that it assumes data generated by a weakly stationary ergodic stochastic process of multivariate normal variables. There are a lot of ways that a real world data for aid distribution might be nonergodic, unstationary, generally distributed or even deterministic!
Verifying that a paper has used a parameter estimation technique like that properly is not a trivial task even for someone who understands GMM quite well. A reader can't be expected to follow what the implications are from reading a study; there is a strong element of trust.
[0] https://en.wikipedia.org/wiki/Generalized_method_of_moments