Note, that the authors make less assumptions than we did in the Generalized Roy Model:
Firstly, they do not assume additive separability of the unobserved components U. Secondly, they do not make any assumptions on how households decide whether to participate. In particular, they make no assumption of optimizing behavior regarding that decision.
To avoid having to put more structure on the model, the authors use three tools: Firstly, they choose an aggregate object of interest, namely the ATE. Secondly, they only look at binary outcome variables which simplifies their analysis. Thirdly, they do not aim for point identification but only derive bounds on the ATE.
The authors face the two classical econometric problems of observing every individual in only one of the two treatments (the evaluation problem) and the selection problem that individuals endogenously decide their treatment status. In addition, the authors also face non-classical measurement error because SNAP participation has been shown to be underreported in surveys and the decision whether or not to report true participation is correlated with observable characteristics.\cite{Bollinger_1997}
Theoretical Analysis
This section gives an introduction to partial identification bounding methods. Firstly, it explains the general procedure to derive possibly informative bounds. It then applies this procedure to the problem of identifying the ATE of SNAP without non-classical measurement error and introduces several assumptions one might be willing to make. The section then shows how the procedure can be adjusted when allowing for non-random misreporting. It concludes with proposing assumptions on the misreporting that can be used to tighten the bounds.