However, this approach also has weaknesses:
Firstly, the approach only allows them to bound the ATE. However, it is questionable how interesting the ATE of the program in its current form is to policy makers: The ATE measures the average difference in health between all eligible households receiving food stamps and the abolition of the program. However, even among Republicans 66% support raises in food stamp benefits. (\citealt{Clement2017}). Thus, the abolition of the program does not seem an interesting quantity to contrast. Instead, policy makers are likely more interested in the effects of adjustments to the program rules, such as changes to the award rules or to the reporting requirements.
Secondly, the approach forces the authors to code all variables as binary. This forces them to give up a lot of important information. This introduces two important weaknesses:
On the one hand, it makes them understate the effect of SNAP. To see why, consider the food insecurity variable: Families answered 18 questions regarding their food insecurity. By coding this variable as binary, they can only report the effect of SNAP on the extensive margin (whether families are food insecure or not). However, all improvements in the intensity of families' food insecurity are disregarded when they do not move families into being food secure.
On the other hand, coding program participation as binary forces them to ignore large differences between participants in the amount of awarded food stamps. To get a feeling for the differences in benefits consider a two person household in 2016: the smallest possible monthly benefit was 16 USD while the largest was above 350 USD (\citealt{Lauffer2017}, p.103). Using this (non-random) variation in treatment intensity could allow a researcher to derive an ATE per dollar of monthly benefits under reasonable assumptions. With more assumptions it might even be possible to estimate a distribution of marginal treatment effects that could be used to inform the debate on whether and which increases in SNAP benefits are effective in improving children's health outcomes.
Regarding the empirical results, it would have been nice if the authors had included two more sets of results:
Firstly, the CPS data was only used for the analysis without measurement error. Given the substantial underreporting in the CPS \cite{C.1997} it would have been interesting to see how the bounds they derived to account for misreporting fare in the presence of such a large degree of underreporting.