Assumption 4: No False Positives
Additionally, the \(\theta\)s can be bounded further, if one is willing to assume a maximal amount of data corruption. Since SNAP data suffers from underreporting this is equivalent to making an assumption on which rates of false positives one allows. Assuming that there are no false positives is motivated by validity studies \cite{Bollinger_1997} that link surveys to administrative records and find that rates of false positive reports are negligibly small for SNAP.

The Data and Institutional Setting

The authors rely mainly on the National Health and Nutrition Examination Survey (NHANES) to evaluate SNAP because of the rich information on children's health. They focus on children from ages 2-17 whose household income is below 130% of its poverty line. These are children that would satisfy the income requirement to be eligible for food stamps. However, there is also a strict wealth requirement, compliance with which the authors cannot verify in their data. As in the theoretical derivation above, SNAP participation is binary, not accounting for differences in the awarded benefits between participating households.
While they do not observe the amount of benefits nor the assets of the household, the NHANES provides them with accurately measured and rich information on the health of the child. In addition to the parents' subjective perception of the child's health, children are measured and weighed by professional nurses and they are tested for anemia.
This yields a sample of 4418 children who were interviewed between 2001 and 2006.
45.6% of these children reported receiving food stamps. There were significant and important differences between participating children and non-participating children as can be seen in the table 1. The comparison shows that recipients are younger, poorer and more food insecure. Nearly one half of all participating households were still food insecure despite the benefits provided by SNAP.  Among the non participating eligible households only one in three was classified as food-insecure. Participating children also seem to be in worse health than their non-participating peers but these differences are not statistically significant.