where f(y) is the density of income Y at y.
The conditional probabilities suffer from the same selection problem as the unconditional probabilities but using the MIV assumption one can bound them and in turn also bound P[H(0)=1], P[H(1)=1].
Assumption 3: Monotone Treatment Response (MTR)
The strictest assumption that the authors consider is that SNAP weakly improves health status:  \(H\left(1\right)\le H\left(0\right)\)
Using this assumption, the ATE cannot be negative. Thus, to some degree this presumes the result. However, this assumption can interact with other assumptions to yield strictly negative upper bounds. It is also well founded as there is a broad consensus among policy makers and researchers that SNAP does not worsen food insecurity. \cite{Bitler_2003}

Accounting for Measurement Error

To account for measurement error, we introduce reported treatment status FS in addition to the actual treatment status FS*. Let Z*=1 denote a correct classification.