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\title{B. Kreider, J. V. Pepper, C. Gundersen and D. Jolliffe:~ Identifying the
Effects of SNAP (Food Stamps) on Child Health Outcomes When
Participation Is Endogenous and Misreported}
\author[1]{Klara Röhrl}%
\author[1]{Ximeng Fang}%
\author[2]{eisenhauer}%
\author[2]{Rocío Baeza}%
\author[1]{Sebastian Becker}%
\affil[1]{University of Bonn}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\maketitle
\endgroup
\sloppy
\section*{Short Summary}
The authors evaluate the Supplemental Nutrition Assistance Program
(SNAP), formerly known as food stamps. They do so by bounding the
Average Treatment Effect (ATE) of the program on the population of
eligible children. They evaluate SNAP this way with respect to food
insecurity, obesity, anemia and subjective health.
The paper has a theoretical and an empirical contribution. The
theoretical contribution is that it extends the literature on partial
identification bounding methods to account for misreporting (in addition
to selection) and derive bounds on the ATE under different sets of
assumptions in this setting. The empirical contribution is that the
authors apply these bounds to SNAP using the National Health and
Nutrition Examination Survey (NHANES) and show that the derived bounds
can be informative.
The authors find evidence that SNAP improves the health outcomes of
eligible children. For anemia and subjective health the bounds point at
substantial health improvements due to SNAP and rule out even small
deteriorations in health. For obesity and food insecurity this is only
the case after excluding the possibility of false positive reports of
SNAP participation.
\small{\tableofcontents}
\newpage
\section{Integration into the Generalized Roy Model Framework}
This section classifies which type of policy evaluations~ this paper
performs, how its model relates to the generalized Roy model, what the
challenges are and how they are addressed.
The task that the authors set for themselves is the first of the three
evaluation tasks set out by~\citealt{Heckman_2007}: They evaluate SNAP in
its existing form on the eligible population.
The population of interest are eligible children. The treatment is
whether the family participates in SNAP or not which is decided
endogenously by the family. The authors start without any assumptions on
the process by which treatment is selected or how health is affected by
observable characteristics, the treatment or unobservables. Relating the
notation of the paper to the notation from the Generalized Roy Model
yields:
\begin{flalign*}
& \text{\textbf{Potential Outcomes}} &\\
Y_1 &= H(1) = f_1(X, Z, U) \\
Y_0 &= H(0) = f_0(X, Z, U) \\
& \text{\textbf{Cost}} \\
C &= c(X, Z, U) \\
& \text{\textbf{Choice}} \\
S &= s(X, Z, U) \\
D &= 1\{S>0\} \\
& \text{\textbf{Observed Outcomes}} \\
Y &= D \cdot H(1) + (1-D) \cdot H(0)
\end{flalign*}
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}
\section{Theoretical Analysis}
{\label{277477}}
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.
\newpage
Partial identification bounding methods use two steps:
\textbf{Step 1:~}Starting from the expression of interest, unobserved
quantities are first decomposed using the Law of Total Probability to
introduce quantities observed in the data and reduce the role of the
remaining unobserved quantities.
\textbf{Step 2:~}One uses well-founded assumptions that allow to further
bound the unobserved quantities that have shown up during the
decomposition.
In the following, these steps are applied to the ATE of SNAP on health
outcomes.
\subsection{Abstracting from Measurement
Error}
{\label{831018}}
Because of binary outcome variables the formula for the ATE is
simplified:
\par\null
\begin{equation}
\text{ATE} = P(H(1)=1) - P(H(0)=1)
\end{equation}
where H(t)=1 denotes the bad health outcome (later the analysis will
look at food insecurity, obesity, anemia and reported subjective
health)~ in treatment t.
Without any data, one can already see that the ATE will lie between -1
and 1. Due to the selection problem neither of the two probabilities can
be directly inferred from the data.
However, applying the law of total probability with respect to the
treatment status FS\textsuperscript{*}, one can decompose each of the
two probabilities:
\begin{align}
P[H(1) = 1] &= P[H(1) = 1 | \text{FS}^*=1] \cdot P[\text{FS}^*=1] \nonumber \\
& + P[H(1) = 1 | \text{FS}^*=0] \cdot P[\text{FS}^*=0] \nonumber \\
\label{eq:decomposition}
P[H(0) = 1] &= P[H(0) = 1 | \text{FS}^*=0] \cdot P[\text{FS}^*=0] \\
& + P[H(0) = 1 | \text{FS}^*=1] \cdot P[\text{FS}^*=1] \nonumber
\end{align}
For both decompositions there is one quantity that cannot be estimated by a simple sample analog from the data: The probability of bad health under the treatment that was not chosen $P[H(1) = 1 | \text{FS}^*=0]$ and $P[H(0) = 1 | \text{FS}^*=1]$. As they are probabilities, each must lie in the [0, 1] interval. In addition, each gets multiplied by the probability of the conditioning event. This yields intervals for the two unconditional probabilities of length $P[\text{FS}^*=0]$ and $ P[\text{FS}^*=1]$, respectively, yielding a total length of the ATE interval of 1. Thus, using the data without any additional assumption we get an interval of length 1 for the ATE. The formulae of its bounds are:
\begin{align}
\nonumber & P[H(1) = 1 | \text{FS}^*=1] \cdot P[\text{FS}^*=1] \\
\nonumber & - (P[H(0) = 1 | \text{FS}^*=0] \cdot P[\text{FS}^*=0] + P[H(0) = 1 | \text{FS}^*=1] \cdot P[\text{FS}^*=1]) \\
\label{eq:ATE_bounds}
& \leq \text{ATE} \leq & \\
\nonumber & P[H(1) = 1 | \text{FS}^*=1] \cdot P[\text{FS}^*=1] + P[H(1) = 1 | \text{FS}^*=0] \cdot P[\text{FS}^*=0] \\
\nonumber & - (P[H(0) = 1 | \text{FS}^*=0] \cdot P[\text{FS}^*=0])
\end{align}
Next, there are assumptions that the authors consider to further bound
the counterfactuals and the unconditional probabilities:
\textbf{Assumption 1: Monotone Treatment Selection (MTS)}
The first assumption puts bounds on the counterfactual conditional
probabilities. Applied to our example the MTS assumption states that
children that participate in SNAP have weakly worse latent health
outcomes than non-participating children:
\par\null
\begin{align*}
P[H(1) = 1 | \text{FS}^*=0] &\leq P[H(1) = 1 | \text{FS}^*=1] \\
P[H(0) = 1 | \text{FS}^*=0] &\leq P[H(0) = 1 | \text{FS}^*=1]
\end{align*}
\textbf{Assumption 2: Income is a Monoton Instrumental Variable (MIV)}
The second assumption is that the probability of bad health decreases
weakly with income. Assuming this allows the derivation of additional
bounds on P{[}H(0)=1{]} and P{[}H(1)=1{]}.
To do so, decompose P{[}H(0)=1{]} and P{[}H(1)=1{]} by applying the Law
of Total Probability with respect to the instrumental variable:
\begin{align*}
P[H(0)=1] = \int P[H(0)=1|Y=y] f(y) dy \\
P[H(1)=1] = \int P[H(1)=1|Y=y] f(y) dy
\end{align*}
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{]}.
\textbf{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}
\subsection{Accounting for Measurement
Error}
{\label{237631}}
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.
Allowing for this, none of the quantities in \ref{eq:decomposition} can be directly inferred from the data.
However, note that the event of true participation $\text{FS*}=1$ can be decomposed into a true positive $\text{FS}^*=1 \wedge \text{Z}^*=1$ and a false negative $\text{FS}^*=1 \wedge \text{Z}^*=0$. Using this decomposition of $\text{FS}^*=t$ each of the eight quantities in \ref{eq:decomposition} can be decomposed by the Law of Total Probability.
The resulting formula for the ATE can be rearranged to get analogous bounds to before:
\begin{align*}
- & P[H(0)=1|FS=0] \cdot P[FS=0] - P[H(1)=0|FS=1] \cdot P[FS=1] + \Theta \nonumber \\
\leq \text{ATE} \leq & P[H(1)=1|FS=1] \cdot P[FS=1] + P[H(0)=0|FS=0] \cdot P[FS=0] + \Theta
\end{align*}
with:
\begin{align*}
\nonumber &\Theta \equiv (\theta^-_1 - \theta^+_0) - (\theta^-_0 - \theta^+_1) \\
\nonumber & \theta^+_j = P(H = j, FS = 1, Z^* = 0) \\
\nonumber & \theta^-_j = P(H = j, FS = 0, Z^* = 0)
\end{align*}
Note that these bounds depend only on observable probabilities,
unobserved probabilities for which we have developed assumptions earlier
and additional unobservable probabilities in \(\Theta\).
To get bounds on the~~\(\theta\)s, the authors use
administrative data to estimate the true participation rate among
eligible children which they calculate to be 50\%. This true
participation rate together with the reported participation rate of
46.5\% yields three bounds on the~\(\theta\)s.
\textbf{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.
\section{The Data and Institutional
Setting}
{\label{459952}}
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.\selectlanguage{english}
\begin{table}
\centering
\caption{{Characteristics of SNAP Eligible Children by Participation}}
\label{tab:descriptives}
\begin{tabular}{lll}
\toprule
Variable & Recipients & Non-Recipients \\
\midrule
Age (in years) & 8.6*** & 9.5 \\
Ratio of income to the poverty line & 0.64** & 0.86 \\
Food-insecure & 0.45** & 0.35 \\
\midrule
Poor or fair health & 0.09 & 0.07 \\
Obese & 0.19 & 0.18 \\
Anemia & 0.013 & 0.010 \\
\bottomrule
\end{tabular}
\end{table}
\section{Results}
{\label{577465}}
This section presents the estimated ATE of SNAP on food insecurity, poor
heath, anemia and obesity. For each outcome variable it first reports
the estimates ignoring misreporting with varying assumptions and then
shows the results allowing for different degrees of misreporting and
different assumptions.~
\subsection{Food Insecurity}
{\label{171261}}
The estimated bounds on the ATE for food insecurity assuming there were
no measurement error are displayed in figure 1 for the NHANES data. The
bound estimates are shown as bars, with whiskers indicating the 95\%
confidence intervals. One can clearly see the identifying power of the
MTS and MIV assumptions in reducing the width of the interval.
Maintaining these two assumptions the data show that the effect of SNAP
on food insecurity is substantial, reducing food insecurity by at least
12\% points and possibly by over 35\% points.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/nhanes-food-insecurity/nhanes-food-insecurity}
\caption{{Bounds for the ATE of SNAP on food insecurity in the NHANES data
{\label{807326}}%
}}
\end{center}
\end{figure}
Since food insecurity is also elicited in the Current Population Survey,
the authors estimate the same bounds using this alternative data set
(figure 2). The results are qualitatively similar but the intervals are
larger such that the MRT assumption is necessary for ruling out that
~SNAP increases food insecurity on average. These larger intervals could
be due to the smaller sample size in the CPS. Another reason could be
the large amount of underreporting present in the CPS (nearly 50\%
according to~\citet{Meyer_2009}).
Since the CPS allows for the construction of instrumental variables that
- if exogenous and valid - could also identify a causal effect of SNAP
on food insecurity, the authors construct two instruments that are
common in the literature that exploit interstate variability in the
implementation of SNAP:
\begin{enumerate}
\tightlist
\item
About one half of states has simplified reporting requirements
\item
About a third of states exempts cars from the asset-test
\end{enumerate}
The authors estimate the ATE from these instrumental variables in two
different ways: Firstly, they estimate it using a linear response model.
Secondly, they follow \citealt{2011} and use the instrumental
variables to non-parametrically bound the ATE from above. These four
estimates are shown as lines in figure 2, where the thinner lines show
the linear response IV estimates and the thicker lines show the
non-parametric upper bound estimates. One can clearly see that at least
one of the assumptions underlying the estimate of the reporting
instrumental variable in the linear response model must be violated
since the estimate lies outside the estimate that relies on no
assumptions.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/cps-food-insecurity-with-IV-3/cps-food-insecurity-with-IV-3}
\caption{{Bounds for the ATE of SNAP on food insecurity in the CPS data
{\label{809709}}%
}}
\end{center}
\end{figure}
Allowing measurement errors widens the bounds on the ATE substantially
as can be seen in figures 3 and 4 for a true participation rate of 50\%
and 70\% respectively. The MIV and MTR assumptions still hold
substantive identifying power. Even more identifying power lies now in
assuming that there are no false positives - especially for a true
participation rate of 50\%. Assuming this leaves the bounds only
slightly larger than the ones without measurement error. Assuming no
false positives is less powerful when the true and reported
participation rate are further apart.
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/food-insecurity-misreporting-p-50/food-insecurity-misreporting-p-50}
\caption{{Bounds for the ATE of SNAP on food insecurity in the NHANES data,
allowing for misreporting with a true participation rate of 50\%
{\label{289635}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/food-insecurity-misreporting-p-70/food-insecurity-misreporting-p-70}
\caption{{Bounds for the ATE of SNAP on food insecurity in the NHANES data,
allowing for misreporting with a true participation rate of 70\%
{\label{423464}}%
}}
\end{center}
\end{figure}
\subsection{Poor Health}
{\label{558024}}
The estimates for the ATE on subjective health are very similar to those on food insecurity ignoring possible measurement error (Fig. 5). However, the results for subjective health are much more robust to allowing for misclassification (Fig. 6 and 7)\footnote{for subjective health, anemia and obesity only the results under the two strongest assumptions were published} and the assumption of no false positives has mostly no noticeable identifying power for this outcome.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/poor-health/poor-health}
\caption{{Bounds for the ATE of SNAP on subjective child health in the NHANES data
{\label{208201}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/poor-health-p-50/poor-health-p-50}
\caption{{Bounds for the ATE of SNAP on subjective child health in the NHANES
data, allowing for misreporting with a true participation rate of 50\%
{\label{350171}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/food-insecurity-misreporting-p-701/poor-health-p-70}
\caption{{Bounds for the ATE of SNAP on subjective child health in the NHANES
data, allowing for misreporting with a true participation rate of 70\%
{\label{936546}}%
}}
\end{center}
\end{figure}
\subsection{Anemia}
{\label{296652}}
The effect of SNAP on anemia is nearly indistinguishable from that on
food insecurity when we abstract from measurement error. After
accounting for misclassification the bounds are only marginally wider
and are basically unaffected by the amount of data corruption one
allows. With a higher true participation rate the interval includes much
more negative values, including that food stamps reduce the incidence of
anemia by 50\% points on average.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/anemia/anemia}
\caption{{Bounds for the ATE of SNAP on anemia in the NHANES data
{\label{444384}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/anemia-p-50/anemia-p-50}
\caption{{Bounds for the ATE of SNAP on anemia in the NHANES data, allowing for
misreporting with a true participation rate of 50\%
{\label{171659}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/anemia-p-70/anemia-p-70}
\caption{{Bounds for the ATE of SNAP on anemia in the NHANES data, allowing for
misreporting with a true participation rate of 70\%
{\label{930913}}%
}}
\end{center}
\end{figure}
\subsection{Obesity}
{\label{339178}}
Assuming measurement errors away, the estimates suggest that SNAP is as
good at reducing the incidence of obesity as it is in reducing the
incidence of anemia. Even without assuming a monotone treatment response
- which one might not wish to uphold for obesity - the effects of SNAP
on obesity appear to be strictly negative (although very small positive
effects cannot be rejected at 95\%).
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/obesity/obesity}
\caption{{Bounds for the ATE of SNAP on obesity in the NHANES data
{\label{122900}}%
}}
\end{center}
\end{figure}
However, allowing for measurement error changes these estimates. The intervals grow much larger\footnote{Keeping in mind that only the results when maintaining the MTS and MIV assumptions are reported when accounting for misreporting} such that effects of 6\% to over 30\% points increases in obesity due to SNAP cannot be rejected depending on which assumptions one wishes to maintain. In contrast to the results for anemia and subjective health, assuming no false positives has noticeable identifying power but not of the same magnitude as for food insecurity.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/obesity-p-50/obesity-p-50}
\caption{{Bounds for the ATE of SNAP on obesity in the NHANES data, allowing for
misreporting with a true participation rate of 50\%
{\label{445013}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/obesity-p-70/obesity-p-70}
\caption{{Bounds for the ATE of SNAP on anemia in the NHANES data, allowing for
misreporting with a true participation rate of 70\%
{\label{647170}}%
}}
\end{center}
\end{figure}
\newpage
\section{Critical Assessment}
Due to the partial identification bounding method, the authors' identification and estimation of the ATE is very clean. The assumptions used to tighten the bounds further are well-motivated. Furthermore, using partial identification allows them to calculate consistent bounds on the ATE that don't require any assumptions, using only frequencies in their data and properties of probabilities in this setting. Such a consistent, assumption-free estimator that is point identified does not exist for observational data.\footnote{If we had a randomized control trial, we would be able to get a point-identified consistent estimator of the ATE. However, point-identified estimators in observational data rely on exogeneity assumptions that can rarely be tested. (For example the independence of errors and the outcome variable conditioning on the regressors in OLS.) There are instances where some of these assumptions can be tested (e.g. in the case of over-identified GMM) but never all - as is the case with the bounds developed here.}
These assumption-free bounds make it possible to transparently show the identifying power of every assumption by reporting how the bounds tighten with each assumption.
\par\null
However, this approach also has weaknesses:
\par\null
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.~
Secondly, the results including measurement error for obesity were only shown with the two strictest assumptions. However, it is exactly obesity where the MTS\footnote{Children that receive food stamps are less often obese than children who don't} and MIV\footnote{The poorer the child's household the more likely that the child is obese.} assumptions are the least innocuous. Therefore, they should have also included the results without these two assumptions in the paper.
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