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\begin{document}

\title{Infant deaths: Notes toward a behavioral model}
\author{Leon Taylor \\
%EndAName
\\
surface mail:\\
Leon Taylor\\
Post Office Box 5452\\
Huntington, West Virginia\\
25703-0452\\
e-mail: Taylorleon@aol.com\\
tel. (304) 523-6637\\
}
\maketitle
\tableofcontents
\listoftables

\begin{abstract}
A woman may choose to become more educated, partly because she wishes to
emulate those around her. A behavioral model suggests that she is less
likely to lose her baby than the less educated by flouting basic rules of
health. Estimates of a three-equation model suggest that low-weight births
are most likely where youths (as well as the adults around them) are most
likely to have dropped out of high school. The impact of the dropout rate on
low-weight births varies from state to state, however. Preliminary estimates
suggest a high marginal cost of health information. [JEL\ I12, I29]
\end{abstract}

\section{Introduction}

Many economic studies of infant death focus on factors that the mother
ostensibly must take as given: Medicaid, paid maternity leave, community
health centers, and access to abortions as well as to neonatal intensive
care.\footnote{%
For studies of these factors, see respectively Currie, Gruber and Fischer
(1995), Winegarden and Bracy (1995), Goldman and Grossman (1988), and
Corman, Grossman and Joyce (1988).} Generally, environmental --- as opposed
to behavioral --- theories of infant death seem to have dominated its
economic study since the early 1800s, when Malthus posited it as a positive
check on population growth.\footnote{%
In ``A summary view of the principle of population,'' Malthus writes that
the pressures on resources due to population growth would ``render the
children unhealthy from bad and insufficient nourishment, which would check
the rate of increase by occasioning a greater proportion of deaths''
(Malthus, page 243). Elsewhere, Malthus also seems to regard infant death as
a preventive check on population growth. His first essay on the principle of
population asserts that, among North American Indians, the miserable toils
of women must ``prevent any but the most robust of infants from growing to
maturity....Misery is the check that represses the superior power of
population and keeps its effects equal to the means of subsistence''
(Malthus, page 82).}

The incidence of low-weight birth greatly influences the rate of infant
death. Indeed, a traditional regression analysis may thereby discount the
importance of social and economic factors. Among the 50 states in 1992, a
simple linear regression on the incidence of low-weight birth accounts for
over half of the variance in the rate of infant death. Adding household
income and the incidence of teenage births as independent variables little
enhances the explanatory power of the equation: That unrestricted form of
the model does not pass an F-test at a level of significance of 1 percent.%
\footnote{%
See Appendix B for details.}

Although environmental factors surely matter, some evidence suggests that
the mother's decisions during and after pregnancy may also affect the
prospects of an infant death. For example, Corman, Joyce and Grossman (1987)
find that beginning prenatal care early is the most cost-effective way to
cut the neonatal death rate for blacks and whites. It would be valuable to
extend our understanding of how and why the mother makes such decisions as
whether to seek out prenatal care. In particular, the impact of her
education on her decisions is of interest, given the general linkage, in
cross-country studies, of a high level of education to a low rate of infant
death (Barro and Jong, 1993).

For instance, it remains possible that the incidence of low-weight birth
itself depends on social and economic factors - especially those that shape
the woman's decisions concerning her pregnancy. This paper explores that
possibility.

The paper develops and tests a model of infant death as an event due in part
to the mother's decisions about education. In the model, the mother
influences the chance for a healthy birth by deciding how much health
information to acquire. If she is poorly educated, then acquiring health
information will cost her a lot, and so she will probably acquire just a bit
of it.

Section 2 presents the model. Section 3 reports the results of testing the
model. Section 4 gives rough estimates, derived from the tests, of the
marginal cost of health information. Section 5 concludes with reflections.

\section{The model}

The paper is to test this idea: Women choose the probability of a healthy
birth in the same way that they make other decisions about the future -- by
choosing the course of action that seems likely to satisfy them the most. To
put it simply, when they try to secure a healthy birth, they consider costs.
More precisely, the model posits that the woman seeks to maximize the sense
of welfare, or utility, that she expects from a healthy birth, net of the
costs of acquiring and using relevant information. Notation follows.

\subsection{Basic functions}

\subsubsection{Probability of an infant death}

The chance of an infant death is $M$, where $0\leq M\leq 1$. This chance
depends, in part, on the mother's access to doctors for prenatal care. The
more information that she has, the more likely it is that she can avoid a
low-weight birth, the probability of which is $W$, $0\leq W\leq 1$. Denote
the information set as $I$, where $0\leq I<\infty $. \ The model posits that 
$\frac{\partial W}{\partial I}<0$. \ This derivative links to the chance of
an infant death, for $M$ depends directly on the chance of a low-weight
birth. Low-weight infants are more likely to die than high-weight infants.
So $\frac{\partial M}{\partial W}>0$.

The model also posits that $\frac{\partial ^{2}M}{\partial W^{2}}>0$,
because a lower-weight birth correlates with both a greater prior belief of
a low-weight birth and with a much greater probability of infant death. The
assertion may bear some explanation. The incidence of a low-weight birth may
be the most conspicuous determinant of infant death, in the sense that one
can estimate a negative correlation between birth weight (of 500 to 4,500
grams) and the probability of infant death more precisely than one can
estimate correlations of other factors with infant death. Indeed, for this
range of birth weight, the rate of infant deaths seems generally to increase
more sharply as weight decreases (MacDorman and Atkinson (1998), Figure 1).%
\footnote{%
For birth weights below 500 grams, infant death is virtually certain; the
rate exceeds 90 percent. For birth weights above 4,500 grams, the rate of
infant death increases with weight.} I postulate that the {\sl ex ante}
probability of a low-birth weight correlates negatively with the {\sl ex post%
} birth weight. That is, I posit that the lower the actual weight, the
greater the prior belief that the birthweight would be below 2,500 grams. To
a degree, health specialists can predict birthweights -- and prematurity,
which correlates with low birth weights -- through sonography. The woman may
also expect a low birthweight because she has observed the births given by
women around her who have followed practices similar to hers.

The more easily that the pregnant woman can reach doctors, the lower the
probability that her baby will die. One may measure her medical access in
terms of the number of doctors in her area. Denote the number of doctors as $%
D$, where $0\leq D<\infty $. Then $\frac{\partial M}{\partial D}<0$. Adding
another doctor to the area's supply will have greater value in an area that
has just a few doctors than in an area that has many doctors. And so every
increase in $D$ will reduce the chance of an infant death, $M$, by less and
less. Thus $\frac{\partial ^{2}M}{\partial D^{2}}>0$.

Summing up these ideas,

\begin{eqnarray}
M &=&M(W,D),  \nonumber \\
\frac{\partial M}{\partial W} &>&0,\frac{\partial ^{2}M}{\partial W^{2}}<0, 
\nonumber \\
\frac{\partial M}{\partial D} &<&0,\frac{\partial ^{2}M}{\partial D^{2}}>0, 
\nonumber \\
\frac{\partial ^{2}M}{\partial W\partial D} &\leq &0.
\end{eqnarray}

One can interpret the cross-partial above in this way: Ready access to
doctors may dampen the rise in the likelihood of an infant death that is due
to a greater likelihood of a low-weight birth. This is because doctors can
recommend ways to avert the death of the baby in the event that it would
weigh little.

\subsubsection{Probability of a low-weight birth}

The likelihood of a low-weight birth, $W$, depends negatively on the data
that the would-be mother has for avoiding such a birth. \ Given her level of
education, however, as she acquires more and more information, she will find
the new data harder and harder to assimilate; for education is the
technology that enables her to acquire information more easily, and that is
held constant. As a result, every additional bit of information is less and
less effective at lowering the likelihood of a low-weight birth. Thus $\frac{%
\partial ^{2}W}{\partial I^{2}}>0$.

The likelihood of a low-weight birth also depends on the access of the
would-be mother to doctors for prenatal care. If she can reach a doctor more
easily, than a low-weight birth is less likely. Thus $\frac{\partial W}{%
\partial D} < 0 $. The value of adding another doctor to the area supply,
however, is smaller when doctors abound than when doctors are scarce. So $%
\frac{\partial^2 W}{\partial D^2} > 0 $.

The model thus posits that

\begin{eqnarray}
W &=&W(I,D),  \nonumber \\
\frac{\partial W}{\partial I}<0, &&\frac{\partial ^{2}W}{\partial I^{2}}>0, 
\nonumber \\
\frac{\partial W}{\partial D}<0, &&\frac{\partial ^{2}W}{\partial D^{2}}>0, 
\nonumber \\
\frac{\partial ^{2}W}{\partial D\partial I} &\leq &0.
\end{eqnarray}

The cross-partial posits this: As the would-be mother learns more about
health from sources other than her doctor, she may duplicate what her doctor
knows. This will reduce his impact on the chance for a low-weight birth.

\subsubsection{Health information}

The amount of health information that the would-be mother obtains and uses
depends on her general education. When I say that she ``obtains''
information about health, I mean that she assembles and assimilates a list
of rules to follow in pregnancy. To assemble the list, she must know where
to turn for information. To assimilate the list, she must know how to
identify good rules -- i.e., rules that, once executed, are most likely to
lead to the results that are claimed for them. The selection, assimilation
and retrieval of information are broad skills that one may acquire in formal
education. When I say that she ``uses'' information about health, I mean
that she follows good rules. To do this, she must have discipline. That she
has an education is evidence that she has discipline since, to acquire
education, one must be able to collect and follow good rules.

Her decision of how much education to acquire depends on the net benefits
that she expects from it. Here, however, she is in a quandary. Lacking
education, she will find it costly to obtain information about the net
benefits of education. So she will rely on folk rules: She will estimate the
benefit and the cost of additional education by observing the amount of
education acquired by the youths and adults that she knows. That few of them
bothered with high school degrees suggests to her that the degrees are worth
little. The lack of education among people she knows also suggests to her a
psychic cost of obtaining education -- social disapproval. By acquiring an
education, she sets herself apart from her group and risks its censure.

The young woman may also infer the value of education from what her
acquaintances decide to do in lieu of an education. Since the uneducated are
less likely to find a job, they are more likely to make choices that center
on the home. Observing such choices suggests to the young woman that an
education is worth little in the sense that it would not benefit her as much
as a simple home life would. Suppose, for instance, that many teens that the
young woman knows have become pregnant. Then she may infer that she would
benefit more from activities related to pregnancy -- such as unconstrained
sex and the anticipation of parenting -- than from education as an
alternative use of her time. Indeed, she may infer that the returns to
education are particularly low if she sees that many women that she knows
decided to have children, or to risk having them, even though they would
likely have to raise them alone. One cost of single parenting is that the
parent has less time for work or hobbies in which she would have gained
directly from her education. A woman who chooses to rear a child on her own
may thus attach little value to education -- and, by implication, to work or
to hobbies. Observing the choice of her friends to become single parents,
the young woman in school concludes that education would not do her much
good, either -- for she considers herself to be like her friends.

More formal expression may aid testing of these ideas. Denote the share of
births to teen mothers out of all births as $T$, where $0\leq T\leq 1$.
Denote the share of births to single mothers out of all births as $S$, where 
$0\leq S\leq 1$. Denote the share of nongraduates out of all adults as $N$,
where $0\leq N\leq 1$. In a given area and at a given time, a rise in $T$, $%
S $ or $N$ will discourage a young woman from getting more education. This
lack of education, in turn, will make it more costly to her to acquire and
use a little more information about health.

Thus the marginal cost of information, $\frac{\partial C}{\partial I}$,
increases as the share of all births to teens ( $T$) increases. That is, $%
\frac{\partial ^{2}C}{\partial I\partial T}>0$. Moreover, as the share of
all births to teens increases, the young woman feels more and more confirmed
in her belief that an education would not be worth much to her. And so she
is more and more likely to drop out of school -- and to later find that
health information is costly to acquire. Then $\frac{\partial ^{3}C}{%
\partial I^{2}\partial T}>0$.

Similarly, a rise in the share of all births to single mothers ($S $) will
boost the cost of acquiring a little more information at an increasing rate.
Thus $\frac{\partial^2 C}{\partial I \partial S} > 0 $ and $\frac{\partial^3
C}{\partial I^2 \partial S} > 0 $.

A rise in the share of nongraduates out of all adults has the same sort of
impact on the cost of acquiring a little more information. Thus $\frac{%
\partial^2 C}{\partial I \partial N} > 0 $ and $\frac{\partial^3 C}{\partial
I^2 \partial N} > 0 $.

In sum, the model posits the cost of health information as the function

\begin{eqnarray}
C & = & C(I;T,S,N),  \nonumber \\
\frac{\partial C}{\partial I} & > & 0, \frac{\partial^2 C}{\partial I^2} > 0,
\nonumber \\
\frac{\partial^2 C}{\partial I \partial T} & > & 0, \frac{\partial^3 C}{%
\partial I^2 \partial T} > 0,  \nonumber \\
\frac{\partial^2 C}{\partial I \partial S} & > & 0, \frac{\partial^3 C}{%
\partial I^2 \partial S} > 0,  \nonumber \\
\frac{\partial^2 C}{\partial I \partial N} & > & 0, \frac{\partial^3 C}{%
\partial I^2 N} > 0.
\end{eqnarray}

The amount of health information used by the woman may influence the
probability that her baby lives or dies. Let $U_{h}$ denote her utility
value in the event that her baby lives; let $U_{m}$ denote her utility value
in the event that her infant dies. $U_{h}$ and $U_{m}$ are parameters, not
functions.

\subsection{Optimization problem}

Given the amount of education that she already has chosen, the would-be
mother decides how much health information ($I$) to acquire and use. Her
goal is to act in a way that will leave her feeling as good as possible
after the birth. So she chooses $I$ to maximize her expected welfare from
the birth. She will thus choose $I$ to maximize

\begin{equation}  \label{L}
L = (1 - M[W,D]) U_h + M[W,D] U_m - C(I; T,S,N)
\end{equation}

Rearranging the first-order condition yields one way to interpret any
solution to the mother's problem:

\begin{equation}
\frac{\partial M}{\partial W}\frac{\partial W}{\partial I}(U_{m}-U_{h})=%
\frac{\partial C}{\partial I}.  \label{foc}
\end{equation}

The condition says this about the would-be mother: To optimize, she will use
information up to the point that the cost of using just a little more will
equal its expected benefit to her in helping her avoid a low-weight birth.%
\footnote{%
The appendix shows that a solution satisfying the first-order condition will
also satisfy the second-order condition. It thus solves the mother's problem.%
}

In other words, the amount of information that she will choose to use is a
function of its benefits and costs to her. Let us express this optimal
amount of information -- the solution to her problem -- as the function $%
I^{\ast }(D,T,S,N)$. Simple comparative statics characterize the function:%
\footnote{%
The appendix has derivations.}

\begin{eqnarray}
\frac{\partial I^*}{\partial D} & \leq & 0,  \nonumber \\
\frac{\partial I^*}{\partial T} & < & 0,  \nonumber \\
\frac{\partial I^*}{\partial S} & < & 0,  \nonumber \\
\frac{\partial I^*}{\partial N} & < & 0.
\end{eqnarray}

These results are simple to interpret if one begins with the earlier
decision of the would-be mother about how much education to acquire. She
will forgo education if a large share of her adult acquaintances have
dropped out of high school ($N$); or if a large share of the births in her
locale occurred to single or teen mothers ($S$ and $T$, respectively). In
turn, the lack of education will make it costly for her to gather and use
health information. So, she will choose to use little health information if
many of her adult acquaintances are dropouts ( $\frac{\partial I^{\ast }}{%
\partial N}<0$); or if many local births were to single or teen mothers ( $%
\frac{\partial I^{\ast }}{\partial S}<0$ and $\frac{\partial I^{\ast }}{%
\partial T}<0$, respectively). She might also forgo learning about health if
she can easily reach doctors ($\frac{\partial I^{\ast }}{\partial D}\leq 0$%
). That is, she might believe that medical treatment and medication would
achieve the same ends as learning health rules on her own. \ She would
substitute the expertise of her doctor for her own knowledge of health rules.

\subsubsection{Optimal functions}

In this simple model, the amount of health information that the would-be
mother chooses to acquire ($I^* $) will help set both the likelihood of a
low-weight birth to her (given by the function $W(I,D) $) and the likelihood
of an infant death (given by the function $M(W,D) $). Solving the model,
then, leads to the three optimal functions

\begin{equation}
I^{\ast }=I^{\ast }(D,T,S,N),  \label{eq1}
\end{equation}

\begin{equation}
W^{\ast }=W^{\ast }(I^{\ast },D),  \label{eq2}
\end{equation}

and

\begin{equation}
M^{\ast }=M^{\ast }(W^{\ast },D).  \label{eq3}
\end{equation}

The model in (\ref{eq1}) - (\ref{eq3}) is behavioral. It presumes that the
mother can affect the chances of a low-weight birth and of an infant death.
But perhaps she cannot really do much about these chances one way or the
other. In that case, a nonbehavioral model will explain them better. The
amount of health information that she chooses to use will not greatly affect
the chances that her baby will have a low weight or that it will die. She
must take those chances as given. In sum, the nonbehavioral model reduces to

\begin{equation}  \label{eq4}
W = W(D)
\end{equation}
and

\begin{equation}  \label{eq5}
M = M(W,D).
\end{equation}
because the amount of health information, $I $, is not relevant to this
approach.

An empirical test of the behavioral model (in (\ref{eq1}) - (\ref{eq3}))
against the nonbehavioral model (in (\ref{eq4}) - (\ref{eq5})) hinges on the
sign of the derivative ${\partial W^*} / {\partial I^*} $. The behavioral
model predicts a positive value for the derivative; the nonbehavioral model
predicts the value of zero.

\section{Tests of the model}

I turn to the estimation. In the system given by (\ref{eq1}) through (\ref
{eq3}), two independent variables -- $W$ and $I$ -- also serve as dependent
variables. They are thus likely to correlate with error terms. Hausman tests
suggest that simultaneity is especially present in the equation to estimate $%
W$.\footnote{%
I regressed an included endogenous variable, DROPOUTS, on the predetermined
variables DOCS, TEEN, NOGRADS and SINGLE. I then included the fitted errors
from this equation in an OLS estimate of the structural equation for LOWT.
The T-value for the coefficient on the fitted error was -7.04, suggesting
the presence of simultaneity in the structural equation. I also regressed
LOWT on the predetermined variables and then included the fitted errors in
an OLS estimate of the structural equation for DEAD. The T-value for the
coefficient on the fitted error was -1.45, $p=.155$.} Ordinary estimation
through least squares could produce biased and inconsistent estimators of
the coefficients (Kelejian and Oates (1981)). Multi-stage estimation seems
appropriate. I will present estimates for the system of three simultaneous
equations in (\ref{eq1}) through (\ref{eq3}).

\subsection{Data}

The dataset draws upon the 50 states of the United States in order to
satisfy two conditions that arise from the model. First: The probability
that a young woman will drop out of school may depend to some degree on the
policies adopted by the public schools. This suggests that observations
should be taken at the same level as the one at which school decisions are
made -- either the district level or the state level. Second: The area
embraced by each observation should be large enough to include, for a
typical woman in that area, the adults who will most influence her
educational decisions. The state is more likely to satisfy this restriction
than the district.

A third reason for using state data may also matter, although possibly less
than the other two reasons, since it does not stem directly from the theory
to be tested. Such variables as the dropout rate among late teens and the
nongraduation rate among adults are estimated with an error that may grow,
relative to the mean of the estimates, as the observation unit shrinks. The
measurement error may thus be relatively larger for data drawn from the
level of the county or the tract than for data drawn from the level of the
state. Where both the dependent and independent variables involve such
measurement errors, the OLS estimator of the response coefficient may yield
estimates that are inconsistent (Pindyck and Rubinfeld, 1991). In the
present instance, it is conceivable that the probability limit of the
estimator may be half of the true value of the response coefficient, when
one measures the variables as deviations from their means.\footnote{%
See appendix \ F for derivations.} Where no suitable instrument is evident,
one may well prefer a small data set that involves small errors for both
variables to a large data set that involves large errors.

The data come from the early 1990s. This gives all schools enough time to
put into effect three sets of federal requirements: First, from Title IX of
the Educational Amendments of 1972, that they let pregnant girls or mothers
attend class in public schools; second, from the Adolescent Pregnancy Act of
1978, that they provide for comprehensive services to pregnant teens
(McCarthy and Radish, 1983); third, that all state Medicaid programs cover
the cost of pregnancy and child birth for women in households with income up
to 133 percent of the poverty line (Currie and Gruber (1997)).

I cannot observe directly the acquisition and use of health information. So
I use, as a proxy for $I$, the dropout rate among high-school students by
state (DROPOUTS). I posit that, where youths tend to drop out of school,
would-be mothers tend to acquire and use little health information. The two
decisions -- to drop out and to get health data -- relate to the earlier
decision about how much education to get. They relate in opposite ways: In
an area where the average person decides to acquire a lot of education, she
is both unlikely to drop out and likely to inform herself keenly about
health. So the decisions to drop out and to get health data respond to the
same factors that affect the earlier decision to get education: the dropout
rate among adults ($N$), the percentage of local births that occurred to
single or teen mothers ($S$ and $T$); and the local supply of doctors ($D$).
Since the decisions to drop out and to get health data respond to these
factors in opposite ways, I posit that ${\partial DROPOUTS}/{\partial x}=-{%
\partial I}/{\partial x}$, where $x\in \{D,T,S,N\}$.

Tables 1 and 2 describe the data. In the three equations estimated, the
dependent variables are the percentage of births below 2,500 grams in 1992
(LOWT); the infant mortality rate (DEAD) in the early 1990s; and the
percentage of those aged 16 to 19 who dropped out of high school in 1990
(DROPOUTS). Tables 7 through 11 suggest the prudence of estimating
simultaneous equations. Of 50 correlations, 23 exceed .5 in absolute value.

The independent variables are the number of doctors per 100,000 residents in
1993 (DOCS); the percentage of births to teen mothers (TEEN) in 1992; the
percentage of adults 25 or older in 1990 who never graduated from high
school (NOGRADS); and the percentage of births that were to unmarried
mothers (SINGLE) in 1992. \vspace{10 mm}

\begin{table}[tbp]
\caption{Description of variables}
\label{describe}
\begin{tabular}{|l|r|r|}
\hline
{\it Variable} & {\it Year} & {\it Description} \\ \hline\hline
DEAD & 1992 & Deaths of infants, younger than 1 yr, \\ 
&  & per 1,000 births \\ \hline
DOCS & 1992 & Number of doctors \\ 
&  & per 100,000 civilian residents \\ \hline
DROPOUTS & 1990 & \% of 16-to-19 year-olds dropping \\ 
&  & out of high school \\ \hline
INCOME & 1992 & Median household money income \\ \hline
LOWT & 1992 & \% of births below 2,500 grams \\ \hline
MDPHC & 1993 & Medicaid spending on personal \\ 
&  & health care per household \\ \hline
NOGRADS & 1990 & \% of those 25 or older \\ 
&  & who weren't high school grads \\ \hline
SINGLE & 1992 & \% of births that were to unmarried mothers \\ \hline
TEEN & 1992 & \% of births that were to teen mothers \\ \hline
\end{tabular}
\end{table}

\vspace{10 mm}

\begin{table}[tbp]
\caption{Descriptive statistics}
\label{descriptive}
\begin{tabular}{|l|r|r|r|r|r|}
\hline
{\it Variable} & {\it Mean} & {\it Std Dev} & {\it Min} & {\it Max} & {\it %
Obs} \\ \hline\hline
DEAD & 8.3880 & 1.3986 & 5.600 & 11.900 & 50 \\ 
INCOME & 31509.0400 & 5300.7474 & 20878.000 & 43374.000 & 50 \\ 
DOCS & 206.1000 & 51.6417 & 130.000 & 361.000 & 50 \\ 
DROPOUTS & 10.3320 & 2.3857 & 4.600 & 15.200 & 50 \\ 
NOGRADS & 23.7140 & 5.6309 & 13.400 & 35.700 & 50 \\ 
TEEN & 12.6800 & 3.4081 & 6.700 & 21.400 & 50 \\ 
SINGLE & 28.5980 & 5.7721 & 15.100 & 42.900 & 50 \\ 
MDPHC & 1107.9828 & 369.5900 & 646.616 & 2697.115 & 50 \\ 
LOWT & 6.8740 & 1.2535 & 4.900 & 9.900 & 50 \\ \hline
\end{tabular}
\end{table}

\subsection{\protect\vspace{10mm}Estimates}

Table 3 sums up linear estimates of the system in (\ref{eq1})-(\ref{eq3}).
Generally, the disparity in coefficient estimates between the 2SLS and 3SLS
versions of the equation for DROPOUTS suggests some correlation of errors
across equations. Of the two estimation procedures, 3SLS is better suited to
removing that correlation (Pindyck and Rubinfeld, 1991).

\begin{table}[tbp]
\caption{Estimates of linearized models}
\label{linear}
\begin{tabular}{|l|r|r|r||r|r|r|}
\hline
\multicolumn{4}{c||}{\bf Two-stage least squares} & \multicolumn{3}{c}{\bf %
Three-stage least squares} \\ \hline\hline
& {\bf MOD1} & {\bf MOD2} & {\bf MOD3} & {\bf MOD1} & {\bf MOD2} & {\bf MOD3}
\\ \hline\hline
{\bf Depvar:} &  &  &  &  &  &  \\ 
{\bf Dead} &  &  &  &  &  &  \\ \hline
{\it Indvars} &  &  &  &  &  &  \\ \hline
{\it Constant} & 3.558 & 3.558 & 3.558 & 3.570 & 3.054 & 3.560 \\ 
t-ratio & 3.473 & 3.473 & 3.473 & 3.485 & 3.035 & 3.475 \\ \hline
{\it Lowt} & 0.913 & 0.913 & 0.913 & 0.913 & 0.947 & 0.913 \\ 
t-ratio & 7.119 & 7.119 & 7.119 & 7.114 & 7.422 & 7.116 \\ 
elasticity & .748 & .748 & .748 & .748 & .776 & .748 \\ \hline
{\it Docs} & -0.007 & -0.007 & -0.007 & -0.007 & -0.006 & -0.007 \\ 
t-ratio & -2.792 & -2.792 & -2.792 & -2.807 & -2.318 & -2.792 \\ 
elasticity & -12.01 & -12.01 & -12.01 & -12.01 & -10.373 & -12.101 \\ 
\hline\hline
{\bf Depvar:} &  &  &  &  &  &  \\ 
{\bf Lowt} &  &  &  &  &  &  \\ \hline
{\it Indvars} &  &  &  &  &  &  \\ \hline
{\it Constant} & 0.584 & 0.584 & -0.418 & 0.581 & -0.336 & -0.422 \\ 
t-ratio & 0.456 & 0.456 & -0.251 & 0.454 & -0.272 & -0.253 \\ \hline
{\it Dropouts} & 0.609 & 0.609 & 0.630 & 0.609 & 0.698 & 0.631 \\ 
t-ratio & 4.971 & 4.971 & 4.998 & 4.973 & 5.919 & 5.001 \\ 
elasticity & .915 & .915 & .947 & .915 & 1.049 & .948 \\ \hline
{\it Docs} &  &  & 0.004 &  &  & 0.004 \\ 
t-ratio &  &  & 0.960 &  &  & 0.961 \\ 
elasticity &  &  & 1.334 &  &  & 1.334 \\ \hline\hline
{\bf Depvar:} &  &  &  &  &  &  \\ 
{\bf Dropouts} &  &  &  &  &  &  \\ \hline
{\it Indvars} &  &  &  &  &  &  \\ \hline
{\it Constant} & 0.428 & 2.596 & 0.428 & -0.442 & 3.062 & 0.595 \\ 
t-ratio & 0.203 & 1.925 & 0.203 & -0.255 & 2.463 & 0.293 \\ \hline
{\it Teen} & 0.445 & 0.275 & 0.445 & 0.387 & 0.153 & 0.378 \\ 
t-ratio & 2.606 & 2.409 & 2.606 & 3.616 & 2.050 & 3.609 \\ 
elasticity & .546 & .337 & .546 & .475 & .188 & .464 \\ \hline
{\it Nograds} & -0.032 & 0.011 & -0.032 & 0.026 & 0.078 & 0.027 \\ 
t-ratio & -0.420 & 0.161 & -0.420 & 0.647 & 1.943 & 0.690 \\ 
elasticity & -.073 & 0.025 & -.073 & .060 & .179 & .062 \\ \hline
{\it Single} & 0.105 & 0.139 & 0.105 & 0.093 & 0.122 & 0.090 \\ 
t-ratio & 1.753 & 2.550 & 1.753 & 2.715 & 3.441 & 2.730 \\ 
elasticity & .291 & .385 & .291 & .257 & .338 & .249 \\ \hline
{\it Docs} & 0.010 &  & 0.010 & 0.013 &  & 0.008 \\ 
t-ratio & 1.320 &  & 1.320 & 2.958 &  & 1.401 \\ 
elasticity & .199 &  & .199 & .259 &  & .16 \\ \hline
\end{tabular}
\end{table}

In Table 3, Model 3 ({\it MOD 3}) is the full model. Model 1 drops the
variable that expresses the abundance of doctors (DOCS) from one equation,
and Model 2 drops it from two equations; for --- compared to other
independent variables in these equations --- the effects of DOCS were
relatively paltry and ambiguous. Table 4 includes estimates of second-order
effects in the 2SLS and 3SLS models.

\subsection{Interpretations}

The tests confirm the importance of nonbehavioral factors. In all runs of
the models, the infant death rate relates negatively to the abundance of
doctors. It also relates positively to the share of all births that are
low-weight. The t-statistics in Table 4, however, suggest that we cannot
reject the possibility that the incidence of low-weight births has no effect
on the rate of infant death when one controls for second-order effects.

Holding constant the incidence of a low-weight birth, a slight rise in the
number of doctors per 100,000 residents leads to a steep drop in the infant
death rate. Conceivably, this may reflect the success of hospitals that have
specialists in postnatal care.

Consideration of the second-order effects in Table 4 suggests that adding
doctors may lower the incidence of infant death at a diminishing rate --
again, holding constant the incidence of a low-weight birth. One can
speculate that there may be diminishing returns to the use of medical
technology, assuming that the use correlates positively with the density of
doctors in a given population. This speculation might help explain why
increasing the incidence of low-weight birth appears to raise the rate of
infant death at an increasing rate. The additional babies of low weight may
have less access than the others to technology, diminishing their chances
for postnatal survival.\footnote{%
Indeed, consider the marginal impact, on the rate of infant death, of
increasing the supply of doctors. This impact is more likely to be greater
than to be lesser if the incidence of low-weight birth is high. In the
equation for {\sl Dead} in Table 4, the coefficient on {\sl LowDocs} is
positive, although the t-statistic suggests that we cannot be sure of that.}

\begin{table}[tbp]
\caption{Estimates of full models}
\begin{tabular}{|l|r|r||l|r|r|}
\hline
& {\bf 2SLS} & {\bf 3SLS} &  & {\bf 2SLS} & {\bf 3SLS} \\ \hline\hline
{\bf Depvar:} &  &  & {\bf Depvar:} &  &  \\ 
{\bf Dead} &  &  & {\bf Dropouts} &  &  \\ \hline
{\it Indvars} &  &  & {\it Indvars} &  &  \\ \hline
{\it Constant} & 17.705 & 14.7377 & {\it Constant} & 0.428 & 0.1716 \\ 
t-ratio & 1.374 & 1.155 & t-ratio & 0.203 & 0.082 \\ \hline
{\it Lowt} & -1.507 & -0.6799 & {\it Teen} & 0.445 & 0.3687 \\ 
t-ratio & -0.447 & -0.204 & t-ratio & 2.606 & 2.220 \\ 
elasticity & -1.235 & -.557 & elasticity & .546 & .452 \\ \hline
{\it Docs} & -0.061 & -0.0587 & {\it Nograds} & -0.0322 & 0.0116 \\ 
t-ratio & -2.397 & -2.348 & t-ratio & -0.420 & 0.160 \\ 
elasticity & -1.499 & -1.442 & elasticity & -.074 & -.0266 \\ \hline
{\it $Docs^2 $} & 0.00005 & 0.00005 & {\it Single} & 0.1046 & 0.1272 \\ 
t-ratio & 1.446 & 1.448 & t-ratio & 1.753 & 2.225 \\ 
elasticity & .253 & .253 & elasticity & .2895 & .352 \\ \hline
{\it $Lowt^2 $} & 0.1045 & 0.0501 & {\it Docs} & .0099 & 0.0076 \\ 
t-ratio & 0.534 & 0.258 & t-ratio & 1.32 & 1.032 \\ 
elasticity & .589 & .282 & elasticity & .197 & .152 \\ \hline
{\it LowDocs} & 0.004 & 0.00399 &  &  &  \\ 
t-ratio & 1.156 & 1.083 &  &  &  \\ 
elasticity & .676 & .674 &  &  &  \\ \hline\hline
{\bf Depvar:} &  &  &  &  &  \\ 
{\bf Lowt} &  &  &  &  &  \\ \hline
{\it Indvars} &  &  &  &  &  \\ \hline
{\it Constant} & -40.463 & -29.150 &  &  &  \\ 
t-ratio & -2.044 & -1.502 &  &  &  \\ \hline
{\it Dropouts} & 6.586 & 4.871 &  &  &  \\ 
t-ratio & 2.537 & 1.907 &  &  &  \\ 
elasticity & 9.899 & 7.321 &  &  &  \\ \hline
{\it Docs} & 0.1398 & 0.0999 &  &  &  \\ 
t-ratio & 1.758 & 1.295 &  &  &  \\ 
elasticity & 4.192 & 2.995 &  &  &  \\ \hline
{\it $Docs^2 $} & -0.000067 & -0.000050 &  &  &  \\ 
t-ratio & -0.842 & -0.660 &  &  &  \\ 
elasticity & -.414 & -.30897 &  &  &  \\ \hline
{\it $Drop^2 $} & -0.1989 & -0.1417 &  &  &  \\ 
t-ratio & -2.465 & -1.784 &  &  &  \\ 
elasticity & -3.089 & 2.201 &  &  &  \\ \hline
{\it DocsDrop} & -0.0114 & -0.0079 &  &  &  \\ 
t-ratio & -2.012 & -1.428 &  &  &  \\ 
elasticity & -3.531 & -2.447 &  &  &  \\ \hline
\end{tabular}
\end{table}

Using the estimates in Table 4, one can infer that an increase in the supply
of doctors may lower the incidence of low-weight birth, but at a diminishing
rate. One cannot place much confidence in the results, however.

One may be more confident that the share of low-weight births relates
positively to the share of youths that drop out of high school. The effect
is rather moderate in the linear model: A 1 percent increase in the
percentage of dropouts leads to roughly a 1 percent increase in the
percentage of low-weight births. Judging from the second-order effects in
Table 4, a rise in the incidence of dropping out may increase the incidence
of low-weight births at a diminishing rate. One can speculate that
additional dropouts may be less likely to have babies.

The share of youths that drop out of high school relates positively to the
share of mothers who are teens. It also relates positively to the share of
mothers who are not married. The ``teen'' effect is the more direct of the
two effects. In an area where many mothers are teens, a female teen in high
school may well feel pressure to drop out herself -- more pressure, in fact,
than she would feel if she simply saw that many mothers were single, for a
teen relates more easily to someone of the same age than to someone of the
same marital status. Since, in this sense, the ``teen'' effect is more
direct than the ``unmarried'' effect, one would also expect it to be the
stronger effect -- and, in most runs, it is. An increase of 1 percent in the
percentage of teen mothers leads to an increase of roughly one-half percent
in the percentage of dropouts. The ``unmarried'' effect, however, also seems
to matter: An increase of 1 percent in the percentage of single mothers
leads to an increase of roughly one-third percent in the percentage of
dropouts.

The direction of the effect on the percentage of dropouts due to the share
of adults who did not graduate from high school is not robust across the
models. In any event, the effect is small. This suggests that potential
dropouts may relate more strongly to their peers in age than to adult role
models.

The share of youths that drop out of high school may relate positively to
the supply of doctors. But the coefficient is not estimated as precisely as
those for teen or single mothers, and the effect is not as large. This is to
be expected. It seems unlikely that teens would attach much weight to their
access to doctors when deciding whether to stay in school.\footnote{%
The supply of doctors might proxy here for the supply of low-skill service
jobs in the state economy. When such jobs abound, teens may want to drop out
of school and go to work.}

\subsection{Impact of low-weight birth}

The main order of business in the testing is to see how factors affect the
probability of low-weight birth and of infant death. Table 4 gives estimates
for second-order approximations of the system in (\ref{eq1}) through (\ref
{eq3}). From the second equation of the 2SLS model, the total impact of the
dropout rate on the rate of low-weight births is positive:

\begin{equation}
{\partial {\it Lowt/}}{\partial {\it Dropouts}}=6.586-1.34\ast 10^{-4}{\it %
Dropouts}.
\end{equation}

Even when the {\it Dropouts} rate is a theoretical 100 (percent), the
derivative remains positive. As the {\it Dropouts} rate increases, the
derivative does diminish, but only slightly. An increase in the rate of
dropouts of one standard deviation (2.3857) leads to a decrease in the rate
of low-weight births of $3.1968\ast 10^{-4}$. That is a decline in the
low-birth rate of about 3 in 10,000.

In comparison, the supply of doctors may have a more substantial impact on
the effect of the dropout rate on the rate of low-weight births. Judging
from the coefficients of the second equation in the 2SLS model, when the
number of doctors drops by a standard deviation (51.6), the derivative ${%
\partial {\it Lowt/}}{\partial {\it Dropouts}}$ increases by .589. That is,
as we move from one state that abounds in doctors to another that lacks
them, an increase in the dropout rate of one percentage point leads to an
additional increase in the rate of low-weight births of nearly one
percentage point.

Broadly, these results suggest that programs to encourage young women to
stay in school, or to complete their general equivalency degrees, may be
most effective in cutting the rate of low-weight births by an absolute
amount in states that already have low rates of low-weight births and that
lack many doctors.

\subsection{Summary of estimates}

Table 5 presents point estimates of derivative values for second-order
approximations of the system (\ref{eq1}) through (\ref{eq3}), evaluated at
the means for variables reported in Table 2. These calculations use the
estimates of the coefficients from the full specifications of the 2SLS and
3SLS models, reported in Table 4. Table 5 presents sign estimates for second
derivatives in equations (\ref{eq1}) and (\ref{eq2}) only, since the
comparative statics do not require the signing of second derivatives in
equation (\ref{eq3}). In scanning the table, the reader should bear in mind
that, for some of these coefficient estimates, the T-tests suggest that the
actual coefficients may not, in fact, differ much from zero. The precision
of the estimates of derivative values should thus be taken as provisional.

Table 5 also compares the signs of the estimated values of the derivatives
to the signs predicted for them. The most striking ``misses'' involve the
second-order impact of the supply of doctors on the rate of low-weight
births.

\begin{table}[tbp]
\caption{Coefficient signs }
\label{signs}
\begin{tabular}{|l|r|r|}
\hline
Coefficient & Estimate & Predicted sign? \\ \hline
\multicolumn{3}{c}{Two-stage least squares} \\ \hline
${\partial Dead} / {\partial Lowt} $ & .761 & Yes \\ 
${\partial^2 Dead} / {\partial Lowt^2} $ & .209 & Yes \\ 
${\partial Dead} / {\partial Docs} $ & -.0096 & Yes \\ 
${\partial^2 Dead} / {\partial Docs^2} $ & .0215 & Yes \\ 
${\partial^2 Dead} / {\partial Lowt \partial Docs} $ & .0043 & No \\ 
${\partial Lowt} / {\partial Drop} $ & .122 & Yes \\ 
${\partial^2 Lowt} / {\partial Drop^2} $ & -.011 & Yes \\ 
${\partial Lowt} / {\partial Docs} $ & -.0058 & Yes \\ 
${\partial^2 Lowt} / {\partial Docs^2} $ & -.00013 & No \\ 
${\partial^2 Lowt} / {\partial Drop \partial Docs} $ & -.0114 & No \\ 
${\partial Drop} / {\partial Docs} $ & 2.040 & Yes \\ 
${\partial Drop} / {\partial Teen} $ & 5.643 & Yes \\ 
${\partial Drop} / {\partial Nograds} $ & -.7636 & No \\ 
${\partial Drop} / {\partial Single} $ & 2.991 & Yes \\ \hline
\multicolumn{3}{c}{Three-stage least squares} \\ \hline
${\partial Dead} / {\partial Lowt} $ & .831 & Yes \\ 
${\partial^2 Dead} / {\partial Lowt^2} $ & .100 & Yes \\ 
${\partial Dead} / {\partial Docs} $ & -.0107 & Yes \\ 
${\partial^2 Dead} / {\partial Docs^2} $ & .0001 & Yes \\ 
${\partial^2 Dead} / {\partial Lowt \partial Docs} $ & .00399 & No \\ 
${\partial Lowt} / {\partial Drop} $ & .315 & Yes \\ 
${\partial^2 Lowt} / {\partial Drop^2} $ & -.2834 & Yes \\ 
${\partial Lowt} / {\partial Docs} $ & -.0023 & Yes \\ 
${\partial^2 Lowt} / {\partial Docs^2} $ & -.0001 & No \\ 
${\partial^2 Lowt} / {\partial Drop \partial Docs} $ & -.0079 & No \\ 
${\partial Drop} / {\partial Docs} $ & 1.566 & Yes \\ 
${\partial Drop} / {\partial Teen} $ & 4.675 & Yes \\ 
${\partial Drop} / {\partial Nograds} $ & 2.751 & No \\ 
${\partial Drop} / {\partial Single} $ & 3.638 & Yes \\ \hline
\end{tabular}
\end{table}

\vspace{10mm}

\bigskip

\begin{table}[tbp]
\caption{Simple correlations: Part 1}
\label{corr1}
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& DEAD & INCOME & DOCS & DROP & NOGR & TEEN \\ \hline
DEAD & 1.0000 & -0.4561 & -0.2497 & 0.3237 & 0.6380 & 0.6594 \\ 
INCOME & -0.4561 & 1.0000 & 0.5051 & -0.2515 & -0.6618 & -0.7435 \\ 
DOCS & -0.2497 & 0.5051 & 1.0000 & -0.1202 & -0.1425 & -0.5620 \\ 
DROPOUTS & 0.3237 & -0.2515 & -0.1202 & 1.0000 & 0.5114 & 0.5960 \\ 
NOGRADS & 0.6380 & -0.6618 & -0.1425 & 0.5114 & 1.0000 & 0.7461 \\ 
TEEN & 0.6594 & -0.7435 & -0.5620 & 0.5960 & 0.7461 & 1.0000 \\ 
SINGLE & 0.5540 & -0.2978 & 0.0476 & 0.5656 & 0.5686 & 0.5439 \\ 
MDPHC & -0.0406 & 0.0426 & 0.5429 & -0.0769 & 0.1918 & -0.2173 \\ 
LOWT & 0.7158 & -0.3089 & 0.0116 & 0.4749 & 0.6693 & 0.6465 \\ \hline
\end{tabular}
\end{table}

\vspace{10mm}

\bigskip 
\begin{table}[tbp]
\caption{Simple correlations: Part 2}
\label{corr2}
\begin{tabular}{|l|r|}
\hline
& SINGLE \\ \hline
DEAD & 0.5540 \\ 
INCOME & -0.2978 \\ 
DOCS & 0.0476 \\ 
DROPOUTS & 0.5656 \\ 
NOGRADS & 0.5686 \\ 
TEEN & 0.5439 \\ 
SINGLE & 1.0000 \\ 
MDPHC & 0.2047 \\ 
LOWT & 0.6617 \\ \hline
\end{tabular}
\end{table}

\bigskip

\begin{table}[tbp]
\caption{Simple correlations: Part 3}
\label{corr3}
\begin{tabular}{|l|r|r|}
\hline
& MDPHC & LOWT \\ \hline
DEAD & -0.0406 & 0.7158 \\ 
INCOME & 0.0426 & -0.3089 \\ 
DOCS & 0.5429 & 0.0116 \\ 
DROPOUTS & -0.0769 & 0.4749 \\ 
NOGRADS & 0.1918 & 0.6693 \\ 
TEEN & -0.2173 & 0.6465 \\ 
SINGLE & 0.2047 & 0.6617 \\ 
MDPHC & 1.0000 & 0.0172 \\ 
LOWT & 0.0172 & 1.0000 \\ \hline
\end{tabular}
\end{table}

\vspace{10mm}

\bigskip

\begin{table}[tbp]
\caption{Probability of no correlation: Part 1}
\label{prob1}
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& DEAD & INCOME & DOCS & DROP & NOGR & TEEN \\ \hline
DEAD & 0.0000 & 0.0009 & 0.0803 & 0.0218 & 0.0000 & 0.0000 \\ 
INCOME & 0.0009 & 0.0000 & 0.0002 & 0.0780 & 0.0000 & 0.0000 \\ 
DOCS & 0.0803 & 0.0002 & 0.0000 & 0.4056 & 0.3235 & 0.0000 \\ 
DROPOUTS & 0.0218 & 0.0780 & 0.4056 & 0.0000 & 0.0001 & 0.0000 \\ 
NOGRADS & 0.0000 & 0.0000 & 0.3235 & 0.0001 & 0.0000 & 0.0000 \\ 
TEEN & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ 
SINGLE & 0.0000 & 0.0357 & 0.7429 & 0.0000 & 0.0000 & 0.0000 \\ 
MDPHC & 0.7793 & 0.7692 & 0.0000 & 0.5957 & 0.1821 & 0.1296 \\ 
LOWT & 0.0000 & 0.0291 & 0.9362 & 0.0005 & 0.0000 & 0.0000 \\ \hline
\end{tabular}
\end{table}

\vspace{10mm}

\bigskip 
\begin{table}[tbp]
\caption{Probability of no correlation: Part 2}
\label{prob2}
\begin{tabular}{|l|r|}
\hline
& SINGLE \\ \hline
DEAD & 0.0000 \\ 
INCOME & 0.0357 \\ 
DOCS & 0.7429 \\ 
DROPOUTS & 0.0000 \\ 
NOGRADS & 0.0000 \\ 
TEEN & 0.0000 \\ 
SINGLE & 0.0000 \\ 
MDPHC & 0.1538 \\ 
LOWT & 0.0000 \\ \hline
\end{tabular}
\end{table}

\bigskip

\begin{table}[tbp]
\caption{Probability of no correlation: Part 3}
\label{prob3}
\begin{tabular}{|l|r|r|}
\hline
& MDPHC & LOWT \\ \hline
DEAD & 0.7793 & 0.0000 \\ 
INCOME & 0.7692 & 0.0291 \\ 
DOCS & 0.0000 & 0.9362 \\ 
DROPOUTS & 0.5957 & 0.0005 \\ 
NOGRADS & 0.1821 & 0.0000 \\ 
TEEN & 0.1296 & 0.0000 \\ 
SINGLE & 0.1538 & 0.0000 \\ 
MDPHC & 0.0000 & 0.9054 \\ 
LOWT & 0.9054 & 0.0000 \\ \hline
\end{tabular}
\end{table}

\subsection{\protect\vspace{10mm}Simulations}

Simulations of the estimated system in (\ref{eq1}) through (\ref{eq3})
suggest that a drop of one percentage point in the dropout rate would lead,
on average, to a short-run drop of .15 in the rate of infant death (or 15
deaths avoided per 100,000 live births) in a state. The precise change,
however, varies from state to state. The predicted drop in the rate of
infant death is greatest for North Dakota (-1.21, or 121 deaths averted per
100,000 live births). The precise change is also sensitive to the number of
stages in the least-squares estimation. Under 2SLS, the predicted mean
change in the rate of infant death is positive (.118, or an additional 11.8
deaths per 100,000 live births). Tables 12 and 13 present the predicted
changes, by state, in the rate of low-weight birth as well as in the rate of
infant death.\footnote{%
Appendix E gives the equations used in the simulations.}

\bigskip 
\begin{table}[tbp]
\caption{Short-run changes in low-weight, death rates}
\label{srchange}
\begin{tabular}{|c|r|r||r|r|}
\hline
& \multicolumn{2}{c||}{2SLS} & \multicolumn{2}{c|}{3SLS} \\ \hline
\multicolumn{1}{|l|}{\it State} & {\it $\Delta $ Lowt} & {\it $\Delta $ Dead}
& {\it $\Delta $ Lowt} & {\it $\Delta $ Dead} \\ \hline\hline
\multicolumn{1}{|l|}{Alabama} & 0.364 & 0.346 & 0.043 & 0.036 \\ 
\multicolumn{1}{|l|}{Alaska} & -0.631 & -0.054 & -0.660 & -0.249 \\ 
\multicolumn{1}{|l|}{Arizona} & 1.354 & 0.821 & 0.743 & 0.546 \\ 
\multicolumn{1}{|l|}{Arkansas} & -0.204 & -0.175 & -0.360 & -0.284 \\ 
\multicolumn{1}{|l|}{California} & 1.799 & 1.234 & 1.049 & 0.912 \\ 
\multicolumn{1}{|l|}{Colorado} & -0.157 & -0.181 & -0.340 & -0.359 \\ 
\multicolumn{1}{|l|}{Connecticut} & 0.654 & 0.797 & 0.216 & 0.278 \\ 
\multicolumn{1}{|l|}{Delaware} & -0.066 & -0.061 & -0.273 & -0.250 \\ 
\multicolumn{1}{|l|}{Florida} & 1.554 & 1.398 & 0.880 & 0.809 \\ 
\multicolumn{1}{|l|}{Georgia} & 1.098 & 1.095 & 0.563 & 0.505 \\ 
\multicolumn{1}{|l|}{Hawaii} & -0.821 & -0.799 & -0.818 & -0.830 \\ 
\multicolumn{1}{|l|}{Idaho} & -0.955 & -0.159 & -0.889 & -0.350 \\ 
\multicolumn{1}{|l|}{Illinois} & 0.253 & 0.258 & -0.050 & -0.050 \\ 
\multicolumn{1}{|l|}{Indiana} & -0.136 & -0.077 & -0.313 & -0.207 \\ 
\multicolumn{1}{|l|}{Iowa} & -2.148 & -0.688 & -1.744 & -0.917 \\ 
\multicolumn{1}{|l|}{Kansas} & -1.016 & -0.580 & -0.944 & -0.660 \\ 
\multicolumn{1}{|l|}{Kentucky} & 0.745 & 0.470 & 0.312 & 0.224 \\ 
\multicolumn{1}{|l|}{Louisiana} & 0.678 & 0.855 & 0.259 & 0.276 \\ 
\multicolumn{1}{|l|}{Maine} & -1.095 & -0.335 & -1.002 & -0.588 \\ 
\multicolumn{1}{|l|}{Maryland} & 1.569 & 2.460 & 0.865 & 1.287 \\ 
\multicolumn{1}{|l|}{Massachusetts} & 0.911 & 1.085 & 0.390 & 0.531 \\ 
\multicolumn{1}{|l|}{Michigan} & -0.385 & -0.324 & -0.497 & -0.422 \\ 
\multicolumn{1}{|l|}{Minnesota} & -1.395 & -0.709 & -1.224 & -0.939 \\ 
\multicolumn{1}{|l|}{Mississippi} & -0.410 & -0.444 & -0.500 & -0.415 \\ 
\hline
\end{tabular}
\end{table}

\bigskip

\begin{table}[tbp]
\caption{Short-run changes in low-weight, death rates, Part 2}
\label{srchange1}
\begin{tabular}{|l|r|r||r|r|}
\hline
\multicolumn{1}{|c}{} & \multicolumn{2}{c||}{2SLS} & \multicolumn{2}{c|}{3SLS
} \\ \hline\hline
{\it State} & {\it $\Delta $ Lowt} & {\it $\Delta $ Dead} & {\it $\Delta $
Lowt} & {\it $\Delta $ Dead} \\ \hline
Missouri & 0.309 & 0.261 & -0.005 & -0.004 \\ 
Montana & -1.437 & -0.608 & -1.240 & -0.739 \\ 
Nebraska & -1.647 & -0.691 & -1.394 & -0.886 \\ 
Nevada & 1.148 & 0.653 & 0.606 & 0.377 \\ 
New Hampshire & -0.441 & -0.196 & -0.540 & -0.374 \\ 
New Jersey & 0.231 & 0.243 & -0.073 & -0.079 \\ 
New Mexico & 0.234 & 0.178 & -0.054 & -0.043 \\ 
New York & 1.160 & 1.644 & 0.573 & 0.811 \\ 
North Carolina & 0.644 & 0.670 & 0.236 & 0.224 \\ 
North Dakota & -2.613 & -0.812 & -2.082 & -1.210 \\ 
Ohio & -0.686 & -0.595 & -0.713 & -0.633 \\ 
Oklahoma & -0.705 & -0.356 & -0.715 & -0.430 \\ 
Oregon & 0.502 & 0.211 & 0.132 & 0.090 \\ 
Pennsylvania & -0.070 & -0.071 & -0.285 & -0.301 \\ 
Rhode Island & 0.919 & 0.821 & 0.416 & 0.429 \\ 
South Carolina & 0.040 & 0.043 & -0.189 & -0.172 \\ 
South Dakota & -1.745 & -0.356 & -1.456 & -0.675 \\ 
Tennessee & 1.139 & 1.263 & 0.586 & 0.591 \\ 
Texas & 0.563 & 0.374 & 0.183 & 0.133 \\ 
Utah & -0.993 & -0.409 & -0.928 & -0.582 \\ 
Vermont & -0.451 & -0.315 & -0.558 & -0.510 \\ 
Virginia & -0.157 & -0.141 & -0.339 & -0.311 \\ 
Washington & 0.139 & 0.067 & -0.129 & -0.094 \\ 
West Virginia & -0.175 & -0.127 & -0.344 & -0.264 \\ 
Wisconsin & -1.504 & -0.779 & -1.295 & -0.908 \\ 
Wyoming & -2.279 & -1.292 & -1.833 & -1.097 \\ \hline
{\it State average} & -0.126 & 0.118 & -0.315 & -0.156 \\ \hline
\end{tabular}
\end{table}

\bigskip

\bigskip

\section{Marginal cost of health information}

Specifying the first-order condition in (\ref{foc}) yields crude and
preliminary estimates of the marginal cost and value of health information
to the pregnant woman. The estimates use $\partial DEAD/\partial LOWT$ for $%
\partial M/\partial W$; and $\partial LOWT/\partial DROPOUTS$ for $\partial
W/\partial I$. To approximate $U_{h}-U_{m}$, the estimates use median
household money income. $U_{h}$ and $U_{m}$ are values; no utility function
is assumed.

Computations are in Table 14. Estimates of marginal cost are sensitive to
the specification of the model, but all amounts are considerable. The
estimates suggest that the would-be mother acquires health information up to
the point that she would be willing to pay more than \$2,900 -- an eleventh
of the annual income of her household -- to forgo acquiring a little more.
It seems that the marginal cost of health information is quite high. Since
the optimizing mother acquires health information until the marginal value
equals the marginal cost, its marginal value -- the gain to the would-be
mother in reducing the chance of an infant death -- is also more than
\$2,900.

While one must be cautious about drawing conclusions from such a simple
model, the high marginal value of information suggests that education would
also be valuable to the woman but that she finds it costly to acquire. The
strength of the peer effects suggests that perhaps the woman drops out of
school because she finds education costly in social terms, not because she
lacks motivation to learn. Though she highly values education, social
pressures powerfully constrain her from completing her degree. This
speculation, while highly tenuous, is somewhat supported by the statistical
results: The share of youths that dropped out of high school related
positively to the share of mothers who were teens and to the share of
mothers who were single. \vspace{10mm}

\begin{table}[tbp]
\caption{Marginal cost of health information}
\label{health}
\begin{tabular}{|l|r|r|r|r|}
Model & $\frac{\partial DEAD}{\partial LOWT} $ & $\frac{\partial LOWT}{%
\partial DROPOUTS} $ & Income & $\approx \frac{\partial C}{\partial I} $ \\ 
\hline\hline
Model 1, 2SLS & .913 & .609 & \$31,509 & \$17,520 \\ 
Model 1, 3SLS & .913 & .609 & \$31,509 & \$17,520 \\ 
Model 2, 2SLS & .913 & .609 & \$31,509 & \$17,520 \\ 
Model 2, 3SLS & .947 & .698 & \$31,509 & \$20,828 \\ 
Model 3, 2SLS & .913 & .630 & \$31,509 & \$18,124 \\ 
Model 3, 3SLS & .913 & .631 & \$31,509 & \$18,152 \\ 
Full Model, 2SLS & .761 & .122 & \$31,509 & \$2,925 \\ 
Full Model, 3SLS & .831 & .315 & \$31,509 & \$8,248 \\ \hline
\end{tabular}
\end{table}

\section{Conclusions and reflections}

While highly tentative, the empirical results suggest two issues for policy
research. First, the results fit the idea that the woman bases her estimates
of the net returns to education on what she sees others do. For example, she
is more likely to drop out of school if she observes that many peers and
adults dropped out, too. But the decisions of adults to drop out, when they
were youths, may have stemmed in part from observing that the peers and
adults of their day had dropped out. And so the impact of an individual
decision may be transmitted over time, from generation to generation.

The empirical results also fit the idea that raising the perceived returns
to education could lower the incidence of infant mortality.

If these provisional interpretations are correct, then one may ask: How can
a family, community or government raise the returns to education that are
perceived by the uneducated?

The results may also suggest that the would-be mother acquires education --
and, by extension, health information -- to the point that she would find it
quite costly to acquire more. If that interpretation is correct, then one
may ask: How can one lower the cost -- especially in terms of social
pressure -- of acquiring education?

\section{Appendix A: Derivations}

\subsection{Second-order conditions}

The second derivative of the function to be maximized ($L$), with respect to
the choice variable, health information ($I$), is

\[
\frac{\partial ^{2}L}{\partial I^{2}}=-\frac{\partial M}{\partial W}\frac{%
\partial ^{2}W}{\partial I^{2}}(U_{h}-U_{m})-\frac{\partial ^{2}M}{\partial
W^{2}}({\frac{\partial W}{\partial I}})^{2}(U_{h}-U_{m})-\frac{\partial ^{2}C%
}{\partial I^{2}}<0.
\]

The strict inequality follows in part from the assumptions that ${\partial^2
W} / {\partial I^2} > 0 $and ${\partial^2 C} / {\partial I^2} > 0 $.

\subsection{Comparative statics}

The mathematics draw upon Chiang (1974). Rewrite ${\partial L}/{\partial I}$
as the function

\[
F(D,T,S,N)=-\frac{\partial M}{\partial W}\frac{\partial W}{\partial I}%
(U_{h}-U_{m})-\frac{\partial C}{\partial I}\equiv 0.
\]

From (\ref{soc}), we have that ${\partial F} / {\partial I} < 0 $. Using the
implicit function theorem, we have that

\[
\frac{\partial F}{\partial D}=-(\frac{\partial ^{2}M}{\partial W\partial D}%
\frac{\partial W}{\partial I}+\frac{\partial M}{\partial W}\frac{\partial
^{2}W}{\partial I\partial D})(U_{h}-U_{m})\leq 0
\]

so

\[
\frac{\partial I^{\ast }}{\partial D}=-\frac{F_{D}}{F_{I}}\leq 0.
\]

Also,

\[
\frac{\partial F}{\partial S}=-\frac{\partial ^{2}C}{\partial I\partial S}<0
\]

so

\[
\frac{\partial I^{\ast }}{\partial S}=-\frac{F_{S}}{F_{I}}<0.
\]

Also,

\[
\frac{\partial F}{\partial N}=-\frac{\partial ^{2}C}{\partial I\partial N}<0
\]

so

\[
\frac{\partial I^{\ast }}{\partial N}=-\frac{F_{N}}{F_{I}}<0.
\]

Finally,

\[
\frac{\partial F}{\partial T}=-\frac{\partial ^{2}C}{\partial I\partial T}<0
\]

so

\[
\frac{\partial I^{\ast }}{\partial T}=-\frac{F_{T}}{F_{I}}<0.
\]

\section{Appendix B: One-equation models}

A linear regression of {\sl Dead} on {\sl Lowt} yields results reported in
Table 15, where $s=0.9868$, $R^{2}=.512$ and adjusted $R^{2}=.502.$ \ The
Durbin-Watson statistic is 1.64.

\begin{table}[h]
\caption{Infant death and low weight}
\begin{tabular}{|l|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} \\ \hline\hline
Constant & 2.8980 & 0.7855 & 3.69 & 0.001 \\ 
Lowt & 0.7987 & 0.1125 & 7.10 & 0.000 \\ \hline
\end{tabular}
\label{lowt}
\end{table}

\bigskip 

The model makes clear that the rate of low-weight birth shapes the rate of
infant death. Still, caveats are in order. First, Mississippi's influence on
the model is palpable; the state had both the highest rate of infant death
(11.9 deaths in every 1,000 live births) and the highest rate of low-weight
birth (9.9 percent of all registered births were below 2,500 grams). Second,
the model seriously mis-predicts the rate of infant death for three states.
For Colorado and Hawaii, it predicts rates that are too high. For Colorado,
the model predicts 9.7; the rate is actually 7.6. For Hawaii, the model
predicts 8.6; the rate is actually 6.3. For South Dakota, the model predicts
a rate that is too low. The model predicts 7.1; the rate is actually 9.3.

A slightly more complex model yields intriguing results. Consider

\begin{equation}
{\sl Dead_{i}}=a_{0}+a_{1}{\sl Income_{i}}+a_{2}{\sl Teen_{i}}+a_{3}{\sl %
Lowt_{i}}+\epsilon _{i}  \label{a1a2}
\end{equation}
where $i$ indexes the states. The results in Table 16 show that the impact
of income is so small, relative to the imprecision of its estimation, that
one cannot discard the possibility that the impact is zero. For the model, $%
s=0.9294$, $R^{2}=.585$ and adjusted $R^{2}=.558.$ \ The Durbin-Watson
statistic is 1.87.

\begin{table}[h]
\caption{Infant death: A fuller model}
\begin{tabular}{|l|r|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} & {\it VIF} \\ \hline\hline
Constant & 4.227 & 1.804 & 2.34 & 0.024 &  \\ 
Income & -0.00003414 & 0.00003978 & -0.86 & 0.395 & 2.5 \\ 
Lowt & 0.5976 & 0.1474 & 4.05 & 0.000 & 1.9 \\ 
Teen & 0.08905 & 0.07713 & 1.15 & 0.254 & 3.9 \\ \hline
\end{tabular}
\label{fuller}
\end{table}

\bigskip 

The standardized form of the model also suggests that income is relatively
unimportant, compared to the impacts of the other independent variables. The
results are in Table 17, where $s=0.6574$. Since standardization reduces the
constant to zero, I have suppressed it here -- and, consequently, I report
no $R^{2}$.

\begin{table}[h]
\caption{Infant death: A standardized model}
\begin{tabular}{|l|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} \\ \hline\hline
Income & -0.1294 & 0.1491 & -0.87 & 0.390 \\ 
Lowt & 0.5356 & 0.1307 & 4.10 & 0.000 \\ 
Teen & 0.2170 & 0.1859 & 1.17 & 0.249 \\ \hline
\end{tabular}
\label{standard}
\end{table}

Despite the statistical and relative insignificance of income in such simple
models, which treat independent variables as little influenced by income,
its impact on infant health may be noticeable. The standardized model
indicates that an increase in median household income by one standard
deviation (\$5,300) leads to a reduction in the rate of infant mortality of
about .13 of a standard deviation, or 18 additional lives saved in every
100,000 live births.

I return now to the regular regressions. As the Variance Inflation Factors
indicate in Table 16, collinearity is not likely to account for the
statistical insignificance of the coefficient on {\sl Income}, although its
simple correlation with {\sl Teen} is somewhat strong (-.744). So one may
wish to test the hypothesis that, in (\ref{a1a2}), $a_{1}=a_{2}=0$.
Calculating the F-statistic for the two models yields

\[
F_{2,47}=\frac{({R^{2}}_{UR}-{R^{2}}_{R})/2}{(1-{R^{2}}_{UR})/47}=\frac{%
(.585-.512)/2}{(1-.585)/47}=4.13.
\]

Critical F-values are 3.23 at the 5 percent level of significance and 5.18
at the 1 percent level. It is hard to reject the idea that a model based on
low-weight birth alone is effective.

The puzzling results concerning income may stem in part from the high
incomes -- and the higher-than-average rates of infant death -- in Alaska
and South Dakota. Omitting those observations from the dataset improves the
fit of the model considerably at the expense of a slight loss of relevant
information. Excluding {\sl Teens} from the model, because of its negative
covariance with {\sl Income}, also improves the fit, at the expense of a
slight downward bias in the {\sl Income} coefficient and a slight upward
bias in the {\sl Lowt} coefficient. Results are reported in Table 18, where $%
s=0.8233.$ \ Also, $R^{2}=.679$ and adjusted $R^{2}=.665$.

\begin{table}[h]
\caption{Infant death: Omitting a control for teens}
\begin{tabular}{|l|r|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} & {\it VIF} \\ \hline\hline
Constant & 5.162 & 1.199 & 4.30 & 0.000 &  \\ 
Income & -0.00007802 & 0.00002460 & -3.17 & 0.003 & 1.1 \\ 
Lowt & 0.8129 & 0.1028 & 7.91 & 0.000 & 1.1 \\ \hline
\end{tabular}
\label{omit}
\end{table}

\bigskip 

Standardized error terms appear to be distributed normally. Under the
Anderson-Darling test, $p=.18$.

Inspection of a graph of the standardized residuals against the fitted
values for the equation revealed no obvious pattern of heteroscedasticity. A
more precise analysis is provided by a linear regression of the error
variances, normalized by their mean, on {\sl Income} and {\sl Lowt}. Given
the normality of the error terms, an application of the Breusch-Pagan test
seemed appropriate here. The test suggested that the error term was
homoscedastic (Pindyck and Rubinfeld (1991)).\footnote{%
Half of the regression sum of squares was 2.807, well below the $\chi ^{2}$
critical value at the 5 percent level for two degrees of freedom (5.99).}

The standardized version of the model suggests that a change in income would
have only 41 percent of the impact on the rate of infant death as a
commensurate change in the incidence of low-weight birth. Results are in
Table 19, where $s=0.5864$.

\begin{table}[h]
\caption{Standardized model without a control for teens}
\begin{tabular}{|l|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} \\ \hline\hline
Income & -0.29482 & 0.09288 & -3.17 & 0.003 \\ 
Lowt & 0.72428 & 0.09163 & 7.90 & 0.000 \\ \hline
\end{tabular}
\label{omitstandard}
\end{table}

Omitting {\sl Teen} from this standardized model introduces a downward bias
in the {\sl Income} coefficient and an upward bias in the {\sl Lowt}
coefficient. Including {\sl Teen} produces the results in Table 20, where  $%
s=0.5929.$

\begin{table}[h]
\caption{Standardized model with a control for teens}
\begin{tabular}{|l|r|r|r|r|}
\hline
{\it Predictor} & {\it Coefficient} & {\it Standard Deviation} & {\it T-ratio%
} & {\it p} \\ \hline\hline
Income & -0.2844 & 0.1584 & -1.80 & 0.079 \\ 
Lowt & 0.7168 & 0.1303 & 5.50 & 0.000 \\ 
Teen & 0.0155 & 0.1890 & 0.08 & 0.935 \\ \hline\hline
\end{tabular}
\label{omitstandard1}
\end{table}

\bigskip 

With the inclusion of {\sl Teen}, the ratio of coefficients for {\sl Income}
to {\sl Lowt} drops from -.4070 to -.3968, a relative change of -2.5
percent. The bias due to excluding {\sl Teen} from the standardized model is
thus small.

Generally, the models suggest that if one treats the share of low-weight
births as an exogenous determinant of the rate of infant death, then the
impact of socioeconomic variables may appear to be relatively small.

\section{Appendix C: Data sources}

DEAD: Deaths of infants under 1 year old, excluding fetal deaths. Excludes
deaths of nonresidents of the U.S. Source: Table 123 of the 1995-96 version
of {\it The statistical abstract of the United States}. Cited sources: U.S.
National Center for Health Statistics, {\sl Vital statistics of the United
States}; and unpublished data.

DOCS: The number of physicians per 100,000 civilian population. Active
non-federal physicians as of January 1, 1993. Excludes doctors of
osteopathy, federal employees, and physicians with unknown addresses.
Source: Table 177 of the 1995-96 version of {\it The statistical abstract of
the United States}. Cited source: American Medical Association, {\sl %
Physician characteristics and distribution in the United States}, Chicago,
annual.

DROPOUTS: Of all persons 16 to 19 years old, the percentage of those who
were not in regular school and had neither completed the 12th grade nor
received a general equivalency degree. Source: Table 242 of the 1995-96
version of {\it The statistical abstract of the United States}. Cited
source: U.S. Bureau of the Census, {\sl 1990 census of population}, CPH-L-96.

INCOME: Median money income of households for 1992, in 1993 dollars. The
deflator used is a variant on the consumer price index. Source: Table 730 of
the 1995-96 version of {\it The statistical abstract of the United States}.
Cited source: U.S. Bureau of the Census, {\sl Current population reports},
P60-188. The census bureau cautions that the Current Population Survey is
designed to gather data primarily on the national level and secondarily on
the regional level. State-level data may be less reliable.

LOWT: Of all registered births, the percentage of births that are less than
2,500 grams (5 pounds, 8 ounces). Excludes births to nonresidents of the
United States. Source: Table 95 of the 1995-96 version of {\it The
statistical abstract of the United States}. Cited source: U.S. National
Center for Health Statistics, {\sl Vital statistics of the United States},
annual; and {\sl Monthly vital statistics report}.

MDPHC: Medicaid spending for personal health care per household. Source:
Derived from Table 153 of the 1995-96 version of {\it The statistical
abstract of the United States}.

NOGRADS: Of all persons 25 years old and over, the percentage who were not
high school graduates. Source: Table 242 of the 1995-96 version of {\it The
statistical abstract of the United States}. Cited source: U.S. Bureau of the
Census, {\sl 1990 census of population}, CPH-L-96.

SINGLE: Of all registered births, the percentage of births to unmarried
mothers. Excludes births to nonresidents of the United States. Source: Table
95 of the 1995-96 version of {\it The statistical abstract of the United
States}. Cited source: U.S. National Center for Health Statistics, {\sl %
Vital statistics of the United States}, annual; and {\sl Monthly vital
statistics report}.

TEEN: Of all registered births, the percentage of births to teenage mothers.
Excludes births to nonresidents of the United States. Source: Table 95 of
the 1995-96 version of {\it The statistical abstract of the United States}.
Cited source: U.S. National Center for Health Statistics, {\sl Vital
statistics of the United States}, annual; and {\sl Monthly vital statistics
report}.

\section{Appendix D: Data}

\bigskip Datasets are in Tables 21 through 24 on following pages.

\begin{table}[tbp]
\caption{Dataset, Part 1}
\label{numbers1}
\begin{tabular}{|l||r|r|r|r|}
\hline
{\it State} & {\it Dead} & {\it Income} & {\it Docs} & {\it Dropouts} \\ 
\hline
Alabama & 10.5 & 26581 & 170 & 12.6 \\ 
Alaska & 8.6 & 43053 & 142 & 10.9 \\ 
Arizona & 8.4 & 30237 & 194 & 14.4 \\ 
Arkansas & 10.3 & 24597 & 162 & 11.4 \\ 
California & 7 & 35948 & 240 & 14.2 \\ 
Colorado & 7.6 & 33456 & 222 & 9.8 \\ 
Connecticut & 7.6 & 42064 & 321 & 9 \\ 
Delaware & 8.6 & 36746 & 209 & 10.4 \\ 
Florida & 8.8 & 28168 & 215 & 14.3 \\ 
Georgia & 10.3 & 29659 & 182 & 14.1 \\ 
Hawaii & 6.3 & 43374 & 244 & 7.5 \\ 
Idaho & 8.8 & 28533 & 131 & 10.4 \\ 
Illinois & 10.1 & 32496 & 230 & 10.6 \\ 
Indiana & 9.4 & 29384 & 168 & 11.4 \\ 
Iowa & 8 & 29603 & 159 & 6.6 \\ 
Kansas & 8.7 & 31254 & 185 & 8.7 \\ 
Kentucky & 8.3 & 24188 & 179 & 13.3 \\ 
Louisiana & 9.4 & 26201 & 201 & 12.5 \\ 
Maine & 5.6 & 30504 & 192 & 8.3 \\ 
Maryland & 9.8 & 38317 & 335 & 10.9 \\ 
Massachusetts & 6.5 & 37447 & 361 & 8.5 \\ 
Michigan & 10.2 & 33233 & 195 & 10 \\ 
Minnesota & 7.1 & 31908 & 232 & 6.4 \\ 
Mississippi & 11.9 & 21186 & 130 & 11.8 \\ \hline
\end{tabular}
\end{table}

\bigskip

\bigskip 
\begin{table}[tbp]
\caption{Dataset, Part 2}
\label{numbers2}
\begin{tabular}{|l||r|r|r|r|}
\hline
{\it State} & {\it Dead} & {\it Income} & {\it Docs} & {\it Dropouts} \\ 
\hline
Missouri & 8.5 & 28180 & 207 & 11.4 \\ 
Montana & 7.5 & 27319 & 169 & 8.1 \\ 
Nebraska & 7.4 & 30948 & 189 & 7 \\ 
Nevada & 6.7 & 32863 & 148 & 15.2 \\ 
New Hampshire & 5.6 & 40617 & 211 & 9.4 \\ 
New Jersey & 8.4 & 40168 & 263 & 9.6 \\ 
New Mexico & 7.6 & 26634 & 190 & 11.7 \\ 
New York & 8.8 & 31981 & 334 & 9.9 \\ 
North Carolina & 10 & 28602 & 198 & 12.5 \\ 
North Dakota & 7.8 & 27766 & 188 & 4.6 \\ 
Ohio & 9.4 & 32344 & 207 & 8.9 \\ 
Oklahoma & 8.8 & 26041 & 153 & 10.4 \\ 
Oregon & 7.1 & 32883 & 210 & 11.8 \\ 
Pennsylvania & 9 & 30777 & 254 & 9.1 \\ 
Rhode Island & 7.4 & 31343 & 271 & 11.1 \\ 
South Carolina & 10.4 & 28404 & 173 & 11.7 \\ 
South Dakota & 9.3 & 27045 & 156 & 7.7 \\ 
Tennessee & 9.4 & 25046 & 210 & 13.4 \\ 
Texas & 7.8 & 28790 & 177 & 12.9 \\ 
Utah & 5.9 & 35276 & 187 & 8.7 \\ 
Vermont & 7.2 & 33736 & 259 & 8 \\ 
Virginia & 9.5 & 39341 & 215 & 10 \\ 
Washington & 6.8 & 34915 & 220 & 10.6 \\ 
West Virginia & 9.2 & 20878 & 182 & 10.9 \\ 
Wisconsin & 7.2 & 34305 & 198 & 7.1 \\ 
Wyoming & 8.9 & 31113 & 137 & 6.9 \\ \hline
\end{tabular}
\end{table}

\bigskip

\begin{table}[tbp]
\caption{Dataset, Part 3}
\label{numbers3}
\begin{tabular}{|l|r|r|r|r|r||}
\hline
{\it State} & {\it Nograds} & {\it Lowt} & {\it Teen} & {\it Single} & {\it %
Mdphc} \\ \hline
Alabama & 33.1 & 8.5 & 18.2 & 32.6 & 811.1888 \\ 
Alaska & 13.4 & 4.9 & 10.9 & 27.4 & 1325.243 \\ 
Arizona & 21.3 & 6.4 & 15 & 36.2 & 869.2676 \\ 
Arkansas & 33.7 & 8.2 & 19.4 & 31 & 1095.756 \\ 
California & 23.8 & 5.9 & 11.8 & 34.3 & 1047.038 \\ 
Colorado & 15.6 & 8.5 & 12 & 23.8 & 697.6912 \\ 
Connecticut & 20.8 & 6.9 & 8 & 28.7 & 1627.036 \\ 
Delaware & 22.5 & 7.6 & 12.4 & 32.6 & 950.3817 \\ 
Florida & 25.6 & 7.4 & 13.5 & 34.2 & 870.9438 \\ 
Georgia & 29.1 & 8.5 & 16.2 & 35 & 1087.712 \\ 
Hawaii & 19.9 & 7.2 & 10 & 26.2 & 936.5079 \\ 
Idaho & 20.3 & 5.5 & 13 & 18.3 & 734.1772 \\ 
Illinois & 23.8 & 7.7 & 12.9 & 33.4 & 1071.611 \\ 
Indiana & 24.4 & 6.7 & 14.1 & 29.5 & 1292.229 \\ 
Iowa & 19.9 & 5.7 & 10.2 & 23.5 & 885.6089 \\ 
Kansas & 18.7 & 6.4 & 12.4 & 24.3 & 797.7178 \\ 
Kentucky & 35.4 & 6.8 & 16.5 & 26.3 & 1176.101 \\ 
Louisiana & 31.7 & 9.4 & 18.1 & 40.2 & 1732.12 \\ 
Maine & 21.2 & 5 & 10.2 & 25.3 & 1520 \\ 
Maryland & 21.6 & 8.3 & 9.8 & 30.5 & 1058.306 \\ 
Massachusetts & 20 & 6 & 7.7 & 25.9 & 1630.858 \\ 
Michigan & 23.2 & 7.5 & 13 & 26.8 & 1104.917 \\ 
Minnesota & 17.6 & 5.2 & 8.1 & 23 & 1309.636 \\ 
Mississippi & 35.7 & 9.9 & 21.4 & 42.9 & 1108.395 \\ \hline
\end{tabular}
\end{table}

\bigskip

\bigskip

\begin{table}[tbp]
\caption{Dataset, Part 4}
\label{numbers4}
\begin{tabular}{|l|r|r|r|r|r||}
\hline
{\it State} & {\it Nograds} & {\it Lowt} & {\it Teen} & {\it Single} & {\it %
Mdphc} \\ \hline
Missouri & 26.1 & 7.3 & 14.5 & 31.5 & 823.1768 \\ 
Montana & 19 & 6 & 11.9 & 26.4 & 1003.115 \\ 
Nebraska & 18.2 & 5.6 & 9.9 & 22.6 & 913.6808 \\ 
Nevada & 21.2 & 7.1 & 12.4 & 33.3 & 646.6165 \\ 
New Hampshire & 17.8 & 5.3 & 6.7 & 19.2 & 1064.439 \\ 
New Jersey & 23.3 & 7.2 & 8 & 26.4 & 1358.577 \\ 
New Mexico & 24.9 & 7.2 & 17 & 39.5 & 1000 \\ 
New York & 25.2 & 7.6 & 9 & 34.8 & 2697.115 \\ 
North Carolina & 30 & 8.4 & 15.4 & 31.3 & 970.8444 \\ 
North Dakota & 23.3 & 5.1 & 9.3 & 22.6 & 1111.57 \\ 
Ohio & 24.3 & 7.4 & 13.6 & 31.6 & 1113.631 \\ 
Oklahoma & 25.4 & 6.7 & 16.8 & 28.4 & 820.9076 \\ 
Oregon & 18.5 & 5.2 & 12.4 & 27 & 810.6961 \\ 
Pennsylvania & 25.3 & 7.2 & 10.5 & 31.6 & 1122.176 \\ 
Rhode Island & 28 & 6.3 & 9.8 & 29.6 & 2103.448 \\ 
South Carolina & 31.7 & 9 & 16.6 & 35.5 & 999.2453 \\ 
South Dakota & 22.9 & 5.2 & 11.4 & 26.6 & 1000 \\ 
Tennessee & 32.9 & 8.5 & 16.9 & 32.7 & 1124.099 \\ 
Texas & 27.9 & 7 & 15.9 & 17.5 & 919.3222 \\ 
Utah & 14.9 & 5.6 & 10.5 & 15.1 & 815.3846 \\ 
Vermont & 19.2 & 5.6 & 8.5 & 23.4 & 1059.361 \\ 
Virginia & 24.8 & 7.4 & 11 & 28.3 & 671.7779 \\ 
Washington & 16.2 & 5.3 & 10.6 & 25.3 & 1070.862 \\ 
West Virginia & 34 & 7.2 & 17.2 & 27.7 & 1524.823 \\ 
Wisconsin & 21.4 & 5.9 & 10.2 & 26.1 & 1135.422 \\ 
Wyoming & 17 & 7.3 & 13.2 & 24 & 778.4091 \\ \hline
\end{tabular}
\end{table}

\section{Appendix E: Simulations}

\subsection{Basic equations and tentative results}

For predictions of the short-run change in the rate of infant death due to a
reduction in the dropout rate of one percentage point, the 2SLS equations
were

\[
\Delta {\sl Lowt}=\Delta {\sl Dropouts}\ast \lbrack 6.586-.3978{\sl Dropout}%
-.0114{\sl Docs}]
\]
and

\[
\Delta {\sl Dead}=\Delta {\sl Lowt}\ast \lbrack -1.507+.209{\sl Lowt}+.004%
{\sl Docs}].
\]

The 3SLS equations were

\[
\Delta {\sl Lowt}=\Delta {\sl Dropouts}\ast \lbrack 4.871-.2834{\sl Dropout}%
-.0079{\sl Docs}]
\]
and

\[
\Delta {\sl Dead}=\Delta {\sl Lowt}\ast \lbrack -.6799+.1002{\sl Lowt}+.00399%
{\sl Docs}].
\]

To calculate the long-run rates of low-weight birth and infant death, I
assumed that, in a steady state, the {\sl Nograds} rate would equal the {\sl %
Dropout} rate. This, in turn, presumes a steady-state distribution of ages
in the population. Tables 25 through 28, below, present the values predicted
by the simulations. On average, the simulations predict that the dropout
rate would fall from 10.3 percent to 9.93 percent but that the rate of
infant death would rise from 8.4 to 8.52 per 1,000 live births. The puzzling
results probably stem from the attempt to hold constant the teen share of
all births ({\sl Teen}), a variable that correlates somewhat strongly with
the nongraduate share of all adults, {\sl Nograds} ( $r=.746$).

\begin{table}[tbp]
\caption{Infant death rate in long-run equilibrium, Part 1}
\label{lre}
\begin{tabular}{c|r|r|r||r|r|r|}
\hline
& \multicolumn{3}{|c|}{Early 1990s} & \multicolumn{3}{c|}{Long run (2SLS)}
\\ \hline
\multicolumn{1}{|l|}{\it State} & {\it Drop} & {\it Lowt} & {\it Dead} & 
{\it Drop} & {\it Lowt} & {\it Dead} \\ \hline\hline
\multicolumn{1}{|l|}{Alabama} & 12.6 & 8.5 & 10.5 & 13.20 & 8.07 & 8.91 \\ 
\multicolumn{1}{|l|}{Alaska} & 10.9 & 4.9 & 8.6 & 9.25 & 6.97 & 8.58 \\ 
\multicolumn{1}{|l|}{Arizona} & 14.4 & 6.4 & 8.4 & 12.41 & 7.79 & 8.40 \\ 
\multicolumn{1}{|l|}{Arkansas} & 11.4 & 8.2 & 10.3 & 13.47 & 8.17 & 9.09 \\ 
\multicolumn{1}{|l|}{California} & 14.2 & 5.9 & 7 & 11.28 & 7.35 & 7.57 \\ 
\multicolumn{1}{|l|}{Colorado} & 9.8 & 8.5 & 7.6 & 10.13 & 7.94 & 8.30 \\ 
\multicolumn{1}{|l|}{Connecticut} & 9 & 6.9 & 7.6 & 9.85 & 7.04 & 6.88 \\ 
\multicolumn{1}{|l|}{Delaware} & 10.4 & 7.6 & 8.6 & 11.07 & 7.99 & 8.45 \\ 
\multicolumn{1}{|l|}{Florida} & 14.3 & 7.4 & 8.8 & 11.76 & 7.62 & 8.03 \\ 
\multicolumn{1}{|l|}{Georgia} & 14.1 & 8.5 & 10.3 & 12.69 & 7.98 & 8.70 \\ 
\multicolumn{1}{|l|}{Hawaii} & 7.5 & 7.2 & 6.3 & 9.72 & 7.85 & 8.06 \\ 
\multicolumn{1}{|l|}{Idaho} & 10.4 & 5.5 & 8.8 & 9.13 & 6.62 & 8.64 \\ 
\multicolumn{1}{|l|}{Illinois} & 10.6 & 7.7 & 10.1 & 11.57 & 7.39 & 7.69 \\ 
\multicolumn{1}{|l|}{Indiana} & 11.4 & 6.7 & 9.4 & 11.09 & 8.47 & 9.29 \\ 
\multicolumn{1}{|l|}{Iowa} & 6.6 & 5.7 & 8 & 8.72 & 6.57 & 8.06 \\ 
\multicolumn{1}{|l|}{Kansas} & 8.7 & 6.4 & 8.7 & 10.00 & 7.99 & 8.67 \\ 
\multicolumn{1}{|l|}{Kentucky} & 13.3 & 6.8 & 8.3 & 11.91 & 8.34 & 9.06 \\ 
\multicolumn{1}{|l|}{Louisiana} & 12.5 & 9.4 & 9.4 & 14.22 & 5.78 & 6.89 \\ 
\multicolumn{1}{|l|}{Maine} & 8.3 & 5 & 5.6 & 9.22 & 7.54 & 8.21 \\ 
\multicolumn{1}{|l|}{Maryland} & 10.9 & 8.3 & 9.8 & 10.95 & 5.31 & 4.94 \\ 
\multicolumn{1}{|l|}{Massachusetts} & 8.5 & 6 & 6.5 & 9.82 & 6.35 & 6.01 \\ 
\multicolumn{1}{|l|}{Michigan} & 10 & 7.5 & 10.2 & 10.61 & 8.15 & 8.73 \\ 
\multicolumn{1}{|l|}{Minnesota} & 6.4 & 5.2 & 7.1 & 8.46 & 7.47 & 7.75 \\ 
\multicolumn{1}{|l|}{Mississippi} & 11.8 & 9.9 & 11.9 & 15.23 & 8.17 & 9.53
\\ \hline
\end{tabular}
\end{table}

\bigskip

\begin{table}[tbp]
\caption{Infant death rate in long-run equilibrium, Part 2}
\label{lrequil}
\begin{tabular}{c|r|r|r||r|r|r|}
\hline
& \multicolumn{3}{|c|}{Early 1990s} & \multicolumn{3}{c|}{Long run (2SLS)}
\\ \hline
\multicolumn{1}{|l|}{\it State} & {\it Drop} & {\it Lowt} & {\it Dead} & 
{\it Drop} & {\it Lowt} & {\it Dead} \\ \hline\hline
\multicolumn{1}{|l|}{Missouri} & 11.4 & 7.3 & 8.5 & 11.84 & 7.76 & 8.24 \\ 
\multicolumn{1}{|l|}{Montana} & 8.1 & 6 & 7.5 & 9.84 & 7.84 & 8.73 \\ 
\multicolumn{1}{|l|}{Nebraska} & 7 & 5.6 & 7.4 & 8.79 & 7.15 & 7.93 \\ 
\multicolumn{1}{|l|}{Nevada} & 15.2 & 7.1 & 6.7 & 10.55 & 8.31 & 9.38 \\ 
\multicolumn{1}{|l|}{New Hampshire} & 9.4 & 5.3 & 5.6 & 7.27 & 5.94 & 6.81
\\ 
\multicolumn{1}{|l|}{New Jersey} & 9.6 & 7.2 & 8.4 & 9.06 & 7.85 & 7.99 \\ 
\multicolumn{1}{|l|}{New Mexico} & 11.7 & 7.2 & 7.6 & 13.57 & 7.03 & 7.84 \\ 
\multicolumn{1}{|l|}{New York} & 9.9 & 7.6 & 8.8 & 11.03 & 5.21 & 4.85 \\ 
\multicolumn{1}{|l|}{North Carolina} & 12.5 & 8.4 & 10 & 12.13 & 7.84 & 8.40
\\ 
\multicolumn{1}{|l|}{North Dakota} & 4.6 & 5.1 & 7.8 & 8.52 & 6.86 & 7.75 \\ 
\multicolumn{1}{|l|}{Ohio} & 8.9 & 7.4 & 9.4 & 11.47 & 7.91 & 8.39 \\ 
\multicolumn{1}{|l|}{Oklahoma} & 10.4 & 6.7 & 8.8 & 12.00 & 8.82 & 9.78 \\ 
\multicolumn{1}{|l|}{Oregon} & 11.8 & 5.2 & 7.1 & 10.51 & 8.03 & 8.48 \\ 
\multicolumn{1}{|l|}{Pennsylvania} & 9.1 & 7.2 & 9 & 10.58 & 7.50 & 7.64 \\ 
\multicolumn{1}{|l|}{Rhode Island} & 11.1 & 6.3 & 7.4 & 10.24 & 7.45 & 7.50
\\ 
\multicolumn{1}{|l|}{South Carolina} & 11.7 & 9 & 10.4 & 12.83 & 8.17 & 8.97
\\ 
\multicolumn{1}{|l|}{South Dakota} & 7.7 & 5.2 & 9.3 & 9.52 & 7.46 & 8.63 \\ 
\multicolumn{1}{|l|}{Tennessee} & 13.4 & 8.5 & 9.4 & 13.03 & 6.79 & 7.39 \\ 
\multicolumn{1}{|l|}{Texas} & 12.9 & 7 & 7.8 & 10.74 & 8.30 & 9.04 \\ 
\multicolumn{1}{|l|}{Utah} & 8.7 & 5.6 & 5.9 & 8.27 & 6.56 & 7.57 \\ 
\multicolumn{1}{|l|}{Vermont} & 8 & 5.6 & 7.2 & 8.94 & 7.84 & 7.99 \\ 
\multicolumn{1}{|l|}{Virginia} & 10 & 7.4 & 9.5 & 10.09 & 7.97 & 8.38 \\ 
\multicolumn{1}{|l|}{Washington} & 10.6 & 5.3 & 6.8 & 9.66 & 7.88 & 8.26 \\ 
\multicolumn{1}{|l|}{West Virginia} & 10.9 & 7.2 & 9.2 & 12.38 & 8.12 & 8.83
\\ 
\multicolumn{1}{|l|}{Wisconsin} & 7.1 & 5.9 & 7.2 & 9.36 & 7.68 & 8.26 \\ 
\multicolumn{1}{|l|}{Wyoming} & 6.9 & 7.3 & 8.9 & 9.85 & 7.62 & 9.05 \\ 
\hline
\multicolumn{1}{|l|}{\it State average} & 10.3 & 6.9 & 8.4 & 10.76 & 7.50 & 
8.13 \\ \hline
\end{tabular}
\end{table}

\bigskip

\bigskip \bigskip 
\begin{table}[tbp]
\caption{Infant death rate in long-run equilibrium, Part 3}
\label{lrequil2}
\begin{tabular}{c||r|r|r|}
\hline
& \multicolumn{3}{c|}{Long run (3SLS)} \\ \hline
\multicolumn{1}{|l||}{\it State} & {\it Drop} & {\it Lowt} & {\it Dead} \\ 
\hline\hline
\multicolumn{1}{|l||}{Alabama} & 12.18 & 8.34 & 9.67 \\ 
\multicolumn{1}{|l||}{Alaska} & 8.65 & 5.86 & 8.47 \\ 
\multicolumn{1}{|l||}{Arizona} & 11.64 & 8.01 & 9.20 \\ 
\multicolumn{1}{|l||}{Arkansas} & 12.35 & 8.46 & 9.84 \\ 
\multicolumn{1}{|l||}{California} & 10.58 & 7.56 & 8.49 \\ 
\multicolumn{1}{|l||}{Colorado} & 9.20 & 7.25 & 8.30 \\ 
\multicolumn{1}{|l||}{Connecticut} & 9.10 & 7.28 & 8.08 \\ 
\multicolumn{1}{|l||}{Delaware} & 10.35 & 7.69 & 8.80 \\ 
\multicolumn{1}{|l||}{Florida} & 11.00 & 7.77 & 8.83 \\ 
\multicolumn{1}{|l||}{Georgia} & 11.84 & 8.16 & 9.42 \\ 
\multicolumn{1}{|l||}{Hawaii} & 8.94 & 7.24 & 8.14 \\ 
\multicolumn{1}{|l||}{Idaho} & 8.19 & 4.99 & 8.37 \\ 
\multicolumn{1}{|l||}{Illinois} & 10.80 & 7.64 & 8.62 \\ 
\multicolumn{1}{|l||}{Indiana} & 10.28 & 7.68 & 9.17 \\ 
\multicolumn{1}{|l||}{Iowa} & 8.03 & 5.37 & 7.87 \\ 
\multicolumn{1}{|l||}{Kansas} & 9.13 & 6.94 & 8.41 \\ 
\multicolumn{1}{|l||}{Kentucky} & 10.83 & 7.95 & 9.27 \\ 
\multicolumn{1}{|l||}{Louisiana} & 13.33 & 7.50 & 8.69 \\ 
\multicolumn{1}{|l||}{Maine} & 8.51 & 6.47 & 7.96 \\ 
\multicolumn{1}{|l||}{Maryland} & 10.09 & 6.73 & 7.37 \\ 
\multicolumn{1}{|l||}{Massachusetts} & 8.94 & 7.12 & 8.02 \\ 
\multicolumn{1}{|l||}{Michigan} & 9.74 & 7.43 & 8.68 \\ 
\multicolumn{1}{|l||}{Minnesota} & 7.75 & 6.37 & 7.41 \\ 
\multicolumn{1}{|l||}{Mississippi} & 14.34 & 8.98 & 10.54 \\ \hline
\end{tabular}
\end{table}

\bigskip

\bigskip \bigskip 
\begin{table}[tbp]
\caption{Infant death rate in long-run equilibrium, Part 4}
\label{lrequil4}
\begin{tabular}{c||r|r|r|}
\hline
& \multicolumn{3}{c|}{Long run (3SLS)} \\ \hline
\multicolumn{1}{|l||}{\it State} & {\it Drop} & {\it Lowt} & {\it Dead} \\ 
\hline\hline
\multicolumn{1}{|l||}{Missouri} & 10.97 & 7.83 & 8.94 \\ 
\multicolumn{1}{|l||}{Montana} & 9.09 & 6.74 & 8.48 \\ 
\multicolumn{1}{|l||}{Nebraska} & 8.04 & 5.94 & 7.64 \\ 
\multicolumn{1}{|l||}{Nevada} & 9.99 & 7.38 & 9.21 \\ 
\multicolumn{1}{|l||}{New Hampshire} & 6.61 & 4.69 & 6.44 \\ 
\multicolumn{1}{|l||}{New Jersey} & 8.38 & 7.12 & 7.93 \\ 
\multicolumn{1}{|l||}{New Mexico} & 12.76 & 7.96 & 9.18 \\ 
\multicolumn{1}{|l||}{New York} & 10.33 & 6.58 & 7.17 \\ 
\multicolumn{1}{|l||}{North Carolina} & 11.20 & 7.93 & 9.10 \\ 
\multicolumn{1}{|l||}{North Dakota} & 7.81 & 5.66 & 7.47 \\ 
\multicolumn{1}{|l||}{Ohio} & 10.65 & 7.78 & 8.89 \\ 
\multicolumn{1}{|l||}{Oklahoma} & 11.01 & 8.11 & 9.66 \\ 
\multicolumn{1}{|l||}{Oregon} & 9.66 & 7.43 & 8.55 \\ 
\multicolumn{1}{|l||}{Pennsylvania} & 9.87 & 7.47 & 8.34 \\ 
\multicolumn{1}{|l||}{Rhode Island} & 9.49 & 7.40 & 8.21 \\ 
\multicolumn{1}{|l||}{South Carolina} & 11.98 & 8.28 & 9.60 \\ 
\multicolumn{1}{|l||}{South Dakota} & 8.84 & 6.31 & 8.43 \\ 
\multicolumn{1}{|l||}{Tennessee} & 12.02 & 7.76 & 8.86 \\ 
\multicolumn{1}{|l||}{Texas} & 9.49 & 7.16 & 8.67 \\ 
\multicolumn{1}{|l||}{Utah} & 7.30 & 5.00 & 7.09 \\ 
\multicolumn{1}{|l||}{Vermont} & 8.15 & 6.98 & 7.80 \\ 
\multicolumn{1}{|l||}{Virginia} & 9.35 & 7.29 & 8.39 \\ 
\multicolumn{1}{|l||}{Washington} & 8.86 & 7.05 & 8.12 \\ 
\multicolumn{1}{|l||}{West Virginia} & 11.29 & 8.07 & 9.35 \\ 
\multicolumn{1}{|l||}{Wisconsin} & 8.65 & 6.67 & 8.04 \\ 
\multicolumn{1}{|l||}{Wyoming} & 9.03 & 6.25 & 8.76 \\ \hline
\multicolumn{1}{|l||}{\it State average} & 9.93 & 7.15 & 8.52 \\ \hline
\end{tabular}
\end{table}

\section{Appendix F: Stochastic errors}

\subsection{\protect\bigskip Errors on both sides of the equation}

The treatment here follows that of Pindyck and Rubinfeld (1991). Let the
observed independent variable be

\[
{x_{i}}^{\ast }=x_{i}+v_{i}
\]
where $x_{i}$ is the true value and $v_{i}$ is a measurement error that is
approximately normal with zero mean and constant variance ${\sigma _{v}}^{2}$%
. Similarly, let the observed dependent variable be

\[
{y_{i}}^{\ast }=y_{i}+u_{i}
\]
where $y_{i}$ is the true value and $u_{i}$ is approximately normal with
zero mean and constant variance. Assume that neither measurement error
correlates with $x_{i}$; with each other; or with itself, in a serial sense.
Finally, let the true equation be

\[
y_{i}=\beta x_{i}.
\]

Then the probability limit of the OLS estimator for $\beta $ -- denote that
estimator as $\hat{\beta}$ -- is

\[
plim\text{ }\hat{\beta}=\frac{\beta }{1+\frac{{\sigma ^{2}}_{v}}{Var(x)}}.
\]

In this paper, $x_{i}$ and ${x_{i}}^{\ast }$ are both rates bounded by 0 and
1. These variables may represent, for example, the true and observed values
of the nongraduation rate. If

\[
\frac{\sigma _{v}^{2}}{Var(x)}\approx 1,
\]
then $plim$ $\hat{\beta}\approx {\beta }/{2}$.

\section{References}

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\end{document}