THE INFANT MORTALITY FERTILITY RELATION
Dan Thomas Rees
B.S., Colorado State University, 1986
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Arts
This thesis for the Master of Arts
Dan Thomas Rees
has been approved for the
Daniel I. Rees
Rees, Dan Thomas (M.A., Economics)
The Infant Mortality Fertility Relation
Thesis directed by Assistant Professor H. Naci Mocan
This paper examines the relation between infant mortality
and fertility rates in the United States in order to test
three conflicting economic theories. These theories are
the Demographic Transition, Malthus1 Population Theory,
and the Modern Economic Theory of Population. The
methodology employed is to conduct Granger-causality
tests between the two series after using the Kalman
filter to arrive at stationarity. It is found that the
bivariate model shows no causal linkages and it is
concluded that the model- is under specified and should be
expanded in future studies.
This abstract accurately represents the content of the
candidate's thesis. I recommend its publication.
2. BACKGROUND AND METHODOLOGY.....................5
Demographic Transition Theory.................5
Malthus' Population Theory....................7
Modern Economic Theory of Population..........7
Unit Root Testing............................13
3. EMPIRICAL IMPLEMENTATION......................18
4. RESULTS..................................... 22
If the intensity of the current political debate is
any indication, there is no national issue more on the
minds of the American public than the fitness of our
current health care delivery system. In defense of their
position, the proponents of radical health care overhaul
claim that the U.S. system has performed poorly when
compared with other Western nations. For those on this
side of the argument there is no statistic more
compelling than the current international ranking of the
U.S. on infant mortality.
Although the overall U.S. infant mortality rate,
defined as the number of infant deaths under one year of
age per 1,000 live births, decreased from 11.9 in 1981 to
8.9 in 1991 (a drop of 25%), it is still suspiciously
high when compared with other Western nations. An
official assessment of the situation is found in the U.S.
Department of Health.and Human Services' Health United
States 1992 (U.S. National Center for Health Statistics).
This report shows that in 1989 the U.S. rate was higher
than twenty-three other countries or territories. In
this year the U.S. rate was nearly 30% higher than that
of West Germany, 60% higher than that of Finland, and
more than twice the level experienced in Japan. Among
the countries that the U.S. trails are all of the other
six "Group of Seven" nations whose average rate is 7.3
compared to 9.8 for the U.S. The next lowest ranking for
a "Group of Seven" country is Italy with an infant
mortality rate of 8.8. Perhaps even more disturbing than
these static comparisons is the direction of the overall
trend for the U.S. in comparison with the rest of the
world. Despite continued decline in yearly infant
mortality rates, the U.S. has lost three places in its
international ranking over the five year period from
How can these alarming figures be explained?
Economists and other social scientists have been asking
this question with increasing frequency in recent years.
In an attempt to better understand this problem increased
attention has been paid to the relationships that exist
between infant mortality and fertility rates. In looking
at the data it is readily seen that both fertility rates
and infant mortality rates in the U.S. have been on a
steady downward path over the past several years.
1 In 1984 the U.S. ranked twenty-first in the world
on infant mortality (U.S. National Center for Health
Figure 1.1 shows the U.S. infant mortality rate from
1956-1985. The figure shows that the U.S. rate has been
steadily declining over the years following a near linear
pattern. The average annual rate of change in infant
mortality rates has actually slowed in recent years,
however. During the period from 1965 to 1981 the average
rate of change was 4.5% per year. In contrast, the
average rate from 1981 to 1988 slowed to 2.5% per year.
Figure 1.2 shows the U.S. fertility rate from 1956-
1985. As with infant mortality, fertility in the U.S.
has been on a downward path. Fertility rates are subject
to much more variation than are infant mortality rates
however. Interestingly, during the 1980's the fertility
rate appears to leave the prior trend and level off. In
fact the fertility level in 1988. (67.2) was virtually
unchanged from that of 1982 (67.3).
By studying the interaction between these two
series, economists hope to be able to better understand
the dynamics affecting the movements of both. This paper
will investigate the dynamic relationship between infant
mortality and fertility using a structural time series
model in hopes of shedding more light on the underlying
factors influencing infant mortality rates in the U.S.
Figure 1.1 U.S. Infant Mortality Rate
Figure 1.2 U.S. Fertility Rate
BACKGROUND AND METHODOLOGY
To investigate the relationships between fertility and
infant mortality rates in the U.S., one must first
examine the three main theories that have been advanced
to explain this relationship. The first of these
theories is known as the theory of demographic
Demographic Transition Theory
Demographic transition theory (see K. Davis
1945) holds that there is a lagged relationship that
exists between infant mortality and fertility. This
theory states that as economies experience a transition
towards more industrialization and urbanization, they
experience a similar demographic transition. That is, as
an economy becomes more industrialized the demographic
characteristics of its population begin to change. Chief
among these changes is a reduction in fertility. This
reduction is a by-product of the improvements in
technology and education that development brings. A more
advanced economy makes medical care more readily
available and is characterized by better nutritional
practices, higher literacy, and less environmental
pollution. These changes have direct negative effects on
the infant mortality experienced in a society. The
overall environment for newborns becomes cleaner and
healthier. It is this reduction in infant mortality
according to this theory that (after an adjustment
period) drives a reduction in the fertility of the
This fertility response is based on the idea that
within households there is an ideal family size. When,
nations experience high infant-death rates they will seek
to keep their fertility rate high enough to compensate in
order to maintain the ideal family size. With fewer
infant deaths such societies will react by reducing their
fertility (after an adjustment period) in order to
continue to maintain their goal of an ideal family size.
Therefore this theory hold that a causal relationship
exists, primarily in developing societies, from infant
mortality to fertility. Reher and Iriso-Napal (1989)
found evidence to support this theory in a study of
historical Spain while Pampel and Pillai (1986) found
similar results in a study of 18 developed nations.
Malthus* Population Theory
Thomas Malthus advanced another theory to explain
the relationship between infant mortality and fertility
in societies (see Malthus 1965). Unlike the demographic
transition theory, this theory of population holds that
causality runs from fertility to infant mortality subject
to a time lag. According to this view, as wages increase
in a society, fertility will also increase as the
economic conditions of families improve and children
become more affordable. This increase, however, leads to
a rise in high risk births, such as those to very young
mothers or those late in their fertility cycle. The
result of this increase in high risk births is an
increase in infant deaths. Here the net result of
increases in wealth are higher fertility and infant
mortality rates. This is due to an elastic demand for
children. Many empirical studies have demonstrated this
type of positive correlation between income and
fertility (Lee and Gan 1989; Koenig, Phillips, Campbell
and D'Souza 1990; Hobcraft, McDonald, and Rutstein 1985).
Modern Economic Theory of Population
Lastly, there is the modern economic theory of
population (see Becker, G. and Lewis, G. 1973; Becker G.
and Barro, R. 1988). This theory differs from the two
discussed above in that it does not specify one-way
causality but instead it hold that influence runs in both
directions. This feedback occurs between infant
mortality and fertility because they are seen as outputs
of a household production decision. That is, experienced
fertility and infant mortality are the result of resource
allocation decisions of the family unit. Parents decide
how much of their available resources they will put
towards medical care, nutrition, and other health inputs.
The number of births experienced by a family and the
health of those babies is then jointly determined by this
production decision process. Feedback is present here
because past birth outcomes will influence future
resource decisions and outcomes. For example, if a
family experiences the death of a child they are likely
to alter their fertility behavior. If the family is
seeking to achieve a certain family size they will
increase their fertility behavior. Likewise infant
mortality can be affected by short birth intervals which
lower the probability of child survival. This theory
suggests that infant mortality and fertility should be
modeled jointly, treating each as endogenous within a
simultaneous system framework. Blau (1986)found results
that support this theory using data for Nicaragua.
Similar findings were also achieved by Holtz and Miller
(1988) and Winegarden (1984).
To summarize, these three theories each suggest
different causality structures between fertility and
infant mortality. Since this is the case it is desirable
to test the available data using time series techniques
to determine which theory, if any, is supported. To
date, few studies have sought to test these theories
within a dynamic framework.
This study improves upon prior research by
accounting for the dynamic mechanism between infant
mortality and fertility and testing causality in both
directions using aggregate data. For the most part,
those studies that have dealt with infant mortality and
fertility have concentrated on the effects in one
direction only. Much of the work thus far has been
focused on the effects of infant deaths on fertility
behavior (Ben-Porath 1976; Schultz 1978; Rosenzweig and
Schultz 1983; Hashimoto and Hongladarom 1980; Sah 1991).
This type of approach imposes unwarranted restrictions on
the model by treating infant mortality as predetermined.
If feedback mechanisms are indeed in place then this type
of analysis will be plagued by simultaneous equation
To avoid this type of bias in the estimation, a
simultaneous equations system which estimates all the
endogenous variables together should be used. When
dealing with a bivariate framework however, this type of
system cannot be identified without adding additional
exogenous variables to the system. Unfortunately,
without very strong evidence to support the addition of
such variables (which is rarely available) this type of
approach can lead to mis-specification. Even if suitable
exogenous variables were available this approach still
suffers from its lack of a dynamic framework. With the
simultaneous equation approach the variables are set up
to affect each other simultaneously and no allowance is
made for variables affecting each other through a time
lag. This suggests the need for a model that will take
into consideration.the dynamic interdependence of the
variables in the system.
The estimation method that will be employed in this
paper to deal with the problems cited above is that of
vector autoregressions (VAR). The VAR methodology is due
to Sims (1980) and as its name implies it consists of
regressing each variable in the system on each of the
other variables (including itself) subject to a lag
structure, as well as on relevant exogenous variables.
This amounts to estimating a system of dynamic reduced
form equations. This method differs from the traditional
econometric approach of specifying the structural
relationships in a model beforehand according to economic
theory. The structural approach necessitates that the
researcher specify a priori the relationships between
variables. In many circumstances however, there is a
degree of uncertainty as to what the underlying structure
should be. Therefore, estimation of a structural model
often imposes restrictions on the data which are
unwarranted. The advantage of the VAR method is that it
is a way to allow the data to determine the structure
because no a priori restrictions are imposed on the model
by the researcher. In this way all of the' feedback
channels between variables is accounted for. This
specification also will fully account for the lagged
influence of one variable upon another because the lag
structure is built into the model. Also since all of the
right hand side variables are lagged (or exogenous) they
are predetermined and can therefore be estimated
efficiently using ordinary least squares (OLS).
The VAR framework also facilitates Granger causality
testing (Granger 1969). Causality in the Granger sense
is defined as follows: A variable X Granger-causes Y if
past values of X can be used to more accurately predict Y
than can be achieved using only past values of Y. This
concept can be illustrated by using the following simple
where X and Y are linear time series and ex and e2 are
independent, identically distributed disturbances (iid).
Here an F-test is.used to test the following hypotheses:
bj=0, j=l...k and dj=0, j=l...k. If the former is
rejected while the latter is not, the conclusion is that
the variable Y Granger causes X. Likewise, X Granger
causes Y when the hypothesis for the coefficient dj is
rejected and the hypothesis for the coefficient bj is
not. If both hypotheses are rejected then the model has
feedback occurring. The last possibility is when both
hypotheses are maintained, which indicates independence
among variables. By performing these tests one can
ascertain if the data being studied support a particular
theoretical causality structure. These type of tests
were implemented by Chowdhury (1988) for thirty-five
developing nations and by Yamada (1985) for developed
countries including the U.S. These papers did not
account for the possibility of a unit root, however.
Unit Root Testing
When working with economic time series one must be
mindful of their stochastic properties. In order to
apply time series modelling techniques such as causality
testing to a given series it must be rendered stationary.
Stationarity means that a series is invariant with
respect to time, i.e. it is in an equilibrium state with
respect its first two moments-its mean and .variance. In
this regard it is important to distinguish between series
that are generated by a trend stationary process (TSP)
and those generated by a difference stationary process
(DSP) Roughly speaking,- a TSP fluctuates around a
deterministic trend. Any shock to the system generating
this type of series will only temporarily cause the
series to leave the trend path. This type of series must
therefore be de-trended in order to make it stationary.
De-trending entails regressing the series under
consideration on a time (linear or quadratic) trend and
then using the residuals in the subsequent analysis.
In contrast, a DSP follows a stochastic path and
shocks will serve to permanently shift the series onto a
new long term path. Put differently, a TSP has a long
memory and will continue on a new shock-induced path
indefinitely. These series are said to contain a unit
root and accordingly must be differenced in order to
become stationary. This type of series is said to be
integrated. The order of integration is the number of
times differencing must be carried out to achieve
stationarity. To distinguish between these two types of
time series, a unit root test was developed by Dickey and
Fuller (Dickey and Fuller 1981). Currently, however, a
disagreement exists among researchers as to whether these
tests are valid, casting doubt on the appropriateness of
their use. To examine the unit root hypothesis consider
the time series yt which is generated by the following
where et is a covariance stationary process with mean
zero, t is a time trend, and a, b, and c are the
parameters. If c
represents an asymptotically stationary AR(1) process
with a linear time trend. If c=l, the model is a random
walk around a linear trend.
Substituting (2) into (1) and rearranging yields the
z0= [a(l-c) +cb]
Equation (2.6) is said to have a unit root if c=l. The
emphasis on unit roots has grown enormously during the
past decade after Dickey and Fuller suggested testing the
unit root hypothesis. Since then, researchers modified
and proposed alternative versions of the original Dickey-
Fuller test (Said, S.E. and Dickey, A.D. 1984; Phillips
P.C.B. 1987; Phillips P.C.B. and Perron,P. 1988).
Given the current controversy about the use of the
Dickey-Fuller test (and its variants), this paper uses a
structural time-series modeling approach to investigate
the trend behavior of infant mortality and fertility.
This method allows us to circumvent unit root tests
during the analysis.
It is hypothesized that the two series in the
following model can be represented as:
yt = |it + et (2.8)
where yt is the value of the series at time t, and pt and
t are the trend and irregular components, respectively.
Within this framework, one can specify a locally linear
trend where the level and the slope of the series are
governed by random walks as follows:
t = Pt-i + S
where i^NID (0, on2) ^t~NID (0, oE2) and E [r|t Â£t]=0.
Changes in iit generate shifts in the level of the trend,
and variations in Â£t produce slope changes. Thus,
equation (9) depicts a flexible formulation of the trend
in yt, which is equivalent to an ARIMA(0,2,1) process.
If oÂ£2= 0, the trend reduces to a random walk with a drift;
i.e. yt is stationary in first differences ( a DSP). If
o^2=0, but o52>0, the trend is still integrated of order
two as the original case. If oI|2=oE2= 0, the model
collapses to a standard regression model with a
deterministic trend (DSP); i.e. pt=p0+pt. The novelty of
estimating this model is that it starts off with a
general trend formulation and investigates whether the
underlying trend dynamics can be reduced to a DSP, or a
TSP. Thus, testing for a unit root using standard
procedures where the null hypothesis is formulated as yt
being a DSP is a special case of the above.
The model employed in this paper will seek to
determine if the data for the U.S. infant mortality and
fertility rates show any dynamic causal relationship,
while accounting for feedback mechanisms. This is
achieved by estimating a bivariate VAR system. Granger
causality tests are then performed to determine whether
causal relationships exist, and if so, to determine the
The data used for this study are monthly U.S. time
series2 on infant mortality and fertility rates from
1956-19853. Both series are taken from the National
2 The data used in the model estimation is converted
from monthly to quarterly for ease of calculation of the
3 The sample period was dependent upon the
availability of monthly infant mortality statistics which
became unavailable beginning in 1986 (See the U.S.
National Center for Health Statistics 1986. Monthly Vital
Statistics Report 35:1).
Center of Health Statistics Monthly Vital Statistics
Report (U.S. National Center for Health Statistics 1957
1986). Fertility is defined as the number of births per
1,000 women between the ages of fifteen and forty-four.
Infant mortality is defined as the number of infant
deaths under one year per 1,000 births.
For both fertility and infant mortality the Kalman
filter was used to compute the underlying trends in the
series. Figure 3.1 depicts the Kalman trend for infant
mortality. Figure 3.2 contains, the Kalman trend for
Figure 3.1 Infant Mortality and Kalman Trend
rigure 3.2 Fertility and Kalman Trend
The bivariate model estimated takes the following form:
*t-i e+E TufvE (3 1 >
i=l j=l Jc=i
Fe'2+E P2i V2^C-,+E *.*% ( 3 2 >
2=1 J=1 Jc=l
where It is the trend deviation for infant mortality rate
at time t, Ft is the trend deviation for fertility rate
at time t4. Qkrepresents three quarterly dummies used to
4 The natural logarithm of both the raw series and
the trend series were taken before deviations were
control seasonality and elt and e2t are white noise error
The lag lengths of each equation were tested using
Sims' likelihood ratio test (Sims 1980):
X2 = (N-C) (In detSu In detSr) (3.3)
where N = the number of restrictions, C = a correction
factor equal to the number of parameters in the
unrestricted equation. Lag lengths of eight, six, and
four quarters were tested. The.null hypothesis of no
difference between eight and six quarters was accepted
and therefore a lag of six quarters is used in the
estimations. The Chi-square statistic was 11.95 with 32
degrees of freedom.
In addition to the above model two other
specifications were run. The first was the same as in
Yamada (1985) with the exception that all races are
analyzed here5. This specification uses logs of the
fertility and infant mortality series and includes a
linear time trend in the equations to control for trend
movements. In addition to this a second specification
was run using a quadratic time trend instead of the
5 Yamada's analysis of the U.S. is restricted to
The results from the causality tests for all three models
are presented in table 4.1.
Table 4.1 F-statistics for Granger-Causality Tests.
Causality Model 1 Model 2 Model 3
Test (with linear (with (with Kalman
trend) quadratic trend
IM -> F6 1.93 1.61 1.44
F -> IM 1.65 2.19*7 1.75
The results of the tests show that with the
exception of model 2, infant mortality and fertility do
not exhibit significant influence over each other for the
period studied. This result differs sharply from those
found in Yamada (1985) where annual data was used for a
6 -> designates unidirectional causality
7 Significant at the 5% level.
different sample period (1923-1977). His findings were
that fertility Granger-causes infant mortality at the 1%
level and that infant mortality Granger-causes fertility
at the 10% level. This finding supports the modern
economic theory of population. The differing result here
could be partly due to a confounding effect of pooling
non-white data with that of the white population.
For comparison purposes a smaller sample that ends
in 1977 (as in Yamada) was used to re-estimate the
equations. The results are shown in table 4.2. As can
be seen the result was that Granger-causality from
fertility to infant mortality was present but causality
running the other direction was not detected. This
suggests that the dynamics behind this relationship could
have begun to change during the decade of the 1980's. As
previously noted both series underwent significant
changes during the period of the 1980's compared with
Table 4.2 F-Statistics for Granger-Causality Tests.
Causality Model 1 Model 2 Model 3
Test (with linear (with (with Kalman
trend) quadratic trend
IM -> F 0.65 0.68 1.44
F -> IM 2.08 2.52*8 3.64**9
0 Significant at the 5% level.
9 Significant at the 1% level.
This paper set out to test the relationship between
infant mortality and fertility in the U.S.using time
series data. The method used here improves upon previous
work in this area by employing the Kalman filter to
construct local linear trends for each series. By doing
so the current controversy over the proper interpretation
of unit root tests is avoided. This type of modeling
allows the data to dictate a trend path which can then be
eliminated by using trend deviations in the model
In addition, VAR modeling was used to allow for
dynamic interrelationships between the data and to avoid
placing unnecessary restrictions on the model. This
method fully accounts for.feedback between variables and
for lagged causality.
No statistical causal relationship was found to
exist between fertility and infant mortality for the U.S.
during the period 1956-1985. Using a smaller sample that
ends in the late 1970's causality from fertility to
infant mortality was detected. This would seem to
indicate that in developed nations where the demographic
transition has been completed, infant mortality rates do
not effect fertility. Perhaps this is due to these
societies having passed some type of threshold for
institutional change. In contrast, it would appear that
household fertility decisions play a major role in
driving birth outcomes.
The lack of significant results here seems to
suggest that the current model is under specified and,
therefore, other variables should be included in the
model. Possible candidates for introduction into the
system are female employment or labor force participation
measures, divorce rates, illegitimacy ratios and income
measures. Another possible extension to this research is
to obtain and test race-specific data in order to more
accurately model the movements of these series among
different demographic segments of the population.
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