ï»¿ WPS6343
Policy Research Working Paper 6343
How Subjective Beliefs about HIV Infection
Affect Life-Cycle Fertility
Evidence from Rural Malawi
Gil Shapira
The World Bank
Development Research Group
Human Development and Public Services Team
January 2013
Policy Research Working Paper 6343
Abstract
This paper studies the effect of subjective beliefs about Model parameters are estimated by maximum likelihood
HIV infection on fertility decisions in a context of high with longitudinal data from the Malawi Diffusion and
HIV prevalence and simulates the impact of different Ideational Change Project, which contain family rosters,
policy interventions, such as HIV testing programs and information on HIV testing, and measures of subjective
prevention of mother-to-child transmission, on fertility beliefs about own HIV status. The model successfully fits
and child mortality. It develops a model of womenâ€™s the fertility patterns in the data, as well as the distribution
life-cycle, in which women make sequential fertility of reported beliefs about own HIV status. The analysis
decisions. Expectations about the life horizon and uses the model to assess the effect of HIV on fertility by
child survival depend on womenâ€™s perceived exposure simulating behavior in an environment without HIV. The
to HIV infection, which is allowed to differ from the results show that the presence of HIV reduces the average
actual exposure. In the model, women form beliefs number of births a woman has during her life-cycle by
about their HIV status and about their own and their 0.15. The paper also finds that HIV testing can reduce
childrenâ€™s survival in future periods. Women update the fertility of infected women, leading to a reduction of
their beliefs with survival to each additional period as child mortality and orphan-hood.
well as when their HIV status is revealed by an HIV test.
This paper is a product of the Human Development and Public Services Team, Development Research Group. It is part
of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy
discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org.
The author may be contacted at gshapira@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
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names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
How Subjective Beliefs about HIV Infection Aect
Life-Cycle Fertility: Evidence from Rural Malawiâˆ—
â€
Gil Shapira
January 30, 2013
JEL classication codes: O15, I10, J13
Keywords: fertility, HIV/AIDS, subjective beliefs
Sector board: Health, Nutrition and Population (HNP)
âˆ—
I thank my dissertation committee members Jere Behrman, Ã?ureo de Paula, Kenneth Wolpin, and
especially my advisor Petra Todd for their dedication and support. I am also grateful to Clement Joubert,
Nirav Mehta, Shalini Roy, and participants of seminars at the University of Pennsylvania, International Food
Policy Research Institute, The World Bank, Tel Aviv University, Haifa University, Hebrew University, Ben
Gurion University, Interdisciplinary Center Herzliya, and University of Washington for helpful comments. I
gratefully acknowledge nancial support from the William and Flora Hewlett Foundation.
The ndings, interpretations and conclusions expressed in this paper are entirely those of the author, and
do not necessarily represent the views of the World Bank, its Executive Directors, or the governments of the
countries they represent.
â€
Development Research Group, The World Bank. Email: gshapira@worldbank.org
1
1 Introduction
Both fertility and HIV prevalence rates in Malawi are among the highest in the world, with
1
the total fertility rate at 5.7 births per woman and the HIV prevalence rate at 10.6 percent.
Malawian women make fertility decisions in an environment characterized by high adult
and child mortality, exacerbated by mother-to-child HIV transmission. Out of a population
of about 15 million, it is estimated that 68,000 die annually from AIDS and that 560,000
2
children under the age of 17 have lost at least one parent to the disease.
There are many policy interventions aimed at reducing HIV in Malawi and other Sub-
Saharan African countries. These include HIV testing programs, information campaigns,
and antiviral distribution programs. Evaluating the eects of such policies on outcomes such
as number of births, child mortality, and orphan-hood requires an understanding of how
women's fertility decisions are aected by the presence of HIV.
An important aspect of the environment in Malawi is that women are typically uncertain
regarding their own HIV status. An infected person can live for many years with no symp-
toms, and testing was not widely available until relatively recently. The median survival
3
time after infection, without treatment, is about 10.4 years. During most of this time, an
4
infected person is in a clinical latency stage and experiences few or no symptoms.
In addition to being uncertain about own HIV status, women often express beliefs about
HIV risk that dier substantially from actual risk. Studies using the Malawi Diusion and
Ideational Change Project data show that individuals in rural Malawi tend to overestimate
both the probability of being HIV-infected (Anglewicz and Kohler, 2009) and the HIV preva-
lence in their community (Anglewicz, 2007). Anglewicz and Kohler (2009) attribute these
high risk assessments to overestimated probabilities of transmission. More than 95 percent
of respondents believe that transmission from a single instance of unprotected intercourse
1 Malawi Demographic and Health Surveys (2010)
2 UNSAID/WHO/UNICEF Epidemiological Fact Sheets (2008)
3 Todd et al. (2007). Estimate for infected adults in Eastern and Southern Africa
4 Morgan et al. (2002), for example, nd median time from infection to AIDS to be 9.4 years and median
time from AIDS to death to be 9.2 months in rural Uganda.
2
with an infected person is highly likely or certain; however, studies estimate that it can be as
5
low as 1 per 1,000 encounters in the absence of an increased viral load (Gray et al., 2001).
Women's perceptions of HIV risk and of their own HIV status aect beliefs about their
own and their children's life expectancy, which in turn may inuence life-cycle fertility
choices. In this paper, I study the determinants of women's reproductive decisions in Malawi,
taking into account uncertainty about HIV status and dierences between perceived and ac-
tual HIV infection risk. I investigate how HIV aects fertility and simulate the impact of
dierent policy interventions, such as HIV testing programs and prevention of mother-to-
child transmission, on fertility and child mortality.
To this end, I develop a dynamic discrete-choice life-cycle fertility model in which ex-
pectations about the life horizon and child survival depend on a perceived infection hazard.
A woman makes annual pregnancy decisions from the time of marriage until she becomes
infecund. She maximizes utility, which depends on her number of children, household con-
sumption, and pregnancies, subject to a per-period budget constraint. The woman faces
uncertainty regarding future income, HIV status, and the survival of herself and her children
in future periods.
A woman's perceived infection hazard is allowed to dier from her actual infection hazard
to reect the misperceptions about HIV risk observed empirically. The perceived hazard rate
for each period is a function of a woman's characteristics, such as her age, region of residence,
marital status, and schooling level. To account for unobservable factors, the hazard rate also
6
incorporates heterogeneity in the form of a discrete number of unobserved types.
Given that HIV is initially asymptomatic, the model assumes that a woman does not
5 The viral load is high in the few weeks following infection and increases again as an infected person
develops AIDS. Infectivity increases with viral load as well as with other conditions, such as the existence of
other STDs. Powers et al. (2008) review the studies estimating HIV infectivity and discuss dierent factors
that increase infectivity. Note however that even a low transmission rate such as 0.001 can translate into
a nontrivial probability of infection during a year of partnership. Gray et al. (2001) nd average frequency
of intercourse to be about 106 acts per year, implying about a 10 percent chance of transmission given this
transmission rate.
6 The probability of being a certain type is a function of the woman's characteristics and follows a multi-
nomial logit specication.
3
observe realizations of the infection process and therefore does not know her HIV status.
Assuming she knows the mortality process associated with HIV infection, however, survival
during each additional period gives her information about her status. Specically, she reduces
her subjective probabilities of having become infected in each of the past periods based on
the fact that she is still alive. According to these probabilities and given the mortality
process, she updates her survival expectations. HIV infection also increases child mortality
probabilities through mother-to-child transmission. In the model, the woman also updates
expectations about the survival of each of her children depending on the probability assigned
to her having been infected at the time of birth.
The dynamic fertility model is estimated using the Malawi Diusion and Ideational
Change Project (MDICP) dataset, a rich longitudinal dataset collected in rural areas of
three dierent districts of the country. The data contain extensive information on more
than 4,000 individuals at the individual and household level. The three sampled regions
vary signicantly in several aspects that are potentially relevant for the analysis, such as
HIV prevalence rates, polygamy rates, and schooling levels. A unique feature of the MDICP
data is that they include measures of subjective expectations regarding a range of outcomes,
including the likelihood respondents assigned to being HIV-infected at the time of the inter-
view. The expectations data were collected using a novel bean-counting method, developed
by Delavande and Kohler (2009), which is appropriate for populations with low levels of nu-
meracy. Delavande and Kohler (2009) nd that the reported subjective expectations follow
basic properties of probabilities and that the assessments of HIV-infection vary meaningfully
with observable characteristics associated with dierent levels of HIV prevalence. I use a
subsample of 1006 married women who were interviewed at least once during the 2006 and
2008 rounds, when family rosters and subjective expectations were collected.
Since 2004, each round of data collection included HIV testing of respondents and the
7
prevalence rate was found to be about seven percent. Although individuals who received
7 HIV prevalence is lower in rural areas than in urban areas. The prevalence in the MDICP sample is
lower than the rural 2004 DHS prevalence because the sample does not include peri-urban areas such as
4
positive test results assign signicantly higher likelihood to being infected two years later
relative to individuals who received negative test results, some individuals who tested positive
later assign a probability of less than one to being infected. I therefore assume that a woman
assigns a probability to the accuracy of the test. Given the test result, the woman updates
the probabilities assigned to infection and the corresponding probabilities of survival.
I structurally estimate the model parameters using maximum likelihood and then use the
estimated model to perform several counterfactual simulations. First, I simulate life-cycle
fertility in an environment with no HIV exposure. The results indicate that the presence of
HIV has an average negative eect on fertility. Overall, women in the no-HIV environment
have on average 0.15 more births over their life-cycle. However, there is heterogeneity in the
eect of the presence of HIV on fertility, with some women decreasing the number of births
and fewer women increasing it. I also simulate the eects of prevention of mother-to-child
transmission and HIV testing programs on fertility outcomes. Although these programs are
not necessarily intended to inuence fertility, they have the potential to aect reproductive
choices by altering beliefs about HIV status, life expectancy, and child survival. Such policy
interventions are implemented in Malawi and are continuously expanding in scope. I nd
these programs to have negligible eects on the overall number of births. However, elimi-
nation of mother-to-child transmission will reduce fertility by infected women who respond
to child mortality by additional pregnancies either because they have high marginal utility
of an additional child or because they assign low likelihood to being infected. I also nd
that HIV testing can reduce fertility by HIV-positive women, leading to a reduction in child
mortality and orphan hood. However, the eect of testing is limited by the partial updating
of beliefs.
trading centers (Obare et al., 2009).
5
1.1 Related Literature
Several studies analyzed the response of fertility in Sub-Saharan Africa to the HIV/AIDS
epidemic and reached mixed conclusions. Young (2005) studies the eect of the HIV/AIDS
epidemic on the welfare of future African generations. He concludes that thanks to a negative
8
eect on fertility, the epidemic, on net, will increase future per capita consumption. He
attributes the reduction in fertility to an unwillingness to engage in unprotected sexual
activity and increasing labor opportunities for women because of scarcity of labor. He shows
empirically a negative relationship between fertility and HIV prevalence using retrospective
fertility histories and seroprevalence in antenatal clinics in South Africa. Using time series
cross-country data on fertility and HIV prevalence rates, Young (2007) nds a similar eect.
Kalemli-Ozcan (2006), using similar types of data, shows that regressing total fertility rate on
HIV/AIDS prevalence can yield both positive and negative eects with dierent estimation
strategies and dierent measures of HIV prevalence.
Later studies use the cross-country data from the latest rounds of the Demographic
and Health Surveys (DHS). These surveys are nationally representative and contain results
of HIV testing of respondents. Fortson (2009) and Juhn et al. (2008) nd lower fertility
rates of HIV-infected women than of HIV-uninfected women; however, they nd no eect
of local prevalence rates on the fertility of uninfected women and an overall insignicant
aggregate eect of HIV on fertility. Fink and Linnemayr (2009), linking historical data from
World Fertility Surveys (WFS) with the DHS data, argue that while HIV does not have
a signicant eect on aggregate fertility levels, it aects dierently women depending on
their educational attainment. They nd that, in the presence of HIV, more educated women
reduce fertility more than uneducated women. Analyzing the Malawi DHS surveys from 2000
and 2004, together with HIV rates obtained from antenatal clinics, Durevall and Lindskog
(2011) conclude that although the HIV/AIDS epidemic has small impact on the number of
8 Accordingto Young (2005), the positive eect of reduces fertility dominates a negative eect of reduced
human capital accumulation by orphans.
6
births a woman experiences it aects timing of fertility. Women in districts with higher HIV
prevalence are more likely to give birth at younger ages and are less likely to do so when
they are over 29 years.
My analysis diers from these studies both in the empirical approach and in the type
of data used in the analysis. The data on individuals' subjective beliefs allow me to ex-
ploit heterogeneity within communities, which the use of data on local HIV prevalence only
does not allow. The structural estimation of a model enables me to perform counterfactual
simulations to assess fertility patterns in dierent environments.
9
Many economists have studied the determinants of fertility in dierent environments.
My analysis is most closely related to studies that model fertility decision-making as a
sequential process (Heckman and Willis (1974)) and studies of fertility in environments with
non-negligible infant and child mortality risk (Wolpin, 1984; Sah, 1991; Mira, 2007). Wolpin
(1984) presents an estimable dynamic discrete-choice fertility model in an environment where
infant survival is uncertain and uses the model to study the response of fertility choices to
experienced infant mortality. Mira (2007) uses a similar modeling framework and extends it
by introducing heterogeneity in infant mortality risk across mothers. Parents in his model
learn about a family-specic component of infant mortality risk throughout their life-cycle.
Fertility choices are inuenced by how the parents adapt to the information received from
infant survival and mortality.
2 Model
2.1 General Setup
I develop a dynamic discrete-choice life-cycle model of woman's fertility decisions in an
environment of exposure to HIV infection. Women maximize subjective expected utility by
making sequential binary fertility choices in a framework similar to that of Wolpin (1984) and
9 Joseph Hotz et al. (1997), Schultz (1997), and Wolpin (1997) survey the literature on fertility.
7
Mira (2007). Specically, a woman makes annual decisions of whether to become pregnant
beginning at the age of her marriage and ending when she becomes infecund at a xed age
F (assumed to be 45). A woman gives birth in the period following the one in which she
became pregnant. If never infected, a woman survives with certainty to age T (assumed
to be 60). An infected woman might die prior to reaching the terminal model period. HIV
infection of a mother at time of birth also increases mortality probabilities of children of ages
zero to three. Given that HIV infection is asymptomatic for the majority of the infection
duration, women cannot observe their HIV status.
Women are heterogeneous with respect to a group of characteristics that are treated as
exogenous and constant determinants of their choices. These characteristics include region
of residence, completed schooling level, the size of the household's land plot, age of marriage,
and whether a woman is married to a polygamous husband. Women are also of dierent
discrete unobserved types, which are incorporated in the model to account for unobservable
permanent factors which might aect preferences as well as exposure to HIV.
2.2 Preferences
Each period, a woman receives a utility ow from household consumption (C ), her number
of children (N ), pregnancy status (p), and a time-varying preference shock ( p ) which is iid
across time and women. The per-period utility function is given by
C (t)Ï†
U (t) = Ï† [1 + exp (Î»1 N (t))] + Î»2,r,e,m,Âµ N (t) + Î»3,r,e,m,Âµ N (t)2 âˆ’ (Î»4,t + p (t)) p(t) âˆ’ (Î»5 + Î»6 t)p(t)p(t âˆ’ 1),
2
p (t) âˆ¼ iidN 0, Ïƒp .
The utility function exhibits constant relative risk aversion (CRRA) in consumption and
includes an interaction term between household consumption and the number of children to
reect consumption being divided among more individuals as the household size increases.
The utility is quadratic in the number of children. The parameters related to preference
for children, Î»2 and Î»3 , are allowed to vary with region of residence (r ), schooling level (e),
8
10
polygamy (m), and unobserved type (Âµ). The utility function also incorporates a non-
pecuniary cost (or benet) associated with being pregnant. This cost includes a stochastic
preference shock as well as a deterministic age-dependant element (Î»4,t ). The cost of preg-
nancy changes if a woman was pregnant in the previous period (Î»5 ). The cost of consecutive
births is allowed to change with the age of women (Î»6 ).
The specication of the model implies perfect control over conception and contraception.
I could have instead introduced a cost of contraceptives, a probability of conception condi-
tional on not wanting to become pregnant and a probability of not conceiving conditional
on trying. Using data on births only, however, I cannot separately identify these elements.
Instead, these elements will be absorbed into the cost of pregnancy (both deterministic and
stochastic) and the preference for children. For example, having the cost of pregnancy change
with age captures physiological factors which vary the propensity to conceive during dierent
stages of a woman's life-cycle. The added cost of consecutive births enables the model to
generate patterns of birth spacing.
2.3 Income and Consumption
It is assumed that households cannot borrow or save, which implies household consumption
(C ) equals the household's income (Y ). The household's income is exogenous and stochastic.
I specify a parsimonious household income function that is appropriate for the context of
subsistence agriculture. I assume that the logarithm of income is distributed as
ln (Y (t)) = Î¸1 + Î¸2 Balaka + Î¸3 Mchinji + Î¸3 Land High + Î¸4 N (t) + Î¸5 t + Î¸6 t2 + y (t),
(1)
2
y (t) âˆ¼ iidN 0, Ïƒy .
Income depends on region of residence (Balaka, Mchinji), size of household's land plot (Land
High), the age of the woman (t), the number of children (N ), and a time-varying income
10 The majority of respondents in each of the three sites of the MDICP dataset are of dierent tribal groups.
The tribes dier in their practices of lineage and residence after marriage which might aect preferences for
children.
9
11
shock ( y ). Realization of the income shock occurs after the fertility decision is made.
Therefore, pregnancy decisions are based on expected income.
2.4 Perceived Infection Hazard
12
A woman's perceived exposure to HIV infection is modeled as a hazard process. Let h(t)
be the probability a woman assigns to getting infected at period t, conditional on being
HIV-negative until then. The perceived hazard rate for period t is given by
1
h(t) = , (2)
1 + exp (âˆ’x(t) Î² )
where x(t) is a vector containing the woman's characteristics, the duration of her marriage,
her age and age squared, and a constant. The parameters related to the constant, age and
13
age squared are allowed to dier for unobserved types.
The perceived unconditional probability of getting infected at period t, P (t), is given by
tâˆ’1
P (t) = h(t) (1 âˆ’ h(k )) . (3)
k=1
2.5 Survival Expectations
A woman is assumed to know the mortality processes associated with HIV infection (for both
adults and children). Survival to each additional period provides her with information about
her HIV status. Specically, she reduces the probability of having gotten infected in each past
period. Given these probabilities and the mortality process, the woman assigns probabilities
to survival to each future period. The woman also updates expectations about the survival of
children of ages zero to three in future periods. Conditional on the probabilities she assigns
11 The Malawi Multiple Indicator Cluster Survey 2006 estimates that 27.5 percent of children aged 5 to 11
are involved in at least one hour of economic work or 28 hours of domestic work per week. A more natural
specication of the income function might take into account the ages of the children; however, I do not keep
track of the ages of children above age 3 in the state space to reduce computational burden.
12 Perceived and actual exposures to HIV infection are treated as exogenous to women's behavior because
there is no variation in self reports of sexual behavior by women in the data. All but very few married
women report having sex with their husbands and not having extramarital partners.
13 The detailed specication of the perceived infection hazard function is presented in the appendix.
10
to having been infected at times of giving birth, she assigns probabilities to the survival of
each of her children to future periods.
Women's beliefs are also updated by receiving an HIV test result. MDICP respondents
did not anticipate being oered HIV testing, and almost all of the respondents agreed to get
tested. Because of these features of the data, I abstract from modeling the decision to get
tested. Instead, I treat the HIV testing as an unanticipated revelation of HIV status. I rst
present the updating of expectations about life horizon without testing and then proceed to
discuss the updating with testing.
2.5.1 Updating Survival Expectations without HIV Testing
Updating probabilities assigned to infection in dierent periods:
Let I (Ï„, t) be the probability a woman assigns to getting infected at period Ï„, conditional
on being alive at t. The probability assigned to having gotten infected at a past or present
period Ï„ is given by
Pr (got infected at Ï„ & alive at t) P (Ï„ )S (Ï„, t)
I (Ï„, t) = = , t â‰¥ Ï„, (4)
Pr (alive at t) 1âˆ’
t
k=1P (k ) (1 âˆ’ S (k, t))
where S (Ï„, t) is the probability a woman who gets infected at period Ï„ survives to period
t. The probability assigned at period t to being HIV-positive is given by the summation of
the probabilities assigned to infection happening in all periods up to t:
t
B (t) = I (k, t) . (5)
k=1
11
Probabilities assigned to survival in future periods:
The probability a woman assigns at time t to being alive at the following period is given by
Pr (got infected at k) Pr (alive at t + 1 | got infected at k & alive at t)
t
Ï€ (t, t + 1) = k=1
+ Pr (not infected at t) (6)
t
= k=1 I (k, t) S (k, Ï„ )
S (k, t) + 1 âˆ’ B (t).
Probabilities assigned to child survival:
âˆ’ +
Let Sc (j ) (Sc (j )) be the probability that a child born to an HIV negative (positive) woman
survives to age j. The probability a woman assigns at period t for a child of age a to survive
to the following period depends on the probability she assigns to having been infected at the
time of the child's birth and is given by
a
Ï€c (t, t + 1) = Pr(mother was HIV-positive at birth)
Ã— Pr (child alive at t + 1 | mother was HIV-pos at birth & child alive at t)
+ Pr(mother was HIV-negative at birth)
(7)
Ã— Pr (child alive at t + 1 | mother was HIV-neg at birth & child alive at t)
+ âˆ’
tâˆ’a Sc (a+1) tâˆ’a Sc (a+1)
= k=1 I (k, t) Sc+
(a)
+ 1âˆ’ k=1 I (k, t) Scâˆ’
(a)
.
2.5.2 Updating Survival Expectations with HIV Testing
HIV testing provides a woman information about her infection status at the time of the
test; however, it does not provide her with any new information about when she might have
gotten infected. I assume women do not use the test result to update their perceived infection
12
hazard process.
Given that some of the women who received positive test results assigned some likelihood
to not being infected after the test, I assume that women assign a probability to the accuracy
of the test result. Specically, she assigns probability ptest to the test result being her
actual status and probability 1 âˆ’ ptest to the test result being uninformative. Therefore, the
probability a woman assigns to being HIV-positive when receiving a test result at period
ttest , denoted as Ë† (ttest ),
B is
Ë† (ttest ) = (1 âˆ’ ptest ) B (ttest ) + ptest 1 {positive test result} ,
B
where B (ttest ) is the probability assigned at period ttest to being HIV-positive without having
been tested.
To compute the subjective life expectancies in periods after ttest , it is necessary to recover
the post-test probabilities women assign to infection in each period. I dene Ë† (t)
P to be the
post-test update of P (t) (dened in Equation 3). These probabilities can be recovered using
the fact that a test result does not provide the woman with any new information about when
she might have gotten infected. Let G(Ï„, ttest ) be the probability assigned to having gotten
infected at Ï„, conditional on being HIV-positive at ttest . This probability does not change
after learning the test result. The after-test probability assigned at ttest to having gotten
infected in a past period Ï„ can be written as
Ë† (Ï„, ttest ) = Pr (got infected at Ï„ | HIV-positive at t) Pr (HIV-positive at t)
I
(8)
Ë† (ttest ) = I (Ï„, t) Ë†
= G (Ï„, ttest ) B B (t) B (ttest ) , Ï„ = 1, ..., ttest .
In addition, similar to Equation 4, the after-test probability assigned at ttest to having gotten
infected in a past period Ï„ is also given by
Ë† (Ï„ ) S (Ï„, t)
P
Ë† (Ï„, ttest ) =
I t , Ï„ = 1, ..., ttest . (9)
1âˆ’ PË† (k ) (1 âˆ’ S (k, t))
k=1
13
By (8) and (9), I get
Ë† (Ï„ ) S (Ï„, t)
P I (Ï„, t) Ë†
t = B (ttest ) , Ï„ = 1, ..., ttest . (10)
1âˆ’ Ë†
P (k ) (1 âˆ’ S (k, t)) B (t)
k=1
I can solve for Ë† (ttest )
Ë† (1), ..., P
P by solving the system of ttest equations with ttest unknowns
(the solution of the system of equations is presented in the appendix). The probability
assigned to infection at a future period t is given by
tâˆ’1
Ë† (t) = h(t)
P Ë† (k ) ,
1âˆ’P t > ttest .
k=1
Given the vector Ë† =
P Ë† (T âˆ’ 1)
Ë† (1), ..., P
P , the perceived probabilities of infection and
survival in periods following the test are constructed as in equations (4) to (7).
2.6 Model Solution
The woman's problem can be formulated as a discrete-choice discrete-time stochastic dy-
namic program. Let â„¦(t) be the state space at time t, consisting of all of the information
relevant to decision-making that the woman has available at that time. Specically, it con-
tains the realized preference shock, the number of living children, ages of young children
at risk of dying, whether she was pregnant in the previous period, and her testing history
(period and result of each test). It also contains her xed characteristics: her region of
residence, schooling level, age of marriage, polygamy status of her husband, and the land
owned by the household. Note that life horizon expectations are fully determined by â„¦(t);
therefore, there is no need to include the expectations in the state space.
Let V (â„¦(t), t) be the value function, that is, the maximized present discounted value
of lifetime utility. Let V f (â„¦(t), t) be the alternative-specic value function, that is, the
value if choice f is taken, with f indicating pregnancy status. The Bellman equation of the
14
optimization problem is
ï£±
ï£´
max V 0 (â„¦(t), t) , V 1 (â„¦(t), t) , t = 1, ..., F âˆ’ 1
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£´
ï£²
V (â„¦(t), t) = EU 0 (t, â„¦(t)) + Î´Ï€ (t, t + 1) E (V (â„¦(t + 1), t + 1 | p(t) = 0, â„¦(t))) , t = F, ..., T âˆ’ 1
ï£´
ï£´
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ï£´
ï£´
ï£´
ï£´
ï£´
ï£³EU 0 (T, â„¦(T )) , t = T
ï£´
ï£´
ï£´
V f (â„¦(t), t) = EU f (t, â„¦(t)) + Î´Ï€ (t, t + 1) E (V (â„¦(t + 1), t + 1 | p(t) = f, â„¦(t))) , f = 0, 1, t = 1, ..., F âˆ’ 1,
where U 1 (t, â„¦(t)) represents a utility ow at state â„¦(t) with pregnancy, and U 0 (t, â„¦(t))
represents the utility ow without pregnancy. The expectation associated with ow utility is
taken over the present income shock. The expectation associated with the next-period value
function is taken over future income and preference shocks as well as over child survival.
V 1 (â„¦(t), t) is strictly increasing in p (t); however, V 0 (â„¦(t), t) is constant in p (t) because
the preference shock enters the utility ow only if the woman becomes pregnant. Let â„¦d (t)
be the set of deterministic elements of the state space, that is, the set without p (t). For any
âˆ—
â„¦d (t) there is a unique critical value (â„¦d (t), t) such that V 1 (â„¦(t), t) = V 0 (â„¦(t), t) . The
solution to the woman's optimization problem is to become pregnant only if the preference
shock is bigger than the corresponding critical value.
2.7 HIV/AIDS and Fertility Outcomes
There are several channels through which the presence of the HIV/AIDS epidemic can aect
fertility outcomes in the framework presented above. These channels are driven by both
actual and perceived exposure to infection. They operate simultaneously and can potentially
have opposing inuences on the timing and number of births as well as on the experienced
child mortality. In terms of the eect of actual HIV infection, the shorter life-span of women's
15
life-cycle will reduce the number of periods during which they can get pregnant and will
therefore have an obvious negative eect on the overall average number of births. The
increase in child mortality, on the other hand, can increase fertility through a replacement
behavior. If there is a decreasing marginal utility from an additional child, the loss of a child
will increase fertility in the periods after the loss.
The impact of the subjective beliefs on fertility outcomes is even more ambiguous. As
described in the previous subsection, a woman compares the expected value of lifetime utility
with and without a current pregnancy. With higher likelihood assigned to being infected, a
woman assigns less likelihood to her and her child's survival and therefore perceives fewer
periods during which she would get utility ows from having the additional child. Given that
the non-pecuniary cost of pregnancy is not aected by beliefs about HIV status, this implies
that the woman would be less likely to get pregnant if she assigns higher probability to being
infected. On the other hand, if a woman has high valuation for children, she might choose to
increase her fertility when she assigns some probability to being infected. She would do so
in expectation that a larger share of her children will not survive. In addition, even without
a change in the total number of births a woman gives, the presence of HIV might change the
timing of her pregnancies. A woman might assign no or small likelihood to being infected
in the present but decide to give birth at younger ages if she perceives a higher likelihood to
being infected in the future. This will depend on her perceived hazard rates throughout the
fertile stage of her life-cycle.
3 Data
3.1 Malawi Diusion and Ideational Change Project
Malawi is a landlocked country located in Southeast Africa. It has a population of about
15 million, comprised of dierent ethnic and religious groups. 81 percent of the population
16
14
lives in rural areas and relies mostly on subsistence agriculture. The Malawi Diusion
and Ideational Change Project (MDICP) data have been collected since 1998 in rural areas
of three districts in Malawi: Balaka in the south of the country, Mchinji in the center, and
15
Rumphi in the north. The dierent rounds of the longitudinal dataset contain extensive in-
formation on more than 4,000 men and women, at the individual, household and community
levels.
A unique feature of the MDICP dataset is that it includes measures of subjective likeli-
hoods respondents assign to being HIV-positive. The subjective expectations were collected
using an elicitation methodology developed by Delavande and Kohler (2009) for a develop-
16
ing country context with low levels of literacy and numeracy. Respondents were provided
with ten beans and a plate. They were asked to allocate dierent number of beans on the
plate to express the likelihood that dierent events will be realized. The respondents were
instructed that zero beans reect certainty that an event will not happen, more beans reect
higher likelihood that an event happens, and that ten beans imply certainty about the event
happening. The likelihood assigned to being HIV-positive is measured by asking: Pick the
number of beans that reects how likely you think it is that you are infected with HIV/AIDS
now. Delavande and Kohler (2009) nd that reported subjective expectations follow basic
properties of probabilities and that the assessments of HIV-infection vary meaningfully with
observable characteristics associated with dierent levels of HIV prevalence.
Since 2004, HIV testing has been oered to all respondents during data collection. The
take-up rates of the tests were high, above 90 percent in all waves. The HIV prevalence rate
in the sample was 6.9 percent Obare et al. (2009). In 2004, test results were available ve to
seven weeks after testing. The testing component of the survey was linked to an experiment
that is described and analyzed by Thornton (2008). Respondents were assigned vouchers
14 data.worldbank.org
15 Detailed information on the Malawi Diusion and Ideational Change Project can be obtained at
http://www.malawi.pop.upenn.edu/.
16 Attanasio (2009) and Delavande et al. (2010) review existing subjective expectations data from devel-
oping countries.
17
for a monetary reward, redeemable upon return to temporary Voluntary Consulting and
Testing (VCT) sites where results where provided. The VCT sites were set up such that all
respondents' homes are within ve kilometers distance from at least one site. Approximately
70 percent of those tested chose to pick up their results. In 2006 and 2008, rapid blood tests
were adopted, eliminating the time delay between testing and provision of results.
For my analysis, I am using a subsample of married women in their rst marriage, who
did not become pregnant from a relationship with men other than their future husbands
prior to marriage. I restrict the sample to women in unbroken rst marriages because I
abstract from modeling any decisions related to marriage or partnership. I also restrict the
sample to include only women who were interviewed in at least one of the 2006 and 2008
rounds. Data collected in these rounds include family rosters, containing information on all
children of respondents, and the elicitation of subjective expectations. I exclude women who
were born before 1960, because the last age of fertility in the model is assumed to be 44 and
2004 is the earliest year included for fertility outcomes. After excluding additional women
for missing information, the estimation sample consists of 1006 women.
Tables 1 to 4 and Figures 1 and 2 provide descriptive statistics for the variables used
in the analysis. The average age of women in the sample is 26.6 in 2004. 29 percent of
the sample resides in Balaka (south), 35 percent in Mchinji (center), and 36 percent in
Rumphi (north). The median number of years of schooling is 5. The women in Balaka
have the lowest schooling levels with 31 percent of women never having attended school and
only three percent having attended some secondary school. The women in Rumphi have
the highest schooling levels, with 99 percent of women ever having attended school and 26
percent some secondary school. The average age of marriage in the overall sample is 17.5
and is similar across regions. Polygamy is most prevalent in Rumphi, with 33 percent of
the women in the sample from that region married to a polygamous husband. The share of
polygamous women is 27 percent in Mchinji and 20 percent in Balaka. As shown in Table
2, HIV testing take-up rates were high in all the rounds: 87.9 percent in 2004, 93.3 percent
18
in 2006, and 94.6 percent in 2008. The percentage of women who tested positive was 2.9
percent in 2004, 3.1 percent in 2006, and 3.7 percent in 2008.
Figure 1 depicts the distribution of the likelihood women assigned to being HIV-infected
in the 2006 and 2008 rounds (pooled), measured by beans on a scale of zero to ten. Fifty
percent of the reports were of zero beans. The percentage of women who chose each category
decreases with the number of beans, except for the 5-bean category. Fewer than one percent
chose the 10-bean category. Table 3 shows the average number of beans. The overall sample
average is 1.52 beans. The average number of beans chosen in each region conforms to the
ranking of HIV prevalence in the general MDICP sample. The averages are 1.78 beans in
Balaka, 1.69 in Mchinji, and 1.16 in Rumphi. The 2004 HIV prevalence rates for these regions
are 7.9 percent in Balaka, 6.4 percent in Mchinji, and 4.4 percent in Rumphi. The table also
shows that average beliefs decrease with schooling and are higher for women married to a
polygamous husband.
Figure 2 depicts the distribution of the likelihood women assigned to being HIV-infected
after receiving HIV test results. Because the testing component of the survey was conducted
after the interview components, these are beliefs reported two years after the tests (the
subsequent round of data collection). Because of low HIV prevalence and high attrition of
infected women, there are only 18 women who report their beliefs two years after receiving
a positive test result. The average number of beans allocated by these women is 4.67 beans,
which is signicantly higher then the average of 1.5 beans reported by women who received
a negative test result. Most of the women who received a positive test result assign at least
some likelihood to not being HIV infected.
I obtain information about births from family rosters that were collected in 2006 and
2008. Respondents were asked to list all of the children ever born to them, but year of
birth is missing for children who died more than two years before the interview. I therefore
use only birth outcomes reported for the years 2004 to 2007. Depending on the year of
marriage, year of birth, and the rounds in which the woman participated, I observe between
19
1 and 4 potential fertility years for women of ages 16 to 45. In total, I observe 3,148 years
of potential fertility with births in 920 of them (29.2 percent). Table 4 shows the annual
birth probabilities of married women by 5-year age groups. The annual birth probabilities
17
decrease from 0.405 for ages 16 to 20 to 0.06 for ages 41 to 45.
3.2 Malawi Second Integrated Household Survey
The MDICP data do not include detailed measures of household consumption. For this
reason, I use the Malawi Second Integrated Household Survey (IHS-2) to impute a better
measure of consumption of households in the MDICP sample. The dataset, gathered by
the Malawi National Statistics Oce in 2004-2005, is part of the World Bank's Living Stan-
dards Measurement Study program. It includes comprehensive data on consumption and
expenditures of households, as well as on local prices. Specically, the dataset includes a
measure of annual consumption expenditure aggregates in real value. The survey was elded
in 26 out of Malawi's 27 districts, including all three MDICP districts. After restricting the
sample to households in the districts covered by MDICP and excluding households without
a woman as a head of the household or as a wife of the head, the estimation sample includes
612 households. 220 of these households are from Balaka, 188 from Mchinji, and 204 from
18
Rumphi.
3.3 Mortality Statistics
I supplement the use of the MDICP data with statistics about survival after infection and
child mortality from other studies that use data from samples with repeated HIV testing
and frequent follow-ups. These data provide more precise information on times of infection
and death than MDICP does. Hallett et al. (2008) estimate probabilities of survival after
infection by tting a Weibull distribution to survival data presented by Todd et al. (2007)
17 The annual birth probability is dened as the proportion of alive and married women of specic age that
give birth.
18 Detailed information on the Malawi Second Integrated Household Survey can be obtained from the
World Bank's website at http://econ.worldbank.org.
20
from 5 studies in Eastern and Southern Africa before highly active antiviral therapy. The
probability of survival to year t conditional on getting infected at year Ï„ is estimated as
2
tâˆ’Ï„
S (Ï„, t) = exp âˆ’ ,
ÏˆÏ„
with ÏˆÏ„ reducing with age of infection. (Parameter estimates and median survival time
are presented in Table 5.) The mortality hazard increases with both duration and age of
infection. The median survival time of an individual infected at ages 15 to 19 is 13.3 years,
while it is 8.4 years for an individual who gets infected at ages 40 to 44.
Data from dierent longitudinal studies with repeated assessments of HIV status of adults
show higher mortality rates of children born to infected women. In these studies, the HIV
status of the children was generally not available. Controlling for dierent background
characteristics, mortality rates of children born to HIV-infected mothers were estimated to
be about three times higher than those for children born to uninfected mothers, with the
eect lasting throughout childhood years (Newell et al., 2004). Crampin et al. (2003) report
child mortality rates by status of mother at birth from a retrospective cohort study with
more than 10 years of follow-up in Karonga district in Northern Malawi. The rates are
reported in Table 6.
4 Estimation
The main estimation sample contains data on 1006 women from the MDICP dataset. The in-
formation on the ith woman consists of up to 4 years of pregnancy choices pi (t), t = tpi , ..., tpi ,
up to 2 reports of subjective assessments of HIV status bi (t), t = tbi , ..., tbi , up to 3 HIV
test results (collected in the 2004, 2006 and 2008 rounds), and a vector of xed characteris-
tics: region of residence, schooling level, year of birth, age of marriage, polygamy status of
husband, and household's land.
I also use an auxiliary sample containing data on 612 households from the IHS-2 dataset.
21
The information on the j th household consists of household annual aggregate consumption
expenditure (yj ), and a vector of household characteristics: region of residence, age of woman
(head or wife of the head of the household), number of children, and household's land.
The rst step of the econometric implementation involves estimating the parameters of
the income function (Equation 1) by ordinary least squares regression using the auxiliary
sample. The rest of the model parameters are estimated using maximum likelihood, tak-
ing the parameters of the income and survival functions as given. The likelihood function
contains the following elements: (1) belief reports probabilities; (2) fertility outcomes proba-
bilities; (3) actual HIV status and survival; and (4) unobserved type probabilities. I proceed
by describing the contribution of each of these elements to the likelihood.
4.1 Beliefs
The perceived infection hazard is estimated using the data on subjective assessments of
HIV-status. Let Bi (t) be woman i's perceived probability of being HIV-positive at period
t and bi (t) be the number of beans she allocates to being HIV-infected in the expectations
elicitation exercise. I make the following two assumptions regarding the relationship between
a woman's belief and the recorded number of beans. First, I assume that beliefs are reported
with some noise. Specically, the reporting error, bi (t), is assumed to be iid across time and
women:
bi (t) âˆ¼ iidN (0, Ïƒb ) .
19
Second, I assume that each discrete bean category corresponds to a probability interval.
A respondent reports the number of beans corresponding to the interval in which her belief
added to the reporting error falls. The reports are assumed to be made according to the
19 As in Delavande and Kohler (2009)
22
20
following rule:
bi (t) = 0, 1 if Bi (t) + bi (t) â‰¤ 0.15,
bi (t) = 2 if 0.15 < Bi (t) + bi (t) â‰¤ 0.25,
.
.
.
bi (t) = 10 if 0.95 < Bi (t) + bi (t).
Let â„¦d
i be the set of initial conditions for woman i. It contains the deterministic (and
observable) elements of the state space. That is, it contains her permanent characteristics
that enter the perceived hazard function. I dene it to also contain the woman's testing
variables (timing and results of tests). Although the woman does not forecast getting tested,
I treat this information as part of the initial conditions for the econometric implementation.
The woman's sequence of life-cycle beliefs are determined given these initial conditions and
her unobserved type, and can be written as i , typei = j .
B t | â„¦d The probability of observing
bi (t), conditional on the set of woman i's initial conditions and her unobserved type is given
by
i , typei =j )
0.15âˆ’B (t | â„¦d
i , typei = j = Î¦
Pr bi (t) = 0, 1 | â„¦d Ïƒb ,
i , typei =j )
0.25âˆ’B (t | â„¦d i , typei =j )
0.15âˆ’B (t | â„¦d
i , typei = j = Î¦
Pr bi (t) = 2 | â„¦d Ïƒb âˆ’Î¦ Ïƒb ,
.
.
.
i , typei =j )
0.95âˆ’B (t | â„¦d
i , typei = j = 1 âˆ’ Î¦
Pr bi (t) = 10 | â„¦d Ïƒb ,
where Î¦ is the cumulative distribution function of the the standard normal distribution.
As shown in Figure 1, the percentage of women reporting each bean category generally
decreases with the number of beans. The percentage of women who report ve beans, 8.78
percent , is higher than the two categories below (3.41 percent report four beans and 6.52
20 Figure 3 shows the distributions of reported beliefs in the 2006 and 2008 rounds. There is a big drop
in the percentage of women choosing the 0-bean category and the number of women who choose the 1-bean
category doubles. My model would not be able to generate this shift. For my empirical analysis, I treat the
0 and 1-bean categories as a single category.
23
percent report three) and is also signicantly higher than the next category (1.59 percent
report six). My model is not likely to capture this pattern. I therefore assume that the
probability interval corresponding to the ve bean category is larger, implying that some of
the women who would allocate 4 or 6 beans to the likelihood of being HIV-positive under
the rule described above, report instead 5 beans. The probability of observing 5 beans is
assumed to be
ï£« ï£¶ ï£« ï£¶
0.65 âˆ’ B t | â„¦d
i , typei = j 0.35 âˆ’ B t | â„¦d
i , typei = j
Pr b(t) = 5 | â„¦d
i , Âµi = j = Î¦
ï£ ï£¸ âˆ’ Î¦ï£ ï£¸.
Ïƒb Ïƒb
4.2 Fertility
The parameters of the utility function are estimated using the data on pregnancy out-
comes. As described in Section 3.6, the solution to a woman's optimization problem is
to become pregnant at period t if the preference shock, pi (t), is bigger than the critical
âˆ—
value (â„¦d
i (t), t, typei ). Conditional on the woman's type and the observable elements of
the woman's state space at time t, the probability of the woman's choice to become pregnant
is given by
i (t), t, typei )
âˆ—
(â„¦d
i (t), typei = j = 1 âˆ’ Î¦
Pr pi (t) = 1 | â„¦d Ïƒp ,
i (t), t, typei )
âˆ—
(â„¦d
i (t), typei = j = Î¦
Pr pi (t) = 0 | â„¦d Ïƒp .
Conditional on the set of initial conditions and the unobserved type, the probability of
observing a sequence of belief reports is independent from the probability of observing a
sequence of fertility outcomes. This is because the set of initial conditions and type map
deterministically into the sequence of life-cycle beliefs.
4.3 Actual Infection Process
Given the assumption of no symptoms, the state space does not contain actual HIV status
beyond any HIV test results that women have received. It is therefore not necessary to recover
the actual hazard process for estimation of the decision-model parameters. For each possible
24
state, the estimated model generates a probability of becoming pregnant conditional on being
alive at that state. I do need an estimate of the actual hazard process for my counterfactual
analysis, however. To study the eect of dierent policy interventions on outcomes of life-
cycle fertility and child mortality, I include the infection and mortality processes in the
simulations.
I assume an actual HIV infection hazard rate with a functional form similar to that of the
perceived infection hazard described in equation (2). Information about the hazard process
is contained in the HIV test results, the age in which the tests were taken, and the ages in
which a woman is last observed (regardless of testing histories). I dene Hi to be a vector of
a woman i's testing and survival information: the oldest age in which she had a negative test
result (if ever), the earliest age at which she received a positive test result (if ever), and the
latest age in which she is observed in the data. The probability of observing a woman with
testing survival history Hi , conditional on her type and observable elements of her initial
state is given by
Pr Hi | â„¦d
i, typei =j .
I present the details of how this probability is computed in the appendix.
4.4 Type Distribution
A multinomial logit specication is used for the type probabilities. The probability that
woman i is of type j is given by
exp(wti Î³j )
Pr typei = j | â„¦d
i = 1+ J
exp(wti Î³k )
, j = 1, ..., J,
k=1
Pr typei = 0 | â„¦d
i = 1+ J
1
exp(wti Î³k )
,
k=1
where wti is a vector of woman's initial conditions, including region of residence, schooling
level, age of marriage, whether she is married to a polygamous husband, year of birth, and
the number of children she had in the rst period observed interacted with the age she was
25
when rst observed. The last term is included to take into account the fact that not all
the women are observed from the time of their marriage. The state in which they are rst
observed depends on past decisions and therefore on their unobserved type.
4.5 Likelihood Function
The contribution of woman i to the sample likelihood is given by
ï£« ï£¶ï£« ï£¶
J tp tb
Li = ï£ Pr pi (t) | â„¦d
i (t), typei = j ï£¸ï£ Pr bi (t) | â„¦d
i, typei =j ï£¸
j =0 t=tp t=tb
Ã— Pr Hi | â„¦d
i, typei =j Pr type = j | â„¦d
i .
5 Estimation Results
5.1 Parameter Estimates
21
The model is t with four unobserved types. Recall that the types can dier with respect to
their preferences for children, the assigned probabilities to the accuracy of HIV test results,
and the perceived and actual exposure to HIV infection. The four types have distinctively
dierent beliefs and characteristics. As seen in Table 12, type 0 women, who represent 28
percent of the sample, have the largest share of women with some secondary education and
the lowest share of women married to polygamist men. They assign no likelihood to being
infected throughout their life cycle. Type 1s, comprising 23 percent of the sample, have
the highest share of women with no schooling and the youngest age of marriage. Type 2s,
comprising only 2.1 percent of the sample, perceive the highest exposure to HIV infection.
On average, women of that group assign probabilities of 0.59 and 0.69 to being infected when
21 Therewere signicant improvements in model t beyond three types. I did not attempt to t the model
with more types because of computational burden.
26
they are 20 and 40 years old respectively. Type 3 women are the biggest group, representing
47 percent of the sample.
Tables 7 to 11 report the parameter estimates and their standard errors. As can be seen
in Table 7, the marginal utilities of additional children are positive for the rst child and
are decreasing with the stock of surviving children for all women. However, the prole of
these marginal utilities varies with region of residence, schooling level, polygamy status and
unobserved type. Relative to women with lower levels of education, women who attended
secondary school have the highest marginal utility for the rst child but the marginal utility
declines at the fastest rate with the number of children. The same is true for women from
Rumphi district relative to women from the other two districts and for type 1 women relative
to the other types.
The childbearing costs, which are assumed to depend only on a woman's age, are esti-
mated to be positive and increasing with age. The cost for the youngest age group (less than
20 years) is 25 percent of that for women of ages 20 to 24 and only about one percent of the
cost for women of ages 40 to 44. The cost of consecutive pregnancies, on the other hand, is
estimated to be decreasing with age.
5.2 Model Fit
To assess model t, I compare the model's prediction of the distributions of reported beliefs
and pregnancy probabilities to the distributions of actual beliefs and pregnancies observed
in the data. I simulate each observed woman 100 times. The simulation starts from the age
a woman is rst observed and takes as given the observable elements of the state space (her
constant characteristics, pregnancy in previous period, and the number and ages of children
younger than four years). For each simulation, I draw an unobservable type from the type
distribution and preferences shocks.
Figures 4 and 5 and Table 13 compare the fertility and belief reporting outcomes predicted
by the model to the actual outcomes observed in the data. Figure 4 shows that the model
27
is able to generate the shape of the reported beliefs distribution. It under-predicts the
proportion of women in the lowest bean category by about 4 percentage points and over-
predicts the proportions in the 2 and 3-bean categories. As described in section 4, I assume
that some of the respondents round and report ve beans instead of four and six. 13.8
percent of belief reports in the data are of the four to six-bean categories; the model predicts
13.6 percent.
Figure 5 depicts the actual and predicted annual pregnancy probabilities for dierent age
groups. The model captures the decline in pregnancy probabilities with age. This pattern
is generated by the diminishing marginal utility of additional children and childbearing
costs, which increase with age. Table 13 shows the actual and predicted annual pregnancy
probabilities by region, schooling level, and polygamy status of the husband.
6 Counterfactual Analysis
Having estimated the structural model parameters, I use the model to perform counterfactual
experiments to quantify the eect of the HIV/AIDS epidemic on fertility outcomes in the
given environment and to evaluate the impacts of dierent policy interventions. I do so by
simulating the women's life-cycle fertility decisions in dierent environments. As with the
previous simulations, performed to assess goodness of t, each woman is simulated 100 times.
However, in the counterfactual simulations I also incorporate infection and mortality. That
is, women are exposed to HIV infection according to the estimated actual infection hazard
process, and, once infected, they are exposed to the mortality process. Survival probabilities
of children of ages zero to three depend on the HIV status of the mother at time of birth
according to the rates in Table 6.
For the counterfactual simulations, I use a younger subsample of the 509 women who were
born after 1978. I exclude the older women to have the composition of the sample, in terms
of characteristics and types, minimally aected by survival to the time in which women are
28
rst observed. The characteristics of the sample used in the counterfactuals is presented in
Table 14. I dene the baseline environment to be with the levels of perceived and actual
hazards of infection estimated by the model, but without any HIV tests. Results of the
baseline environment simulation are presented in Panel A of Table15. The table includes
the average beliefs and infection rates at dierent points of the women's life-cycle, number
of life-cycle births as well as the average number of child mortality experienced by a woman.
The results are shown for the full sample as well as by type.
To assess how fertility outcomes would have been dierent were there no HIV, I simulate
fertility in an environment with no HIV (and no beliefs about HIV). These results are
presented in Panel B of Table 15. Comparison of fertility outcomes in the two environments
indicates that HIV has a negative average eect on fertility for the given group of women.
The average number of births during a woman's life in the no-HIV environment is 7.22 in
22
comparison to 7.07 in the baseline environment. Although the total amount of births is
lower in the baseline environment, the total incidents of child mortality is higher by seven
percent as the percentage of children not surviving to age beyond age 4 increases to 17.5
percent from 15.8 percent in the presence of HIV.
Eighty-ve percent of the women will experience the same number of births in both
environments, three percent will have higher fertility in the HIV environment and twelve
percent will experience fewer pregnancies. The women who have higher fertility in the
presence of HIV are of two groups. The rst group includes women who are infected with
HIV but assign relatively low probability to being infected. They respond to experienced
child mortality (due to mother to child transmission) by having additional pregnancies. The
second group is of women who have high valuation of children and would increase fertility
early in their life-cycle in anticipation that some of their children might die. However, the
overall average reduction in fertility in the presence of HIV is largely due to the shorter
life-span of women infected with HIV as well as women that choose to reduce fertility as
22 The
life-cycle number of births is higher than the national total fertility rate estimated at 6 pregnancies
per woman because fertility rates are higher in rural areas of Malawi.
29
they assign lower probabilities to their own as well as their children survival in the future.
There is heterogeneity in the eect of HIV on fertility by women's characteristics. The
average dierence for type 0 women is only 0.03. Because they do not assign any likelihood to
being infected, it is completely due to the increase in mother and child mortality by infected
women. Type 1s experience an average of 0.17 fewer pregnancies in the presence of HIV
although this group has the highest share of women (8 percent) who actually increase fertility
in that environment. Type 3s are the most aected, with a dierence of 0.2 pregnancies.
Women from Balaka, with the highest HIV prevalence and highest average likelihood women
assign to being infected, will have on average 0.31 more births in the no-HIV environment
while women from Mchinji and Rumphi will have on average 0.08 and 0.1 more births
respectively.
The fertility outcomes are aected by perceived as well as actual exposure to HIV. As
expected, the dierence in the number of births in the two environments depends not only
on whether a woman got infected during the fertile stage of her life cycle but also on when in
the life cycle she got infected. Women who got infected by age 25 (5.1 percent of the sample)
experience 1.57 less births in the baseline environment relative to the no-HIV environment.
Women who got infected by age 45 (11.6 percent of the sample) experience a reduction of
0.84 births and women who did not get infected by the end of their fertile stage, experience
an average dierence of 0.06 births.
To separate the eect of women's shorter life-span on the number of births from the other
channels through which HIV can aect fertility (child mortality and survival expectations),
I count pregnancies in the no-HIV environment which occurred only in periods in which
women would have been alive in the baseline environment. This brings the average number
of births down to 7.1, only 0.03 higher than the number in the baseline environment. During
these periods, women of types 0 and 1 experience fewer births in the no-HIV environment,
by 0.01 and 0.08 respectively. Type 2s and 3s experience more births. Type 2s, who have the
highest beliefs about being infected, experience on average 0.19 more births and Type 3s,
30
the largest group in the sample, experience 0.1 more births. Regardless of whether women
decrease or increase the number of births in the presence of HIV, most of the divergence in
fertility rates happens in the second half of the women's fertile stage of the life-cycle. This
is because older women assign higher likelihood to being infected, they are more likely to
actually be infected, and the marginal utility from additional child reduces with the number
of children.
In Panel C of Table 15 I present results from a simulation of an environment in which
there is no mother-to-child transmission. This simulation can be thought of as an assessment
23
of the potential eect of provision of antiviral treatment that prevents such transmission.
In the simulation, child mortality probabilities of children born to HIV-infected women are
equated to those of children born to HIV-negative women. The average number of life-cycle
births in this environment is 7.05, a slight dierence from the 7.07 births experienced in the
baseline environment. Although the reduction in expected and experienced child mortality,
less than one percent of women will change the number of births they give.
To study the impact of HIV testing on fertility outcomes, I simulate life-cycle fertility
in counterfactual scenarios in which the respondents are oered a single test in dierent
points in their life cycle: at the time of their marriage (age 17 on average), age 25 and age
35. The impact of testing on outcomes depends on how women update their beliefs after
learning their test result. This updating depends on the discrepancy between the beliefs and
actual HIV status and on the accuracy women assign to test results. The extent to which
testing will aect outcomes is expected to be limited given the estimated low probabilities
most women assign to the accuracy of test results (presented in Table10). In addition, the
impact of testing also depends on the extent to which the updated beliefs alter the relative
valuation of women's choices. This can vary with age and the number of children a woman
has as life horizon shortens and marginal utility from additional children reduced. It is also
important to keep in mind that while the test result gives information about one's status
23 Inhigh-income countries, the rate of mother-to-child transmission has been reduced to less than one
percent (unaids.org).
31
in the present, women's decisions are also aected by perceived exposure to infection in the
future. A woman receiving a negative test result, for example, might not change by much
her survival expectations if she believes she is in high risk of getting infected in the periods
just after the test.
The results of simulating testing at age of marriage and ages 25 and 35 are presented in
Panels D, E, and F respectively. 1.3 percent of women are simulated to be infected at the
time of their marriage, 4.1 percent at age 25 and 5.7 percent at age 35. At ages 25 and 35,
women have already given 3.3 and 6.1 births on average. All three simulations result in a
negligible eect on the average number of births and child mortality women experience over
their life-cycle. As expected, type 0s and 1s, who assign probabilities of zero and 0.03 to
the accuracy of the test result, do not alter their choices. Type 2s all receive negative test
results. However, given the probability of 0.21 they assign to the accuracy of the test results
and their high perceived risk of infection, the total average number of births they give does
not change. Type 3s, who assign the highest probability of 0.76 to the accuracy of the test
results, reduce their fertility after receiving a positive test result. The group that received
a positive test result at the time of marriage reduced the average number of births by 0.42,
from 4.98 to 4.56. The women of type 3 who receive a positive test result at ages 25 and
35 have 0.39 and 0.21 fewer births. The smaller eect at age 35, relative to that when tests
were given in earlier ages, is partially due to the fact that mortality from HIV accelerates
with age, leaving the women fewer periods during which to make fertility choices.
Finally, in Panel G of Table 15 I present a simulation of a scenario in which women are
oered a test at age 25. Unlike the former testing simulations, women perceive the test result
as accurate and fully update their beliefs after receiving a test result. There is a negligible
eect on the overall average number of births and experienced child mortality. There is very
little impact on the fertility behavior of women who receive a negative test result. Women
who receive a positive test result, however, see an average reduction of 0.19 in the number
of births. HIV-positive women of type 0 and 1, who do not reduce fertility in the results
32
presented in Panel E, have 0.07 and 0.05 less births in this environment. Type 3s who receive
a positive test result have 0.57 fewer births than in the baseline environment.
7 Conclusion
In this paper, I specied and structurally estimated a dynamic model of fertility in an
environment with high HIV prevalence. My analysis takes into account uncertainty about
HIV status and the discrepancies between perceived and actual exposure to HIV infection.
Women's perceptions about their exposure to HIV infection and about their own HIV status
aect their beliefs about their own and their children's life expectancy. Beliefs about life
expectancies, in turn, aect fertility choices by changing the prole of expected lifetime
utilities associated with each choice. I estimate the model parameters by maximum likelihood
with longitudinal data from the Malawi Diusion and Ideational Change Project, which
contain measures of subjective beliefs about own HIV status. The model ts well the fertility
patterns in the data, as well as the distribution of reported beliefs about own HIV status.
Model simulations are informative about how HIV aects fertility and about the impact
of policies aimed at reducing HIV, such as HIV testing programs and provision of antiviral
therapy, on fertility and child mortality. Results show that the presence of HIV reduces
fertility, both for women who are infected and who are not infected. The presence of HIV
reduces the average number of life-cycle births by 0.15. I also nd that HIV testing is eective
at reducing fertility of infected women, leading to a reduction in child mortality and orphan
hood; however, the partial updating of beliefs following a test mitigates this eect. I also
nd that prevention of mother-to-child transmission through dissemination of antiviral drugs
would have a negligible eect on the average number of life-cycle births, although it reduces
the incidence of child mortality.
33
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36
A Tables and Figures
Table 1: Descriptive Statistics
All Balaka Mchinji Rumphi
Variable (South) (Center) (North)
Region: Balaka 0.29 - - -
Mchinji 0.35 - - -
Rumphi 0.36 - - -
Education: No school 0.14 0.31 0.15 0.01
Primary 0.73 0.66 0.8 0.73
Secondary 0.12 0.03 0.05 0.26
Land>1 hectare 0.44 0.19 0.53 0.55
Polygamy 0.27 0.2 0.27 0.33
Age of Marriage 17.54 17.02 17.56 17.93
(2.25) (2.28) (2.03) (2.35)
Age 2004 26.63 25.99 25.75 28.01
(8.09) (8.44) (7.21) (8.45)
Number of observations 1006 355 288 363
Table 2: Test Take-up and Percentage Tested Positive by Year of Test
2004 2006 2008
% N % N % N
Took Test 87.9% 620a 93.28% 759a 94.6% 741a
Tested positive 2.94% 545b 3.11% 708b 3.71% 701b
a
Number of women who were oered to take a HIV test for the given year
b
Number of women tested for the given year
37
Table 3: Reported Belief about Own HIV Infection, Measured in Beans: 2006 and 2008
pooled
Mean sd N
All 1.52 2.13 1640
Region Balaka 1.78 2.16 462
Mchinji 1.69 2.24 565
Rumphi 1.16 1.96 613
Schooling No school 1.77 2.2 230
Primary 1.55 2.18 1214
Secondary 1.06 1.67 196
Polygamy Mono 1.35 1.35 1188
Poly 1.95 2.44 452
Age â‰¤26 1.41 2.02 658
>26 1.59 2.21 982
Figure 1: Belief Distribution, 2006 and 2008 pooled
60
50
40
Percentage
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10
Num ber of Beans
38
Figure 2: Beliefs by Test Result, 2006 and 2008 pooled
Table 4: Annual Pregnancy Probabilities
Age Group
16-20 21-25 26-30 31-35 36-40 41-45
All Prob. 0.405 0.377 0.307 0.273 0.211 0.06
N 412 778 698 539 370 351
Region Balaka Prob. 0.414 0.398 0.323 0.31 0.281 0.086
N 162 226 155 142 121 81
Mchinji Prob. 0.369 0.357 0.32 0.267 0.177 0.082
N 130 319 291 172 113 85
Rumphi Prob. 0.433 0.382 0.282 0.253 0.176 0.038
N 120 233 252 225 136 185
Schooling None Prob. 0.375 0.492 0.27 0.263 0.31 0.085
N 24 61 89 95 84 82
Primary Prob. 0.395 0.366 0.315 0.281 0.183 0.053
N 332 596 520 381 262 247
Secondary Prob. 0.482 0.372 0.292 0.238 0.167 0.045
N 56 121 89 63 24 22
Polygamy Mono Prob. 0.412 0.388 0.309 0.285 0.23 0.077
N 354 605 511 396 235 196
Poly Prob. 0.362 0.335 0.299 0.238 0.178 0.039
N 58 173 187 143 135 155
39
Table 5: Parameter Estimates of the Survival-after-Infection Function
Age Group
15-19 20-24 25-29 30-34 35-39 40-44 45-49
ÏˆÏ„ , Weibull scale parameter 16.0 15.4 14.1 12.1 11.0 10.1 7.9
Median survival years after infection 13.3 12.8 11.7 10.0 9.1 8.4 6.6
Estimated by Hallett et al. (2008)
Table 6: Child Mortality Rates per 1000 Person-Year, by Mother HIV status
Child's Age
0 1 2 3-4
Mother HIV-negative at birth 115 26 18 8
Mother HIV-positive at birth 331 128 87 41
Source: Crampin et al. (2003)
Figure 3: Belief Distribution, by Year
70
60
50
Percentage
40
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10
Number of Beans
2006 2008
40
Table 7: Maximum-Likelihood Parameter Estimates: Preferences
Parameter Description Estimate SE
Ï† CRRA parameter 1.066 0.051
Î»1 Child-consumption interaction -0.166 0.052
Î»2 - type 0 N (t), Number of children, type 0 767.4 1282.58
Î»2 - type 1 N (t), Number of children, type 1 1501 1150.40
Î»2 - type 2 N (t), Number of children, type 2 698.8 3475.82
Î»2 - type 3 N (t), Number of children, type 3 1082 1206.99
Î»2 - Balaka N (t), Balaka shifter 161 450.07
Î»2 - Rumphi N (t), Rumphi shifter 637 563.72
Î»2 - primary N (t), Primary shifter -393.6 624.67
Î»2 - secondary N (t), Secondary shifter 196.6 785.85
Î»2 - poly N (t), Polygamy shifter -251.5 420.21
Î»3 - type 0 N (t)2 , type 0 shifter -55.08 98.12
Î»3 - type 1 N (t)2 , type 0 shifter -136.9 85.09
Î»3 - type 2 N (t)2 , type 0 shifter -24.78 302.23
Î»3 - type 3 N (t)2 , type 0 shifter -24.76 89.28
Î»3 - balaka N (t)2 , Balaka shifter 17.31 46.05
Î»3 - rumphi N (t)2 , Rumphi shifter -57.22 61.50
Î»3 - primary N (t)2 , Primary shifter 6.743 64.29
Î»3 - secondary N (t)2 , Secondary shifter -72.04 86.07
Î»3 - poly N (t)2 , Polygamy shifter 0.7026 43.84
Î»4 Pregnancy, p(t) 133.6 4966.52
Î»4 - age 20-24 p(t), age 20-24 shifter 400.3 706.72
Î»4 - age 25-29 p(t), age 25-29 shifter 2,432 679.71
Î»4 - age 30-34 p(t), age 30-34 shifter 3,653 816.29
Î»4 - age 35-39 p(t), age 35-39 shifter 5,624 1132.3
Î»4 - age 40-44 p(t), age 40-44 shifter 10,680 2073.07
Î»5 Consecutive pregnancy, p(t)p(t âˆ’ 1) 8,511 1799.57
Î»6 Consecutive-age interaction, p(t)p(t âˆ’ 1)t -218.7 92.89
Ïƒp Standard deviation of preference shock 6,698 1291.56
Î´ Discount factor 0.8675 0.073
Table 8: Ordinary Least Squares Parameter Estimates: Income Function
Parameter Description Estimate SE
Î¸1 Constant 10.619 0.135
Î¸2 Balaka -0.0717 0.046
Î¸3 Mchinji -0.092 0.049
Î¸4 Land High 0.4295 0.039
Î¸5 Number of children, N (t) 0.0499 0.011
Î¸6 Period, t 0.0118 0.007
Î¸7 Period squared, t2 -0.0002 0.0001
Ïƒy Standard deviation of income shocks 0.536407 0.065
41
Table 9: Maximum-Likelihood Parameter Estimates: Infection Hazard Function, Perceived
and Actual
Perceived Actual
Parameter Estimate SE Estimate SE
Constant, type 0 -8.05 1E+07 -12.6 2.448
Constant, type 1 -4.298 0.815 -9.611 1.652
Constant, type 2 0.1935 5.164 -17.51 1340.9
Constant, type 3 -8.311 2.420 -11.38 2.085
Period, type 0 -0.08 2E+07
Period, type 1 -0.0414 0.123
Period, type 2 -0.2406 0.281
Period, type 3 0.1208 0.225
Period squared, type 0 -1.88 9E+06
Period squared, type 1 0.0004 0.004
Period squared, type 2 0.0057 0.009
Period squared, type 3 -0.001 0.006
Period 0.1587 0.223
Period squared -0.0053 0.007
Duration of marriage 0.0662 0.053 0.0171 0.099
Primary 0.6926 0.392 2.231 0.672
Secondary 0.5547 0.885 2.891 1.054
Land > 1 hectare 0.1973 0.213 0.3524 0.441
Polygamy 0.3766 0.296 0.5221 0.571
Balaka 0.126 0.319 2.245 0.609
Rumphi -2.48 5.180 1.824 1.072
Table 10: Maximum-Likelihood Parameter Estimates: Other Parameters Related to Beliefs
Parameter Description Estimate SE
ptest , type 0 Test result accuracy, type 0 0 21.236
ptest , type 1 Test result accuracy, type 1 0.0301 0.115
ptest , type 2 Test result accuracy, type 2 0.2084 0.193
ptest , type 3 Test result accuracy, type 3 0.7668 0.222
Ïƒb Standard deviation of reporting error 0.25 0.009
Table 11: Maximum Likelihood Parameter Estimates: Type Distribution Parameters
Type 1 Type 2 Type 3
Parameter Estimate SE Estimate SE Estimate SE
Constant -1.748 2.614 -5.543 30.05 -3.207 2.593
Balaka -1.275 1.297 -0.39 29.87 -1.685 1.324
Rumphi -3.775 2.453 5.68 29.13 -2.887 2.086
Primary -0.8293 1.412 -3.453 3.112 1.31 1.744
Secondary -1.93 2.569 -10.28 345.8 -0.4232 2.914
Polygamy 1.688 1.112 1.793 1.011 1.03 1.128
Age of marriage 0.2263 0.242 -0.0397 0.296 0.3614 0.225
Year of birth 0.2532 0.154 0.125 0.197 0.2245 0.151
N (t) Ã— t -0.0011 0.011 -2E-05 0.019 -0.0036 0.010
42
Table 12: Predicted Selected Characteristics by Unobserved Type
Type 0 Type 1 Type 2 Type 3
Region Balaka 0.27 0.46 0.02 0.24
Mchinji 0.10 0.42 0.01 0.46
Rumphi 0.63 0.11 0.97 0.30
Schooling None 0.18 0.29 0.07 0.05
Primary 0.62 0.62 0.93 0.85
Secondary 0.20 0.09 0.001 0.10
Polygamous 0.25 0.32 0.6 0.25
Year of Birth 1971 1981 1975 1979
Age of Marriage 17.3 17.1 16.9 17.9
Prob. assigned to being HIV-positive :
Age 20 0 0.16 0.59 0.01
Age 30 0 0.29 0.65 0.03
Age 40 0 0.39 0.69 0.13
Share of sample 0.28 0.23 0.02 0.47
Figure 4: Model Fit: Actual and Predicted Reported Beliefs
Figure 5: Model Fit: Pregnancy Probabilities, by Age Groups
43
Table 13: Model Fit: Actual and Predicted Pregnancy Probabilities
Age Group
15-19 20-24 25-29 30-34 35-39 40-44
Region Balaka A 0.414 0.398 0.322 0.312 0.305 0.086
P 0.408 0.381 0.336 0.305 0.250 0.117
N 162 226 155 138 105 58
Mchinji A 0.369 0.357 0.32 0.268 0.192 0.104
P 0.366 0.350 0.308 0.282 0.231 0.082
N 130 319 291 168 99 67
Rumphi A 0.433 0.382 0.282 0.256 0.183 0.043
P 0.444 0.393 0.290 0.238 0.1743 0.0468
N 120 233 252 223 131 161
Schooling None A 0.375 0.492 0.27 0.264 0.343 0.085
P 0.474 0.388 0.308 0.307 0.254 0.107
N 24 61 89 91 70 59
Primary A 0.395 0.366 0.315 0.283 0.195 0.063
P 0.397 0.367 0.307 0.268 0.207 0.062
N 332 596 520 375 241 208
Secondary A 0.482 0.372 0.292 0.238 0.167 0.045
P 0.428 0.388 0.309 0.228 0.177 0.036
N 56 121 89 63 24 22
Polygamy Mono A 0.412 0.388 0.309 0.286 0.243 0.083
P 0.414 0.378 0.323 0.280 0.225 0.083
N 354 605 511 392 214 156
Poly A 0.362 0.335 0.299 0.241 0.19 0.046
P 0.355 0.350 0.264 0.241 0.196 0.053
N 58 173 187 137 121 130
A = actual, P = predicted, N = number of observations
44
Table 14: Selected Characteristics of Subsample Used for Counterfactual Simulations
All By Type
Variable 0 1 2 3
Type 0 0.11 - - - -
1 0.32 - - - -
2 0.02 - - - -
3 0.55 - - - -
Region Mchinji 0.38 0.05 0.44 0.01 0.43
Balaka 0.31 0.21 0.43 0.01 0.27
Rumphi 0.31 0.73 0.13 0.98 0.3
Schooling None 0.08 0.05 0.2 0.01 0.02
Primary 0.76 0.48 0.69 0.99 0.85
Secondary 0.15 0.47 0.10 0 0.12
Polygamous 0.2 0.15 0.25 0.53 0.18
Number 50,900 6,459 16,472 850 27,119
Table 15: Counterfactual Simulations
By Type
All 0 1 2 3
Panel A: Baseline (HIV, without any HIV tests)
Probability assigned to being HIV-positive:
Age 17 0.04 0 0.11 0.50 0.003
Age 25 0.09 0 0.23 0.64 0.02
Age 35 0.16 0 0.35 0.65 0.07
HIV prevalence:
Age 17 0.01 0.004 0.03 0 0.006
Age 25 0.04 0.01 0.10 0 0.02
Age 35 0.06 0.02 0.13 0 0.03
Number of life-cycle births 7.07 6.13 6.28 6.47 7.72
Child Mortality 1.24 1.03 1.22 1.06 1.29
Panel B: No HIV
Number of life-cycle births 7.22 6.16 6.44 6.61 7.90
Child Mortality 1.14 0.99 1.02 1.06 1.24
Panel C: No mother-to-child transmission
Number of life-cycle births 7.05 6.13 6.24 6.46 7.72
Child Mortality 1.13 1.00 1.00 1.06 1.23
45
All Type 0 Type 1 Type 2 Type 3
Panel D: HIV test at period of marriage
Probability assigned to being HIV-positive:
Age 17 0.04 0 0.11 0.44 0.004
Age 25 0.09 0 0.23 0.59 0.02
Age 35 0.16 0 0.35 0.63 0.07
Number of life-cycle births 7.07 6.13 6.28 6.47 7.72
Child Mortality 1.24 1.03 1.22 1.05 1.29
Panel E: HIV test at age 25
Probability assigned to being HIV-positive:
Age 25 0.09 0 0.23 0.51 0.02
Age 35 0.15 0 0.34 0.58 0.07
Number of life-cycle births 7.07 6.13 6.28 6.47 7.72
Child Mortality 1.24 1.03 1.22 1.06 1.29
Panel F: HIV test at age 35
Probability assigned to being HIV-positive:
Age 35 0.13 0 0.34 0.52 0.04
Number of life-cycle births 7.07 6.13 6.28 6.46 7.73
Child Mortality 1.24 1.03 1.22 1.06 1.29
Panel G: HIV test at age 25, assigning full accuracy to test results
Probability assigned to being HIV-positive:
Age 25 0.04 0.01 0.10 0 0.02
Age 35 0.14 0.01 0.29 0.42 0.07
Number of life-cycle births 7.06 6.13 6.27 6.50 7.72
Child Mortality 1.23 1.03 1.22 1.05 1.29
46
B Specication of the infection hazard function (Section
3.4)
The perceived probability of getting infected at period t, conditional on not getting infected
before is given by
1
h(t) = ,
1 + exp (âˆ’x(t) Î² )
where
x (t) Î² = Î²1 + Î²2,Âµ t + Î²3,Âµ t2 + Î²4 Marriage Duration + Î²5 Primary
+Î²6 Secondary + Î²7 Land High + Î²8 Poly + Î²9 Balaka + Î²10 Rumphi.
The parameters related to age and age squared are allowed to vary with unobserved type
(Âµ). As described in section 4.3, the same specication is used for the actual infection hazard
process, with h replaced by Ëœ
h and Î² replaced by Ëœ.
Î²
47
C Updating beliefs after testing (Section 3.5)
I want to solve for Ë† (Ï„ ) , Ï„ = 1, ..., ttest
P from the system of equations presented in Equation
10:
PË† (Ï„ ) S (Ï„, ttest )
Ë† (ttest ) ,
= G (Ï„, ttest ) B Ï„ = 1, ..., ttest
ttest Ë†
1âˆ’ k=1 P ( k ) (1 âˆ’ S ( k, ttest ))
Isolating Ë† (Ï„ ) :
P
Ë† (Ï„ ) S (Ï„, ttest ) = 1 âˆ’ ttest Ë† (ttest ) ,
Ë† (k ) (1 âˆ’ S (k, ttest )) G (Ï„, ttest ) B
P k=1 P Ï„ = 1, ..., ttest
ttest Ë† (ttest )
G(Ï„, ttest )B
Ë† (Ï„ ) = 1 âˆ’
P Ë† (k ) (1 âˆ’ S (k, ttest ))
P
k=1,k=Ï„ Ë† (ttest ) ,
S (Ï„, ttest )+(1âˆ’S (Ï„, ttest ))G(Ï„, ttest )B
Ï„ = 1, ..., ttest
Introducing notation:
Ë† (ttest )
G(Ï„, ttest )B
1. Ïˆ (Ï„, ttest ) = Ë† (ttest ) ,
S (Ï„, ttest )+(1âˆ’S (Ï„, ttest ))G(Ï„, ttest )B
Ï„ = 1, ..., ttest
2. Î¶ (Ï„, ttest ) = Ïˆ (Ï„, ttest ) (1 âˆ’ S (Ï„, t)) , Ï„ = 1, ..., ttest
ttest âˆ’1
3. Num(Ï„ ) = Ïˆ (Ï„, ttest ) i=1,i=Ï„ (1 âˆ’ Î¶ (i, ttest ))
1 ttest âˆ’1 ttest âˆ’1
4. Den (Ï„ ) =1âˆ’ 2 k=1 Î¶ (k, ttest ) i=1,i=k Î¶ (i, ttest )
ttest âˆ’1 ttest âˆ’1 ttest âˆ’1
+2
3 k=1 Î¶ (k, ttest ) i=1,i=k Î¶ (i, ttest ) m=1,m=k,m=i Î¶ (m, ttest )
ttest âˆ’1
âˆ’ Â· Â· Â· + [âˆ’1]ttest ttest âˆ’2
ttest âˆ’1 i=1 Î¶ (i, ttest )
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The solution to the system of equations is given by:
Ë† (Ï„ ) =
P Num(Ï„ )
, Ï„ = 1, ..., ttest âˆ’ 1
Den(Ï„ )
Ë† (ttest ) = Ïˆ (ttest , ttest ) 1 âˆ’ ttest âˆ’1 Ë† (k ) (1 âˆ’ S (k, ttest )) ,
P k=1 P Ï„ â‰¥ ttest
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D Probabilities of Actual HIV and Survival History (Sec-
tion 4.3)
Let Ëœ (t)
h be the actual HIV infection hazard rate. The functional form is assumed to be
similar to that of the perceived infection hazard described in equation (2). The probability
of getting infected at t, conditional on being HIV-negative until then, is given by
Ëœ (t) = 1
h .
Ëœ
1 + exp âˆ’x(t) Î²
The unconditional probability of getting infected at period t is given by
tâˆ’1
P Ëœ (t)
Ëœ (t) = h Ëœ (k ) .
1âˆ’h
k=1
Information about the hazard process is contained in the HIV test results, the age in
which the tests were taken, and the ages in which a woman is last observed (regardless of
testing histories). Let ti represent the age in which a woman is last observed (and is therefore
âˆ’
known to be alive at that age). Let ti be the oldest age in which a woman is observed to get
a negative test result. Let t+
i be the youngest age a woman is observed to receive a positive
test result. Let Hi = tâˆ’ +
i , t i , ti be the vector of observed HIV history of woman i, with tâˆ’
i
+
(ti ) equaling zero if a woman is never tested negative (positive). Women's HIV histories
belong to one of the following 4 categories:
1. Never tested:
The probability of observing a woman who was never tested alive at ti , conditional on
her initial conditions and unobserved type, is given by
ti
Pr Hi = 0, 0, t | â„¦d t =1âˆ’ Ëœ k | â„¦d , 1 âˆ’ S (k, ti ) .
i, typei = j = Pr alive at P i typei =j
k=1
2. Tested only negative:
The probability of observing a woman who was only tested negative and is alive at ti ,
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conditional on her initial conditions and unobserved type, is given by
Pr Hi = tâˆ’ , 0, t | â„¦d
i, typei = j = Pr negative at tâˆ’ , alive at t
tâˆ’ Ëœ (k) t Ëœ k | â„¦d , k Ëœ j | â„¦d ,
= k=1 1âˆ’h 1âˆ’ k=tâˆ’ +1 h i typei =j j =tâˆ’ +1 1âˆ’h i typei =j 1 âˆ’ S k, t
tâˆ’ Ëœ (k) âˆ’ t Ëœ k | â„¦d ,
= k=1 1âˆ’h k=tâˆ’ +1 P i typei =j 1 âˆ’ S k, t .
3. Tested only positive:
The probability of observing a woman who if infected by t+ and alive at t, conditional
on her initial conditions and unobserved type, is given by
Pr Hi = 0, t+ , t | â„¦d
i, typei =j = Pr got infected before t+ , alive at t
t+
= Ëœ k | â„¦d ,
P typei = j S k, t .
i
k=1
4. Seroconverter:
The probability of observing a woman who is known to be HIV-negative until tâˆ’ ,
positive by t+ , and alive at t tâˆ’ < t+ â‰¤ t , conditional on her initial conditions and
unobserved type, is given by
i , typei = j = Pr got infected between t and t , alive at t
Pr Hi = tâˆ’ , t+ , t | â„¦d âˆ’ +
t+ Ëœ k | â„¦d , typei = j S k, t .
= k=tâˆ’ P i
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