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Content
ESSAYS ON FAMILY PLANNING POLICIES
by
Fei Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
August 2014
Copyright 2014 Fei Wang
1
Acknowledgements
The most correct decision that I made six years ago is to have Professor John Strauss as
my advisor. He is a big name in Development Economics, and his mentoring is as
incomparable as his academic achievements. He has been giving uncountable advice,
from how to think, to how to word. I am going to start my academic career soon, and
will also have my own students. I think the best way of saying thanks to John is to
become someone like him, an accomplished scholar and a conscientious mentor.
I also greatly appreciate the advice of many professors, including but not limited
to Caroline Betts, Eileen Crimmins, Richard Easterlin, Cheng Hsiao, Adriana Lleras-
Muney, Roger Moon, Jeff Nugent, Anant Nyshadham, Geert Ridder, Guofu Tan,
Guillaume Vandenbroucke, and Simon Wilkie.
Department of Economics is a warm family. Without Young Miller and Morgan
Ponder, my life would be a mess. I enjoy fighting on with Will Kwon and Jaime Meza,
the other two graduating students of John. I am also excited to collaborate with Robson
Morgan, Brijesh Pinto, and Malgorzata Switek on academic projects. Knowing all my
friends in the department, too many to be listed, is one of the most amazing experiences
in my life.
This paper is particularly for my parents. They were exposed to the Great Famine
as young kids, to the Cultural Revolution as teenagers, and to the One-Child Policy as
parents. Without them, I myself would never have chances to make the Quantity-
Quality Trade-off Hypothesis a reality.
2
Table of Contents
Acknowledgements ........................................................................................................................ 1
List of Figures ................................................................................................................................... 3
List of Tables ..................................................................................................................................... 4
Abstract .............................................................................................................................................. 5
Chapter 1 Introduction ................................................................................................................ 6
Chapter 2 History of China's Family Planning Policies ................................................... 12
2.1 Period 0: Without family planning policies (1949-1962) .............................................. 15
2.2 Period 1: Mild and narrowly implemented family planning policy (1963-1970) 17
2.3 Period 2: Strong and widely implemented family planning policy (1971-1979) . 18
2.4 Period 3: One-child policy (since 1980) .............................................................................. 20
Chapter 3 Static Analysis ........................................................................................................... 23
3.1 Review of measures of China’s family planning policies .......................................... 24
3.2 Data ................................................................................................................................................ 29
3.3 Empirical strategy and policy measurement .................................................................... 33
3.4 Empirical results ........................................................................................................................ 40
3.4.1 Estimation results .............................................................................................................................. 40
3.4.2 Partial effects of family planning policies on the number of births .......................... 42
3.4.3 Simulated over-cohort fertility under several scenarios of policy history ............ 45
3.4.4 Policy effects by schooling and gender of the first birth ................................................ 49
3.4.5 Sensitivity of results to different policy measures ............................................................ 53
3.5 Conclusion ................................................................................................................................... 63
Chapter 4 Dynamic Analysis .................................................................................................... 64
4.1 Literature review on the application of duration analysis.......................................... 65
4.2 Data ................................................................................................................................................ 67
4.3 Duration model and policy measurement ........................................................................ 70
4.3.1 Duration model .................................................................................................................................. 70
4.3.2 Policy measurement ......................................................................................................................... 77
4.3.3 Other covariates ................................................................................................................................. 78
4.4 Empirical results ........................................................................................................................ 80
4.4.1 Estimation results .............................................................................................................................. 80
4.4.2 Policy effects on the probability of having a certain number of births ................... 84
4.4.3 Heterogeneous policy effects by urban-rural and ethnicity ......................................... 88
4.4.4 Policy effects on fertility decline by cohort ........................................................................... 90
4.4.5 Model without frailty .................................................................................................................... 100
4.5 Conclusion ................................................................................................................................ 101
Chapter 5 Conclusion ............................................................................................................... 102
Reference ...................................................................................................................................... 104
Appendix A ................................................................................................................................... 110
Appendix B ................................................................................................................................... 112
3
List of Figures
Figure 1.1 Total fertility rate of China by year .................................................................... 7
Figure 2.1 Overall, urban, and rural total fertility rates of China by year .................... 14
Figure 2.2 Overall, Han, and non-Han total fertility rates of China by year ................ 15
Figure 3.1 Number of births by cohort of mothers, CHNS versus census ................... 33
Figure 3.2 Probability of conception by age ...................................................................... 37
Figure 3.3 Intensity of exposure to period 1, 2 and 3 policies by cohort of women .... 38
Figure 3.4 Number of births by cohort of women ............................................................ 46
Figure 3.5 Predicted number of births under different policy histories ....................... 47
Figure 3.6 Predicted number of births with actual schooling and no schooling ......... 49
Figure 3.7 Probability of conception by age, separately plotted for urban Han women,
rural Han women, urban non-Han women, and rural non-Han women ............. 54
Figure 3.8 Probability of conception by age, separately plotted for cohorts 1950 or older,
cohorts 1951-60, cohorts 1961-70, and cohorts 1971 or younger ............................ 55
Figure 4.1 Distribution of durations between two births ................................................ 74
Figure 4.2 Comparison between predicted probabilities and actual fractions, by cohort
of women ........................................................................................................................ 86
Figure 4.3 Predicted probability of childlessness under different policy histories ..... 92
Figure 4.4 Predicted probability of having exactly 1 birth under different policy histories
.......................................................................................................................................... 93
Figure 4.5 Predicted probability of having exactly 2 births under different policy
histories .......................................................................................................................... 94
Figure 4.6 Predicted probability of having exactly 3 births under different policy
histories .......................................................................................................................... 95
Figure 4.7 Predicted probability of having 4 or more births under different policy
histories .......................................................................................................................... 96
Figure 4.8 Actual number of births, and the lower and upper bounds of predicted
number of births, by cohort of women ...................................................................... 98
Figure 4.9 Predicted number of births under different policy histories ....................... 99
4
List of Tables
Table 2.1 Secular and cross-sectional variations of birth quota ..................................... 22
Table 3.1 Descriptive statistics of selected variables for static analysis ........................ 31
Table 3.2 OLS regressions of number of births on determinants ................................... 40
Table 3.3 Effect of each policy on the number of births of each group of women ...... 43
Table 3.4 OLS regressions of number of births on determinants, by women’s schooling
and gender of the first birth ................................................................................................. 50
Table 3.5 Effect of each policy on the number of births of each group of women, by
women’s schooling and gender of the first birth ............................................................. 52
Table 3.6 OLS regressions of number of births on determinants, using different
probabilities of conception to construct policy measures ............................................... 56
Table 3.7 Effect of each policy on the number of births of each group of women, by the
probability of conception used to construct policy measures ........................................ 58
Table 3.8 OLS regressions of number of births on determinants, using incomplete
measures or measures lacking heterogeneity ................................................................... 60
Table 3.9 Effect of each policy on the number of births of each group of women, using
incomplete policy measures or measures lacking heterogeneity .................................. 62
Table 4.1 Number and fraction of women with certain number of birth spells .......... 68
Table 4.2 Descriptive statistics of selected variables for spells 1 to 4 ............................ 69
Table 4.3 Estimation results of the duration model ......................................................... 80
Table 4.4 Specification tests: 3-point frailty, 2-point frailty, or no fraity? ..................... 83
Table 4.5 Comparison between predicted probabilities and actual fractions, full and
subsamples ............................................................................................................................. 85
Table 4.6 Policy effects on the probability of having a certain number of births ........ 88
Table 4.7 Policy effects on the probability of having a certain number of births, by
urban-rural and ethnicity ..................................................................................................... 89
Table 4.8 Policy effects on the probability of having a certain number of births, models
without frailty versus with 3-point frailty....................................................................... 100
5
Abstract
This paper estimates the effect of family planning policies on fertility in China. Both
static and dynamic analyses are conducted. The static analysis estimates the policy
effects on the number of births that a woman has ever had, using a cross-sectional
sample of ever-married women with their birth records, from the China Health and
Nutrition Survey (CHNS). The dynamic analysis explores the relationship between
family planning policies and the likelihood of childbearing over time, using a panel
sample of CHNS that records women’s births in each year since age 15. The dynamic
analysis applies a multiple-spell mixed-proportional hazard model where the
unobserved individual heterogeneity is non-parametrically estimated, as suggested by
Heckman and Singer (1984). Both static and dynamic analyses show that, family
planning policies could explain about half of the fertility decline between cohorts 1943
and 1972 in the sample. Both analyses also improve policy measures adopted by
previous studies, and make them reflect more complete policy history, and capture
more heterogeneity of policy exposure. Particularly, the static analysis shows that,
different policy measures could lead to substantially different results, which highlights
the importance of measuring family planning policies appropriately.
6
Chapter 1 Introduction
After World War II, family planning programs, aiming to lower birth rates, started to
prevail in the developing world, triggered by popular beliefs that rapid population
growth could obstruct the economic development of developing countries (Bongaarts,
Mauldin, and Phillips, 1990; Lapham and Mauldin, 1985; Szreter, 1993). Soon
afterwards, fertility rates substantially dropped in many developing countries during
the 1970s and 1980s, particularly in Asia and Latin America (Bongaarts, Mauldin, and
Phillips, 1990; Lapham and Mauldin, 1985). The concurrence of family planning
programs and demographic transitions has spawned a large number of studies that
attempted to explore their relations.
Such coexistence also emerges in China, the most populous country and the
largest developing economy. China initiated family planning policies in 1963. The
policies evolve over three periods, from a mildly and narrowly enforced policy in the
1960s, to a widely and strongly implemented policy in the 1970s, and eventually
transformed to the harshest one-child policy in 1980. In parallel, Figure 1.1 illustrates
China’s total fertility rates from 1949 to 2001. During the 1950s and 1960s, fertility rates
stayed at high levels, except for a dip in 1959-1961 caused by a great famine. Since 1970,
fertility rates have dived from 6 to nearly 2 within just 10 years, and continued to
decline after 1980.
1
1
The most recent officially released total fertility rate, 1.18 in 2010, failed to convince many scholars who believed the
true figure should have been higher. Nevertheless, it has been a consensus that China’s current fertility rate is below
the replacement level.
7
Figure 1.1 Total fertility rate of China by year
Note: Data originally came from national surveys conducted by China’s statistical authorities, and were
collected by Yang (2004, pp. 264– 265).
This paper explores to what extent China’s family planning policies could
explain its fertility transition. Previous studies fail to reach an agreement on the issue.
Many papers conclude that family planning policies explained a sizable fraction of
China’s fertility decline. However, other studies argue that the policy effects had been
overstated.
The disagreement partly results from different ways of measuring family
planning policies. Furthermore, the policy measures generally have shortcomings.
Previous studies might set up policy measures based on incomplete policy history, or
apply endogenous measures to estimation. Moreover, their measures in general ignore
8
people’s heterogeneous exposure to policies. This paper tries to unify and improve
previous measures. First, this paper creates policy measures by taking advantage of
more complete policy variations over time and across people. Second, policy measures
will be mainly constructed on mothers’ birth cohort, which is exogenous. Third, this
paper heterogenizes policy measures by the length of mothers’ policy exposure and the
age at which they were exposed to policies. Such heterogeneity is characterized based
on women’s probability of conception by age. Estimation results are robust to the use of
different probabilities of conception.
Using the improved policy measures, this paper first estimates policy effects on
the total number of births that a woman has ever had. This static analysis uses a cross-
sectional sample of ever-married women from the China Health and Nutrition Survey
(CHNS). The sample records the entire birth history of a woman up to the year when
she was surveyed, as well as other demographic and socioeconomic variables.
Regressing the number of births on family planning policies and other
determinants, the static analysis draws several conclusions. First, with actual policy
exposure, the predicted number of births dropped by about 2 from cohort 1943 to
cohort 1972. While had there been no family planning policies, the number of births
would have declined by around 1 for the same cohort span. Therefore, family planning
policies could explain about half of the fertility decrease between cohorts 1943 and 1972
in the sample. As a comparison, women’s schooling could explain about 10% of the
fertility decline for the range of cohorts. Second, policy effects tend to be smaller for
more-educated women or a woman whose first birth is a son, than less-educated
9
women or a woman whose first birth is a daughter, probably because the former desire
fewer births than the latter, and thus receive lighter pressure from the policies. Third,
given the data and empirical specifications, using defective policy measures that are
adopted by previous studies may generate substantially different policy effects.
Because such static analysis is incapable to capture the dynamic process of
childbearing and other factors, the paper continues to explore the policy effect on
fertility with dynamic analysis, and adjusts the improved policy measures for the new
framework.
The dynamic analysis uses a panel sample, which is constructed from the cross-
sectional sample for the static analysis, and records women’s births and other
characteristics in each year since age 15.
The dynamic analysis applies a multiple spell duration model to evaluate policy
effects for the first four birth spells. The fertility outcome is a dummy variable
indicating whether a woman had a birth in some year. The duration model expresses
the probability of having a birth in some year as a non-linear function of observed
characteristics and unobserved individual heterogeneity. The observed variables
include family planning policies and other demographic and socio-economic factors
that have been considered to be related to fertility. Their coefficients are estimated
parametrically and are assumed to change over birth spells. The unobserved individual
heterogeneity is assumed to follow a mass-point distribution, and is estimated non-
parametrically. Specifically, the number of mass points and their locations and
10
probabilities are all estimated from the data, as suggested by Heckman and Singer
(1984).
Based on the estimated model, this paper derives the probability of childlessness,
having exactly 1, 2, 3, and 4 or more births. The predicted probabilities match well with
actual fractions in the data. By turning on and off the policy period by period, this paper
further calculates the probability difference with and without some period of policy.
Being exposed to the one-child policy reduces the probability of having exactly 2, 3, and
4 or more births by 12.4%, 72.1%, and 37.1%, respectively, and correspondingly
increases the probability of childlessness and having exactly 1 birth by 105.5% and
187.6%. Earlier periods of policy present similar patterns, but smaller effects. The
dynamic analysis further examines the policy effects by residential location and
ethnicity. Generally, the policy has stronger effects for urban women and ethnic
majorities than for rural women and minorities. These results are consistent with the
policy history.
Moreover, the dynamic analysis simulates fertility rates by birth cohort of
women under different policy histories. Conclusions are similar to the static analysis.
Family planning policies could explain about 40%-50% of the fertility decline between
cohort 1943 and 1972 in the sample. However, without any policy, fertility would still
have demonstrated a downward trend over cohorts.
In addition to family planning policies, other individual characteristics have
shown noticeable impact. Better-educated women tend to substantially decrease their
likelihood of childbearing. If a woman’s first birth is a son, she would be much less
11
likely to have a large number of births, which manifests the strong son preference in
China.
The whole paper is arranged as follows. Chapter 2 introduces the history of
China’s family planning policies. Chapter 3 conducts the static analysis. Chapter 4
continues with the dynamic analysis. Chapter 5 concludes.
12
Chapter 2 History of China's Family Planning Policies
China's Population and Family Planning Law
2
indicates:
The State adopts a comprehensive measure to control the size and raise the general
quality of the population. The State relies on publicity and education, advances in science and
technology, multi-purpose services and the establishment and improvement of the reward and
social security systems in carrying out the population and family planning programs.
The comprehensive measure, plainly speaking, chiefly comprises propaganda,
service and birth quota. Propaganda endeavors to convince the public that family
planning benefits the nation’s development and their own welfare. Local clinics offer
free contraceptives and low-priced family planning medical service. Being an exclusive
feature, birth quota limits the number of births per married couple. Those who comply
with the birth quota are rewarded, while violators are penalized.
According to the stringency and enforcement of the policy measures, China's
family planning policies can be segmented into four periods: the period without policies
(1949-1962), the period with mild and narrowly implemented policies (1963-1970), the
period with strong and widely enforced policies (1971-1979), and the period with the
harshest one-child policy (since 1980).
In each period, policies are tougher for urban people than for rural people,
because the ideologies of big families and son preference have been more deep-rooted
2
http://english.gov.cn/laws/2005-10/11/content_75954.htm. This paper studies family planning policies in the
People's Republic of China.
13
in rural areas. Moreover, policies are stricter for Han people
3
than for non-Han people
as family planning is less urgent for the latter whose fraction in the population is small.
The rest of this chapter introduces details for the policy evolution over periods,
4
and the urban-rural and ethnic policy differences, putting emphasis on birth quota.
Figure 2.1 and Figure 2.2 will further help visualize the policy history. Both figures
incorporate the overall total fertility rate (TFR) of China by calendar year. Figure 2.1
also shows urban and rural TFRs, and Figure 2.2 adds TFRs for Han and non-Han
people. Vertical dotted lines segment the policy periods.
5
3
Han is the major ethnicity of China. The 2010 Census of China indicated that 91.51% were Han Chinese
(http://en.wikipedia.org/wiki/Sixth_National_Population_Census_of_the_People's_Republic_of_China).
4
For convenience, number the four periods with 0, 1, 2 and 3.
5
The two figures will show the consistency between TFRs and policy history, rather than prove the causal effects of
policies on TFRs.
14
Figure 2.1 Overall, urban, and rural total fertility rates of China by year
Note: The overall total fertility rate is cited from Yang (2004, pp. 264-265). The urban and rural total
fertility rates are cited from Yang (2004, pp. 134, 135 and 139). Data originally come from various national
surveys conducted by China's statistical authorities. The left, middle and right dotted lines mark the years
of 1963, 1971 and 1980, and segment the whole history into four periods.
15
Figure 2.2 Overall, Han, and non-Han total fertility rates of China by year
Note: The overall total fertility rate is cited from Yang (2004, pp. 264-265). The Han (ethnic majority) and
non-Han (ethnic minority) total fertility rates are cited from Yang (2004, pp. 145 and 150). Data originally
come from various national surveys conducted by China's statistical authorities. The left and right dotted
lines mark the years of 1971 and 1980, the starting years of the period 2 and 3 family planning policies.
2.1 Period 0: Without family planning policies (1949-1962)
On the eve of the foundation of the People's Republic of China, the supreme leader Mao
Zedong publicly argued that China preferred a large population,
6
which also fit China's
traditional ideology of family size, Duo Zi Duo Fu (more children, more happiness).
Moreover, China was deeply influenced by a birth-encouraging policy of the Soviet
6
Mao said: “A large population is preferred in China. No matter how large it is, we can always handle it with
production…Human being is the most valuable resource of the world…Human can create any miracle.” (Yang, 2004,
pp. 43)
16
Union.
7
Consequently, in 1949-1953, China curbed measures of birth control and
financially subsidized large families.
8
Challenged by rapid population growth, in 1954, China began to terminate or
relax certain restrictions on birth control.
9
Meanwhile, a few public intellectuals, for
instance, Shao Lizi and Ma Yinchu, publicly advocated family planning.
10
Thereafter,
the knowledge of birth control spread through public media.
11
However, family
planning policies were not officially conceived.
In 1958, with the onset of the Great Leap Forward campaign that aimed to use
China's vast population to rapidly transform the country from an agrarian economy
into a modern communist society,
12
endorsing family planning was politically
incorrect.
13
The campaign was followed by a great famine (1959-1962), which caused a
dramatic TFR decline, from 5.679 in 1958 to 3.287 in 1961.
14
Under such circumstances,
family planning was rarely discussed. When the famine ended, women started to make
up fertility and the TFR rose back to 6.023 in 1962 and even shot up to 7.502 in 1963.
15
Figure 2.1 and Figure 2.2 illustrate this history. Figure 2.1 further shows that urban and
7
A mother bearing and raising a large family in the Soviet Union would be awarded an honorary title “Mother
Heroine” (Yang, 2004, pp. 44). Also see http://en.wikipedia.org/wiki/Mother_Heroine.
8
China strongly restricted sterilization and abortion, and strictly controlled the production and sale of contraceptives
which were prohibited from import (Yang, 2004, pp. 44-45).
9
In 1954, China removed restrictions on contraception and the sale of contraceptives, and relaxed restrictions on
abortion. However, sterilization was still under strong control (Yang, 2004, pp. 47-48).
10
Shao and Ma both supported contraception and late marriage, but disagreed on abortion (Yang, 2004, pp. 48-50,
52).
11
The knowledge of birth control only spread to some cities of some provinces (Yang, 2004, pp. 50, 53, 54, 58).
12
http://en.wikipedia.org/wiki/Great_Leap_Forward.
13
Yang (2004, pp. 59).
14
Yang (2004, pp. 61).
15
Yang (2004, pp. 61-62).
17
rural TFRs co-moved in the period, the urban TFR being slightly lower. Data on Han
and non-Han TFRs for this period were not found.
2.2 Period 1: Mild and narrowly implemented family planning policy (1963-1970)
In 1962, the Chinese government issued an instruction about the implementation of
family planning on December 18
th
, 1962, known as the No. [62] 698 document, which
marked the start of China's family planning policies.
16
The period 1 policy set a population growth target,
17
approved late marriage,
18
established family planning institutions,
19
and disseminated family planning
knowledge and technology.
20
Specific policies varied by province. For instance,
Shandong's policy could be summarized as “one (child) is not few, two are just right,
three are too many”. Shanghai's policy advocated that a married couple should not
have more than three children, birth spacing should be at least four years, and a
woman's age of having the first birth should exceed 26.
21
This paper assumes the birth
quota in period 1 to be 3. Although bearing more than three children was not
mandatorily prohibited, having a large family would result in political or social
pressure because the period 1 policy was promoted primarily through effective political
16
Yang (2010, pp. 27). Because the document was released in the end of 1962, I assume it came into effect from 1963.
17
The annual population growth rate targets were 2%, 1.5%, and 1% for the Recovery period (1963-1965), the third
Five-year plan (1966-1970), and the fourth Five-year plan (1971-1975), respectively (Yang, 2004, pp. 62).
18
The Ministry of Health proposed that, the marriage age for male and female should be 28 and 25, respectively. In
practice, age of late marriage varied by province (Yang, 2004, pp. 62, 68).
19
The national family planning institution was established in 1964 and local agencies have been founded
successively since 1963 (Yang, 2004, pp. 65).
20
Contraceptive knowledge and tools were available in local clinics. Restrictions on abortion and sterilization were
basically removed (Yang, 2004, pp. 65-67).
21
Yang (2004, pp. 68).
18
or social movements. Economic incentives were also established. For example, families
abiding by family planning policies would be subsidized in wages, housing and
commodities.
22
The period 1 policy was implemented for urban Han people only.
23
Figure 2.1 is
consistent with this history. On one hand, urban TFR started to decline in period 1;
while on the other hand, rural TFR remained high. As 80% of the population was in
rural areas,
24
the overall TFR stayed at high levels in this period.
The Cultural Revolution, launched in 1966, paralyzed family planning
institutions and disrupted the implementation of family planning policies. However,
the policy was not abolished and the urban TFR remained at low levels.
25
2.3 Period 2: Strong and widely implemented family planning policy (1971-1979)
Concerned about the negative impact of the Cultural Revolution, the Chinese
government issued a report on family planning in 1971, known as the No. [71] 51
document, to restate the importance of family planning. The report signified that family
planning policies recovered from the Cultural Revolution and stepped into a new
stage.
26
22
Yang (2004, pp. 64, 74).
23
Urban Han people living in the five autonomous regions were not exposed to family planning policies until the
beginning of 1970s (Yang, 2004, pp. 144-145). The five autonomous regions, Xinjiang, Inner Mongolia, Tibet, Ningxia,
and Guangxi, are provincial administrative areas where non-Han people agglomerate. See
http://en.wikipedia.org/wiki/Autonomous_regions_of_China for more details.
24
Yang (2004, pp. 69).
25
The urban TFR kept falling in this period expect in 1968. Moreover, family planning institutions started to restore
in 1969 (Yang, 2004, pp. 75).
26
Yang (2004, pp. 73).
19
Like period 1, the period 2 policy also comprised a population growth target and
contraceptive support.
27
Moreover, the policy became nationally uniform, known as
“late, long, few”. “Late” means late marriage and childbearing. The recommended
minimum age of marriage was 25 for men and 23 for women; women were suggested
having births after 24. “Long” means the birth spacing should be at least three years.
“Few” means a married couple could at most have two children.
28
The period 2 policy was stronger than the period 1. First, a married couple could at
most have two children in period 2, while three children were allowed, though
discouraged, in period 1. Second, the enforcement of period 2 policy was stricter. Mao
Zedong, the supreme leader, backed family planning policies in period 2,
29
and thus,
greatly bolstered its enforcement. The mode of agricultural production in this period
also discouraged large families. Farming jobs were centrally and equally assigned to
adult peasants, and they were not able to flexibly work more and earn more, in order to
raise more children.
30
In 1971, the policy began to spread to the urban Han people living in the five
autonomous regions
31
and to all rural Han people.
32
However, population growth
targets differed between urban and rural areas. The urban population annual growth
27
In 1971, the policy proposed that urban and rural population growth rates should reach down to 1% and 1.5% by
1975 (Yang, 2004, pp. 72). In 1978, the policy proposed that the national population growth rate should go below 1%
by 1981 (Yang, 2004, pp. 74). From 1974 on, 14 contraceptive pills or tools were supplied without charge (Yang, 2004,
pp. 76).
28
The policy was first implemented in some parts of China, and then was extended to the whole nation in 1973
(Yang, 2004, 73).
29
For example, in 1974, Mao said: “We must control the population.” (Yang, 2004, pp. 73)
30
Yang (2004, pp. 80, 135).
31
Yang (2004, pp. 144, 145).
32
Yang (2004, pp. 77-79).
20
rate was set to be 1% and the rural growth rate was 1.5%.
33
Figure 2.1 and Figure 2.2
have shown consistent facts. In Figure 2.1, the rural TFR started to drop in 1971, but was
higher than the urban TFR. Decreasing urban and rural TFRs also pulled down the
overall TFR. In Figure 2.2, the Han TFR dropped after 1971 and the TFR gap between
Han and non-Han people widened.
34
The urban or rural non-Han people who were
living outside the five autonomous regions were also affected by the period 2 policy.
2.4 Period 3: One-child policy (since 1980)
As a natural evolution of the period 2 policy, the one-child policy was conceived in 1979
and was officially launched in 1980.
35
The one-child policy, as its name suggests, allows
a married couple to have only one child, particularly designed for Han families. This
policy is stricter than previous versions.
The strictness of the one-child policy is also enhanced by its enforcement.
Previous policies were mainly implemented with political, social, or administrative
measures. In 1978, family planning policies, for the first time, appeared in the
Constitution and more details were added to the 1982 amended Constitution. Since late
1980s, the central and local governments have successively legislated family planning.
36
33
Yang (2004, pp. 72).
34
The non-Han TFR also declined, even though non-Han people were not officially constrained by family planning
policies in period 2 (Yang, 2004, pp. 143-145), implying externalities of family planning policies or the impact of other
factors.
35
In September 1980, the CPC central committee published an open letter to expound the necessity of the one-child
policy. This event was usually considered as the starting point of the one-child policy (Yang, 2004, pp. 86).
36
Yang (2004, pp. 161).
21
Legal measures, such as monetary penalties and subsidies,
37
have ensured the effective
enforcement of the one-child policy.
In early 1980s, the one-child policy was successfully implemented for urban Han
families, but received large resistance from rural Han people.
38
Subsequently, in the
mid-1980s, the one-child policy was relaxed for rural Han families and they were
allowed to have a second child if certain conditions were met. For example, they could
have the second birth when the first child was a daughter.
39
Figure 2.1 shows that both
urban and rural TFRs stayed at low levels, but with a gap.
In 1982, the policy started to cover most non-Han people, but in more relaxed
forms. In general, an urban non-Han family could conditionally have two children and
a rural non-Han couple might be allowed to have three or even more children. For the
ethnic groups with small population sizes, the policy is even further relaxed.
40
In Figure
2.2, the Han TFR remained low and the non-Han TFR substantially dropped in 1980-
1989. The Han and non-Han TFR gap, though smaller, still exist.
The secular and cross-sectional variations of the birth quota are summarized in
Table 2.1.
37
Families with only one child would receive the one-child subsidy. While families which illegally had more births
would have to pay fines. Fines are multiple times of local annual income, but standards are different across provinces.
For example, Beijing women who illegally have the second birth would have to pay fines that are three to ten times of
the local average annual income. McElroy and Yang (2000), and Li and Zhang (2009) discuss relevant topics.
38
Yang (2004, pp. 86).
39
Yang (2004, pp. 87). This case reflects strong son preference in rural areas.
40
Yang (2004, pp. 146-148).
22
Table 2.1 Secular and cross-sectional variations of birth quota
Period 1 (1963-70) Period 2 (1971-79) Period 3 (since 1980)
Urban Han
Mild policy allowed
but discouraged three
children*
Strong policy allowed
but discouraged two
children
One-child policy
allows only one child
Rural Han No restriction
Milder policy than the
urban Han version
One-child policy
conditionally allows
two children
Urban non-Han No restriction** No restriction**
Specific policy
conditionally allows
two children
Rural non-Han No restriction No restriction**
Specific policy
conditionally allows
three or even more
children
Note: * The urban Han people living in the five autonomous regions were not exposed to family planning
policies in period 1.
** Part of the non-Han people might be affected by family planning policies, particularly when they were
living outside the five autonomous regions.
Over time, the birth quota and its enforcement get stronger. Within each period,
the birth quota is more stringent for urban and Han people, than for rural and non-Han
people. The evolution of propaganda and service has not been as sharp as the birth
quota, but their secular and cross-sectional patterns are similar. Throughout this paper,
I will treat the history of the birth quota as the history of family planning policies.
41
41
I will mainly use urban-rural and ethnicity to capture the cross-sectional variations of family planning policies.
Policies also varied across provinces, but information is insufficient to specify provincial policy differences,
particularly for early periods. Fines differentials could have been used as an alternative, but data are not available for
the period 1 and 2 policies either. Moreover, recent policy changes are not considered in the paper. For example, if
one of a married couple is the only child, the couple could have a second birth.
23
Chapter 3 Static Analysis
This chapter analyzes the effect of family planning policies on the number of births that
a woman has ever had, using a cross-sectional sample of ever-married women. This
analysis is static.
The contribution of family planning policies to China’s fertility transition has
been under debate. Many papers conclude that family planning policies explain a
sizable portion of China’s fertility decline, including Lavely and Freedman (1990), Yang
and Chen (2004), and Li, Zhang and Zhu (2005). However, other studies argue that the
impact of family planning policies on fertility has been overstated (Cai (2010), McElroy
and Yang (2000), Narayan and Peng (2006), and Schultz and Zeng (1995)).
One source of the disagreement is that those studies measure family planning
policies differently. This analysis attempts to highlight the importance of measuring
family planning policies, and tries to unify and improve the policy measures adopted
by previous studies. Chapter 3.1 reviews the static analyses on the fertility effect of
China’s family planning policies, underscoring the shortcomings of their policy
measures. Chapter 3.2 introduces the data used in this chapter. Chapter 3.3 shows
model specifications, and the way of constructing policy measures. Chapter 3.4 displays
and interprets the empirical results. Chapter 3.5 concludes.
24
3.1 Review of measures of China’s family planning policies
Quantitative analyses of family planning policies require appropriate policy measures.
Previous studies mainly construct policy measures with general demographic variables,
based on secular and cross-sectional policy variations. Such measures are named
“constructed measures” in the paper.
Previous constructed measures are generally incomplete, sometimes endogenous,
and usually lack heterogeneity. An incomplete measure ignores part of the policy
variations and tends to underestimate the policy effect. Some constructed measures
reflect a part of secular policy variations, but fail to take cross-sectional variations into
account. Yang and Chen (2004) use the 1992 Household and Economy Fertility Survey
(HESF) sample to assess the fertility effect of family planning policies. They apply the
year dummies of being married, from 1970 to 1989, to capture policy effects for different
marriage cohorts. Narayan and Peng (2006) use time series data and models to estimate
policy effects on fertility. Their policy measures are time dummies for two periods,
1970-1979 and 1980-2000. Similarly, Edlund, Li, Yi and Zhang (2008) measure policies
with a dummy variable of being exposed to the one-child policy one year prior to a
mother’s childbearing.
Some studies utilize cross-sectional policy variations, but fail to capture secular
evolution. Cai (2010) uses a county level cross-sectional data of Jiangsu and Zhejiang,
collected from the 2001 statistical yearbooks of the two provinces and the 2000 census
compilations, to estimate the effect of family planning policies on fertility. He measures
25
policies with the percentage of population with agricultural hukou
42
and the percentage
of Han population in each county.
More studies take both secular and cross-sectional policy variations into account,
but either omits a part of the urban-rural or ethnic variations or a part of the policy
change over time. Li, Zhang and Zhu (2005) apply a difference-in-difference approach
to assess the impact of the one-child policy on fertility. The treatment and control
groups are Han and non-Han people. The pre-treatment and post-treatment samples
are taken from the 1982 and 1990 census, respectively. Li and Zhang (2007) use a
provincial panel data involving 28 provinces over 20 years (1978–1998) to estimate the
effect of birth rate on economic growth. In their first stage regressions, they use the
percentage of non-Han people in each province/year to instrument the birth rate. Li
and Zhang (2009) also adopt a difference-in-difference approach in the first stage
regressions, similar to Li, Zhang and Zhu (2005). Qian (2009) uses an individual level
cross-sectional sample from the 1990 census and the 1989 CHNS to test the quantity-
quality trade-off hypothesis. In the first stage regressions, she instruments the number
of an individual’s siblings with a triple interaction of the individual’s gender, year of
birth and region of birth. This identification strategy is essentially difference-in-
difference-in-difference. First, under the one-child policy, rural parents can have the
second birth if the first is a girl. Second, such policy relaxation varies by region. Third, if
42
Hukou is a household registration system in China. In general, urban and rural people have non-agricultural and
agricultural hukou, respectively. Rural people who temporarily migrate to urban areas generally keep their
agricultural hukou as before.
26
the individual was born in early years, his/her parents may not be influenced by the
one-child policy. Islam and Smyth (2010) use the 2008 China Health and Retirement
Longitudinal Survey (CHARLS) data to estimate the effect of number of children on
parental health. In the first stage regressions, similar to Qian (2009), they instrument the
number of children with a triple interaction of a rural dummy, child’s gender and
child’s birth period. Compared to Qian (2009), they do not take advantage of the policy
variations among rural regions, but instead, utilize the urban-rural policy difference.
Banerjee, Meng and Qian (2010) use an individual level cross-sectional data, collected in
the 2008 Urban-Rural Migration in China and Indonesia Survey (RUMiCI), to study the
impact of number of children on parental saving behaviors. In their first stage
regressions, they instrument the number of children with a dummy variable indicating
if a child was born after 1971, starting year of the period 2 policy, and its interaction
with the child’s gender. Wu and Li (2012) use an individual level panel sample with five
waves from the CHNS data to assess the effect of family size on maternal health. In
their first stage regressions, they construct a time variable about the one-child policy,
interact it with urban dummy and Han dummy, and used them to instrument family
size.
Some measures are endogenously constructed, which may bias the effect of
family planning policies on fertility or invalidate policy variables as instrumental
variables for fertility. As reviewed above, Yang and Chen (2004) use year dummies of
being married to capture secular policy variations. Similarly, Banerjee, Meng and Qian
(2010), Edlund, Li, Yi and Zhang (2008), Islam and Smyth (2010), and Qian (2009) take
27
advantage of actual timing of childbearing to measure policy exposure. However, the
timing of marriage and childbearing is endogenous and might be correlated with
unobserved factors related to fertility.
In addition to suffering from endogeneity, those measures also lack
heterogeneity. For example, a 20-year-old woman and a 40-year-old woman may both
bear a child during the one-child policy. If a single dummy variable is used to measure
the policy exposure, then the value of exposure for both women would be 1. However,
intuitively, the younger woman should have larger exposure to the policy because she
will be affected by the policy nearly through out her entire childbearing period, while
the older woman has almost physiologically finished childbearing during the policy
period. Wu and Li (2012) construct a more heterogeneous measure, which is
proportional to the length of time exposed to the policy, but is not as precise as it could
be. For example, if a woman is exposed to the policy between 20 and 30 years of age and
another is exposed between 30 and 40 years of age, their measures will be assigned with
the same value. However, the first woman arguably has a larger exposure because 20-30
is the peak interval of childbearing, while 30-40 is not. The Wu-Li measure does not
consider such heterogeneity.
This chapter will try to make the constructed measures more complete,
exogenous and heterogeneous. Further, Chapter 3.4.5 shows how the results may
change if alternative policy measures are used.
Other than constructed measures, a few studies also use specific measures for
family planning policies.
28
Specific measures directly come from data that contain specific information on
family planning policies. For example, Schultz and Zeng (1995) use individual level
cross-sectional data for rural areas of three provinces in China, which were collected in
the 1985 In-Depth Fertility Survey (IDFS), to assess the effect of local family planning
and health programs on fertility. Family planning policies are measured by the
availability of a family planning service station, a family planning outreach worker, a
doctor or nurse, and a local clinic in a rural village.
43
McElroy and Yang (2000) use
household level cross-sectional data for rural areas across ten provinces, which were
collected in the 1992 Household Economy and Fertility Survey (HESF), to estimate the
intensity of county-level family planning policies on the number of children per family.
The HESF sample contains county-level monetary penalties imposed on “over-quota”
births and they are used to measure the county-level policy intensity. Li and Zhang
(2008) use an individual level cross-sectional data, collected in the 1989 CHNS, to study
how birth behaviors of a woman are affected by the birth behaviors of her neighbors. In
the first stage regressions, they measure the one-child policy with community-level
monetary penalties on “over- quota” births and subsidies for one-child families, and
used them to instrument the fertility of neighbors, similar to McElroy and Yang (2000).
Huang, Lei and Zhao (2014) find that the one-child policy could explain over one third
of the increase in twin births in China since 1970s, because parents may have registered
single births as twins in order to avoid monetary penalties of the one-child policy,
43
The availability was measured by dummy variables. The interactions of the dummy variables were also controlled
for in regressions.
29
under which having twins is legal. They measure the one-child policy with provincial
level fines from 1979 to 2000 for 30 provinces.
Specificity is one of the most notable advantages of such measures as they are so
detailed that they can hardly be contaminated by irrelevant factors. However, detailed
information of monetary penalties and subsidies prior to the one child policy period is
even not existent. Therefore, this chapter will not discuss specific measures.
3.2 Data
The section uses the birth history data from the China Health and Nutrition Survey
(CHNS).
44
The ongoing CHNS is one of the most widely used micro-data about China.
Conducted by an international team, the CHNS collects information on household and
individual economic, demographic, and social variables, particularly the factors about
health and nutrition. Surveys were conducted in 1989, 1991, 1993, 1997, 2000, 2004, 2006,
2009, and 2011 across twelve provinces.
45
A large group of interviewees have been
followed longitudinally.
The CHNS surveys ever-married women, below 52 years of age,
46
about their
birth history, in 1991, 1993, 2000, 2004, 2006, 2009, and 2011. A woman may be tracked
wave-wise. The CHNS team combined the birth history data of all waves, kept only the
44
More information about the CHNS can be found on the 29official website:
http://www.cpc.unc.edu/projects/china.
45
Before wave 2000, the survey covered eight provinces: Guangxi, Guizhou, Henan, Hubei, Hunan, Jiangsu,
Liaoning, and Shandong. Heilongjiang was included in wave 2000 and thereafter. Beijing, Chongqing, and Shanghai
were further included in wave 2011.
46
The surveyed women were under 50 in wave 1991. Although only the women under 52 (or 50) should be surveyed,
a few women in the sample were above the supposed age during the survey. I kept those observations to enlarge the
sample, after checking their validity.
30
latest wave of record for each woman who has ever been survey, and released the
refined cross-sectional data online in 2013.
47
The cross-sectional data contains the birth
history of a woman up to the latest wave of survey for her. I restrict the birth history
data to women aged 15 or above during the survey. To rule out extreme cases, I further
drop the women who ever had births below 15 or above 49 from the data.
48
The birth history data includes the date of birth, gender, living arrangement, and
date of death of every child that a woman has ever had and allows us to map the
history of family planning policies onto the entire childbearing process. Other
demographic and socioeconomic variables can be found from other modules of the
CHNS. For currently married women, the information of their husband can be obtained.
Only ever-married, but not all women were asked about their birth history
because marriage is a pre-condition for childbearing in China, both traditionally and
legally. Based on the data used in the paper, the proportion of non-marital childbearing
is below 5% and has no rising trend over cohorts, which is different from what Hotz,
Klerman and Willis (1997) present about non-marital childbearing in the U.S.
49
Table 3.1 shows descriptive statistics of selected variables for ever-married
women in the sample.
47
The data is named “m12birth”, and was released in January 2013 on the official website of the CHNS. The data
doesn’t contain the information of survey year. Therefore, I merged the data to other ever-married women data (for
example, the marriage history data for the same set of women) with the information of survey year, and mapped the
latest wave to each woman in the birth history data.
48
Only 0.3% women were dropped.
49
They point out, in the U.S, less than 6% of births were out-of-wedlock in 1963, while this proportion rose to 30% in
1992.
31
Table 3.1 Descriptive statistics of selected variables for static analysis
Birth Cohort of Ever-married Women
Full sample 1950 or older 1951-60 1961-70 1971-80 1981 or younger
Number of births 1.70 3.11 2.03 1.48 1.23 0.87
(1.12) (1.43) (1.07) (0.79) (0.66) (0.61)
0-1 birth (%) 53.9 9.5 35.2 60.0 74.0 89.3
(49.9) (29.3) (47.8) (49.0) (43.9) (30.9)
2-3 births (%) 39.2 56.2 56.5 38.1 25.4 10.5
(48.8) (49.6) (49.6) (48.6) (43.6) (30.7)
4 or more births (%) 6.9 34.3 8.3 2.0 0.5 0.2
(25.3) (47.5) (27.6) (13.9) (7.1) (3.9)
Age at survey 40.81 48.68 48.14 41.82 33.57 25.66
(8.87) (3.38) (4.81) (6.39) (4.48) (2.91)
Urban (%) 36.2 30.4 37.0 38.7 36.6 31.6
(48.0) (46.0) (48.3) (48.7) (48.2) (46.5)
Han (%) 88.4 85.2 88.6 89.8 87.2 89.6
(32.1) (35.6) (31.8) (30.3) (33.4) (30.5)
Coast (%) 27.7 26.8 23.8 28.3 29.2 33.1
(44.8) (44.3) (42.6) (45.1) (45.5) (47.1)
No schooling (%) 17.0 49.6 29.0 8.9 5.2 1.7
(37.6) (50.0) (45.4) (28.5) (22.2) (12.9)
Primary school (%) 18.3 29.0 18.7 18.2 15.7 9.6
(38.7) (45.4) (39.0) (38.6) (36.4) (29.5)
Middle school (%) 35.6 14.0 27.3 41.2 43.4 45.8
(47.9) (34.7) (44.6) (49.2) (49.6) (49.9)
High school (%) 20.6 6.6 22.3 23.9 20.8 22.6
(40.5) (24.9) (41.6) (42.7) (40.6) (41.9)
College (%) 8.5 0.8 2.7 7.8 15.0 20.3
(27.9) (9.0) (16.3) (26.8) (35.7) (40.3)
N 7105 863 1648 2375 1564 655
Note: Standard deviations are in parentheses. Coast indicates whether a woman lives in the east coast of
China, including Beijing, Shandong, Jiangsu and Shanghai in the sample. Primary school, middle school, high
school and college indicate the highest level of schooling.
Means and standard deviations are shown for the full sample, and for different
birth cohorts of women. The sample comprises 7105 women. They on average have 1.7
children, over half having no more than 1 birth. They were on average 41 years old at
survey. 36% women live in urban areas, 88% women are Han Chinese, and 28% women
32
live on more-developed east coast of China. Over half women did not obtain high
school education or above.
Over birth cohorts, the total number of children ever born to a woman decreases
from about 3 to below 1. The faction of women having 0 or 1 birth sharply rises. This
eye-catching fertility decline should be partly attributed to unfinished childbearing of
young cohorts. However, fertility transition remains substantial for cohorts older than
1970 who were on average above 40 and had essentially completed childbearing by the
survey.
The fraction of urban women increases and then decreases over cohorts. The first
half reflects urbanization, and the second half implies that urban young cohorts are less
likely to be married and thus less likely to appear in the sample. The proportion of
women with primary school education is smaller over cohorts, and the proportion of
better-educated women increases greatly.
CHNS is not nationally representative as it underrepresents the northwest
population of China. Figure 3.1 plots the average number of births by birth cohort of
women, obtained from the CHNS sample and the one percent sample of China’s 1990
census, to examine how representative the CHNS sample is in terms of fertility change
over cohorts.
33
Figure 3.1 Number of births by cohort of mothers, CHNS versus census
The CHNS sample has shown similar patterns of fertility change with the census
sample. The census fertility level drops below the CHNS sample for the youngest
cohorts, because those women were still far from completing childbearing in 1990.
3.3 Empirical strategy and policy measurement
In a demand model of fertility (e.g., Hotz, Klerman and Willis, 1997), a married couple
maximizes their utility by choosing the number (and the quality) of children and
34
consumption subject to budget and time constraints. Then, the demand function for the
number of children, n, can be expressed as
n = N(p, w, I, θ), (3.1)
where p is a vector of various prices which directly or indirectly affect n; w is the wage
of mothers (the price of mothers’ time); I is the household non-labor income; and θ is a
vector of attributes that affect n, including parental preferences, technologies, parental
fecundity, etc.
China’s family planning policies can enter the demand function through various
channels. For example, birth quota raises the price of high-order births; family planning
service lowers the price of contraceptives; propaganda shifts parental preferences. With
appropriate policy measures that integrate different channels, the demand function can
be expressed as
n = N(FPP, p, w, I, θ), (3.2)
where FPP is a vector of policy measures.
Easterlin and Crimmins (1985) propose a different analytical framework for
fertility, which specifies three channels through which various factors affect the number
of children ever born: the demand for children, the supply of children and fertility
regulation. Equation (3.2) also matches well with their framework and all function
35
arguments can be mapped onto the three channels. For example, parental fecundity
influences the supply of children; prices and income affect the demand of children;
family planning policies are fertility regulation. Other than the arguments specified
earlier in this section, their supply channel highlights the survival rate (or mortality rate)
of children, which could be added to θ.
As the policy effects on n is the major interest of the chapter, other variables will
be reduced to exogenous variables. In other words, a reduced-form equation will be
estimated, as in Equation (3.3).
𝑛 𝑖 = 𝛼 + ∑ ( 𝛽 𝑗 0
𝐹𝑃𝑃 𝑗𝑖
+ 𝛽 𝑗 1
𝐹𝑃𝑃 𝑗𝑖
× 𝑈𝑟𝑏𝑎𝑛 𝑖 + 𝛽 𝑗 2
𝐹𝑃𝑃 𝑗𝑖
× 𝐻𝑎𝑛 𝑖 )
𝑗 =1,2,3
+𝛿 𝑈𝑟𝑏𝑎𝑛 𝑖 + 𝜃𝐻𝑎𝑛 𝑖 + ∑𝛾 𝑘 𝑋 𝑘𝑖 𝑘 + 𝜂 𝑐 + 𝘀 𝑖 ,
(3.3)
In this equation, i indicates woman i. 𝑛 𝑖 is the number of children ever born to
woman i. 𝐹𝑃𝑃 𝑗𝑖
measures woman i’s exposure to the period j policy. As China’s family
planning policies differ by urban-rural and ethnicity, 𝐹𝑃𝑃 𝑗𝑖
is further interacted with an
urban dummy and a Han dummy. 𝑋 𝑘 involves a set of variables of women and their
husbands, such as schooling dummies, province dummies, and dummies of age at
survey. 𝜂 𝑐 captures cohort variables, including a cohort linear trend, 5-year cohort
dummies,
50
interactions of 5-year cohort dummies and urban dummy, interactions of 5-
50
Estimation results are robust if the 5-year cohort dummies here are replaced with a more precise specification, for
example, 3-year cohort dummies.
36
year cohort dummies and Han dummy, and interactions of 5-year cohort dummies and
province dummies.
𝐹𝑃𝑃 𝑗𝑖
is defined as below:
𝐹𝑃𝑃 𝑗𝑖
= ∑ 𝑝 ( 𝑎 )
𝑎 𝑒𝑗𝑖
𝑎 =𝑎 𝑠𝑗𝑖 .
(3.4)
𝑎 represents age. 𝑎 𝑠𝑗𝑖
and 𝑎 𝑒𝑗𝑖 are woman i’s age when period j policy started and ended.
According to the policy history, 𝑎 𝑠𝑗𝑖
and 𝑎 𝑒𝑗𝑖 are defined as
𝑎 𝑠 1
= 1963 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 , 𝑎 𝑒 1
= 1970 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 ;
𝑎 𝑠 2
= 1971 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 , 𝑎 𝑒 2
= 1979 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 ;
𝑎 𝑠 3
= 1980 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 , 𝑎 𝑒 3
= 𝑠𝑢𝑟𝑣𝑒𝑦 𝑦𝑒𝑎𝑟 − 𝑏𝑖𝑟𝑡 ℎ 𝑦𝑒𝑎𝑟 .
(3.5)
𝑝 ( 𝑎 ) measures the probability of conception at age a, with support from age 15 to
49. Figure 3.2 illustrates 𝑝 ( 𝑎 ) , which is calculated based on birth records of all women in
the sample.
37
Figure 3.2 Probability of conception by age
Note: Plotted based on all women in the sample. For any age below 15 or above 49, the probability is 0.
Chapter 3.4.5 will show that empirical results are robust to the 𝑝 ( 𝑎 ) calculated
based on subsamples of women. Figure 3.3 illustrates 𝐹𝑃𝑃 1
, 𝐹𝑃𝑃 2
, and 𝐹𝑃𝑃 3
by birth
cohort of women.
38
Figure 3.3 Intensity of exposure to period 1, 2 and 3 policies by cohort of women
Note: Each point represents the mean of 𝐹𝑃𝑃 𝑗𝑖
within a birth cohort.
Women born in 1940s, 1950s, and after 1960 are mostly affected by the period 1,
period 2, and period 3 policy, respectively.
Policy exposure defined in Equation (3.4) is more heterogeneous than previous
measures. First, longer a woman is exposed to a policy, greater 𝐹𝑃𝑃 𝑗𝑖
is likely to be.
Second, if a woman is exposed to a policy during her peak age of childbearing, 𝐹𝑃𝑃 𝑗𝑖
tends to be greater than those exposed to the policy at non-peak age of childbearing.
Moreover, 𝐹𝑃𝑃 𝑗𝑖
is essentially a function of birth cohorts, which are exogenous.
39
Equation (3.3) has considered policy variations over the three periods, and policy
differences between urban and rural areas, and between Han and non-Han people,
therefore the policy measurement is more complete than previous measures.
Variables p, w, I and θ are assumed to be largely characterized by 𝑋 𝑘 and 𝜂 𝑐 . For
example, prices, infant mortality rate and technologies exhibit certain patterns over time,
and the patterns might differ by region and ethnicity. Therefore, linear cohort trend, 5-
year cohort dummies, and interactions of 5-year cohort dummies and various group
dummies (urban dummy, Han dummy, and province dummies) could essentially
capture those variables. Moreover, wage and household income are largely determined
by schooling and age of women and their husband. All other uncontrolled factors go to
the error term 𝘀 𝑖 , and are assumed to be uncorrelated with controlled variables.
To complete the empirical specification, I add the exposure to the great famine
(1959-1962). This variable is defined similarly to policy exposure, with 𝑎 𝑠𝑗𝑖
and 𝑎 𝑒𝑗𝑖 the
age when the great famine started and ended. On one hand, this variable can capture
the fertility drop caused by the famine; on the other hand, as fertility make-up after the
famine triggered the period 1 policy, this variable helps deal with the issue of
endogenous program placement.
40
3.4 Empirical results
3.4.1 Estimation results
Table 3.2 shows OLS estimation results, with standard errors in parentheses. Standard
errors are clustered at the primary sampling unit/five-year birth cohort level.
51
Table 3.2 OLS regressions of number of births on determinants
Dependent variable: Number of births a woman has ever had
(1) (2) (3)
Policy 1 -1.222 -1.278 -2.308
(1.469) (1.476) (1.414)
Policy 1 × Urban -1.325 -0.507 -1.491
(0.688)* (0.981) (0.788)*
Policy 1 × Han 0.822 0.917 1.251
(1.034) (1.046) (0.974)
Policy 1 × Urban × Han -0.880
(0.708)
Policy 2 -1.890 -1.895 -3.521
(1.325) (1.351) (1.376)**
Policy 2 × Urban -1.129 -0.849 -0.475
(0.384)*** (0.553) (0.461)
Policy 2 × Han 0.112 0.146 0.869
(0.642) (0.703) (0.694)
Policy 2 × Urban × Han -0.293
(0.516)
Policy 3 -2.385 -2.350 -3.170
(1.212)** (1.214)* (1.283)**
Policy 3 × Urban -0.180 -0.119 0.049
(0.160) (0.189) (0.177)
Policy 3 × Han -0.132 -0.133 0.062
(0.193) (0.195) (0.211)
Policy 3 × Urban × Han -0.056
(0.100)
Urban 1.397 1.432 1.098
(0.724)* (0.725)** (0.868)
Han (women) 0.699 0.664 -1.059
(1.022) (1.027) (0.969)
Han (women’s husband) 0.448
51
Each primary sampling unit is an urban or suburban neighborhood in a city, or a town or village in a county. The
total number of clusters is 2,256.
41
(0.559)
Famine -1.470 -1.509 -3.620
(1.678) (1.681) (1.683)**
Schooling of women
Primary school -0.252 -0.253 -0.265
(0.044)*** (0.044)*** (0.050)***
Middle school -0.485 -0.484 -0.499
(0.041)*** (0.041)*** (0.047)***
High school -0.711 -0.709 -0.669
(0.044)*** (0.044)*** (0.052)***
College or above -0.818 -0.816 -0.713
(0.046)*** (0.046)*** (0.056)***
Schooling of women’s husband
Primary school -0.013
(0.060)
Middle school -0.016
(0.056)
High school -0.118
(0.058)**
College or above -0.181
(0.061)***
Province dummies Yes Yes Yes
For women
Cohort linear trend Yes Yes Yes
Five-year cohort dummies Yes Yes Yes
Age dummies Yes Yes Yes
Five-year cohort dummies * Urban dummy Yes Yes Yes
Five-year cohort dummies * Han dummy Yes Yes Yes
Five-year cohort dummies * Province
dummies
Yes Yes Yes
For women’s husband
Cohort linear trend No No Yes
Five-year cohort dummies No No Yes
Age dummies No No Yes
Five-year cohort dummies * Urban dummy No No Yes
Five-year cohort dummies * Han dummy No No Yes
Five-year cohort dummies * Province
dummies
No No Yes
P value for significance of policies 0.0025 0.0036 0.0386
R squared 0.5388 0.5392 0.5684
N 7105 7105 5922
Note: Standard errors, in parentheses, are clustered at the primary sampling unit/five-year birth cohort
level. * p<0.1; ** p<0.05; *** p<0.01. The base group for five-year cohort dummies consists of the cohorts
42
older than 1941. Cohort 1986 and younger cohorts belong to the same group.
The dependent variable is the number of births a woman has ever had. Column
(1) is the baseline regression. Column (2) further controls for triple interactions of policy
exposure, urban dummy and Han dummy, in order to capture more cross-sectional
policy variations. As compared to column (1), column (3) further includes variables of
women’s husband. The information of women’s husband may be missing because of,
for example, being divorced or widowed, so the number of observations has decreased
sizably.
Through all columns, variables related to family planning policies are generally
negative, implying that policies reduce the number of births, and policy effects are
stronger for urban and Han people than for rural and non-Han people. Tests confirm
that family planning policies are jointly significant in all regressions. Partial effects of
family planning policies on the number of births will be calculated in Chapter 3.4.2.
Moreover, schooling has strong and robust effects through columns. Higher the
schooling level is, greater the effects of schooling would be. In column (3), schooling of
women’s husband tends to further reduce the number of children.
3.4.2 Partial effects of family planning policies on the number of births
Based on the estimation results of Table 3.2, Table 3.3 calculates the effect of each period
of family planning policy on the number of births that a woman has ever had, i.e., the
fertility difference between the case that a woman was exposed to one policy and the
43
case that the woman was not exposed to that policy in the period of that policy,
provided that other variables, including the other two policies, are constant.
Table 3.3 Effect of each policy on the number of births of each group of women
Note: The partial effect of policy i (i = 1, 2, 3) for a group of women is the difference between the
predicted number of births of those women under actual exposure to policy i and the women’s predicted
number of births under null exposure to policy i, other variables, including the other two policies,
keeping constant. * p < 0.1; ** p < 0.05; *** p < 0.01.
Table 3.3 consists of three panels, derived from columns (1) to (3) of Table 3.2. In
each panel, women are categorized into four groups, and family planning policies are
segmented into three periods, as in Table 2.1.
Along each group (row), the number below “Policy j” (j = 1, 2, 3) indicates the
change in the number of births of a woman in that group because of her exposure to the
Panel 1: Derived from regression (1) of Table 3.2
Policy 1 Policy 2 Policy 3
Urban Han -0.082 -0.462** -2.012**
Rural Han -0.024 -0.288 -1.775**
Urban non-Han -0.070 -0.469** -1.897**
Rural non-Han -0.103 -0.385 -1.515**
Panel 2: Derived from regression (2) of Table 3.2
Policy 1 Policy 2 Policy 3
Urban Han -0.083 -0.459** -1.982**
Rural Han -0.022 -0.283 -1.750**
Urban non-Han -0.049 -0.426** -1.825**
Rural non-Han -0.108 -0.386 -1.493*
Panel 3: Derived from regression (3) of Table 3.2
Policy 1 Policy 2 Policy 3
Urban Han -0.121* -0.497** -2.282**
Rural Han -0.064 -0.429** -2.192**
Urban non-Han -0.104** -0.621*** -2.308**
Rural non-Han -0.195 -0.717** -2.014**
44
period j policy, keeping other factors constant. For example, in panel 1, the number -
0.288, corresponding to “Rural Han” and “Policy 2”, means, because a rural Han
woman has been exposed to the period 2 policy, her number of children has reduced by
0.288, keeping all the other variables (including the other two policies) constant.
Take the -0.288 again as an example. It is calculated in the following way:
(-1.890 + 0.112) × mean(exposure to policy 2 | urban = 0, Han = 1)
= (-1.890 + 0.112) × 0.162 = -0.288,
where -1.890 and 0.112 are the coefficients of “Policy 2” and “Policy 2 × Han”, and 0.162
is the average exposure to the period 2 policy for the rural Han women. In Panel 2, the
calculation considers the triple interactions. Symbols ***, **, and * denote statistical
significance at 1%, 5%, and 10% levels, respectively.
The three panels exhibit some common features. First, for any group of women,
policy effects get greater and greater over periods. The effects also become more and
more statistically significant. Second, within each period of policy, policy effects for
urban women are stronger than rural women.
52
For policy 3, effects for Han women are
52
In the partial effects derived from the baseline regression, the difference of policy effects between urban and rural
women are statistically significant for policies 1 and 2, but insignificant for policy 3.
45
greater than non-Han women;
53
but it doesn’t always hold for policy 1 and 2. These
features are consistent with what Table 2.1 implies.
Panels 1 and 2 have very similar results. Panel 3 has shown greater effects, but
the secular and cross-sectional patterns are consistent with panels 1 and 2.
3.4.3 Simulated over-cohort fertility under several scenarios of policy history
This part attempts to answer to what extent family planning policies have explained
fertility decline over birth cohorts. Figure 3.4 illustrates the number of births by birth
cohort of women. Figure 3.4 does not show the women born before 1940, because the
sample size is too small. For this part, I will focus on cohorts 1943 to 1972, because
cohort 1943 is the starting point of fertility decline, and cohort 1972 is the youngest
cohort whose average age was above 35 at survey. Figure 3.2 shows that, women have
basically completed childbearing around 35.
53
The difference is not statistically significant.
46
Figure 3.4 Number of births by cohort of women
Figure 1.1 implies that China’s fertility decline by calendar year can be
segmented into the pre-1980 period and post-1980 period. In the former period, only the
period 1 and 2 policies could affect women’s fertility. In the latter period, the period 3
policy is active. Therefore, I first simulate the number of births over cohorts by
assuming there has been no family planning policy. Then, I simulate the number of
children by assuming only period 1 and 2 policies have ever existed, and the period 2
policy has been always on stage since it started in 1971 (no period 3 policy). Finally, I
47
further include the period 3 policy and simulate the number of births under the actual
exposure to all three periods of policies. Simulations are plotted in Figure 3.5.
Figure 3.5 Predicted number of births under different policy histories
The Y and X axes are simulated number of births and birth cohorts of women.
The short dashed line represents the predicted number of births had there been no
family planning policy, which is calculated by assuming all policy exposure to be 0 in
regression (1) of Table 3.2. The long-dashed line assumes women have only been
exposed to the period 1 policy and the long-lasting period 2 policy. The solid line
48
illustrates the predicted number of births under women’s actual exposure to all the
three periods of policies.
Without family planning policy, the number of births decreases from about 4.75
to around 3.75. With the actual period 1 policy and long-lasting period 2 policy, fertility
goes down from about 3.75 to about 1.75. Further inclusion of the period 3 policy pulls
down the fertility curve, but not by much. Therefore, roughly speaking, family planning
policies have explained about half of the fertility decline over these cohorts.
One caveat should be noted. The long-dashed line in Figure 3.5 tells a different
story as compared to Table 3.3. Table 3.3 decomposes the contribution of each period of
policy under the actual policy history, while the long-dashed line is plotted based on a
counterfactual history of the period 2 policy. The reason is that, when exploring policy
effects over cohorts, the range of cohorts (cohorts 1943 to 1972) is not long enough to
capture all possible effects of period 1 and 2 policies. Therefore, Figure 3.5 is not
concluding that the period 1 and 2 policies have explained half of the fertility decline in
actual history, but is saying that half of the fertility decline over these cohorts would
have been attributed to the period 1 and 2 policies had the period 2 policy been always
effective and had the period 3 policy been never launched.
Mother’s schooling is often considered as an important determinant of fertility
transition. As a comparison, Figure 3.6 predicts the number of births over cohorts under
actual schooling (solid line), and assuming women’s schoolings are all 0.
49
Figure 3.6 Predicted number of births with actual schooling and no schooling
Without schooling, the fertility level drops from about 3.75 to 1.75; while under
actual schooling, it decreases from about 3.5 to about 1.25. Therefore, schooling has
explained about 10% of the over-cohort fertility decline.
3.4.4 Policy effects by schooling and gender of the first birth
Family planning policies may have different effects for women with different
characteristics. This part heterogenizes the effect of family planning policies on the
number of births by women’s schooling and by the gender of the first birth.
50
Table 3.4 regresses the number of births on policy exposure and other variables
separately for the more-educated and less-educated women (columns (1) and (2)), and
separately for the women whose first child is son versus daughter (columns (3) and (4)).
The cut-off point of schooling is 9 years, which represents the completion of middle
school.
Table 3.4 OLS regressions of number of births on determinants, by women’s schooling
and gender of the first birth
(1) (2) (3) (4)
Schooling ≥ 9
years
Schooling <
9 years
First birth is
son
First birth is
daughter
Policy 1 1.914 0.818 1.691 1.103
(0.973)** (1.013) (0.939)* (0.755)
Policy 1 × Urban -1.054 -0.706 -0.286 -0.749
(0.446)** (0.683) (0.378) (0.591)
Policy 1 × Han -0.261 -0.707 -0.335 -0.160
(0.769) (0.818) (0.831) (0.538)
Policy 2 0.848 0.210 1.632 0.071
(0.491)* (0.792) (0.688)** (0.582)
Policy 2 × Urban -0.427 -0.793 -0.342 -1.017
(0.289) (0.560) (0.269) (0.530)*
Policy 2 × Han 0.105 -1.103 -1.272 -0.303
(0.411) (0.629)* (0.651)* (0.448)
Policy 3 0.127 0.022 0.522 -0.028
(0.289) (0.549) (0.442) (0.365)
Policy 3 × Urban -0.156 -0.508 -0.033 -0.720
(0.180) (0.394) (0.178) (0.381)*
Policy 3 × Han 0.306 -0.734 -0.796 -0.261
(0.227) (0.413)* (0.447)* (0.258)
Urban -0.243 0.206 -0.636 0.075
(0.320) (0.767) (0.313)** (0.629)
Han -0.955 1.875 1.403 0.296
(0.482)** (0.904)** (0.832)* (0.562)
Famine 0.307 -1.103 0.123 1.204
(2.062) (1.464) (1.640) (1.917)
Years of schooling -0.193 -0.041
(0.041)*** (0.025)
51
Squared years of schooling 0.006 -0.001
(0.002)*** (0.003)
Primary school -0.297 -0.270
(0.054)*** (0.061)***
Middle school -0.529 -0.473
(0.049)*** (0.056)***
High school -0.686 -0.768
(0.052)*** (0.060)***
College or above -0.732 -0.886
(0.053)*** (0.062)***
Province dummies Yes Yes Yes Yes
Cohort linear trend Yes Yes Yes Yes
Five-year cohort dummies Yes Yes Yes Yes
Age and its square Yes Yes Yes Yes
Cohort trend * Urban dummy Yes Yes Yes Yes
Cohort trend * Han dummy Yes Yes Yes Yes
Cohort trend * Province
dummy
Yes Yes Yes Yes
P value for significance of
policies
0.0324 0.0467 0.0045 0.1131
P value for equal coefficients
test
(1) = (2): 0.0000 (3) = (4): 0.0000
R squared 0.3642 0.4606 0.5523 0.5206
N 4360 2745 3456 3200
Note: Standard errors, in parentheses, are clustered at the primary sampling unit/five-year birth cohort
level. * p<0.1; ** p<0.05; *** p<0.01. Years of schooling and its square, instead of schooling level dummies,
are controlled for in regressions (1) and (2), to ensure the model specifications are identical in the two
columns.
The empirical specifications are different from the baseline regression of Table
3.2 (column (1)) on two aspects. First, I control for years of schooling and its square
instead of categories of schooling in the first two columns, because women in columns
(1) and (2) belong to exclusive schooling categories, and controlling for years of
schooling could make the two regressions more comparable. Second, I control for age
and its square instead of age dummies, and control for interactions of cohort linear
52
trend and other variables instead of interactions of five-year cohort dummies and other
variables, because the sample size of each regression has been cut down significantly.
Table 3.4 also shows the p values for equal coefficients tests (Chow tests) between
columns (1) and (2), and between columns (3) and (4). Test results support that
estimation results are significantly different between the women with different
characteristics.
Table 3.5 further calculates the partial effects of policies for each regression of
Table 3.4, using the method for Table 3.3.
Table 3.5 Effect of each policy on the number of births of each group of women, by
women’s schooling and gender of the first birth
Panel 1: Schooling >= 9 years
Policy 1 Policy 2 Policy 3
Urban Han 0.013 0.062 0.224
Rural Han 0.023** 0.079*** 0.350*
Urban non-Han 0.006 0.022 -0.025
Rural non-Han 0.025** 0.085* 0.101
Panel 2: Schooling < 9 years
Policy 1 Policy 2 Policy 3
Urban Han -0.078 -0.502** -0.651**
Rural Han 0.013 -0.229 -0.414
Urban non-Han 0.006 -0.165 -0.289
Rural non-Han 0.125 0.064 0.011
Panel 3: First birth is son
Policy 1 Policy 2 Policy 3
Urban Han 0.055 0.003 -0.226
Rural Han 0.080* 0.057 -0.198
Urban non-Han 0.045 0.201* 0.358
Rural non-Han 0.145* 0.341** 0.337
Panel 4: First birth is daughter
Policy 1 Policy 2 Policy 3
Urban Han 0.009 -0.202* -0.760*
Rural Han 0.059 -0.041 -0.204
53
Urban non-Han 0.010 -0.160 -0.563
Rural non-Han 0.094 0.015 -0.018
Note: Panels 1 to 4 are derived from regressions (1) to (4) of Table 3.4. The partial effect of policy i (i = 1, 2,
3) for a group of women is the difference between the predicted number of births of those women under
actual exposure to policy i and the women’s predicted number of births under null exposure to policy i,
other variables keeping constant. * p<0.1; ** p<0.05; *** p<0.01.
Compare panels 1 and 2, and we can conclude that less-educated women have
been influenced more by family planning policies than more-educated women. One
explanation is that, less-educated women tend to have more births, and thus would face
greater restrictions from family planning policies. Comparison of panels 3 and 4 imply
that, women whose first child is daughter have been affected more by family planning
policies than those whose first child is son. One explanation is that, due to son
preference, women who have a daughter at first tend to have more births in order to
obtain a son, and thus would face more restrictions from family planning policies.
3.4.5 Sensitivity of results to different policy measures
Policy measures are constructed using the probability of conception by age, which is
computed based on the birth records of all women in the sample. Ideally, one should
obtain the probability by using the women who have never been exposed to family
planning policies. However, the sample doesn't include such women. But I could
calculate the probability based on different groups of women, categorized by their
cohorts, living in urban or rural areas, and ethnicity. Particularly, the probability
derived from the oldest cohorts, or from rural non-Han women, should be close to the
54
probability of conception for the women who have never been exposed to family
planning policies.
Figure 3.7 illustrates the probability of conception by age, for urban Han women,
rural Han women, urban non-Han women, and rural non-Han women, respectively. All
probability distributions are similar.
Figure 3.7 Probability of conception by age, separately plotted for urban Han women,
rural Han women, urban non-Han women, and rural non-Han women
55
Figure 3.8 plots the probability of conception by age, for cohorts 1950 or older, cohorts
1951-60, cohorts 1961-70, and cohorts 1971 or younger, respectively. The probability
distributions of older cohorts have thicker tails than younger cohorts.
Figure 3.8 Probability of conception by age, separately plotted for cohorts 1950 or older,
cohorts 1951-60, cohorts 1961-70, and cohorts 1971 or younger
Table 3.6 reruns the baseline regression using policy measures defined on the
different probabilities of conception by age.
Table 3.6 OLS regressions of number of births on determinants, using different probabilities of conception to construct
policy measures
(1)
Distr. of
all
(2)
Distr. of
urban Han
(3)
Distr. of
rural Han
(4)
Distr. of
urban
non-Han
(5)
Distr. of
rural non-
Han
(6)
Distr. of
cohort
1950 or
older
(7)
Distr. of
cohorts
1951-60
(8)
Distr. of
cohorts
1961-70
(9)
Distr. of
cohort
1971 or
younger
Policy 1 -1.222 -1.159 -1.239 -1.365 -1.172 -1.011 -1.162 -1.219 -1.189
(1.469) (1.419) (1.476) (1.458) (1.533) (1.597) (1.459) (1.397) (1.440)
Policy 1 × Urban -1.325 -1.219 -1.360 -1.314 -1.307 -1.154 -1.110 -1.351 -1.342
(0.688)* (0.675)* (0.684)** (0.647)** (0.749)* (0.805) (0.717) (0.610)** (0.656)**
Policy 1 × Han 0.822 0.696 0.861 0.821 0.864 0.770 0.714 0.739 0.810
(1.034) (1.023) (1.023) (0.978) (1.128) (1.221) (1.083) (0.900) (0.983)
Policy 2 -1.890 -1.911 -1.857 -1.927 -1.936 -1.983 -2.058 -1.673 -1.773
(1.325) (1.278) (1.337) (1.363) (1.350) (1.382) (1.283) (1.301) (1.311)
Policy 2 × Urban -1.129 -1.065 -1.135 -1.151 -1.201 -1.264 -1.071 -1.018 -1.088
(0.384)*** (0.379)*** (0.382)*** (0.389)*** (0.409)*** (0.436)*** (0.395)*** (0.359)*** (0.370)***
Policy 2 × Han 0.112 0.180 0.065 0.082 0.195 0.275 0.326 -0.070 0.025
(0.642) (0.627) (0.640) (0.666) (0.682) (0.718) (0.645) (0.585) (0.616)
Policy 3 -2.385 -2.423 -2.341 -2.409 -2.453 -2.485 -2.534 -2.096 -2.236
(1.212)** (1.171)** (1.224)* (1.246)* (1.219)** (1.237)** (1.169)** (1.210)* (1.204)*
Policy 3 × Urban -0.180 -0.157 -0.188 -0.186 -0.187 -0.173 -0.151 -0.200 -0.180
(0.160) (0.149) (0.167) (0.159) (0.150) (0.141) (0.139) (0.192) (0.172)
Policy 3 × Han -0.132 -0.119 -0.139 -0.131 -0.126 -0.122 -0.111 -0.159 -0.143
(0.193) (0.181) (0.200) (0.194) (0.188) (0.183) (0.172) (0.215) (0.204)
Urban 1.397 1.382 1.384 1.376 1.454 1.443 1.382 1.250 1.342
(0.724)* (0.729)* (0.719)* (0.711)* (0.746)* (0.776)* (0.747)* (0.689)* (0.712)*
Han 0.699 0.736 0.691 0.746 0.654 0.649 0.709 0.775 0.705
(1.022) (1.033) (1.011) (1.007) (1.064) (1.109) (1.069) (0.954) (0.995)
Famine -1.470 -1.400 -1.479 -1.430 -1.526 -1.774 -1.395 -1.308 -1.483
(1.678) (1.549) (1.753) (1.656) (1.641) (1.651) (1.475) (1.910) (1.803)
Primary school -0.252 -0.252 -0.252 -0.252 -0.252 -0.252 -0.251 -0.253 -0.252
(0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.043)*** (0.044)*** (0.044)*** (0.044)***
Middle school -0.485 -0.485 -0.485 -0.486 -0.484 -0.484 -0.484 -0.487 -0.486
57
(0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)***
High school -0.711 -0.711 -0.711 -0.712 -0.710 -0.709 -0.709 -0.713 -0.712
(0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)***
College or above -0.818 -0.818 -0.818 -0.818 -0.817 -0.816 -0.817 -0.820 -0.818
(0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)***
Province dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes
Cohort linear
trend
Yes Yes Yes Yes Yes Yes Yes Yes Yes
Five-year cohort
dummies
Yes Yes Yes Yes Yes Yes Yes Yes Yes
Age dummies Yes Yes Yes Yes Yes Yes Yes Yes Yes
Five-year cohort
dummies × Urban
dummy
Yes Yes Yes Yes Yes Yes Yes Yes Yes
Five-year cohort
dummies × Han
dummy
Yes Yes Yes Yes Yes Yes Yes Yes Yes
Five-year cohort
dummies ×
Province dummies
Yes Yes Yes Yes Yes Yes Yes Yes Yes
P value for
significance of
Yes Yes Yes Yes Yes Yes Yes Yes Yes
policies 0.0025 0.0049 0.0018 0.0021 0.0027 0.0030 0.0047 0.0022 0.0018
R squared 0.5388 0.5386 0.5388 0.5387 0.5388 0.5388 0.5387 0.5387 0.5388
N 7105 7105 7105 7105 7105 7105 7105 7105 7105
Note: Standard errors, in parentheses, are clustered at the primary sampling unit/five-year birth cohort level. * p<0.1; ** p<0.05; *** p<0.01
Column (1) constructs policy measures with the probability for all women, which is
identical to the baseline regression in Table 3.2. Columns (2) to (5) construct policy
measures with the probability of conception of urban Han women, rural Han women,
urban non-Han women, and rural non-Han women, respectively. Columns (6) to (9)
construct policy measures using the probability of conception of cohorts 1950 or older,
cohorts 1951-60, cohorts 1961-70, and cohorts 1971 or younger. Coefficients are fairly
robust though columns.
Table 3.7 derived partial effects of policies for columns (1) to (9) of Table 3.6,
similar to Table 3.3 and Table 3.5. Partial effects are very similar through the nine panels.
Table 3.7 Effect of each policy on the number of births of each group of women, by the
probability of conception used to construct policy measures
Panel 1: Use the probability of childbearing of all women
Policy 1 Policy 2 Policy 3
Urban Han -0.082 -0.462** -2.012**
Rural Han -0.024 -0.288 -1.775**
Urban non-Han -0.070 -0.469** -1.897**
Rural non-Han -0.103 -0.385 -1.515**
Panel 2: Use the probability of childbearing of urban Han women
Policy 1 Policy 2 Policy 3
Urban Han -0.075 -0.440** -2.029**
Rural Han -0.027 -0.278 -1.803**
Urban non-Han -0.060 -0.456** -1.918**
Rural non-Han -0.093 -0.394 -1.546**
Panel 3: Use the probability of childbearing of rural Han women
Policy 1 Policy 2 Policy 3
Urban Han -0.086 -0.471** -1.983**
Rural Han -0.024 -0.293 -1.744**
Urban non-Han -0.075 -0.473** -1.867**
Rural non-Han -0.108 -0.378 -1.483*
Panel 4: Use the probability of childbearing of urban non-Han women
Policy 1 Policy 2 Policy 3
59
Urban Han -0.088 -0.474** -2.033**
Rural Han -0.033 -0.298 -1.791**
Urban non-Han -0.074* -0.476** -1.919**
Rural non-Han -0.115 -0.392 -1.530*
Panel 5: Use the probability of childbearing of rural non-Han women
Policy 1 Policy 2 Policy 3
Urban Han -0.071 -0.446** -2.065**
Rural Han -0.017 -0.270 -1.820**
Urban non-Han -0.063 -0.459** -1.947**
Rural non-Han -0.092 -0.381 -1.565**
Panel 6: Use the probability of childbearing of cohort 1950 or older
Policy 1 Policy 2 Policy 3
Urban Han -0.059 -0.441** -2.081**
Rural Han -0.013 -0.261 -1.843**
Urban non-Han -0.054 -0.457** -1.961**
Rural non-Han -0.076 -0.385 -1.592**
Panel 7: Use the probability of childbearing of cohorts 1951-60
Policy 1 Policy 2 Policy 3
Urban Han -0.065 -0.424** -2.108**
Rural Han -0.024 -0.269 -1.880**
Urban non-Han -0.053 -0.455** -1.994**
Rural non-Han -0.086 -0.415 -1.624**
Panel 8: Use the probability of childbearing of cohorts 1961-70
Policy 1 Policy 2 Policy 3
Urban Han -0.100 -0.468** -1.812**
Rural Han -0.033 -0.298 -1.578*
Urban non-Han -0.081* -0.459** -1.694*
Rural non-Han -0.118 -0.351 -1.316*
Panel 9: Use the probability of childbearing of cohorts 1971 or younger
Policy 1 Policy 2 Policy 3
Urban Han -0.088 -0.469** -1.902**
Rural Han -0.025 -0.293 -1.673**
Urban non-Han -0.076 -0.466** -1.788**
Rural non-Han -0.108 -0.368 -1.415*
Therefore, while constructing policy measures, results are not sensitive to the
choice of probability of conception by age.
However, if one adopts policy measures that are incomplete or lack
heterogeneity, results may differ substantially.
60
Table 3.8 studies this issue. Column (1) is the baseline regression. Column (2)
omits the period 1 policy and column (3) omits both period 1 and 2 policies. Column (4)
defines policy exposure as a dummy variable that indicates whether a woman used to
be exposed to a policy between age 15 and 49. While constructing policy measures,
column (5) replaces the probability of conception with a uniform distribution over age
15 to 49 (probability at each age is
1
35
), which only considers the length of time of policy
exposure.
Table 3.8 OLS regressions of number of births on determinants, using incomplete
measures or measures lacking heterogeneity
(1)
Baseline
regression
(2)
Omitting
policy 1
(3)
Omitting
policy 1 and
2
(4)
Using
policy
dummies
(5)
Assuming
uniform
policy
exposure
Policy 1 -1.222 -2.693 -3.161
(1.469) (1.222)** (2.859)
Policy 1 × Urban -1.325 0.748 -1.678
(0.688)* (0.653) (1.615)
Policy 1 × Han 0.822 3.620 4.426
(1.034) (1.749)** (2.384)*
Policy 2 -1.890 -0.877 -0.347 -4.647
(1.325) (0.644) (0.237) (2.532)*
Policy 2 × Urban -1.129 -0.648 -0.075 -2.606
(0.384)*** (0.367)* (0.100) (0.848)***
Policy 2 × Han 0.112 -0.107 0.363 2.568
(0.642) (0.588) (0.244) (1.780)
Policy 3 -2.385 -1.381 -0.256 1.954 -2.978
(1.212)** (0.441)*** (0.292) (0.576)*** (1.812)
Policy 3 × Urban -0.180 0.012 0.185 -0.054 -0.250
(0.160) (0.163) (0.147) (0.097) (0.135)*
Policy 3 × Han -0.132 -0.184 -0.156 -3.238 -0.051
(0.193) (0.195) (0.188) (1.519)** (0.208)
Urban 1.397 0.702 0.594 1.828
61
(0.724)* (0.637) (0.636) (0.778)**
Han 0.699 0.784 0.755 -0.644
(1.022) (0.915) (0.902) (1.162)
Famine -1.470 -0.712 0.377 0.758 -0.824
(1.678) (1.292) (1.228) (1.186) (1.502)
Primary school -0.252 -0.252 -0.253 -0.253 -0.250
(0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)***
Middle school -0.485 -0.484 -0.487 -0.488 -0.481
(0.041)*** (0.041)*** (0.041)*** (0.041)*** (0.041)***
High school -0.711 -0.710 -0.712 -0.713 -0.704
(0.044)*** (0.044)*** (0.044)*** (0.044)*** (0.044)***
College or above -0.818 -0.819 -0.824 -0.825 -0.815
(0.046)*** (0.046)*** (0.046)*** (0.046)*** (0.046)***
Province
dummies
Yes Yes Yes Yes Yes
Cohort linear
trend
Yes Yes Yes Yes Yes
Five-year cohort
dummies
Yes Yes Yes Yes Yes
Age dummies Yes Yes Yes Yes Yes
Five-year cohort
dummies *
Urban dummy
Yes Yes Yes Yes Yes
Five-year cohort
dummies * Han
dummy
Yes Yes Yes Yes Yes
Five-year cohort
dummies *
Province
dummies
Yes Yes Yes Yes Yes
P value for
significance of
policies
0.0025 0.0046 0.3091 0.0118 0.0000
R squared 0.5388 0.5382 0.5366 0.8596 0.5395
N 7105 7105 7105 7105 7105
Further, Table 3.9 derives the partial effects of policies from Table 3.8. Compare
panels 2 and 3 to panel 1, we can conclude that, if earlier policies are omitted, policy
effects would be systematically underestimated, because policy effects in that case
would only be the additional effects to earlier policies. If policy dummies are adopted
62
(panel 4), partial effects would also greatly deviate from panel 1. If only the length of
time of policy exposure is taken into account (panel 5), partial effects would also be
different, but the difference is not large.
Table 3.9 Effect of each policy on the number of births of each group of women, using
incomplete policy measures or measures lacking heterogeneity
Panel 1: Derived from baseline regression
Policy 1 Policy 2 Policy 3
Urban Han -0.082 -0.462** -2.012**
Rural Han -0.024 -0.288 -1.775**
Urban non-Han -0.070 -0.469** -1.897**
Rural non-Han -0.103 -0.385 -1.515**
Panel 2: Missing policy 1
Policy 1 Policy 2 Policy 3
Urban Han - -0.259*** -1.158***
Rural Han - -0.159*** -1.103***
Urban non-Han - -0.237** -1.012***
Rural non-Han - -0.179 -0.877***
Panel 3: Missing policy 2
Policy 1 Policy 2 Policy 3
Urban Han - - -0.169
Rural Han - - -0.290*
Urban non-Han - - -0.053
Rural non-Han - - -0.163
Panel 4: Use policy dummies
Policy 1 Policy 2 Policy 3
Urban Han 0.364 -0.029 -1.338
Rural Han 0.220 0.008 -1.283
Urban non-Han -0.387 -0.199 1.901***
Rural non-Han -0.882** -0.178 1.954***
Panel 5: Assume uniform policy exposure by age
Policy 1 Policy 2 Policy 3
Urban Han -0.013 -0.428** -2.061*
Rural Han 0.046 -0.195 -1.860*
Urban non-Han -0.118 -0.607*** -1.995*
Rural non-Han -0.165 -0.500* -1.732
63
3.5 Conclusion
This chapter estimates the effect of China’s family planning policies on the number of
births that a woman has ever had, based on a cross-sectional sample of ever-married
women with records of birth history.
This chapter focuses on unifying and improving previous measures of China’s
family planning policies. Previous measures generally capture part of the policy history,
often stand on endogenous variables, and usually ignore the heterogeneity of policy
exposure. To address these issues, this chapter takes advantage of the secular policy
variations along the entire three policy periods, and the cross-sectional policy
differences between urban and rural areas, and between Han and non-Han people.
Moreover, the policy measures consider women’s length and age intervals of policy
exposure, so that they are more heterogeneous.
Using the improved measures, the partial effects of policies show consistent
patterns with the policy history. Simulations further find that, family planning policies
could explain about half of the fertility decline between cohort 1943 and 1972, while
women’s schooling, which has been considered to be an important determinant of
fertility change, could only explain about 10% of the fertility decline over the same
cohort range. Moreover, policy effects are stronger for less-educated women and
women whose first birth is a girl, because these women tend to have more births had
family planning policies not existed. Finally, if earlier periods of policies are omitted,
policy effects would be underestimated. Policy effect may also differ substantially if
policy measures lack heterogeneity.
64
Chapter 4 Dynamic Analysis
Static analysis can only capture the effect of family planning policies on the total
number of births, but is not able to explore the effect of policies over the dynamic
process of childbearing, particularly the effect on the timing of childbearing (Hotz,
Klerman, and Willis 1997).
This chapter extends the study from static analysis to dynamic analysis. The data
set used in this chapter is a panel sample constructed from the cross-sectional sample
for the static analysis. Each observation of the panel sample records a woman’s fertility
behavior and other characteristics in one year. This chapter applies a multiple-spell
duration model to evaluate policy effects for the first four birth spells. The duration
model uses a mixed proportional hazard function. The function describes non-
monotonic duration dependence for fertility behavior, takes into account effects of
family planning policies and various individual characteristics, and includes
unobserved individual heterogeneity which is non-parametrically specified, as
suggested by Heckman and Singer (1984). This chapter adopts the improved policy
measures for the static analysis, and adjusts them for dynamic analysis.
This chapter is arranged as follows. Chapter 4.1 reviews literature that studies
fertility, particularly the effect of family planning policies on fertility, using duration
analysis. Chapter 4.2 describes the data used for duration analysis. Chapter 4.3
introduces the duration model and policy measures. Chapter 4.4 shows empirical
results. Chapter 4.5 concludes.
65
4.1 Literature review on the application of duration analysis
Duration analysis is used to analyze what determines the duration of a state. In social
sciences, duration models have been applied to the studies of labor economics that
focus on the determinants of unemployment duration. For example, Ham and Rea Jr
(1987) and Katz and Meyer (1990) both study the effect of unemployment benefits on
unemployment duration.
Duration analysis has also been applied to other fields, particularly in birth
behaviors (Arroyo and Zhang 1997). For example, Newman and McCulloch (1984)
estimate the risk of births with various duration model specifications. Olsen and
Wolpin (1983) estimate the effect of child mortality on fertility.
However, not many studies on family planning programs have adopted duration
models. A few studies use duration models to study China’s family planning policies.
Ahn (1994) estimates the effect of one-child policy on the possibility of having the
second and third births in three provinces of China. Li and Choe (1997) examine the
effect of one-child policy on the likelihood of having the second birth as well as the
duration before the second birth. Poston (2002) studies the effect of the gender of past
births on future births. Poston also controls for a measure of the one-child policy. These
papers generally ignore the policies prior to the one-child policy. Moreover, when
studying multiple birth spells, they assume independence among spells. This paper
dynamically estimates the effects of all three periods of policies, and allows for flexible
between-spell correlations.
66
Other than the papers about China’s family planning policies, Angeles, Guilkey,
and Mroz (2005) jointly estimate how education, marriage and fertility are determined
and affect each other in the presence of family planning programs in Indonesia.
Hashemi and Salehi-Isfahani (2013) estimate the effect of family planning programs on
fertility in Iran. This paper will contribute more evidence to the branch of literature.
One of the most important econometric issues is to specify duration dependence
and individual heterogeneity for duration models. Ideally, both duration dependence
and heterogeneity should be flexibly specified. But Baker and Melino (2000) warn that,
models with non-parametrically specified duration dependence and heterogeneity may
substantially bias estimates. They find that non-parametrically specifying one of the two
parts would be enough to generate convincing results. This paper non-parametrically
specifies individual heterogeneity, and assumes duration dependence to have quadratic
forms, which can capture some non-monotonic patterns.
Heckman and Singer (1984) propose an algorithm to the non-parametrical
estimation of the individual heterogeneity. They let the heterogeneity follow a discrete
distribution where the number, locations and probability of support points all need to
be estimated. Specifically, they start by maximizing the likelihood function with the
heterogeneity with 2 points; after the model with M-points heterogeneity is estimated, a
new model with the heterogeneity of (M + 1)-points is estimated, until the likelihood
cannot be increased. In a Monte Carlo analysis, they find estimated coefficients
following such procedures are close to their true values. This paper adopts their method.
67
Baker and Melino (2000) show that, such procedures may lead to over-searching
problems: too many support points for the heterogeneity could largely bias estimates.
Instead, they propose a new criterion that on one hand maximizes the likelihood
function but on the other hand penalizes too many support points. This paper also
applies their criterion, and the estimated results remain the same.
4.2 Data
The dynamic analysis in this paper relies on a panel sample, which is constructed from
the CHNS cross-sectional sample used for the static analysis. Observations of the panel
data record each woman’s information at each age in years. A woman’s observations
span from age 15 to the age of being censored, or age 49, whichever is less.
The childbearing life of a woman is segmented by her birth behaviors. The first
birth spell, or spell 1, is from age 15 to the age at the first birth. Similarly, spell j + 1
indicates the time interval between the j’th birth and the (j + 1)’th birth.
Through out the chapter, I use i to denote a woman, j to represent a spell, and k
to indicate a year within a spell. Within spell j of woman i, 𝑘 = 1, 2, … , 𝐾 𝑖𝑗
where 𝐾 𝑖𝑗
is
the observed length of duration in years of that woman-spell.
The fertility outcome variable 𝑦 𝑖𝑗𝑘 is a dummy variable indicating whether
woman i has new births during the k’th year of spell j. It equals 0 if woman i remains in
spell j, after k years stay in that spell. If 𝑦 𝑖𝑗𝑘 equals 1, woman i exits spell j during the
k’th year of that spell, and then enters spell j + 1 in the following year.
68
Detailed descriptive statistics of the original cross-sectional sample can be found
in Chapter 3.2. Table 4.1 describes the number and percentage of women who have a
certain number of birth spells. Women with exactly j spells are almost equivalent to
those with j − 1 births, because most women did not have births during their last year in
the sample.
54
Table 4.1 also shows the number and percentage of women who are
censored before age 49.
Table 4.1 Number and fraction of women with certain number of birth spells
Number of women %
Number of women
censored before age 49
% women censored
before age 49
1 spell 412 5.8 378 91.7
2 spells 3419 48.4 2908 85.1
3 spells 1977 28.0 1368 69.2
4 spells 789 11.2 397 50.3
5 spells 294 4.2 141 48.0
6 or more spells 166 2.4 58 34.9
Total 7057 100.0 5250 74.4
Only 5.8% women have exactly 1 spell, and 48.4%, 28.0% and 11.2% women have
exactly 2, 3 and 4 spells, respectively. Women with exactly 1, 2, 3 and 4 spells comprise
93.4% of the sample. This chapter only analyzes birth behaviors for the first four spells.
In other words, the whole birth history of 93.4% women is included for analysis, and for
the rest 6.6%, their first four spells are considered, and they are censored in the year of
54
The percentage of women having births during their last year in the sample is only 1%.
69
having the fourth birth.
55
Overall, 74.4% of women were younger than 49 in the year of
survey.
Table 4.2 shows descriptive statistics of selected variables for the first four spells.
Variables in each spell are summarized based on the women who experienced that spell.
Table 4.2 Descriptive statistics of selected variables for spells 1 to 4
Spell 1 Spell 2 Spell 3 Spell 4
Average duration (years) 9.10 3.80 3.30 3.30
Variance of duration 10.70 8.10 5.60 5.10
End up with births (%) 94.3 47.6 37.4 35.6
Urban (%) 36.3 35.8 24.0 19.9
Han (%) 88.4 88.4 85.6 81.7
Years of schooling 8.20 8.10 6.10 4.50
Coastal province (%) 27.8 27.7 18.0 14.4
Born in 1950 or before (%) 11.9 12.4 23.6 43.0
Born in 1951-60 (%) 23.2 24.2 32.7 35.0
Born in 1961-70 (%) 33.5 34.0 29.1 16.7
Born in 1971-80 (%) 22.2 21.9 12.4 4.8
Born in 1981 or after (%) 9.3 7.5 2.2 0.5
First birth is son (%)
51.9 46.3 43.0
Number of women 7057 6521 3157 1226
92.4% women end the first spell with births, and their average waiting time
before the first birth is about 9 years. Over spells, the fraction of women having births at
the end of spells dropped, as well as their duration before births.
Urban women, Han women, more educated women and women living in the
more developed coastal provinces are less likely to appear in high order birth spells.
Older cohorts of women tend to have more births than younger cohorts. More
55
This paper assumes that parameters vary across spells. Sample size of spells beyond 4 is too small to estimate a
different set of parameters, and therefore this paper only focuses on the first four spells. I also tried to include all
spells, but assume the spells beyond the fourth spell share the same parameters with spell 4. This generates similar
results.
70
interestingly, the percentage of women whose first birth is a son decreases over spells,
implying that a woman whose first birth is a daughter tends to have more births, which
reflects son preference in China (Das Gupta et al., 2003; Jensen and Chintan, 2003).
4.3 Duration model and policy measurement
4.3.1 Duration model
Duration analysis focuses on what affects the duration of a state. It has been widely
used for studies on, for instance, unemployment durations (for example, Ham and Rea
Jr (1987), Katz and Meyer (1990)). Many other fields, including birth behaviors, have
also adopted this method (Arroyo and Zhang 1997).
Technically, duration analysis may start with specifying a hazard function θ(t),
56
which illustrates the rate of exiting some state (or some birth spell in this paper) at time
point t after having stayed in the state for t. Here t is a continuous measure of time.
By definition,
𝜃 ( 𝑡 )= 𝑙𝑖𝑚 𝛥𝑡 →0
𝑃𝑟𝑜𝑏 ( 𝑡 ≤ 𝑇 < 𝑡 + ∆𝑡 |𝑇 ≥ 𝑡 )
∆𝑡 =
𝑓 ( 𝑡 )
1 − 𝐹 ( 𝑡 )
(4.1)
where T indicates durations, and F(t) and f(t) are the distribution function and density
function of T. Conversely,
56
Natural scientists usually start with specifying the distribution function of duration T, F(t), constructing likelihood
functions based on observed durations, and estimating parameters of the distribution function using maximum
likelihood methods. As social scientists, particularly economists, prefer to study people’s decisions, hazard functions
would be a more natural start (Van den Berg 2001). But the two ways are technically equivalent, reflected by the
equivalence of Equations (4.1) and (4.2).
71
𝐹 ( 𝑡 )= 1 − 𝑒𝑥𝑝 ( − ∫ 𝜃 ( 𝑠 ) 𝑑𝑠 𝑡 0
) (4.2)
Hazard functions are assumed to differ over spells, specified as 𝜃 𝑗 ( 𝑡 ) , with j =
1, ..., J. Further, woman i’s hazard rate in spell j not only depends on the elapsed time t,
but also relies on her observed time-varying characteristics 𝒘 𝑖 𝐻 ( 𝑡 ) , time-invariant
variables 𝒛 𝑖 and her unobserved heterogeneity 𝑣 𝑖 which is also called frailty in duration
analysis. Therefore, hazard functions can be expressed as 𝜃 𝑗 ( 𝑡 |𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 , 𝑣 𝑖 ) , where
𝒘 𝑖 𝐻 ( 𝑡 )= {𝒘 𝑖 ( 𝜏 ) |0 ≤ 𝜏 < 𝑡 }, representing the entire history of 𝒘 𝑖 ( 𝜏 ) from time point 0 to t.
One of the most popular specifications for 𝜃 𝑗 ( 𝑡 |𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 , 𝑣 𝑖 ) is the Mixed
Proportional Hazard (MPH) function (Van den Berg 2001), i.e,
𝜃 𝑗 ( 𝑡 |𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 , 𝑣 𝑖 )= 𝜓 𝑗 ( 𝑡 ) 𝜙 ( 𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 ; 𝜷 𝑗 ) 𝑣 𝑖 (4.3)
𝜓 𝑗 ( t) , the baseline hazard, is a function of elapsed time for spell j. 𝜙 ( 𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 ; 𝜷 𝑗 ) is a
function of individual characteristics and allows them to have spell-specific coefficients.
Two more assumptions are needed (Wooldridge 2010).
Assumption 1. Only contemporaneous covariates matter.
𝜃 𝑗 ( 𝑡 |𝒘 𝑖 𝐻 ( 𝑡 ) , 𝒛 𝑖 , 𝑣 𝑖 )= 𝜃 𝑗 ( 𝑡 |𝒘 𝑖 ( 𝑡 ) , 𝒛 𝑖 , 𝑣 𝑖 ) .
72
Assumption 1 is strong. However, it would be much less restrictive if
contemporaneous time-varying variables are constructed to accumulate past
information, which is the case in this paper. For simplicity, let [𝒘 𝑖 ( 𝑡 ) , 𝒛 𝑖 ] = 𝒙 𝑖 ( 𝑡 ) .
Assumption 2. Time-varying variables are locally constant within one year.
𝒙 𝑖 ( 𝑡 )= 𝒙 𝑖 ( 𝑘 ) 𝑖𝑓 𝑘 − 1 ≤ 𝑡 < 𝑘 .
Theoretically t is continuous, but actual data are usually grouped. For example,
one time unit in this paper is one year. Assumption 2 is crucial to simplify analyses for
grouped data.
Under the two assumptions, hazard functions can be connected to observed
fertility outcomes 𝑦 𝑖𝑗𝑘 .
𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘 = 1|𝑦 𝑖𝑗 1
= 0, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
= 0)
= 𝑃𝑟𝑜𝑏 ( 𝑘 − 1 ≤ 𝑇 < 𝑘 |𝑇 ≥ 𝑘 − 1)
=
𝐹 ( 𝑘 )− 𝐹 ( 𝑘 − 1)
1 − 𝐹 ( 𝑘 − 1)
= 1 − 𝑒𝑥𝑝 (− ∫ 𝜓 𝑗 ( 𝑠 ) 𝜙 ( 𝒙 𝑖 ( 𝑠 ) ; 𝜷 𝑗 ) 𝑣 𝑖 𝑘 𝑘 −1
𝑑𝑠 )
= 1 − 𝑒𝑥𝑝 (− (∫ 𝜓 𝑗 ( 𝑠 )
𝑘 𝑘 −1
𝑑𝑠 )𝜙 ( 𝒙 𝑖 ( 𝑡 ) ; 𝜷 𝑗 ) 𝑣 𝑖 )
(4.4)
73
All terms in the first three lines of Equation (4.4) are conditional on 𝒙 𝑖 ( 𝑡 ) and 𝑣 𝑖 .
The second line translates birth decisions into durations. The third line simply uses the
conditional probability formula. Equation (4.2) is plugged in to generate the fourth line.
Assumptions 1 and 2 apply for the last two lines.
Figure 4.1 illustrates the probability of having the j’th birth by the duration since
the (j − 1)’th birth (j = 1, 2, 3, 4). The probability of having the first birth is bell-shaped
over duration, reaching the peak in the ninth year. The probabilities of having the
second, third and fourth births all rise first and then fall over duration, with peaks in
the second year. As the probabilities of childbearing by duration exhibit non-monotonic
patterns, ∫ 𝜓 𝑗 ( 𝑠 )
𝑘 𝑘 −1
𝑑𝑠 could simply parametrically specified as 𝑒𝑥𝑝 ( 𝑎 𝑗 𝑘 2
+ 𝑏 𝑗 𝑘 + 𝑐 𝑗 ) .
57
57
Extreme values at the tail of quadratic functions may substantially impact estimation. I also try to specify the
baseline hazard as a mixture of quadratic function and step function: one-step function for the tail and quadratic
function for the rest. The iterated estimation with such baseline hazard and frailty did not converge. But the
estimation with such baseline hazard and no frailty converged, and the estimation results are similar to those with
quadratic baseline hazard and no frailty, which implies extreme values at the tail may not be a big issue.
74
Figure 4.1 Distribution of durations between two births
(a) Duration before the first birth (b) Duration between birth 1 and 2
(c) Duration between 2 and 3 (d) Duration between 3 and 4
Further, assume 𝜙 ( 𝒙 𝑖𝑗𝑘 ; 𝜷 𝑗 )= 𝑒𝑥𝑝 ( 𝒙 𝑖𝑗𝑘 𝜷 𝑗 ) , which is a common specification.
Then, Equation (4.4) becomes
𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘 = 1|𝑦 𝑖𝑗 1
= 0, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
= 0) (4.5)
75
= 1 − 𝑒𝑥𝑝 ( −𝑒𝑥𝑝 ( 𝑐 𝑗 + 𝑏 𝑗 𝑘 + 𝑎 𝑗 𝑘 2
+ 𝒙 𝑖 ( 𝑘 ) 𝜷 𝑗 + 𝑙𝑛 ( 𝑣 𝑖 ) ) ) .
Lastly, 𝑙𝑛 ( 𝑣 𝑖 ) is non-parametrically specified, and is assumed to follow a mass-
point distribution, as suggested by Heckman and Singer (1984). Assume its distribution
has M supporting points, with locations 𝑞 1
, ⋯ , 𝑞 𝑀 and probabilities 𝑝 1
, ⋯ , 𝑝 𝑀 . All p’s
and q’s are unknown parameters such that ∑ 𝑝 𝑚 𝑀 𝑚 =1
= 1 and 𝔼 ( 𝑙𝑛 ( 𝑣 𝑖 ) )= 0. Heckman
and Singer (1984) show that such frailties improve parametrically specified ones in
uncovering true coefficients of interest. In practice, one starts with 2 points of support,
and estimates the locations and probabilities of the two supporting points. Then, one
estimates three supporting points and if the third point generates a larger likelihood, it
is accepted. Such steps continue until the likelihood function cannot be further
improved. The procedure of estimating the distribution of frailty is further discussed in
Appendix A.
An intuitive way of understanding the model is to start with its underlying
model, as follows.
𝑦 𝑖𝑗𝑘 ∗
= 𝑐 𝑗 + 𝑏 𝑗 𝑘 + 𝑎 𝑗 𝑘 2
+ 𝒙 𝑖 ( 𝑘 ) 𝜷 𝑗 + 𝑙𝑛 ( 𝑣 𝑖 )+ 𝘀 𝑖𝑗𝑘 . (4.6)
𝑦 𝑖𝑗𝑘 ∗
is woman i’s utility gained from having the j’th birth in the k’th year after her
last birth. It is determined by elapsed time in spell j, observed characteristics, frailty,
76
and error term 𝘀 𝑖𝑗𝑘 . By construction, observed birth decisions 𝑦 𝑖𝑗𝑘 can be linked to 𝑦 𝑖𝑗𝑘 ∗
in the following way.
𝑦 𝑖𝑗𝑘 = {
1 𝑖𝑓 𝑦 𝑖𝑗𝑘 ∗
≥ 0
0 𝑖𝑓 𝑦 𝑖𝑗𝑘 ∗
< 0
. (4.7)
Assume 𝘀 𝑖𝑗𝑘 follows the standard type I extreme value distribution, then
Equation (4.5) is obtained.
𝑙𝑛 ( 𝑣 𝑖 ) is a common factor over woman i’s spells, so it captures some between-
spell correlations, which is crucial for a multi-spell model. A more general way is to
specify frailties for each woman-spell, 𝑣 𝑖𝑗
, and allow 𝑣 𝑖𝑗
’s to be correlated for woman i.
This paper will try this frailty in the future.
The likelihood function of woman i is
ℒ
𝑖 = ∑ 𝑝 𝑚 (∏ ∏ 𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘 = 1|𝑦 𝑖𝑗 1
, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
, 𝑞 𝑚 )
𝑦 𝑖𝑗𝑘 𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘
𝐾 𝑖𝑗
𝑘 =1
𝐽 𝑖 𝑗 =1
𝑀 𝑚 =1
= 0|𝑦 𝑖𝑗 1
, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
, 𝑞 𝑚 )
1−𝑦 𝑖𝑗𝑘 )
(4.8)
where
77
𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘 = 1|𝑦 𝑖𝑗 1
, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
, 𝑞 𝑚 )
= 1 − 𝑒𝑥𝑝 ( −𝑒𝑥𝑝 ( 𝑐 𝑗 + 𝑏 𝑗 𝑘 + 𝑎 𝑗 𝑘 2
+ 𝒙 𝑖 ( 𝑘 ) 𝜷 𝑗 + 𝑞 𝑚 ) ) .
(4.9)
and
𝑃𝑟𝑜𝑏 ( 𝑦 𝑖𝑗𝑘 = 0|𝑦 𝑖𝑗 1
, … , 𝑦 𝑖𝑗 ( 𝑘 −1)
, 𝑞 𝑚 )
= 𝑒𝑥𝑝 ( −𝑒𝑥𝑝 ( 𝑐 𝑗 + 𝑏 𝑗 𝑘 + 𝑎 𝑗 𝑘 2
+ 𝒙 𝑖 ( 𝑘 ) 𝜷 𝑗 + 𝑞 𝑚 ) ) .
(4.10)
Then, the likelihood function for all women is ℒ = ∏ ℒ
𝑖 𝐼 𝑖 =1
. An implicit
assumption is that given covariates and frailties, women’s birth decisions are
independent over spells, and between each other.
4.3.2 Policy measurement
Woman i’s exposure to the period r policy (r = 1, 2, 3) at age a is defined as a
𝑃𝑜𝑙𝑖𝑐𝑦 𝑖 𝑟 ( 𝑎 )= ∑ 𝑝 ( 𝑎𝑔𝑒 )× 𝐼 ( 𝑎𝑔𝑒 ∈ [𝑎 𝑖 𝑟𝑆
, 𝑎 𝑖 𝑟𝐸
])
𝑎 𝑎𝑔𝑒 =15
(4.11)
Woman i’s exposure to policy r started at age 𝑎 𝑖 𝑟𝑆
, and ended at age 𝑎 𝑖 𝑟𝐸
. For
example, a woman born in 1960 started to be exposed to the period 2 policy (1971–1979)
at age 15, and stopped being exposed to the policy at age 19; a woman born in 1950
78
started to be exposed to the period 3 policy (1980–present) at age 30, and stopped being
exposed at the end of her last year in the sample. Indicator function I(.) tells whether
woman i was exposed to policy r at some age.
Exposure indicators are further weighted by the probability of conception at age
a, p(a), as in the static analysis.
Lastly, woman i’s exposure to the period r policy at age a is constructed as the
accumulation of all her past exposure to the policy. It implies that, even though a policy
ended, it would still affect women’s birth decisions in the following, reflecting the long-
run effect of family planning policy.
58
Assumption 1 is less restrictive with such policy
measures. Policy measures are further interacted with urban dummy and Han dummy
in the model.
4.3.3 Other covariates
Other than policy measures, the model further controls for exposure to the 1959–1961
great famine. Famine exposure is similarly constructed, by letting 𝑎 𝑖 𝑟𝑆
and 𝑎 𝑖 𝑟𝐸
be the
starting and ending age of being exposed to the famine.
59
The model also controls for the urban dummy and Han dummy alone. They are
assumed to be time-invariant. The urban dummy is defined according to the residential
58
Examples of long-run effects include permanently changed fertility preference, infertility due to sterilization, etc.
59
The famine officially ended before 1962, followed by the fertility makeup in 1962. Therefore, 𝑎 𝑖 𝑟𝐸
is assumed to be
the age in 1962.
79
location reported in women’s most recent interviews.
60
However, urban women might
live in rural areas before, and rural women might come from urban areas. Either case
would underestimate the urban-rural gap of the policy effects, given that the policy was
designed to be enforced more stringently in urban areas. But even though, the
estimated gap is still fairly large.
Women’s years of schooling were reported in their most recent surveys, and their
schooling history is deduced by assuming that they started schooling at age 6. The
model then controls for three dummies for women’s highest education levels: finishing
primary school, finishing middle school and finishing high school and above, treating
not finishing primary school as the base group.
61
This paper, like many other studies, assumes education is exogenous to fertility
decisions. It would be less restrictive if both education and fertility are assumed to be
endogenous and are jointly estimated, as in Angeles, Guilkey, and Mroz (2005).
Many other important determinants of fertility are not available, including
various prices directly or indirectly related to having and raising children, the history of
income and wealth, infant mortality rates, and so on. Instead, this paper controls for a
number of variables that aim to proxy these omitted variables. They are birth cohort
trend of women and its square, regional dummies,
62
cohort trend × urban dummy,
60
The urban dummy can also be defined on household registration types, usually known as hukou types. But nearly
20% women did not report their hukou types.
61
Denote years of schooling by S. Finishing primary school, middle school and high school correspond to 6 ≤ S < 9, 9
≤ S < 12 and S ≥ 12. The base group means S < 6.
62
The twelve provinces in the sample are grouped to four regions: Northeast China, East China, South Central China,
and Southwest China. The division of regions is purely geographic. Heilongjiang and Liaoning belong to the
80
cohort trend × Han dummy, cohort trend × coastal province dummy,
63
age of women
and its square, and a dummy indicating whether the first birth is a son.
64
4.4 Empirical results
4.4.1 Estimation results
Table 4.3 shows the coefficients of selected variables estimated from the mixed
proportional hazard duration model. Coefficients are assumed to differ by birth spell.
Robust standard errors are shown in parentheses. Three stars, two stars and one star
symbolize statistical significance at levels of 1%, 5% and 10%.
Table 4.3 Estimation results of the duration model
Birth spell
1 2 3 4
Policy 1 -3.875 -2.108 -0.508 2.596
(0.595)*** (0.779)*** (2.955) (1.562)*
Urban × Policy 1 0.018 0.615 -0.291 -1.531
(0.440) (0.401) (0.510) (0.842)*
Han × Policy 2 0.756 0.420 0.945 -0.889
(0.524) (0.433) (0.628) (1.137)
Policy 2 -3.104 -1.399 -0.568 1.516
(0.389)*** (0.723)* (3.157) (1.660)
Urban × Policy 2 0.166 -0.788 -0.739 -1.504
(0.201) (0.234)*** (0.322)** (0.663)**
Han× Policy 2 -0.209 -0.695 -0.203 -0.694
(0.282) (0.302)** (0.385) (0.759)
Policy 3 -2.697 -2.539 -2.170 0.512
Northeast China, Beijing, Shanghai, Shandong and Jiangsu belong to the East China, Hubei, Henan, Hunan and
Guangxi belong to the South Central China, and Chongqing and Guizhou belong to the Southwest China.
63
Beijing, Shanghai, Jiangsu and Shandong are coastal provinces, representing relatively developed regions.
Liaoning and Guangxi are by the ocean, too, but they are usually not regarded as developed provinces.
64
As all women entered the sample at age 15, age is perfectly collinear with duration in the first birth spell.
Therefore, age and its square are not controlled for in spell 1. Gender of the first birth is also controlled for in spell 2
and beyond.
81
(0.428)*** (0.715)*** (3.286) (2.088)
Urban × Policy 3 0.352 -0.812 -0.914 -1.210
(0.216) (0.195)*** (0.423)** (1.054)
Han × Policy 3 0.574 -0.500 -0.024 0.100
(0.299)* (0.257)* (0.449) (1.263)
Urban 0.293 -0.260 -0.595 -0.459
(0.194) (0.289) (0.576) (0.964)
Han 0.245 -0.181 -0.217 1.006
(0.238) (0.358) (0.713) (1.470)
Famine -5.372 -3.306 0.719 1.446
(0.736)*** (1.012)*** (1.728) (1.527)
Primary school -0.122 -0.286 -0.228 -0.089
(0.055)** (0.054)*** (0.080)*** (0.139)
Middle school -0.380 -0.569 -0.479 -0.080
(0.046)*** (0.052)*** (0.086)*** (0.157)
High school or above -0.976 -1.347 -0.421 -1.144
(0.054)*** (0.085)*** (0.153)*** (0.489)**
Duration 1.028 0.284 0.248 0.322
(0.034)*** (0.027)*** (0.071)*** (0.100)***
Squared duration -0.026 -0.023 -0.028 -0.037
(0.001)*** (0.002)*** (0.007)*** (0.011)***
Birth year trend -0.010 0.021 0.046 -0.050
(0.014) (0.022) (0.059) (0.097)
Squared birth year
trend
0.000 -0.001 -0.001 0.000
(0.000) (0.000)*** (0.001) (0.001)
Age 0.796 0.349 -0.352
(0.193)*** (0.917) (0.379)
Squared age -0.013 -0.008 0.002
(0.003)*** (0.013) (0.005)
Birth 1 is son -0.460 -0.426 -0.314
(0.039)*** (0.059)*** (0.096)***
Constant -6.636 -12.239 -6.428 3.998
(0.348)*** (2.687)*** (13.391) (6.613)
Region dummies Yes Yes Yes Yes
Cohort trend ×
urban dummy
Yes Yes Yes Yes
Cohort trend × Han
dummy
Yes Yes Yes Yes
Cohort trend × coast
province dummy
Yes Yes Yes Yes
Chi-squared statistic
for 0 policy effect
(df = 9)
122.1 131.8 75.5 17.6
P value for 0.0000 0.0000 0.0000 0.0400
82
0 policy effect
N 177473 177473 177473 177473
As the hazard function can be expressed as 𝑙𝑛 ( 𝜃 𝑗 ( 𝑘 |𝑥 𝑖 ( 𝑘 ) , 𝑣 𝑖 ) )= 𝑐 𝑗 + 𝑏 𝑗 𝑘 +
𝑎 𝑗 𝑘 2
+ 𝒙 𝑖 ( 𝑘 ) 𝜷 𝑗 + 𝑙𝑛 ( 𝑣 𝑖 ) , the coefficients in Table 4.3 can be interpreted as the semi-
elasticity of hazard rates to independent variables.
The first nine coefficients express the effects of the three policies on fertility.
Policy effects cannot be directly read from the coefficients. The partial effect of each
period of policy on the probability of having a certain number of births will be shown in
the next subsection.
Table 4.3 shows Chi-squared statistics for the joint statistical significance of
policy variables for each birth spell, as well as corresponding p values. In the first three
spells, the policy variables are strongly significant; in the fourth spell, they are
statistically significant at 5% level.
All three schooling dummies exhibit negative and statistically significant effects
on fertility, except that in the last column only finishing high school education shows
significant effect. Moreover, the effect of education is generally stronger as education
levels rise.
The coefficients of duration and its square are significant in all spells, and
support inverse-U shapes for duration dependence.
Women’s age also shows an inverse-U pattern in spell 2; while in spells 3 and 4,
age and its square are not significant determinants. Lastly, if a woman’s first child is a
83
son, she is much less likely to have more births. The son preference effect persists
through all birth spells.
The estimated frailty has three supporting points: 𝑞 1
̂ = −3.6644, 𝑞 2
̂ = −1.148,
and 𝑞 3
̂ = 0.31603 , with probabilities 𝑝 1
̂ = 0.044, 𝑝 2
̂ = 0.0962, and 𝑝 3
̂ = 0.8598.
Table 4.4 tests whether the 3-point frailty is different from no frailty or a 2-point
frailty. Panel A implies that the model with 3-point frailty is significantly different from
that without frailty. Similarly, Panel B shows that the model with 3-point frailty is
significantly different from the model with 2-point frailty.
Table 4.4 Specification tests: 3-point frailty, 2-point frailty, or no fraity?
Panel A. 3-point frailty versus no frailty
Log likelihood of estimated model with 3-point frailty -31881.939
Log likelihood of estimated model without frailty -32050.923
Log likelihood ratio statistic 337.96892
P value (Chi-squared distribution, df = 4) 0.0000
Panel B. 3-point frailty versus 2-point frailty
Log likelihood of estimated model with 3-point frailty -31881.939
Log likelihood of estimated model with 2-point frailty -31901.675
Log likelihood ratio statistic 39.471251
P value (Chi-squared distribution, df = 2) 0.0000
Panel C. Hannan-Quinn Information Criterion
Estimated model without frailty -32297.461
Estimated model with 2-point frailty -32152.575
Estimated model with 3-point frailty -32137.203
Baker and Melino (2000) pointed out that, frailty with too many supporting
points may bias estimation. Traditionally, the number of supporting points for frailty
are chosen by maximizing log likelihood. They applied Hannan-Quinn Information
Criterion by imposing a term which penalizes too many parameters. The revised
84
criterion maximizes 𝑙𝑛 ℒ − 𝑐𝑝 , where 𝑙𝑛 ℒ is the log likelihood, 𝑐 = 𝑙𝑛 ( 𝑙𝑛𝐼 ) (I is the
number of women in the sample), and p is the number of parameters in the model.
Panel C lists the values of 𝑙𝑛 ℒ − 𝑐𝑝 for estimated models without frailty, with 2-point
frailty, and with 3-point frailty, respectively. With this criterion, the model with 3-point
frailty is still the optimal choice.
4.4.2 Policy effects on the probability of having a certain number of births
This part calculates policy effects on the probability of having a certain number
of births. Given the estimates of the first four birth spells, I can derive the probabilities
of having exactly 0, 1, 2, and 3 births, and the probability of having 4 or more births.
Computation details are in Appendix B.
I first compare the derived probability of having a certain number of births with
the actual fraction of women having that certain number of births in the sample, to
check how well the estimated model fits the data. Table 4.5 shows the comparison. In
general, the derived probabilities fit the actual factions very well.
Table 4.5 Comparison between predicted probabilities and actual fractions, full and subsamples
1 birth 2 births 3 births 4 or more births
actual predicted actual predicted actual predicted actual predicted Number of women
Full sample 0.486 0.485 0.280 0.290 0.112 0.107 0.065 0.056 7057
Urban Han 0.640 0.641 0.200 0.211 0.066 0.063 0.022 0.021 2357
Rural Han 0.414 0.413 0.328 0.334 0.128 0.125 0.080 0.069 3883
Urban non-Han 0.461 0.521 0.270 0.246 0.108 0.100 0.088 0.038 204
Rural non-Han 0.357 0.322 0.290 0.329 0.186 0.166 0.126 0.118 613
Middle school or above 0.632 0.623 0.229 0.238 0.050 0.048 0.014 0.013 4354
Below middle school 0.250 0.261 0.363 0.374 0.212 0.202 0.147 0.126 2703
Coastal province 0.637 0.630 0.205 0.213 0.064 0.063 0.027 0.023 1963
Inner province 0.427 0.429 0.309 0.320 0.131 0.124 0.080 0.069 5094
First birth is son 0.569 0.516 0.275 0.289 0.100 0.096 0.056 0.045 3458
First birth is daughter 0.456 0.436 0.321 0.311 0.139 0.128 0.083 0.074 3198
Figure 4.2 displays how well the predicted probabilities of having exactly 1, 2, 3,
and 4 or more births match the actual fractions by women’s birth cohort. Generally, the
goodness of fit is satisfactory, except for the youngest cohorts in (a) and the oldest
cohorts in (c).
65
Figure 4.2 Comparison between predicted probabilities and actual fractions, by cohort
of women
(a) Having exactly 1 birth (b) Having exactly 2 births
65
The probability of childlessness is not shown for checking the goodness of fit, because it can be obtained simply by
subtracting other probabilities from 1. If other predicted probabilities match well, so would the probability of
childlessness.
87
(c) Having exactly 3 births (d) Having 4 or more births
All probabilities above were derived under actual exposure to family planning
policies. Next, I turn off the period r policy (r = 1, 2, 3), and calculate the probability of
having a certain number of births had women not been exposed to policy r, all the other
variables, including the other two policies, remaining unchanged. Further, the
difference between the probabilities with policy r on and off is defined as the partial
effect of the period r policy on the probability of having a certain number of births.
We would expect that, with exposure to a policy, a woman tends to reduce her
probability of having a large number of births, and correspondingly increase the
probability of having a small number of births. Table 4.6 shows this pattern. Numbers
in squared brackets are percent changes.
88
Table 4.6 Policy effects on the probability of having a certain number of births
Period 1 policy Period 2 policy Period 3 policy
no birth 0.003 0.007 0.031
[4.5%] [13.2%] [105.5%]
1 birth 0.004 0.041 0.318
[0.9%] [9.2%] [187.6%]
2 births 0.000 0.014 -0.041
[0.0%] [4.9%] [-12.4%]
3 births -0.016 -0.054 -0.275
[-12.8%] [-33.6%] [-72.1%]
4 or more births 0.009 -0.008 -0.034
[18.1%] [-12.5%] [-37.1%]
Note: Percent changes are in squared brackets.
Exposure to the period 3 policy reduces the probabilities of having exactly 2
births, 3 births, and 4 or more births by 0.041, 0.275, and 0.034 (or by 12.4%, 72.1%, and
37.1%), respectively. On the other hand, exposure to policy 3 increases the probabilities
of childlessness and having only 1 birth by 0.031 and 0.318 (or by 105.5% and 187.6%).
The effects of policy 1 and 2 have shown similar patterns, but the effect of the period 3
policy is greater than period 2, and the effect of the period 2 policy is larger than period
1.
4.4.3 Heterogeneous policy effects by urban-rural and ethnicity
Table 4.7 shows policy effects on the probability of having a certain number of
births for urban Han, rural Han, urban non-Han and rural non-Han women.
89
Table 4.7 Policy effects on the probability of having a certain number of births, by
urban-rural and ethnicity
Policy 1 Policy 2 Policy 3
Childlessness Urban Han 0.003 0.009 0.023
[6.9%] [25.8%] [121.4%]
Rural Han 0.003 0.009 0.020
[10.1%] [31.5%] [126.4%]
Urban non-Han 0.002 0.008 0.037
[4.2%] [17.0%] [177.6%]
Rural non-Han 0.006 0.011 0.023
[17.3%] [35.6%] [122.5%]
Policy 1 Policy 2 Policy 3
Having exactly 1 birth Urban Han 0.00800 0.0900 0.425
[1.2%] [17.0%] [218.2%]
Rural Han 0.00400 0.0340 0.263
[1.2%] [10.2%] [254.1%]
Urban non-Han 0.00500 0.0400 0.338
[1.0%] [9.1%] [234.2%]
Rural non-Han 0.00500 0.0150 0.181
[2.0%] [6.1%] [209.3%]
Policy 1 Policy 2 Policy 3
Having exactly 2 births Urban Han -0.002 -0.001 -0.107
[-0.6%] [-0.4%] [-31.2%]
Rural Han -0.003 0.024 0.019
[-0.7%] [7.2%] [5.5%]
Urban non-Han 0.010 0.054 0.062
[3.7%] [23.3%] [27.5%]
Rural non-Han 0.018 0.034 0.079
[5.6%] [10.8%] [29.7%]
Policy 1 Policy 2 Policy 3
Having exactly 3 births Urban Han -0.006 -0.068 -0.281
[-7.2%] [-47.5%] [-78.9%]
Rural Han -0.023 -0.064 -0.282
[-13.3%] [-30.3%] [-65.6%]
Urban non-Han -0.014 -0.065 -0.337
[-10.0%] [-34.1%] [-72.9%]
Rural non-Han -0.060 -0.089 -0.267
[-23.3%] [-31.3%] [-57.6%]
Policy 1 Policy 2 Policy 3
Having exactly 4 or more births Urban Han -0.003 -0.030 -0.060
[-10.0%] [-53.6%] [-69.9%]
90
Rural Han 0.018 -0.003 -0.020
[25.1%] [-3.0%] [-18.8%]
Urban non-Han -0.003 -0.038 -0.099
[-6.3%] [-43.7%] [-67.0%]
Rural non-Han 0.030 0.029 -0.016
[25.0%] [24.5%] [-9.6%]
Note: Percent changes are in squared brackets
Similar to Table 4.6, family planning policies generally reduce the probabilities of
having exactly 3 and 4 or more births, and raise the probabilities of childlessness and
having only 1 birth. Moreover, the effect of policy 3 is stronger than policy 1 and 2.
Table 4.7 reveals more information. Through all panels, policy effects are
generally stronger for urban women than for rural women. Moreover, policies tend to
lower the probability of having exactly 2 births for urban and Han women, but tend to
increase the probability for rural and non-Han women. In contrast, policy effects in
reducing the probability of having exactly 3 births tend to be greater for rural and non-
Han women than for urban and Han women. It implies that rural or non-Han women
desire more births than urban or Han women, so that they may respond to policy
exposure more actively at higher order of births.
4.4.4 Policy effects on fertility decline by cohort
This part tries to answer to what extent family planning policies could explain China’s
fertility transition, based on the estimation results of the duration model.
Similar to Chapter 3.4.3, I first predict the probability of having a certain number of
births over cohorts had women never been exposed to any policy (long-dashed lines).
91
Second, turn on the period 1 and 2 policy only, assuming the period 2 policy has been
enforced all the time since it started in 1971, and predict the probability of childbearing
over cohorts (short-dashed lines). Third, further let the period 3 policy start, and predict
the probability under actual exposure to all policy phases (solid lines).
Figure 4.3 shows that, without any policy, the probability of childlessness would
gradually rise from nearly 0 to 0.02. Policies level up the probability for all cohorts. The
period 1 and long-lasting period 2 policy displays larger effects than the period 3 policy,
but their gap is small in magnitude.
92
Figure 4.3 Predicted probability of childlessness under different policy histories
Figure 4.4 shows, without any policy, the probability of having exactly 1 birth
increases from about 0 to nearly 0.2 over cohorts. With the period 1 and long-lasting
period 2 policy, the probability rises from 0 to nearly 0.5. After the period 3 policy starts,
the probability increases from 0 to over 0.6.
93
Figure 4.4 Predicted probability of having exactly 1 birth under different policy histories
Figure 4.5 shows that, without any policy, the probability of having 2 births rise
from 0.2 to 0.4 over cohorts. The period 1 and long-lasting period 2 policy levels up the
probability for older cohorts while pulls it down for younger cohorts.
94
Figure 4.5 Predicted probability of having exactly 2 births under different policy
histories
In Figure 4.6, the probability of having exactly 3 births had women not been
exposed to any policy drops from about 0.5 to 0.4 over cohorts. Appearance of the
period 1 and long-lasting period 2 policy decreases the probability by about 0.2 through
all cohorts. The period 3 policy further pulls down the probability by about 0.1 for
younger cohorts.
95
Figure 4.6 Predicted probability of having exactly 3 births under different policy
histories
In Figure 4.7, the curves before cohort 1950 are difficult to interpret. Starting
from cohort 1950, the probability of having 4 or more births decreases from 0.15 to 0.05
over cohorts. Family planning policies further pull down the probability.
96
Figure 4.7 Predicted probability of having 4 or more births under different policy
histories
To summarize the five figures over, first, for older cohorts, family planning
policies reduce the probability of having 3 or more births, and increase the probability
of having 2 or fewer children. While for younger cohorts, policies decrease the
probability of having 2 or more births and increase the probability of childlessness and
having only 1 birth. Second, the period 3 policy in general has larger effects than the
97
period 1 and 2 policy. Third, even though without any policy, the probability of having
a certain number of births are still moving towards lower fertility levels over cohorts.
It is also of interest to investigate policy effects on the number of births. The
expected number of births for a cohort of women can be expressed as 𝔼 ( 𝑛 )= 𝑃 1
+ 2𝑃 2
+
3𝑃 3
+ ∑ 𝑖 𝑃 𝑖 𝑖 ≥ 4
, where 𝑃 𝑖 (i = 1, 2, ...) indicates the probability of having exactly i births. 𝑃 1
,
𝑃 2
and 𝑃 3
have been estimated from the model, but 𝑃 𝑖 (i ≥ 4) is not available. Therefore, I
rewrite the equation as 𝔼 ( 𝑛 )= 𝑃 1
+ 2𝑃 2
+ 3𝑃 3
+ 𝑃 ≥ 4
𝔼 ( 𝑛 |𝑛 ≥ 4) , where 𝑃 ≥ 4
is the
probability of having 4 or more births which has been estimated, and 𝔼 ( 𝑛 |𝑛 ≥
4) measures the expected number of births had women had 4 or more births. 𝔼 ( 𝑛 |𝑛 ≥ 4)
cannot be derived either, but we could set a lower bound and an upper bound for
𝔼 ( 𝑛 |𝑛 ≥ 4) . Apparently, its lower bound is 4. We set the upper bound to be 9, the
maximal number of births in the sample.
Therefore, we could calculate the lower bound and upper bound of 𝔼 ( 𝑛 ) . Figure
4.8 plots the actual number of births by cohort, as well as the lower and upper bounds
of the expected number of births by cohort. The actual line lies between the two bounds
for most cohorts, and is particularly close to the lower bound.
98
Figure 4.8 Actual number of births, and the lower and upper bounds of predicted
number of births, by cohort of women
Following similar procedure, Figure 4.9 plots the predicted number of birth in
various scenarios of policy exposure, for both the lower bound and upper bound. This
figure ranges from cohorts 1943 to 1972.
99
Figure 4.9 Predicted number of births under different policy histories
(a) Lower bound (b) Upper bound
The solid line represents the predicted fertility without any family planning
policy, the short-dashed line means the predicted number of births had women been
only exposed to the period 1 and long-lasting period 2 policy, and the long-dashed line
predicts the number of births under actual exposure to all policy phases.
In (a), without family planning policies, the number of births declines from about
3 to around 2.25; while with policies, it drops from 3 to about 1.5. This implies that
family planning policies would explain about half of the between-cohort fertility decline.
In (b), without policies, fertility decreases from 4 to 2.5; with policies, it goes down from
4 to 1.5. Policies would explain about 40%. Therefore, we could conclude that policies
could explain 40-50% of the fertility decline over cohorts. However, fertility would still
have declined greatly had there been no family planning policy.
100
4.4.5 Model without frailty
It is believed that a duration model without individual heterogeneity would bias
estimation. Table 4.8 compares policy effects derived from the model with 3-point
frailty and the one without frailty.
Table 4.8 Policy effects on the probability of having a certain number of births, models
without frailty versus with 3-point frailty
Period 1 policy Period 2 policy Period 3 policy
no birth no frailty 0.003 0.006 0.036
[4.1%] [9.7%] [129.2%]
3-point frailty 0.003 0.007 0.031
[4.5%] [13.2%] [105.5%]
1 birth no frailty 0.005 0.040 0.285
[1.1%] [9.2%] [148.8%]
3-point frailty 0.004 0.041 0.318
[0.9%] [9.2%] [187.6%]
2 births no frailty -0.002 0.009 -0.053
[-0.6%] [3.2%] [-15.2%]
3-point frailty 0.000 0.014 -0.041
[0.0%] [4.9%] [-12.4%]
3 births no frailty -0.018 -0.052 -0.245
[-13.8%] [-32.3%] [-69.1%]
3-point frailty -0.016 -0.054 -0.275
[-12.8%] [-33.6%] [-72.1%]
4 or more births no frailty 0.012 -0.002 -0.023
[25.7%] [-3.9%] [-28.9%]
3-point frailty 0.009 -0.008 -0.034
[18.1%] [-12.5%] [-37.1%]
It turns out that, the two models generate fairly similar policy effects. But this
conclusion only holds in the context of this paper, and does not have generalized
implications.
101
4.5 Conclusion
This chapter estimates the effect of China’s family planning policies on dynamic fertility
behaviors, using a panel sample constructed from the cross-sectional sample for the
static analysis. The dynamic analysis explores the dynamic process of fertility that the
static analysis is not able to investigate. Moreover, the dynamic analysis adopts similar
policy measures that are used by the static analysis.
The dynamic analysis uses duration model. The duration model expresses the
probability of childbearing in a year as a non-linear function of women’s observed
characteristics, and an individual heterogeneity, or frailty. The frailty is assumed to
follow a discrete distribution whose number, location and probability of support points
are all estimated. Eventually, the estimated frailty has three support points.
Based on the estimates, I calculate the probability of childlessness, and the
probabilities of having exactly 1, 2, 3, and 4 or more births. The predicted probabilities
are close to the actual fractions in the sample. Further, by turning on and off each policy,
I derive the effect of each policy on the probability of having a certain number of births.
The one-child policy decreases the probabilities of having 2, 3, and 4 or more births, and
increases the probabilities of childlessness and having only 1 birth. Earlier policies have
similar patterns but smaller magnitudes. Simulations further show that, family
planning policies could explain about 40%-50% of the fertility decline between cohort
1943 and 1972, which is consistent with the conclusion of the static analysis.
102
Chapter 5 Conclusion
In many developing countries, family planning programs were accompanied by fertility
decline. But evidence for their relations is mixed, including the most populous and
largest developing country, China.
This paper estimates the effect of family planning policies on fertility in China.
Both static and dynamic analyses are conducted. The static analysis estimates the policy
effects on the number of births that a woman has ever had, using a cross-sectional
sample of ever-married women with their birth records, from the China Health and
Nutrition Survey (CHNS). The dynamic analysis explores the relationship between
family planning policies and the likelihood of childbearing over time, using a panel
sample of CHNS that records women’s births in each year since age 15. The panel
sample is constructed from the cross-sectional sample used for the static analysis. The
dynamic analysis applies a multiple-spell mixed-proportional hazard model estimated
semi-parametrically. Particularly, the unobserved individual heterogeneity in the model
is non-parametrically estimated, as suggested by Heckman and Singer (1984).
Static and dynamic analyses share a couple of common conclusions. First, family
planning policies could explain about half of the fertility decline between cohorts 1943
and 1972 in the sample. Second, women with higher education and women whose first
birth is a son tend to have fewer births.
Both static and dynamic analyses also improve policy measures adopted by
previous studies, and make them reflect more complete policy history, and capture
103
more heterogeneity of policy exposure. Particularly, the static analysis shows that,
different policy measures could lead to substantially different results, which highlights
the importance of measuring family planning policies appropriately.
104
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Appendix A
One can start with a proportional hazard model without frailty, and then use estimated
coefficients as initial values for estimating the model with a two-mass-point distributed
frailty.
One can arbitrarily choose initial values for the locations and probabilities of the
two mass points, such that the expectation of frailty is 0. In this paper, the initial
locations are −1 and exp(0.4), with probabilities
𝑒𝑥𝑝 ( 0.4)
1+𝑒𝑥𝑝 ( 0.4)
and
1
1+𝑒𝑥𝑝 ( 0.4)
, respectively.
After the model with M-mass-point (M ≥ 2) distributed frailty converges, search
for the location of the (M + 1)’th point from a starting location to an ending location step
by step. The range of searching region should be large enough to include possible
locations of the new point. In this paper, the region is set to be from −100 to 100, with
step length 0.01. Then, place a small mass (for example, 0.05) onto each point within the
searching region. If the maximum increase in likelihood is greater than 0, the location
corresponding to this maximum is used as the initial value of the new location,
otherwise the procedure stops as adding a new point does not improve the likelihood.
In this paper, supporting points of the two-mass-point frailty are estimated to be
-2.2849 and 0.20745, with probabilities 0.0832 and 0.9168, respectively. Location of the
third point is determined to be −1 with initial probability 0.05. Adding this new point
increases log likelihood by 0.0799, the maximum increase among all points in the
searching region. Then, fix the location and probability of the first point, and
correspondingly adjust the location and probability of the second point such that
111
probabilities sum up to 1 and expectation of frailty is 0. Thus, initial values for
estimating three-mass-point frailty are determined.
Supporting points of the three-mass-point frailty are finally estimated to be -
3.6644, -1.148, and 0.31603, with probabilities 0.044, 0.0962, and 0.8598, respectively. The
maximum increase in log likelihood is negative (-0.0118) when searching for the fourth
point. Therefore, the frailty eventually has three supporting points.
112
Appendix B
Assume woman 𝑖 appears in the sample from age 15 to 𝐴 𝑖 (𝐴 𝑖 is the minimum
between 49 and her age at the last interview). According to the model, woman 𝑖 ’s
probability of having a birth in the 𝑘 ∗
th year of spell 𝑗 , conditioning on that she did not
have births before that year in that spell, is 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ∗
) ) , where 𝜃 𝑖𝑗
( . ) is the hazard
function for spell 𝑗 conditioning on 𝒙 𝑖 and 𝑣 𝑖 .
66
Then the probability of not having a
birth that year, conditioning on not having births before that year, is 𝑒𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ∗
) ) .
Given that woman 𝑖 has had 𝑗 − 1 births, her probability of having the 𝑗 th birth
in the 𝑘 ∗
th year after the (𝑗 − 1)th birth is
(∏ 𝑒𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ) )
𝑘 ∗
−1
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ∗
) ) )= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖𝑗
( 𝑘 )
𝑘 ∗
−1
𝑘 =1
) ( 1 − 𝑒 𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ∗
) ) ) ,
which measures the probability of not having any birth in the first 𝑘 ∗
− 1 years of spell
𝑗 , but having a birth in the 𝑘 ∗
th year.
Similarly, given that woman 𝑖 has had 𝑗 − 1 births, her probability of not having
more births in her rest life is
∏ 𝑒𝑥𝑝 ( −𝜃 𝑖𝑗
( 𝑘 ) )
𝐾 𝑖𝑗
𝑘 =1
= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖𝑗
( 𝑘 )
𝐾 𝑖𝑗
𝑘 =1
),
where 𝐾 𝑖𝑗
corresponds to her last year in the sample.
Based on the two probabilities above, probabilities of having a certain number of
births can be calculated as follows.
66
Through this appendix, I omit 𝒙 𝑖 and 𝑣 𝑖 from hazard functions for simplicity.
113
(a) Probability of childlessness
∏ exp( −𝜃 𝑖 1
( 𝑘 ) )
𝐴 𝑖 −15+1
𝑘 =1
= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 1
( 𝑘 )
𝐴 𝑖 −15+1
𝑘 =1
).
(b) Probability of having exactly 1 birth. Assume the birth occurs at age 𝑎 𝑖 1
, then the
probability of not having the birth before 𝑎 𝑖 1
but having it at 𝑎 𝑖 1
is
𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 1
( 𝑘 )
𝑎 𝑖 1
−15
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 1
( 𝑎 𝑖 1
− 15 + 1) ) ) .
Having exactly 1 birth further requires having no more births after 𝑎 𝑖 1
, whose
probability is
∏ 𝑒𝑥𝑝 ( −𝜃 𝑖 2
( 𝑘 ) )
𝐴 𝑖 −𝑎 𝑖 1
𝑘 =1
= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 2
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 1
𝑘 =1
).
Therefore, the probability of having exactly 1 birth at age 𝑎 𝑖 1
is the multiplication
of the two probabilities above:
𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 1
( 𝑘 )
𝑎 𝑖 1
−15
𝑘 =1
) ( 1 − 𝑒 𝑥𝑝 ( −𝜃 𝑖 1
( 𝑎 𝑖 1
− 15 + 1) ) ) 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 2
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 1
𝑘 =1
).
As 𝑎 𝑖 1
may range from 15 to 𝐴 𝑖 , the probability of having exactly 1 birth is
∑ (𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 1
( 𝑘 )
𝑎 𝑖 1
−15
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 1
( 𝑎 𝑖 1
− 15 + 1) ) ) 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 2
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 1
𝑘 =1
)) .
𝐴 𝑖 𝑎 𝑖 1
=15
(c) Probability of having exactly 2 births. Assume woman 𝑖 has the first birth at age 𝑎 𝑖 1
and the second birth at 𝑎 𝑖 2
(𝑎 𝑖 1
< 𝑎 𝑖 2
). Then, similarly, the probability of having the first
birth at 𝑎 𝑖 1
is
114
𝑃𝑟 ( 𝑎 𝑖 1
)= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 1
( 𝑘 )
𝑎 𝑖 1
−15
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 1
( 𝑎 𝑖 1
− 15 + 1) ) ) .
Further, the probability of having the second birth at 𝑎 𝑖 2
is
𝑃𝑟 ( 𝑎 𝑖 2
)= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 2
( 𝑘 )
𝑎 𝑖 2
−𝑎 𝑖 1
−1
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 2
( 𝑎 𝑖 2
− 𝑎 𝑖 1
) ) ) .
The probability of not having more births after age 𝑎 𝑖 2
is
∏ 𝑒𝑥𝑝 ( −𝜃 𝑖 3
( 𝑘 ) )
𝐴 𝑖 −𝑎 𝑖 2
𝑘 =1
= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 3
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 2
𝑘 =1
).
Therefore, the probability of having exactly 2 births at age 𝑎 𝑖 1
and 𝑎 𝑖 2
,
respectively, is
𝑃𝑟 ( 𝑎 𝑖 1
) 𝑃𝑟 ( 𝑎 𝑖 2
) 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 3
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 2
𝑘 =1
).
As 𝑎 𝑖 1
ranges from 15 to 𝐴 𝑖
− 1, and 𝑎 𝑖 2
is from 𝑎 𝑖 1
+ 1 to 𝐴 𝑖 , then the probability
of having exactly 2 births is
∑ ∑ 𝑃𝑟 ( 𝑎 𝑖 1
) 𝑃𝑟 ( 𝑎 𝑖 2
) 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 3
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 2
𝑘 =1
) .
𝐴 𝑖 𝑎 𝑖 2
=𝑎 𝑖 1
+1
𝐴 𝑖
− 1
𝑎 𝑖 1
=15
(c) Probability of having exactly 3 births. Assume the three births occur at age 𝑎 𝑖 1
, 𝑎 𝑖 2
and 𝑎 𝑖 3
(𝑎 𝑖 1
< 𝑎 𝑖 2
< 𝑎 𝑖 3
). Similarly, the probability of having exactly 3 births is
∑ ∑ ∑ 𝑃𝑟 ( 𝑎 𝑖 1
) 𝑃𝑟 ( 𝑎 𝑖 2
) 𝑃𝑟 ( 𝑎 𝑖 3
) 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 4
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 3
𝑘 =1
)
𝐴 𝑖 𝑎 𝑖 3
=𝑎 𝑖 2
+1
,
𝐴 𝑖 −1
𝑎 𝑖 2
=𝑎 𝑖 1
+1
𝐴 𝑖
− 2
𝑎 𝑖 1
=15
where 𝑃𝑟 ( 𝑎 𝑖 1
) and 𝑃𝑟 ( 𝑎 𝑖 2
) were derived above, and
115
𝑃𝑟 ( 𝑎 𝑖 3
)= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 3
( 𝑘 )
𝑎 𝑖 3
−𝑎 𝑖 2
−1
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 3
( 𝑎 𝑖 3
− 𝑎 𝑖 2
) ) ) .
(d) Probability of having 4 or more births. Similarly, the probability of having 4 or more
births is
∑ ∑ ∑ ∑ 𝑃𝑟 ( 𝑎 𝑖 1
) 𝑃𝑟 ( 𝑎 𝑖 2
) 𝑃𝑟 ( 𝑎 𝑖 3
) 𝑃𝑟 ( 𝑎 𝑖 4
)
𝐴 𝑖 𝑎 𝑖 4
=𝑎 𝑖 3
+1
𝐴 𝑖 −1
𝑎 𝑖 3
=𝑎 𝑖 2
+1
,
𝐴 𝑖 −2
𝑎 𝑖 2
=𝑎 𝑖 1
+1
𝐴 𝑖
− 3
𝑎 𝑖 1
=15
where 𝑎 𝑖 4
is the age for the fourth birth, and
𝑃𝑟 ( 𝑎 𝑖 4
)= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 4
( 𝑘 )
𝑎 𝑖 4
−𝑎 𝑖 3
−1
𝑘 =1
) ( 1 − 𝑒𝑥𝑝 ( −𝜃 𝑖 4
( 𝑎 𝑖 4
− 𝑎 𝑖 3
) ) ) .
In order to calculate the probability of having exactly 4 births, I need further to
know
∏ 𝑒𝑥𝑝 ( −𝜃 𝑖 5
( 𝑘 ) )
𝐴 𝑖 −𝑎 𝑖 4
𝑘 =1
= 𝑒𝑥𝑝 (− ∑ 𝜃 𝑖 5
( 𝑘 )
𝐴 𝑖 −𝑎 𝑖 4
𝑘 =1
).
But birth spells beyond 4 are not included for estimation, so the probability of
having exactly 𝑛 (𝑛 = 4, 5, ...) births is not computable.
In order to reduce computation burden, the probability of having 4 or more
births is simply calculated by 1 − 𝑃 0
− 𝑃 1
− 𝑃 2
− 𝑃 3
, where 𝑃 𝑛 is the probability of
having exactly 𝑛 births (𝑛 = 0, 1, 2, 3).
Each probability calculated above is conditional on one value of frailty. The final
probability is the expectation of probabilities with different values of frailty, i.e,
𝑃 𝑛 = ∑ 𝑝 𝑚 𝑃 𝑛 ( 𝑞 𝑚 ) ,
𝑀 𝑚 =1
116
where the frailty follows a mass-point distribution with supporting points 𝑞 𝑚 and
probabilities 𝑝 𝑚 (𝑚 = 1, . . . , 𝑀 ).
The probabilities are calculated for each woman. Average probabilities of a
group of women can be directly obtained by averaging probabilities for the women in
that group.
Calculation of standard errors of such probabilities would be super tedious, and
is not considered by the paper.
Abstract (if available)
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Asset Metadata
Creator
Wang, Fei
(author)
Core Title
Essays on family planning policies
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Economics
Publication Date
06/23/2014
Defense Date
05/12/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
China,family planning policies,fertility,OAI-PMH Harvest
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Strauss, John A. (
committee chair
), Crimmins, Eileen M. (
committee member
), Nugent, Jeffrey B. (
committee member
), Ridder, Geert (
committee member
)
Creator Email
wangfei.econ@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-422475
Unique identifier
UC11285968
Identifier
etd-WangFei-2576.pdf (filename),usctheses-c3-422475 (legacy record id)
Legacy Identifier
etd-WangFei-2576.pdf
Dmrecord
422475
Document Type
Dissertation
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Wang, Fei
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(contributing entity),
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(collection)
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Tags
family planning policies
fertility