1. Explaining delayed entry in marriages- A Hazard Model Approach
Chinmay Sharma- Empirical Microeconomics
Abstract
Marriage trends in the USA over the past few decades have witnessed subsantial changes. In particular,
the median age at which both men and women are marrying has substantially increased by approximately
6 years over the time period 1960-2010. In the same time period, there have a been a variety of institutional
changes that have aected the marriage decision as well. The aim of this paper is to try and explain the
factors that explain the age at marriage and their interactions with the institutional changes that have
taken place in American society. This paper adopts a hazard model approach and attempts to model the
duration of singlehood.
Contents
1 Introduction 2
1.1 Some Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review and Institutional Changes 5
3 Theoretical Framework 6
3.1 Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Data 9
4.1 Age at First Marriage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Marital Statuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Wage and Salary Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 Gender . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.5 Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.6 Race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Methodology 11
5.1 Modelling the hazard function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.1 Constant Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.2 Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.1.3 Log-Logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1.4 Generalized Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1
2. 5.2 Forming the Log-Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Results 16
6.1 Constant Hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.3 Log-Logistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4 Selecting Between Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.5 Testing For Structural Change (Chow Type of Test) . . . . . . . . . . . . . . . . . . . . . . . 22
6.6 Using 1970 as a breakpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.7 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion 26
8 Appendix 26
8.1 Appendix A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.3 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.4 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.5 Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.6 Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.6.1 Testing for the Appropriateness of the Weibull Model . . . . . . . . . . . . . . . . . . 29
8.6.2 Testing for the Appropriateness of the Constant Model . . . . . . . . . . . . . . . . . . 30
8.6.3 Testing for the Appropriateness of the Generalized Gamma Model . . . . . . . . . . . 30
9 References 30
1 Introduction
Marriage as an institution has been witnessing a lot of changes over the course of the past 50-60 years.
The structure of families is changing around the world- people are cohabiting more these days, fertility has
declined, marriage rates have decreased and divorce rates have increased. There are many economic impli-
cations of the changes in the structure of marriages and families in general. For instance, the demographic
pyramid changes given the decreases in fertility. Moreover, tax revenues may be aected by fewer individuals
marrying, given that secondary earners are taxed at very high marginal tax rates (Stevenson and Wolfers,
2007). The following picture is illustrative of the changes in marriage and divorce trends in the United States:
2
3. g
Figure 1: Source (Stevenson and Wolfers, 2007)
In the picture above, the marriage rate is dened as new marriages per thousand people and the divorce
rate is dened as the new divorces per thousand people. As can be seen, the marriage rate has been
steadily declining since the 1970s and the divorce rates have been steadily increasing over the course of the
same time period.
1 Goldsten et al (2001) point towards the fact that it is not necessary that people are
marrying less, but just that they are marrying at a later stage.
1.1 Some Explanations
These trends are perhaps best understood in a Beckerian framework, whereby individuals form marriages
because they are complements in home and market production. As a result, it is pareto optimal for
individuals to form a union and maximize total production. Stevenson and Wolfers (2007) state that such
complementaries are in fact reducing, and this can explain the trends that we see above. First, Greenwood
and Vanderbrouke (2005) allude to a study in Indiana which reported that the fraction of women that spent
more than 4 hours on housework reduced to 14% in 1999 from 97% in 1924. This reduction in hours spent
in home production has been attributed to technological progress and can be thought of as reducing the
complementarity gains from marriage.
2 Second, female labor force participation has also been steadfastly
increasing, which has implications for the meaning of household specialization. (Stevenson and Wolfers,
2007). Third, there have been sociological changes to American society, whereby individuals live together
without actually getting married. Cohabitation has become a stepping stone before marriage (Stevenson
and Wolfers, 2007) and there has been a marked increase in the number of individuals cohabiting. Fourth,
there have been quite a few institutional changes that have immediate implications for marriages. The
legalization of abortion, abolition of anti-miscegenation laws and institutionalization of no-fault divorce
have potential consequences for trends in marriages. The changing structure of American families is now
more stark then ever and shows no signs of slowing down. This study will focus on another empirical
1Upon a closer examination, divorce rates have decreased after the 1980s. But this trend is plausibly in part attributable to
the decline in marraige rates themselves.
2In principle, one can think of technological progress as making men and women closer substitutes in home production.
3
4. regularity that holds in American data- the delay in marriage timing by both men and women. The
following picture paints a clearer image of this phenomenon:
Figure 2: Source: CPS(http://www.pewsocialtrends.org/2011/12/14/barely-half-of-u-s-adults-are-married-
a-record-low/)
It can be seen from the graph above that the median age of marriage for women increased from 20.3 in
1960 to 26.5 in 2010. This increase is extremely substantial and shows the changes in marriage trends. This
graph basically is the motivation for the present study. This study wishes to examine the factors that explain
the age of rst marriage. The age of rst marriage can also be thought of as the duration of singlehood
3.
This study does not consider explicitly the decision to marry, but rather the length of time an individual
remains single. As a result, survival analysis will be used in order to model the duration of singlehood.
There have been a wide range of factors that the literature points towards that can aect the decision to
marry. An investigation of these factors can lend insights into why we are witnessing the trends discussed
above. For instance, if factors that explain age at rst marriage themselves are changing, then the decision of
when to marry will be aected as well. Futhermore, there have been several institutional changes in American
society which have also been identied as being critical in changing the structure of marriages. Stevenson
and Wolfers (2007) discuss at length a few of these institutional changes: the birth control pill, laws on
abortion, no-fault divorce laws and laws allowing marriage between races have been identied as having a lot
of impact on marriage structures. This study will also exploit the implementation of some of these policies
and examine if these institutional changes have interacted with factors determining the age of marriage by
testing for structural changes in the parameters of models using Chow-type tests. This paper will focus on the
time period 1967-1971 given data constraints. This is in itself not problematic, because a lot of the changes
were happening at around this time period. The rest of the paper is organized as follows: Section 2 discusses
the relevant literature on the determinants of marriage and discusses some of the institutional changes that
took place. Section 3 presents a basic model of demand for marriage and derives some testable predictions
3Duration of singlehood here is dened as the length of time between birth and the rst marriage
4
5. while Section 4 discusses the dataset that will be used. Section 5 presents the estimation methodology ,
Section 6 dicusses the results obtained and Section 7 concludes.
2 Literature Review and Institutional Changes
This section is devoted to listing the various determinants of marriage that have been commonly cited in
both the economics and sociology literature. If individuals marry because the gains from marriage are higher
than the gains from remaining single (Becker, 1973), then any factor which aects the perceived gains from
marriage can be responsible for the trends that have been described above. I refer to these gains as perceived
only because individuals ex ante do not know about the future value of remaining married, and this introduces
uncertainty. At the extensive margin, Spreitzer et al (1974) nd that if men have poor relationships with
their close family members, they tend to marry less.This has been explained through the lens of family of
orientation where individuals form expectations about what marriage should be like. Their study also notes
that men that come from more democratic families tend to marry more. Furthermore, as cohabitation is
becoming more and more socially accepted and commonplace, couples can wait longer before they choose to
marry. Moreover, there are important dierences in marriage trends by race in the US. For instance, 25% of
African Americans in the 1950-55 age cohort had not married by the age of 45 compared to 10% of whites.
These sociological reasons have an eect both on the decision to marry and when to marry.
According to Carter and Glick (1970, quoted in Spreitzer et al ,1974), it is those individuals who are at
the bottom and the top of the education distribution that are least likely to get married, albeit for dierent
reasons. The individuals who nd themselves at the bottom of the education distribution are probably unable
to nd a spouse that is willing to marry them whereas those at the top of the distribution are unable to
nd someone worthy of marrying them. Stevenson and Wolfers (2007) further note that individuals that
are educated are 3 percentage points more probable to get married by the age of 45 than individuals that
are not educated.
4 Moreover, more educated individuals are also less likely to divorce. Many studies note
the eect of education on marriage: the interesting result from these studies is the dierential impact of
education by gender. For instance, more educated men are more likely to get married sooner whereas more
educated women are more likely to remain single longer(Goldstein et al, 2001). Of course, as individuals
are receiving more and more education, they will delay the timing into marriage. Isey and Stevenson (2007)
note that women who are highly educated have delayed marriage more than lesser educated ones. Of course,
increased education necessarily reduces time for both home and market production, thereby reducing the
complementarity gains from marriage. Guner, Caucutt and Knolwes (2002) note that the changes in the
patterns of women's education is responsible for about 30% of the delay in fertility. All in all, the choice to
invest in education has immediate consequences on the age to marry.
As mentioned above, recent years have seen a surge in female labor force participation rates. Isey and
Stevenson (2007) note that in 2007, only 36% of women with young children did not have a job compared to
70% in 1970. The increase in female labor force participation is now a well documented one, and as Guner,
Caucutt and Knowles (2002) note, changes in women's labor market conditions have direct implications
for marital decisions, because they change the bargaining power of wives relative to their husbands. In
her famous study, Oppenheimer (1988) states that age of marriage basically depends on the age at which
individuals nd themselves in stable careers. The intuition is straightforward, given that women are become
increasingly independent (nancially), they can aord to spend more time being selective and nding a
suitable match.
The Beckerian model points towards the gains from marriage as resting on the complementarities of
4Here, education is dened as completion of college
5
6. home and market production. However, insofar as women are increasing market production, the gains from
complementarity are reducing as well. It is not ex ante clear if the eect of income on marriage timing
should be the same for women and men. Under the assumption that men's time in home production is not
substitutable for a wife's production, a woman who earns low wages can basically marry an individual with
higher income and make up for the low market wages by specialized home production. A woman that earns
high wages, however, has a high opportunity cost of staying at home and is therefore lesser likely to spend
time in home production. Moreover, given that she earns high wages, she can be more selective about the
man she chooses to marry. Using Oppenheimer's line of reasoning, given that information is imperfect at
the time of marriage, an individual can only evaluate the earnings potential of the partner. However, if men
can be thought of as specializing in market production, then men that earn higher wages should be able to
get married sooner. This reasoning has important implications: it points towards the fact that men who
earn higher wages relative to other men face dierent incentives and situations compared to women who earn
higher wages relative to other women. I will explore this in more detail in the empirical section.
It is clear that sociological, education and occupation related factors can have eects on the entry to
marriage. Also, given that in recent years, trends in these factors have been changing, they can perhaps
explain the trends in marriage delay as well. At the same time that such changes were taking place, there
were a lot of changes that took place in the American legal system that altered the gains from marriage. It is
not inconceivable that these changes also interacted with the factors that are generally understood to aect
the decision to marry. First, in 1960, the Food and Drug Administration approved the birth control pill and
by the year 1965, more than 40% of unmarried women who resorted to some form of contraception were
taking the pill. Initially, only married women were eligible to obtain the pill. In the year 1976 , however,
the Einsenstadt v.Baird case resulted in all women being eligible to obtain the pill. This basically resulted
in a decrease in out of wedlock births, thereby decreasing the number of marriages that took place because
of unwanted births (Stevenson and Wolfers, 2007). Second, in 1970, the the Roe v. Wade case resulted in
abortion being legalized nationwide in the US. This along with the birth control pill resulted in a decrease
in the likelihood of unwanted pregnancies. Naturally, one can think of both of these factors as having an
eect on marriage decisions. Apart from potentially representing a change in the structure of marriage
dynamics, this policy can have dierential eects by income levels. For women who a higher income prole
and therefore a higher opportunity cost of staying at home, the increased choice over fertility gives them
more 'freedom' in choosing when to marry compared to a woman with a lower earnings prole. Third, in
1967, anti-miscegenation laws were abolished in the US via the Loving v. Virginia ruling. This type of
institutional change has denite implications for marriage decisions by race. Unconstrained by the law, one
can imagine that individuals now have a greater choice set on whom to marry, and this can change the age
of rst marriage.
Given the predictions by the literature on the topic, the various factors that I will look at in my study as
determinants of the duration of singlehood are basically education, income levels, gender, race, interactions
of gender and income and interactions of education and gender. Moreover, I will also look at how the eect of
these factors have changed over time specically around the time period that the aforementioned institutional
changes were eected. The next section describes a basic theoretical model which will be useful in deriving
some predictions.
3 Theoretical Framework
In order to derive some testable predictions, a simple model of demand for marriage will be considered. The
main insights in the model will be guided from some of the existing studies in the literature. For purpose of
exposition, I do not make distictions between men and women.
6
7. Consider an innitely lived represantative individual whose preferences are represented by the following
utility function:
U = u(c, l, m)
wherein the agent derives utility from c (consumption), l (leisure) and m (number of marriages) and
u : R3+
→ R. The standard conditions for the utility functions apply to each of its arguments:
Ux 0, Uxx 0
The consumer's problem is basically as follows:
max
ct,ltmt
∞
t=0
βt
u(ct, lt, mt)
I consider mt as the number of marriages to be a continuous variable so as to make the problem dierentiable.
In this sense, this model is not exactly equivalent to the problem at hand.
5 However, in principle, the demand
for marriages can be thought of as a decision when to marry, whereby an individual can choose to marry
if the demand for marriage is high and remain single if this demand is low. The agent's time and budget
constraint is the following:
lt + ht = T
and
At+1 = R[At + w(e)ht − ct − pmm]
In the problem above, the agent divides her time between leisure and hours worked. The wage the indi-
vidual receives depends on her level of education e, and the agent buys consumption goods (the numeraire)
and marriage goods at price pm. Whatever is not consumed is saved in an asset which commands a net
interest rate of R. The intuition here is that although individuals enjoy marriage, they also have to pay for
it. This payment could be thought of in principle as psychological costs (monetized), costs of bearing/raising
children and expenditure on marriage goods that the individual would generally not buy if single. Moreover,
to guarantee an interior solution, it is further asumed that
lim
x→0
Ux = ∞,
lim
x→∞
Ux = 0
for each of the arguments of the utility function.
Given that the problem is stationary, it can be written in recursive form with lt, ct,and mt as control
variables and At as the state variable. The Bellman equation for the problem (conditional on an agent's
education level) is then:
Ve1
(A) = max
c,n,h
{u(c, l, m) + βVe1
(A )}
We can substitute out A to obtain:
Ve1
(A) = max
c,n,h
{u(c, l, m) + βVe1
(R[A + wt(e)(T − l) − c − pmm])}
5If m was an indicator variable, then writing the problem in recursive form would entail transforming it to an optimal
stopping problem,with agents decided the optimal period of marriage.
7
8. Deriving the First Order Necessary conditions is straightfoward:
c : uc(c, l, m) − β
∂Ve1
∂A
(A ) = 0
l : ul(c, l, n) − β
∂Ve1
∂A
(A )wt(e) = 0
and
m : um(c, l, n) − β
∂Ve1
∂A
(A )pm = 0
The above equations along with the budget constraint characterize optimal behavior of the agent. In order
to generate testable predicitons and comparative statics., it is useful to derive expressions for the Marginal
Rates of Substitution (MRS) between consumption and marraige and leisure and marriage.
3.1 Proposition 1
Combining the FONCs above, we have
MRSc,m =
um(c, l, m)
uc(c, l, m)
= pm
The intuition of the above MRS is straightforward.
Consider that the price of marriage increases, such that pc pc. To preserve equality, and given the fact
that the felicity function is concave, it must be that the number of marriages relative to consumption goods
is now lessser than before. Any factor which increase pm will reduce the number of marriages in equilibrium.
3.2 Proposition 2
Deriving the MRS between leisure and marriages is also similar (conditional on the level of education).
MRSl,m =
ul(c, l, m)
um(c, l, m)
=
wt(e)
pm
Now, consider an increase in w, w . Using the same reasoning as above, ceteris paribus, the individual
decreases leisure relative to marriages and works longer (as the opportunity cost of staying at home increases).
This implies that individuals that earn higher wages will marry lesser.
3.3 Proposition 3
As is assumed in the labor literature, let us take us as given the fact that
∂w
∂e 0.More educated individuals
will earn higher wages on the labor market. For any two individuals with education levels e, e , e e,
w(e ) w(e). Under the assumption of a representative agent such that all agents in the economy have the
same utility function, the MRS sugggests that the more education individual will marry lesser compared to
a lesser educated one. More educated individuals will demand lesser marriage than lesser educated ones.
8
9. 3.4 Implications
Given the three propositions above, any factors that directly or indirectly aect the price of marriage, the
wage rate or the education level will have implications for marriage behavior. Consider for instance rst, an
individual who earns higher wages. For this individual, the cost of leisure increases relative to marriage (or
in other words, the returns to working increases). As a result, relative to marriages, this indivdual decreases
amount of time spent on leisure (and more time working). A similar argument holds for an individual who
chooses higher levels of education. Finally, what is lesser obvious is how to interpret institutional changes
in this context. Consider rst, for instance the legalization of abortion. Prior to the enactment of this law,
some individuals would get married because of births that took place out-of-wedlock. However, with increased
control over fertility, perceived gains from getting married go down. This is because part of the utility from
getting married would plausibly come from the stigmatizing eects of having a child out-of-wedlock. With
this component now lower, um(c, l, m) is now lower ∀c, l, m ceteris paribus. Looking at the MRS, individuals
consume lesser marriage relative to consumption goods. A similar line of reasoning applies to the introduction
of the birth control pill. Finally, let us take the example of no-fault divorce laws. Given that no-fault divorce
laws palpably increase the probability of a divorce, the price of getting married is aected by this greater
uncertainty. As a result, one can think of this law as increasing pm. According to the model, individuals will
then choose to get married lesser.
4 Data
Given that I will be working with American data, I will use the Integrated Public Use Microdata Series
(IPUMS) website which is basically a set of data from the years 1962-2014 from the March Current Popula-
tion Survey (CPS). This is basically a nationally representative survey which is a joint eort of the Census
Bureau and the Bureau of Labor Statistics. The CPS has evolved over time, from being a survey that col-
lected questions predominantly regarding unemployment, to now asking a whole range of questions regarding
demographic, labor force characteristics and personal level characteristics. The CPS records microdata for
individuals, and oers identiers, thus making it possible to match individuals to their households. I have
information on highest level of education completed, hourly salary, gender and race. I have more that 6
million observations for the time period that I am considering.
4.1 Age at First Marriage
This variable is basically the main variable of interest. Given that I will be working with hazard functions,
the age at rst marriage can of course be thought of as the duration of singlehood. The purpose of this paper
is to try and explain the duration of singlehood. The interviewers hired by the CPS asked individuals about
their age the time of rst marriage. If a person has not been married, he/she is assigned a value of 00 or
Not in Universe. For such individuals, I will replace their values with their age, and then specify that they
have an incomplete spell in order to obtain their contribution to the log-likelihood (explained in detail below)
This variable is available from the years 1967 -197., and I am therefore constrained to choosing data from
between this time period. This is not particularly problematic, given that most of the institutional changes
that occured happened around this time period. The descriptive statistics for this variable are given below:
9
10. Figure 3: Summary Statistics
4.2 Marital Statuse
I need the marital status variable as variable that indicates failure. For individuals who have never married,
they form an incomplete spell, whereas for those who have married sometime in their life, their spell has
been completed. The marital status variable available on the CPS website basically has 6 categories of an
individual's marriage status. A tabulation of the dierent statuses is provided in Appendix C. To denote an
incomplete spell, I use the variable never married/single and all the other categories represent individuals
that married at some stage in their life. The variable I construct takes on the value 1 if the individual belongs
to any of the other 5 categories and 0 if the individual has never married.
4.3 Wage and Salary Income
The choice between hourly and weekly wages has been long debated in the literature. The obvious advantage
of using hourly wages is that these can be though as more accurate returns to skill, given that they are
independent of the amount of time spent working. Consider for instance an individual who earns a very low
hourly wage but works overtime. He/she will have an inated annual income, which could mask the true
return to skill. The problem with using hourly wages in this scenario is that these are not available for all
the years in question. As a result, if I wish to construct an estimate of the hourly wage, I will basically have
to construct it in the following way:
ˆHourly Wagei =
Annual Incomei
Weeks Workedi × Hours Worked Per Weeki
Unsurprisingly, this measure can induce a lot of measurement error, as all three variables in question may
be measured with error. As a result, I restrict the analysis to the Wage and Salary Income variable, which
measures the respondent's pre-tax wage income recorded for the previous year. Instead of using this variable
directly, I scale it by $1000, because it is hard to conceive of a unit increase in wage aecting the decision
to marry signicantly (at least economically speaking). The estimated coecients will then correspond to
changes in hazard rates given a $1000 change in salary income. The variable INCWAGE is a measure of the
wage and salary income.
4.4 Gender
As noted in the literature review section, males and females could have dierent incentives to form unions,
and could therefore exhibit dierential timings of marriages. As a result, it is necessary to include gender
10
11. as a covariate test this claim empircally. In the CPS, the variable SEX basically reports the respondent's
gender. I will create a dummy variable that will take the value 1 if the individual is female.
4.5 Education
According to the litearture, the eect of education is dierent for men and women. For men, a higher level
of education, might increase their bargaining power, and could therefore mitigate the delay that undertaking
education might have on the age of marriage. For women, on the other hand, increases in education would
imply increases in earnings, and this might disincentivize them from actually marrying early. The education
variable that I will use is the EDUC variable, which reports the highest level of education completed by the
respondent. Following the literature, I construct the education variable based on those individuals who are
enrolled it/or who have a college degree and those who do not. Appendix A shows the tabulation of the
education variable along with the sample frequency for the pooled dataset.
4.6 Race
In the literature review discussed above, many studies have noticed the dierence in the age at rst marriage
between the dierent races in the US. I basically control for race by using the RACE variable, which basically
contains 26 dierent codes (corresponding to dierent races). A tabulation of the codes is given in Appendix
B. I create two dummy variables which take on the values 1 if the individual is of African American origin
and 1 if the origin belongs to any other race except African American and Caucasian. The base case is then
Caucasian.
5 Methodology
5.1 Modelling the hazard function
The broad aim of this paper is to try and explain the duration of singlehood. A naturally suited methodology
to go about this type of analysis is duration analysis. Out of my sample of n individuals every year, I have
information on the length of times that they have spent before marriage (if they are indeed married) and
their ages if they have not. I therefore have t1...tn duration data for individuals. There are two conceptually
appealing reasons why I wish to use duration models. The rst is the fact that a lot of the observations I have
will be censored from above: a lot of individuals have never gotten married in the rst place- they form an
incomplete spell. As a result, if I just take the sample average of these durations, there will be a downward
bias. The second is the fact that I wish to learn about duration dependence, i.e. how are an individual's
prospects altered the longer he/she stays single? Drawing a close parallel to the unemployment literature, it
is not inconceivable that staying single for very long can have adverse signalling eects, which can further
decrease your chances of exiting singlehood. I will rst dene the hazard of marriage at time t as:
λ(t) = P(Duration of Singlehoodi = t|Duration of Singlehood ≥ t) =
P(Duration of Singlehoodi = t)
P(Duration of Singlehood ≥ t)
Basically, the hazard of marriage at time t is the conditional probability that individual i will terminate her
spell of singlehood and get married at time t , conditional on being single till at least t. Of course, it is of
interest to look at how the hazard rate changes as a function. Specic hazard functional forms allow me to
get an estimate of
∂h(t)
∂t , which is duration dependence.
11
12. Below, I present the smoothed unconditional hazard estimate ( smoothed version of the Kaplan-Meier
hazard estimate) for the pooled sample (1967-2011):
Figure 4: Hazard Function (Estimated)
As can be seen above, the hazard function is non-monotonic, and basically peaks at around age 27.
After this point, the hazard function monotonically decreases, potentially illustrative of negative duration
dependence. This means that the hazard of marrying is highest at age 27, or that the probability of marriage
is highest at this age, conditional on being single till at least then. A potential explanation for the decline in
the hazard function after that is the signalling hypothesis: individuals that stay single for longer may give
a bad signal to a potential pool of partners, hence prolonging their singlehood spell even more. Of course,
one must bear in mind that this is an estimated hazard function for the entire sample, and masks all the
structural changes that I wish to examine. The ip side of the smoothed hazard estimate is the Nelson-
Aalen cumulative hazard estimate.The graph below plots the Nelson-Aalen estimate:
12
13. Figure 5: NA Cumulative Hazard
Before conditioning on covariates, I would like to present hazard functions by dierent categories. Ap-
pendix E shows the estimated hazard functions by sex, whereas Appendix E shows hazard functions by
receiving college education or not.
6
As the aim of this paper is to try and explain the duration of singlehood, I will now condition this function
on a variety of covariates as described in the theory section. The expression for the hazard function will then
be:
hi(t, x) = λ(t)F(educationiβ1, incomeβ2, sexβ3, raceiβ4, education × sexβ5, wage × sexβ6)
which is basically a Cox Proportional Hazard (PH) model. The baseline hazard function is λ(t) which is
then scaled by the systematic part given dierences across individuals. The advantage of using the Cox
PH model is the fact that it leaves the functional form of the baseline hazard unspecied. Therefore, this
specication can be thought of as being semi-parametric, as a functional form needs to be specied for the
systematic part of the hazard. Given the form of the PH hazard function, two dierent individuals (with
dierent characteristics) will have hazard rates that are proportional for all time periods. I would also like
to discuss the choice of the parametric form for the hazard function.
Dierent choices result in dierent shapes. A common one used is a constant hazard, where the hazard
of marriage is assumed to be constant for all time periods. This is not intuitively appealing; the hazard of
6One should think of these as just descriptive, because they do not net out the eects of other variables.
13
14. marriage at age 1 is denitely dierent from the hazard of marriage at age 20! Methodologically as well,
one just has to look at the unconditional hazard function to ascertain that this assumption is not really
reasonable. I will basically be considering a few functional forms for the hazard specication. If I am to
assume that the underlying unconditional hazard function to a degree can serve as a rough guide to modelling
the conditional hazard function, then the Weibull or the Exponential hazard functions may not be the best,
as they do not allow for changing shapes of the hazard function. However, this way to select conditional
hazard functions may not necessarily be the best because the underlying hazard function actually changes
with covariates. (Parametric Models, NYU Lecture) I will also estimate hazard functions of the log-logistic
and generalized gamma functional forms, and the advantages of each are described in turn below. I will
then perform tests to see which hazard function is the most appropriate. The following are the ones I will
estimate:
5.1.1 Constant Hazard
The starting point of my analysis will be to assume that the hazard rate is constant over time. Although
extremely restrictive, this can prove to be a useful starting point. In this case, I dene the hazard of marrying
at any time period t as:
h(t, X) = λ
In order to ensure that the hazard rate remains non-negative, I will basically model the conditional hazard
as an exponential function of the covariates. I can then estimate the following conditional hazard model:
h(t, X) = λi = exp(Xiβ)
Given the form of the hazard function, the density is trivially derived as f(t) = h(t)S(t) = λexp(−λt). Given
this functional form (and under the assumption of complete spell-inow sampling data type, an assumption
which will be used to illustrate the log-likelihoods in this section) the log-likelihood of the sample can be
expressed as:
LN =
N
i=1
{λi exp(−λit)}
with λi dened as above. (Parametric Models, NYU Lecture)
5.1.2 Weibull
The Weibull distribution is a more exible form to estimate the hazard function. The Weibull function is
basically a two-parameter generalization of the exponential function (Llull,2011), and it is nests the Constant
Hazard as a special case. The Weibull function allows for duration dependence, but the only problem is
that the duration dependence can either be positive or negative. In other words, it can only allow for
a monotonically increasing or decreasing hazard function. Again, letting λi = exp(Xiβ), the conditional
hazard function is given by:
h(t, X) = λp(λt)p−1
λ 0
where basically λ denotes the location parameter and p is the parameter that determines how the hazard
function behaves with respect to time. Again, the log-likelihood of the sample can be conveniently expressed
as:
14
15. LN =
N
i=1
{λp(λt)p−1
exp(−λt)p
}
In this specication, if p 1, then the hazard is decreasing in time, if it is the converse, then the hazard is
increasing in time and if it is exactly 1, then the model reduces to an exponential one. (This is visible from
the hazard rate). (Parametric Models, NYU Lecture)
5.1.3 Log-Logistic
The Log-Logistic model, akin to the Weibull, has two parameters, γ and λ, whereby γ denes the shape
and λ determines the location. The advantage of the log-logistic model is that it actually allows for hazard
functions that are non monotonic. The hazard function in this case is :
h(t, X) =
λ
1
γ
i t[ 1
γ −1]
γ[1 + (λit)
1
γ ]
The sample log-likelihood to be estimated is then expressed as:
LN =
N
i=1
{
λ
1
γ t[ 1
γ −1]
γ[1 + (λit)
1
γ ]2
×
1
1 + (λit)
1
γ
}
and where λ = exp −(Xiβ). In the case of the log-logistic model, dierent values of γ determine the shape of
the conditional hazard. For instance, if ˆγ 1,then we get a hazard that rises initially, and then falls. Con-
versely, if ˆγ ≥ 1, we obtain a conditional hazard function that decreases in time. Again, if the unconditional
hazard can serve as a rough guide to selecting a parametric functional form, then the log-logistic is probably
more appropriate than the Constant or the Weibull specication. (Parametric Models, NYU Lecture)
5.1.4 Generalized Gamma
The nal model that I will estimate is the Generalized Gamma Model. The Generalized gamma model allows
a great deal of exibility for hazard rates. Moreover, the Generalized Gamma model nests within it (amongst
others), the constant and Weibull specication. The density of this distribution is given by:
f(t) =
λp(λt)pκ−1
exp(−λt)p
Γ(k)
After the estimation procedure, I then perform tests to see which distribution is implied. The form of the
tests are discussed in the results section below.
5.2 Forming the Log-Likelihood
To form the hazard function, I will use the variable Age at rst Marriage (which corresponds to the duration
of singlehood). The data that I will be able to use is from the years 1967 -1971 because this variable is
only available for this time period range. In order to describe my data as survival data on STATA, I need to
basically specify a failure variable, i.e a variable that is able to indicate a complete spell. As mentioned above,
I use the marital status variable to denote an incomplete spell. I basically have two types of observations:
15
16. individuals who never married and individuals who married at some point (and therefore, age at rst marriage
is their duration of singlehood). For the latter type, the contribution to the likelihood is f(ti|xi). For the
former, their contribution to the likelihood is basically Pr(T ≥ ¯ti|xi) = 1 − F( ¯agei|xi) where f(.) and F(.)
correspond to the pdf and cdf of the function used. As a result, the log likelihood can therefore be compactly
expressed as follows:
LN =
N
i=1
{wi ln f(tI|xi) + 1 − wi ln(1 − F( ¯agei|xi)}
where wi = 1(Completed Spelli)
6 Results
6.1 Constant Hazard
I rst present the regression results for the Constant Hazard Model with all the data pooled. All the
dierent hazard specications are variants of the Cox PH model, where baseline hazard is to be estimated
non-parametrically. The systematic part of the constant hazard by default is the exponential function. The
following is the output that I obtain on STATA:
Figure 6: Constant Hazard: Output
There are a few results worth mentioning. Females basically have a higher conditional hazard of leaving
singlehood than men, i.e., their expected duration is of staying single is lesser than that of men. It is
16
17. important to note however, that this result can be driven by the fact that I have pooled all the observations
together, and there might have been some structural changes not captured in this specication. Historically
speaking, the average age of women at marriage has always been less than that of men. However, as can be
seen from Figure 2, although the median age of marriage is increasing for men, that of women is increasing
faster, hence closing the age gap. The following gure is illustrative of this result (not conditional on any
other characteristics):
Figure 7: Comparison of hazard functions by Sex
One can see from the graph above that the hazard of marriage is maximized for females much earlier than
that of men. Moreover, it declines rather rapidly as well.
Second, relative to other races, African Americans have a lower hazard of marriage, i.e. their expected
duration of staying single is higher than that relative to other races. This is actually consistent with previous
studies as well, as has been demonstrated in the literature review section. With regards to the college variable
(which is an indicator that takes on the value 1 if the individual is at least college educated), the results show
that college educated individuals basically have a lower hazard of marriage. This is in line with Proposition 3
from the theoretical model developed in Section 3. Although this result makes sense according to the theory,
it is hard to know ex ante which is the mechanism that is driving this result.
If one thinks of human capital investments and home production as independent time-consuming activities,
then choosing to undertake human capital investments leaves one lesser time to engage in home production
17
18. (for example, marriage). This strain on limited time could cause individuals to shift away from marriage
while studying. However, this is another force at work. More educated individuals arguably have higher
bargaining power in the marriage market, ceteris paribus. As a result, they can choose a) be more picky (in
which case this eect reinforces the previous one), or b) nd someone they want to get married to earlier,
given that they have a greater set of potential partners (in which case, this eect countervails the previous
one). For men, it seems that both eects work in the same direction. Of course, it is of interest to see if this
eect is dierent for women! The coecient on the interaction term shows that for females, being educated
makes them delay marriage more than it does men. This eect is statistically signicant at conventional
levels, and shows that women who are educated indeed can spend much more time being picky about
their potential partners (given that time taken to complete college education is constant). The limitation of
the theoretical model was the fact that it did not make a distinction between men and women. Specifying
dierent functional forms of utility forms would be more realistic and could deliver dierent predictions for
men and women .
Lastly, I also wish to look at the eect of wages on the hazard of marriage. It seems that higher wages
result in the an increase in the hazard function, that is, a lower expected duration of staying single.This
result is not in line with Proposition 3 derived in theoretical section which states that higher income would
lead to a decrease in demand for marriage. However, traditional bargaining-power type arguments suggest
thatt individuals that make higher wages will have a bigger set of potential partners to choose from. The
interesting result again here is the interaction of the wage variable with gender. The coecient on wages for
females, although signicant in totality, is signicantly attenuated. For women then, higher wages are not as
important a determinant for age of marriage than it is for men. I now present estimates for the other models.
6.2 Weibull
The following is the output I obtain from STATA for the Weibull specication. Again, an advantage of the
Weibull model is that it allows me to model duration dependence:
18
19. Figure 8: Weibull Model Estimation
The results are qualitatively similar to the constant hazard model, with an important exception. In this
specication, one can see that college educated males have a higher hazard of marriage than non-college
educated men. This result is the opposite to that in the previous specication. However, the interaction term
for women shows that this eect is completely overturned! College educated women, relative to non -college
educated women, have a much lower hazard of marriage, statistically signicant at conventional levels. This
reversal in signs clearly alludes to an eect that is dierent for men and women. Economically speaking, the
costs and benets associated with marriage are perhaps perceived dierently by men and women. Finally,
as described above, the Weibull function allows for duration dependence. If the estimated parameter p 1,
then the hazard is increasing monotonically with time, and conversely, if p 1,then the hazard is decreasing
with time. Surprisingly, from the output above, one can see that p has been very precisely estimated to be
greater than 1. This means that there is positive duration dependence, or that as one is single for longer,
one's chances of marrying increase. This is not in line with the unconditional hazard estimate presented
above.
6.3 Log-Logistic
As mentioned above, this specication is also an attractive alternative to the Weibull and the constant hazard
specication. This is due to the fact that the log-logistic specication allows for non monotonicities in the
hazard function. The following is the output that I get from STATA:
19
20. Figure 9: Log-Logistic Estimation
The signs of the coeceint are opposite in values as compared to the previous ones. This is because of the
fact that the model has been estimated in Accelerated Failure Time Format. In this case, we have basically
estimated a log-linear regression of the form:
ln(T) = Xβ + z, , u = z/σ, u ∼ Logistic
In the above, σ is the scale factor. The interpretation of coecients is the same as a log-linear model. This
implies that relative to men, women's expected duration in singlehood decreased by approximately 17%. This
is in line with previous results. For men, the eect of an increases in wage is to lower the expected duration of
singlehood, an eect that is much lesser for women. Finally, college education for men increases the expected
duration of singlehood, and eect that is much more pronounced for women. This is an empirical consistency
that is basically true for all specications! As mentioned above, the log logistic specication includes the
shape parameter γ, which if less than one, is indicative of a conditional hazard function that rises initially,
and then falls with time. This is exactly what can be seen from the results above. I fail to reject the following
test:
H0 : γ ≤ 1
HA : not H0
20
21. 6.4 Selecting Between Models
There are various ways one can go about selecting between the three models that I have estimated above.
A puritan way to do this would be by comparing the log likelihood and then seeing which model maximizes
the log likelihood. According to this approach, I would select the Log Logistic specication from above. This
is actually not really that surprising the log logistic allows for a conditional hazard function much like the
unconditional one shown .Another approach would be to estimate a generalized gamma model, which nests
the Weibull and the Constant Hazard, and then test to see which one is implied. The following is the output
I obtain from STATA:
Figure 10: Generalized Gamma Model
As can be see, the generalized gamma model is also estimated in accelerated time format. This is akin
to running a log-linear regression of durations.Now, to select between models, I perform the following three
tests:
H0 : κ = 1
HA : not H0
If I fail to reject the null, basically the Weibull distribution is the appropriate one.
The second test that I perform is as follows:
H0 : κ = p = 1
21
22. HA : not H0
If I fail to reject his more restrictive test, then the Constant Hazard is implied.
Finally, I can test if the gamma distribution is the most pertinent.
7 To test for its relevance, I would
basically test:
H0 : p = 1
HA : not H0
After I perform the tests on STATA, (output show in the Appendix), I basically obtain that that reject the
null for all three tests! It seems that statistically speaking, none of these models is accurate. It is no surprise
that I have rejected the null so resoundingly in all cases. This is because of the fact that the shape and
location parameters have been so precisely estimated that their condence intervals are extremely small. A
problem with the generalized gamma model is the fact that it can only help to choose between the models
that it contains. As can be seen, in this case, it is not possible to choose between the dierent models
based on tests. The alternative then is to look at the log-likelihood, which suggests that I should choose the
generalized gamma model. While testing for the structural breaks, only the Generalized Gamma model will
be used.
6.5 Testing For Structural Change (Chow Type of Test)
6.6 Using 1970 as a breakpoint
To test for Structural breaks, I will resort to a Chow-type of test. This test presupposes the knowledge of
a breakpoint, and then tests if the slope coecients have changed signicantly at the breakpoint (viz-a-viz
time interactions with all variables). As the generalized gamma model is already in a log -linear format,
the test is straightforward. In order to have sucient data points before the change point, I choose to use
the 1970 law of the legalization of abortion as a structural turning point of the model. This is because of
the fact that legalization of abortion has consequences on the perceived benets and costs of marriage. For
instance, it can postpone marriage because of a decrease in shotgun marriages, as discussed in the theoretical
framework section above . The cost of being sexually active reduces, and partners can choose to be more
selective and not marry just because of unwanted pregnancies. In order to test for the structural break, I
divide the sample in two parts:
t1 = 1967, 1968, 1969
t2 = 1970, 1971
The generalized Gamma model in accelerated failure time format will basically look like:
ln(T) = Xβ + DX × β + z, , u = z/σ, u ∼ Log Gamma
where D = 1 if the time period under consideration is 1970 or more recent, and where X is my covariate
list from above. The following is the output I obtain on STATA:
7A point I would like to make here is the fact that the log likelihood is actually maximized under the generalized gamma
specication.
22
23. Figure 11: Structural Break (Generalized Gamma Output)
There are some results that warrant comment here. First, the eect of college for men, although still
increasing the duration of singlehood, does so lesser. The opposite eect is true for women- after 1970, the
eect of college education on the duration of singlehood for women is now greater than was before 1970, i.e.,
the slope has become steeper. Second, after 1970, females are still on average getting married earlier relative
to men, but the dierence has reduced (by approximately 0.2%). Finally, the eect of wage on duration of
singlehood for men (although still negative) has reduced (by about 16%). This alludes to the fact that for
men, wage is now becoming less important of a determinant for marriage than before. (For men, wages in
both cases decrease duration of singlehood). For women, pre-reform, wages decreased duration of singlehood
by approximately 2% and this eect has decreased to 1.8% after reform. One must bear in mind that these
changes are really speaking not economically signicant.
6.7 Discussion of Results
The theoretical model predicted that increased (monetized) costs of marriages, higher wages and higher
education levels would all lead to a decrease in the demand for marriages. Some of the results suggest
otherwise, mostly coming from heterogenous impacts for men and women:
23
24. 1. Across all specications, women marry earlier than men.
2. The predictions of Proposition 2 are matched by the data: college education increases the duration of
singlehood (but this eect is stronger for women)
3. The predictions of Proposition 3 are not matched by the data. Higher income leads to a lower expected
duration of singlehood (this eect is weaker for women)
4. The predictions of Proposition are silent on the structural test results. This is because although the
theory predicts a decrease in demand for marriages, it does not predict how the parameters on education
and income change as a result of these changes. However, the decrease in the age gap between males
and females alludes to the fact that women are generally lowering their demand for marraiges, i.e. they
are marrying later.
Putting all these results together, one can extrapolate as to why we are witnessing such a change in marriage
trends in the data. A rst explanation is the increase in college attainment by both men and women. The
following graph is illustrative of this fact:
Figure 12: Rising College Education
The results suggests that college education increases the duration of singlehood. As the graph above
suggests, more and more individuals are getting educated. As a result, the overall average age of marriage
will be increasing as well in the population. Another way to explain the rising age at marriage is to see the
trends in out of wedlock births. The de-stigmatization of out-of-wedlock births can conceivably have an eect
on the decision to marry . The following image shows trends in the out of wedlock births in the USA over
the past few decades:
24
25. Figure 13: Trends in Out of Wedlock Births
As can be clearly seen in the graph above, out-of-wedlock births as a fraction of all briths have been
increasing steadily. This trend could be illustrative of the fact that it is now becoming more and more
acceptable to have children without marrying. Finally,a study by Loughran and Zissimopolos notes that
conditional on working, marriage negatively aects the wage prole of women. Given that women are
rational and know about such wage depressing eects of marriage, it is plausible that they choose to delay
marriage as long as possible.
25
29. 8.5 Appendix E
Figure 18: Hazard Function by College
8.6 Appendix E
8.6.1 Testing for the Appropriateness of the Weibull Model
Figure 19: Weibull Test
29
30. 8.6.2 Testing for the Appropriateness of the Constant Model
Figure 20: Constant Hazard Test
8.6.3 Testing for the Appropriateness of the Generalized Gamma Model
Figure 21: Gamma Test
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31
