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I. Introduction

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The empirical study of the relationship
between fertility and labour supply is crucial for testing the existing
theories that link the family and the labour market. Up to now, most of the
evidence found points out a negative correlation between fertility and female
labour supply, but many of these results are esteemed to be blurry as far as
the problem of endogeneity of fertility is not solved:

“… it has proven difficult to find enough
well-measured exogenous variables to permit cause and effect relationships to
be extracted from correlations among factors such as the delay of marriage,
decline of childbearing, growth of divorce, and increased female labour force
participation” (Robert J. Willis, 1987 p. 74).

Indeed, there are good
reasons to believe that fertility and labour supply are jointly determined,
thus preventing from extracting any causal interpretation: The fact that
fertility is both used in the literature as an explanatory variable and as a
dependent variable of labour force is compelling. Fortunately, this problem can
be tackled through diverse ingenious instrumental variable (IV) strategies, such
as that of Lundborg, P., Plug, E., and Rasmussen, A. W. (2017), who analyse the
effect of childbearing on labour market outcomes among women with similar
working histories that become mothers for
the first time through in vitro fertilization (IVF). Because the success of
such treatment can be regarded as a random result from nature, it becomes
plausible to compare the labour-market outcomes of those women who effectively
gave birth after the treatment to those of who did not.

Similarly, Cristia J. P. (2008)
focuses on women who sought for medical ‘advice and testing’ to get pregnant for the first time as well. Indeed, this
can also be regarded as a hypothetical experiment in which women seeking for
help are randomly assigned a baby by nature, allowing the author to compare the
labour-market outcomes of those women who were successful in getting pregnant to
those of who were not. In contrast with these two studies, however, in this
paper we will explore the causal relation from fertility to labour supply
following the alternative IV strategy based on the sibling sex mix in families with two or more children put forward by
Angrist, J. D. & Evans, W. N. (1998), with a subset of the same data base
used by the authors, to recreate some of their central estimates and
corroborate their findings.

Although standard household
theories predict that the labour-market consequences of having a first child
are stronger than those of having additional children, this difference in focus
allows us to ‘exploit’ the fact that parents prefer a mixed-sex composition of
their children, instead of having offspring of the same sex. More
interestingly, it has been observed that parents who have siblings of the same
sex are more likely to go on in having an additional child. And because the sex
mix is a randomly assigned factor, given by nature, a ‘dummy’ variable that
indicates whether or not the sex of the second child matches that of the first,
same sex, can be used as an
instrument for further gestation or ‘fertility’ among women with at least two
children (for the sake of simplicity and conciseness, we leave aside the use of
twinning as an alternative instrument
in this paper, which the authors find to deliver similar results). Thus, our
instrument captures the effect of moving from the second to the third child on
women’s labour supply.

II. The Data

In this study we use a
subset of the same Census Public Use Micro Samples (PUMS) of 1980 data set used
by Angrist, J. D. & Evans, W. N. (1998), but we leave aside the 1970 and
1990 samples originally considered by the authors, who detect a substantial
decline in fertility and an increase in women’s labour supply throughout the
period. By contrast, we focus exclusively in the causality effect from
fertility to labour supply in one particular period, which is the central
question of this work. Additionally, we also ignore the subsample of ‘married’
women considered by them; which leaves us only with the analysis of a subset of
the subsample of all women with two or more children contained in the 1980 PUMS
(recall that we are interested only in the marginal effect on labour supply of
moving from the second to the third child), which consists of 355,356 observations.

The variables and their
descriptive statistics are provided in Table 1, where the covariate of our main
interest is the binary variable Morethan2
children (as indicator of ‘fertility’, the endogenous variable), and Samesex is its instrument. As we will
see ahead, the two components of the latter, 2Boys and 2Girls, are
also shown. Demographic and labour supply variables are also included in the
lower half. Notice that among all women of the sample, 40.19%
had a third child.

III. Sex-Mix and Fertility

We can model parents’
sex-mix preferences and utility in the following way: Suppose a couple has
already  children, and they decide on the additional
number of children they want to have, . Because parents prefer a
mixed-sex composition of their offspring, having already a same-sex composition
reduces the utility from  and increases, at turn, the marginal utility
of . Thus, under these
circumstances, parents are more likely to decide to have an additional child.

Accordingly, Table 2 reports
different estimations of the effect of ‘sex-mix’ on fertility that reveal this
phenomenon. Recall here, however, that we are only interested in women with two or
more children. Thus, Table 2 shows the relationship between the fraction of
women who have a third child and the sex-mix of the first two children. Specifically,
women are divided into four groups according to the sex composition of their
offspring: two boys, two girls, one boy and one girl, and simply, two children
of the same sex. The last row displays the difference between the same-sex and
mixed-sex group averages.

Table 2 allows to infer that
women with two children of the same sex are noticeably more likely to go on in
having a third child than women with one boy and one girl. Concretely, 43.18% of the mothers with same sex children go on
in having a third child, while a markedly lower 37.13%
of mothers with mixed-sex children decide to have another one.

IV. Fertility and Labour Supply

A.    Wald Estimator

Because our Samesex instrumental variable is
essentially randomly assigned by nature, we can safely extract a causal
interpretation from the regression of fertility on labour supply. Consider the
following bivariate regression model:

Where  is labour supply (Workedforpay) or any of our other measures of labour-market
outcomes described in Table 1, and  is our endogenous fertility measure, Morethan2. As usual, we denote as  our Samesex
binary instrumental variable, and we define the estimator for binary
instrumental variables (), also called the Wald
Estimator, as:

In which  is the mean of  for the observations where  is equal to one, and the other terms are
defined in an analogue manner. Here, the numerator captures the relationship
between  and , while the denominator
captures that of  and . Thus, any effect of  on  is attributable to the effect of  on . This is, the  estimates the average effect of  on  for those women whose fertility () has been affected by the
sex-mix () of their offspring.

This  can be easily obtained by running a simple regression
of the endogenous variable with the instrument (as we will see, similar to a
first-stage estimation without covariates in the two-stage least-square, 2SLS,
estimation framework), to obtain the denominator, and a regression of the
outcome variable and the instrument to obtain the numerator (notice that both
the numerator and denominator are ‘scalars’), and then dividing one scalar over
the other (also, a procedure equivalent to running the second-stage estimation
without covariates in the 2SLS method). Effectively, the first column of Table
3 reports the components of  separately, showing in the first row the
denominator of the Wald estimate, , where it can be seen that
the effect of Samesex on Morethan2 is equal to 0.0605 (which is the same as the difference between the same-sex and
mixed-sex group averages reported in the last row of Table 2); and in the
remaining rows, different estimations of the numerator,  (one for each labour-market outcome),
suggesting that indeed, additionally to having more children than women with
one boy and one girl, women with two children of the same sex present a lower
labour supply. Specifically, the Wald estimates reported in the second column,
obtained from dividing the numerator by the denominator, indicate that having
more than two children decreased the supply of labour (the Workedforpay variable) by 13.89 (-0.0084/0.0605)
percentage points, weeks worked by 6.456,
hours worked by XX, and labour income by $2,273.666 per year.

B.     Two-Stage Least-Squares

We now try a different approach
to the problem by using the two-stage least-squares (2SLS) estimator. While the
Wald Estimator allowed us to identify the effect of fertility on labour supply,
the 2SLS estimator allows us to relate our labour-market outcomes (Workedforpay, Hoursweek, and Labourincome)
to fertility controlling for a list of other exogenous covariates, which
include Age, Age at first birth, Familyincomelog,
and Education. However, at this point
we deviate somewhat from the original authors’ estimation in that we treat Familyincomelog as a covariate instead
of a dependent variable, for it is more likely that the mothers’ family wealth
determines how prone they are to participate in the labour market, and not the
other way around (i.e. if a mother’s family is relatively rich, she might be
less urged to work while childbearing than a mother from a poor family). This
is made evident by the insignificant effects found by the authors when treating
this variable as an outcome variable. At the same time, we also make emphasis
on the role of mothers’ education because several theories put forward the idea
that the impact of fertility on labour supply varies with the years of
schooling, and there is some empirical evidence in the same sense, showing that
the more educated women’s labour supply is more sensitive to fertility than the
labour supply of the less educated women (Gronau, R. 1986).

Another advantage of using
2SLS is that it also permits us to control for any ‘secular’ additive effects
of childbearing as we use the Samesex
instrument. Indeed, because Samesex
is an interaction term comprising the sex of the first two children, it is
potentially correlated with the sex of either child, which can ultimately be a
problem if the sex of offspring affects in some way parents’ attitude towards
the labour market (see Angrist, J. D. & Evans, W. N. (1998) for a
proof). Thus, we can add the Boy1st (S1)
and Boy2nd (S2) regressors
described in Table 1 to eliminate the possibility of an omitted-variables bias
arising from these sources. We can then write the following regression model
linking the labour supply and labour-market outcomes with the endogenous fertility
variable, , the vector of other
exogenous variables, , and the additive effects
for the sex of each child, as:

Now, the first-stage
equation relating the endogenous Morethan2
variable to the sex-mix is:

Where  is the first-stage effect of the instrument on
. A variant of this approach
also used by the authors exploits the possibility of formulating an over-identified
model by decomposing the Samesex
instrument into two separate indicators: 2Boys
and 2Girls. To see this more clearly,
realize how we can express our instrument as:

Where S1 and S2
are, as we know, our indicators for male firstborn and second-born children, Boy1st and Boy2nd (notice how the instrument renders cero if both S1 and S2 are of different sex, and one if they are of the same
sex), from where we can extrapolate the two separate instruments: 2Boys, S1S2, and 2Girls,
(1-S1)(1-S2), also
reported in Table 1. This over-identification strategy is advantageous because
we might expect any bias arising from the so called ‘secular’ effects of child
sex on labour supply to be different for each of these two instruments, while
the effect of childbearing can be expected to be independent of whether Samesex equals to 2Boys or 2Girls. In this formulation,
however, since S1i, S2i, S1iS2i, and (1-S1i)(1-S2i) are linearly dependent, we
must drop one of this variables to avoid perfect multi-collinearity problems, in
which case we choose to withdraw S2i.
Hence, the following alternative regression model using the two separate
instruments can be specified:

Where the first-stage
regression is now:

Table 4 reports the results
of the first-stage estimations for both the just-identified and the
over-identified regression models. We can see, on the one hand, that mothers
with two children of the same sex are 6.2% more
likely to have a third one; and on the other hand, that the effect of 2Girls on fertility is higher than that
of 2Boys, suggesting that parents are
more willing to persist in having children until they can have a boy. As for
our covariates of particular interest here, we can see that having a higher
level of education reduces the likelihood of having a third child by 2%, and that greater family wealth slightly
increments this probability (this can interpreted as mothers from rich families
being less worried about the economic difficulties of childbearing).

At this point, however, we
would like to know something about the validity of our instrumental variables.
For this purpose, we can perform a test for the strength of our instruments in
each first-stage regression (in both the just-identified model, where Samesex instrument was used, and in the
overidentified model, where we used the 2Boys
and 2Girls instruments). When testing
for the ‘strength’, we are actually interested in the correlation between the
endogenous variable Morethan2 and
each of our instruments (indeed, the ‘first-stage condition’ means that the
instruments(s) considered should bring some knowledge to the endogenous
variable). Effectively, this correlation is measured by the first-stage Partial R2, where in the case of the
just-identified model is equal to 0.0043,
and in the overidentified model is 0.0044.
This result is interesting because it proves a small correlation, although the
first-stage F-statistic in each of
the models is sufficiently large to reject the null-hypothesis that the
instruments are ‘weak’ (see
appendix 2).

Having checked this aspect of our instruments’
validity, we can now run the regression of the effect of  on the
different labour-market outcomes using both the just-identified and the
over-identified models. Simultaneously, we conduct simple ordinary
least-squares (OLS) regression to compare the results and have an insight of
the magnitude of the bias arising from our endogenous variable. The results are
presented in Tables 5 through 7.

As can be seen, when using
the single Samesex instrument (the
just-identified model), having a third child reduces the probability of
participating in the labour market by around 11.5 percentage
points, the number of hours per week worked by 8-9
per year, and the amount of earnings by more than $1,745.
Likewise, when using the over-identified model, the probability of
participating in the labour market falls by around 10.6 percentage points, the number of hours worked by 7-8, and the earnings by almost $1,610. Thus, although the first-stage estimations
suggested that mothers of two boys are less likely than mothers of two girls to
have a third child, the 2SLS estimates in Table 5 allow us to infer that
separating our Samesex instrument
into its two components doesn’t change the magnitude of the coefficients very much
and, in the end, the same conclusions are reached. Finally, the bias arising
from the endogeneity of our fertility variable (Morethan2) clearly overestimates the
effects of fertility on the labour-market outcomes, as can be appreciated when
comparing the OLS with the 2SLS results described above. As for our other
covariates of interest, we see that both education and family income yield the
expected results: The former has a positive effect in all of the three
labour-market outcomes, incrementing the labour supply (Workedforpay) by 3.1%, hours
worked by 5-6, and labour income by $795-$798. Conversely, family income reduces
labour supply by 2.7%, hours worked by 7-8, and the mothers’ labour income by $757-$758 (recall these are all mothers of at
least two children, and they are more likely to abandon the labour-market if
they have the economic support from wealthy relatives).

V. Concluding Remarks

We have seen that both the
Wald and the 2SLS estimates consistently confirm the thesis that increasing
fertility (moving from the second to the third child) reduces women’s
participation in the labour market. Effectively, on the one hand, these results
are in line with those found in the cited literature, but they appear to be
less ‘harsh’. For example, while Lundborg et. al. (2017) calculate a reduction
of working hours per week of 5.9, our estimates show a reduction of only 4.5 working hours, and as Cristia J. P. (2008) finds that
‘having a first child younger than one year reduces female employment by 26
percentage points’, our calculations only account for a 10.6%-11.5% decrease. This is due to the fact that our IV
strategy focuses on the effect of the third-born child instead of the
first-born child, which is expected by the standard household theories to have
a lower impact on the labour supply of women.

On the other hand, however, the effects
calculated in this paper (big or small) shouldn’t be over-dimensioned. In
effect, referring to the calculations made by Angrist, J. D. & Evans,
W. N. (1998) in their ‘Table 1’, we can observe that the probability of having more than two children
decreased by around 15.7 percentage points between 1970 and 1980; while, at the
same time, the participation of women in the labour market rose by about 13.2
percentage points in the same period. If we then use the 2SLS estimation of the
impact of fertility on labour supply (Workedforpay)
using the Samesex instrument reported
in the upper part of the central column of Table 5 (-0.115),
we can deduce that declining fertility accounted for an increase in
labour-market participation of roughly 1.8 percentage
points (0.157×0.115). Thus, our study also lets
us conclude that, although fertility has a significant negative impact on
labour supply, the increase in the labour-market participation rate has been so
substantial that declining fertility only accounts for a small fraction of the
whole change. 


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