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Empirically probing the quantity–quality model


This paper etimates the causal effects of family size on girls’ education in Mexico, exploiting prenatal son preference as a source of random variation in the propensity to have more children within an instrumental variables framework. It finds no evidence of family size having an adverse effect on education. The paper then weakens the identification assumption and allows for the possibility that the instrument is invalid. It finds that the effects of family size on girls’ schooling remain extremely modest at most. Families that are relatively large compensate for reduced per-child resources by increasing maternal labour supply.

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  1. 1.

    There is an abundant literature showing that parents with large families invest less in children’s education than parents with small families, but much of this evidence is non-causal. Schultz (2008) provides a review.

  2. 2.

    Li et al. (2008) and Rosenzweig and Zhang (2009) find evidence consistent with the quantity–quality model, whilst Qian (2009) finds a positive effect of an additional child on school enrolment. Other than these studies, work that estimates the effects of family size on children’s education generally relates to developed countries and generally shows no or only very weak evidence of a quantity–quality trade-off (Black et al. 2005; Cáceres-Delpiano 2006; Conley and Glauber 2006—all for the US; Angrist et al. 2010 for Israel).

  3. 3.

    Angrist and Evans (1998) also defend the validity of the same-sex instrument for the US; Rosenzweig and Wolpin (2000) on the other hand find evidence of economies of scale in India.

  4. 4.

    The asset index is computed by aggregating indicators for whether or not a household owns 10 assets including blender, fridge, gas stove and radio, among others.

  5. 5.

    We use a linear specification in this paper, given that the instrumental variables are binary.

  6. 6.

    Sex composition was first used as an instrument for family size by Angrist and Evans (1998) and has since been applied by others such as Angrist et al. (2010) and Conley and Glauber (2006). These studies use same-sex births as the instrument, whether all male or all female; Lee (2008) on the other hand uses all-female births. Another commonly used instrument is twin births. Rosenzweig and Zhang (2009) highlight a number of concerns underlying the validity of this instrument (including differential birth endowments and birth intervals of twins versus singletons). The most likely direction of ensuing bias of the IV estimates is positive (Behrman et al. 1994; Rosenzweig and Zhang 2009): we conducted an analysis using twin births as an instrument and found some positive IV estimates, leaving us with concerns that these issues may indeed be relevant in our context but are unfortunately not possible to investigate further given the available data. Finally, a third type of instrument exploits variation in implementation and enforcement of fertility policies as an instrument. Qian (2009) and Wu and Li (2012) use variation in the implementation and enforcement of China’s One-Child Policy to identify causal effects of family size on children’s education and maternal health, respectively.

  7. 7.

    Similar correlations have been found in contexts with son-biased fertility preferences. See for example Rosenblum (2013) for India.

  8. 8.

    In using only all-female births as our instrument, the reader may be concerned that we are using just one part of the variation induced by sex composition preferences. For completeness, Table 12 in the Appendix reports results from the analysis using sex composition (either same-sex births or all-male and all-female births) as an instrument for family size. As can be seen from the table, the all-female instrument has considerably more power in the first stage and results are primarily driven by it.

  9. 9.

    We condition on the first n − 1 births being female as the instrument is preference for at least one son.

  10. 10.

    One reason for this is that children of the nth birth may be of different sexes; another reason is to avoid any selection bias arising from families that have children after a male birth being different from those that do not.

  11. 11.

    Though the importance of birth order for education choices has been highlighted in the literature (Black et al. 2010, 2011; Rosenzweig and Zhang 2009), as we will see, we find little evidence of heterogeneity in the effects of family size by birth order in the sample considered here.

  12. 12.

    Most localities were chosen on the basis of having been graded with a high degree of marginalisation on the basis of the 1995 Census data.

  13. 13.

    This is the vast majority (92 %) of those aged less than 18. For the remaining 8 %, it is the case that the mother is deceased or in another household.

  14. 14.

    To check whether this is the case, we follow Dahl and Moretti (2008) and test whether sex composition affects the probability of maternal divorce and parental cohabitation. We find a very small statistically significant positive (∼0.1 %) correlation between all-female composition and maternal divorce (relative to an all-male composition) and a small statistically significant negative correlation with parental cohabitation.

  15. 15.

    Though we could potentially retain them in the sample when we consider the outcomes of second- and third-borns, a reason for not doing so is that we have some concerns about coding birth orders for households with children above age 18. Note that we also drop households that reported more than one household head (0.03 %) and households (1.5 %) with suspect data, mainly the reporting of implausible ages.

  16. 16.

    These latter two levels are the ones that children of our age range should have achieved (for instance, Mexican children would complete lower secondary school by age 14 if they started primary school at age 6 and progressed through without repeating any grades). Note also that all of these outcomes are measured at a particular point in time between ages 12 and 17 and are thus not necessarily indicators of completed schooling.

  17. 17.

    At the basic education level, participation in private education in Mexico is low, at 10 %, and is not relevant for the poor population considered here.

  18. 18.

    Schultz (2004) and Behrman et al. (2005) document lower secondary school enrolment amongst girls than boys in the communities comprising the sample for PROGRESA, justifying the premium for girls in the subsidy. However, it should be noted that there is a sizeable literature attributing any differences between the sexes to availability of schools/distance to schools/marriage markets rather than preferences for boys’ schooling per se.

  19. 19.

    Though there are no fees for public schools, direct costs of schooling include purchasing textbooks, stationary and school uniforms and transportation to and from school. Note also that the opportunity cost of schooling is increasing with age, which may explain the observed patterns.

  20. 20.

    Whilst some of the existing literature pools males and females, we eschew from doing this (in line with Ponczek and Souza 2012). We have two reasons for this. First, we have just shown that there are strong son preferences in the population, where an all-female composition is more than twice as likely (and in one case, almost 10 times as likely!) to induce families to continue their childbearing compared to an all-male composition (Table 2). Second, pooling restricts the causal effect of family size on education to be the same for both sexes. It is however well known that, particularly in developing country contexts, due to economic reasons such as higher costs of sending females to school and/or lower returns for females in the labour market (e.g. Airola and Juhn 2005; Attanasio and Binelli 2010) or social norms supporting preferences for sons (e.g. Deaton and Subramaniam 1996; Oster 2009; Chakravarty 2010), the gender of a child plays an important role in parents’ human capital investment decisions and processes.

  21. 21.

    The sex ratio at birth in Mexico has historically stood around this level: it was 1.01 between 1990 and 1994 (Parazzini et al. 1998), and the Mexican Demographic and Health Surveys suggest that the sex ratio of all children ever born stood at 1.03 in 1987 (Arnold 1992).

  22. 22.

    Three of our outcome variables, school enrolment, primary school completion and lower secondary completion, are binary: thus, we use LPM estimation in this case. For convenience, we use the term OLS throughout the text.

  23. 23.

    This follows Angrist and Imbens (1995).

  24. 24.

    By ‘completed family size’, we mean completed as at the time of the survey.

  25. 25.

    Whilst we do not explicitly consider non-linear effects of family size in this paper (Mogstad and Wiswall 2010), our use of different instruments affecting different margins of increase in family size allows us to see whether there is any evidence of non-linearities in the effects of family size on children’s education.

  26. 26.

    The figures in the columns give the relative likelihood that compliers have the characteristic listed in column A. For instance, a figure of 0.75 means that the population of ff compliers is 3/4 as likely to have a non-qualified father compared to the overall population.

  27. 27.

    We also experimented with pooling parities by instrument instead. So for instance, for the fff instrument, we estimated a specification in which we pooled first- and second-borns in families with at least three children in which the first two are female. Whilst an advantage of this alternative is that we need not impose the assumption that the causal impact of family size on education is the same across all instrument-specific subsamples (and so across all margins of increase in family size), we fail to boost precision sufficiently by pooling in this manner. These results are available on request.

  28. 28.

    Exceptions include Rosenzweig and Wolpin (2000) and Rosenzweig and Zhang (2009), who provide direct evidence on the likely validity of same-sex and twin instruments, respectively. Angrist et al. (2010) address the issue mainly by comparing twins and sex composition estimates, as the omitted variables bias associated with each type of instrument should act differently.

  29. 29.

    Though these are the most commonly cited concerns in the literature, we acknowledge that other unobserved factors might result in a negative correlation between the instrument and the error term in the structural equation, thereby biasing downward the IV estimates. For instance, the quality of marriage might be lower in all-female households, which may be adverse for children’s schooling (see for instance Brown and Flinn 2011); another example concerns the role of socialisation at home, which differs depending on sibling composition—girls with brothers tend to have more ‘masculine’ traits perhaps because brothers encourage girls to be more assertive and outspoken (Koch 1955). So it is plausible to expect that, if assertiveness is associated with better success at school, girls with sisters will do less well in school (see Butcher and Case 1994).

  30. 30.

    Recent UNESCO statistics for Mexico show that 98 % of girls and 98 % of boys are in primary school and 74 % of girls and 71 % of boys are in secondary school (UNESCO 2011), evidence from Parker and Pederzini (2000) shows that the gender gap in education in Mexico has fallen substantially over the last 30 years, to the extent that females and males below the age of 20 no longer display significant differences in educational attainment, as measured by years of schooling. Duryea et al. (2007) analyse the educational gender gap in Latin America and the Caribbean and find that the most striking differences are across income groups and not gender.

  31. 31.

    Angelucci et al. (2010, 2012) document the importance of extended family networks for this population in making schooling choices (particularly in response to the PROGRESA grant) and in providing support following adverse events.

  32. 32.

    The PROGRESA evaluation sample includes ∼24,000 households in 506 villages, who were interviewed on eight occasions over the period 1997–2007. We are unable to match households in the PROGRESA evaluation sample to those in our sample as different household identifiers are used in the two datasets.

  33. 33.

    Data on clothing are not available for our main sample, ENCASEH. We pool post-programme data from surveys in October 1998 and May 1999, from control villages only, to ensure the analysis is not contaminated by any potential programme effects. We retain families where the first-born child is below 18 years old—not just 12–17 years of age as in main analysis—to boost sample sizes. Compared to our main sample, families here have fewer children on average; parents are also on average younger and more educated.

  34. 34.

    The sample pools families with at least two children where the first is a male, families with at least three children where the first two are male and families with at least four children where the first three are male.

  35. 35.

    As further reassuring evidence, we re-emphasise that there is no relation between all-female births and any of the covariates in our model—see Table 3. Another salient point is that whilst we cannot control for savings in our data, results are robust to the inclusion or exclusion of proxies for resources (mother’s education, household assets, home and land ownership).

  36. 36.

    If it is positive, we only obtain one-sided bounds.

  37. 37.

    This is clear from corollary 1 of Nevo and Rosen (2012), which implies that, in our case, \(\beta _{Z\ast }^{\mbox {IV}} >\beta ^{\mbox {OLS}}\). Moreover, the corollary shows that the larger the correlation between the instrument and the endogenous regressor, the greater the improvement of \(\beta _{Z\ast }^{\mbox {IV}} \) over \(\beta ^{OLS}\) and thus the tighter the lower bound.

  38. 38.

    We use the parity-pooled sample given the considerable gains in precision as discussed in Section 4.2

  39. 39.

    Another possibility—and one raised by Angrist et al. (2010)—is that households use public funds to smooth the shock to fertility. The main candidate in our context is the PROGRESA programme, providing mainly subsidies for school attendance. However, the data used for the analysis in this paper relate to the period before PROGRESA was introduced (indeed our data were collected in order to identify households eligible for the subsidy—see Section 3.1).

  40. 40.

    The first-stage equation for the sample of families with at least four children follows a similar pattern. Note also that the second-stage specifications corresponding to Eqs. 1316 are the same as Eq. 1 in the main text.

  41. 41.

    In Eq. 14, we must drop \(m_{2i}\) since {\(m_{1i}\),\(m_{2i}\), \(ff_{i}\), \(mm_{i}\)} are linearly dependent.


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We are grateful to conference participants at the European Economic Association Annual Conference, Milan 2008, the Latin American and Caribbean Economic Meetings, Medellin 2010 and the Royal Economic Society Conference, London 2011; also to Orazio Attanasio, Valérie Lechene, Marta Mier y Terán, David Phillips, Nancy Qian, Adam Rosen, Marta Rubio-Codina and Marcos Vera-Hernández for the useful discussions; to two anonymous referees for the useful suggestions; and to Marta Mier y Terán for providing access to the data. All errors are the responsibility of the authors. We are grateful to the Economic and Social Research Council for funding this work through research grant RES-000-22-2877.

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Correspondence to Emla Fitzsimons.

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Responsible editor: Junsen Zhang



Using sex composition as an instrument for family size

In this section, we present results using sex composition as an instrument for family size. The analysis follows closely upon Angrist et al. (2010). The sex composition instrument can be defined in two ways, both of which are used (separately) below: (1) a single indicator for whether the first n births are of the same sex, regardless of whether male or female, and (2) two indicators—one for an all-female composition and one for an all-male composition.

We conduct this analysis on the first \(n-1\) children from families with at least n births, separately for \(n = 2, 3, 4\). These samples are labelled \(2+, 3+\;\text {and}\;4+\), respectively, below. For illustrative purposes, we show the first-stage specifications for \(n = 2+\) and for \(n = 3+\).Footnote 40

The first-stage specifications for the sample of first-borns in families with at least two children (\(2+\) families) are

$$\begin{array}{@{}rcl@{}} F_{i}=\alpha_{0} +\alpha_{1} X+\beta_{1} m_{1i} +\beta_{2} m_{2i} +\pi Z+\xi_{i} \end{array} $$
$$\begin{array}{@{}rcl@{}} F_{i}=\alpha_{0} +\alpha_{1} X+\beta_{1} m_{1i} +\eta_{f} ff_{i} +\eta_{m} mm_{i} +\xi_{i} \end{array} $$

where Eq. 13 uses as an instrument an indicator for whether the first two births in i’s family are of the same sex, Z, and Eq. 14 uses as an instrument an indicator for the first two births being female (\(ff_{i})\) and the first two births being male \((mm_{i})\). Note also that \(m_{1i}(m_{2i})\) is an indicator for a male first (second) birth in i’s family.Footnote 41

The first-stage specifications for the sample of families with at least three children (\(3+\) families) are

$$\begin{array}{@{}rcl@{}} F_{i}&=&\alpha_{0} +\alpha_{1} X_{i} +\beta_{1} m_{1i} +\beta_{2} m_{2i} +\beta_{3} m_{3i} +\alpha_{2} s_{12i} +\pi Z+\beta k_{i} +\xi_{i} \end{array} $$
$$\begin{array}{@{}rcl@{}} F_i&=&\alpha_{0} +\alpha_{1} X_{i} +\alpha_{2} m_{1i} +\eta_{1} ff_{i} +\eta_{2} mm_{i} +\eta_{3} \left({1-s_{12i} } \right)\times m_{3i}\\ &&+\eta_{4} fff_{i} +\eta_{5} mmm_{i} +\beta k_{i} +\xi_{i} \end{array} $$

where Z in Eq. 15 is an indicator for whether the first three births in i’s family are of the same gender and \(fff_{i}\) and \(mmm_{i}\) (in Eq. 16) are indicators for the first three births being male and female in i’s family, respectively. In addition, we control in both specifications for the sex composition of earlier births in Eq. 15 through the terms \(m_{1i}\), \(m_{2i}\) and \(m_{3i}\) as defined already, and \(s_{12i}\), which is an indicator that the first two births in i’s family are of the same gender, and in Eq. 16, through the terms \(m_{1i}\), \(ff_{i}\), \(mm_{i}\) and \((1-s_{12i}) \times m_{3i}\). This means that the parameter \(\pi \) in Eq. 15 estimates the difference in family size between families with a succession of three same-sex children (fff or mmm) and those with a specific mixed-sex composition (two same-sex children followed by a different-sex child, i.e. ffm or mmf, consistent with the conditioning in Section 2). The parameter \(\eta _{4}(\eta _{5})\) in Eq. 16 captures the difference in family size between those with an ffm (or mmm) composition relative to a specific mixed composition—ffm(mmf).

The top panel of Table 12 shows the first-stage estimates for the \(2+, 3+\; \text {and}\; 4+\) samples. It shows that families where the first n children are of the same sex have an additional 0.06–0.089 children compared to families with a mixed-sex composition. However, when we allow for differential effects by gender (as per Eq. 16; columns 3, 6 and 9), it becomes clear that this correlation is driven primarily by families with an all-female composition. In all cases, the correlation for the all-female composition is at least twice that of the all-male instrument (and almost seven times for the ff and mm instruments), highlighting the strong son preferences in this population.

Table 12 Effect of family size on education (mixed-sex and same-sex instruments)

Turning to the second-stage estimates shown in the lower panels of Table 12 in the Appendix, we see that the manner in which the instrument is specified matters importantly for the identified effect, particularly in the \(2+\) case (columns 2 and 3). The reason for this is that the second-stage TSLS estimate is equivalent to a weighted average of instrument-specific causal effects (i.e. TSLS estimates computed using a particular instrument on its own), where the weights depend on the relative magnitudes of the first stages (Imbens and Angrist 1994; Angrist and Imbens 1995). So the TSLS estimates reported in column 3, where the first-stage coefficient for the ff instrument is almost seven times as large as that for the mm instrument, are driven primarily by causal effects for the ff compliers. By contrast, the TSLS estimates displayed in column 2 are computed in a manner that weights equally the causal effects for ff and mm compliers, thereby generating different estimates to those in column 3.

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Fitzsimons, E., Malde, B. Empirically probing the quantity–quality model. J Popul Econ 27, 33–68 (2014).

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  • Fertility
  • Education
  • Instrumental variables
  • Latin America

JEL Classification

  • I20
  • J13
  • J16