The Impact of Family Composition on Educational Achievement

A. Data

B. Demand for Multiple Sons—Son-Preferring Stopping Rules

The Taiwanese have a long tradition of pro-male bias, for cultural and economic reasons. Confucianism—the grounding philosophy in Taiwan, Japan, Korea, and imperial China—dictates social statutes and provides rationales for the subordination of women to men, within a strict family hierarchy. According to Confucianism, family line and wealth should be transmitted from father to son, irrespective of ability, except in cases where there is no direct male line. In return, sons and their spouses assume responsibility for taking care of the parents if they are too infirm to work. In contrast, daughters move out of the family household at the time of marriage. These social norms have acted as old-age social security for the elderly for centuries in the form of extended families composed of sons (and their spouses, if married), unmarried daughters, parents, and grandparents. Although old-age social security (not based on employment) in Taiwan began in 2008, extended families (even if they do not live together) are still the primary source of support for the elderly. Thus, the demand for old-age social security is more likely to be met by having more sons.

Statistics suggest that a firstborn son reduces family size. Other observed family backgrounds being equal, firstborn males have 0.26 fewer siblings than firstborn females, as the top panel of Table 2 shows. This is over 10 percent of the average sibsize of all families (2.47). The effect of having a firstborn son on sibsize is greater among families in rural areas (Columns 4–5) or with less well-educated parents (Columns 6–9). Because a firstborn son significantly reduces the chances of having a second child, the families with a firstborn son or firstborn daughter with the same number of siblings are not comparable because the former group probably has preferences for larger families. Thus, our analysis separates 434,729 firstborn sons from 416,315 firstborn daughters out of all families with two or more children.

We report the demand for multiple sons in the middle and bottom panels of Table 2, where we estimate the effect of a change in sibling gender composition on sibsize, conditional on observed family backgrounds. Taiwanese parents not only prefer sons to daughters but also prefer multiple sons to mixed-sex composition—unlike American families, who prefer mixed-sex composition, as documented by Angrist and Evans (1998). This tendency gets stronger if the mother or the father is less educated or if the child was born in a rural area. The first two columns of Model I show that having a son, regardless of the birth order, decreases sibsize by 0.43. Because the result indicates that birth order is not important in explaining the demand for sons in our data, we further use Model II where we focus on the impact of sibling gender composition on sibsize, leaving out the factor of birth order. The estimates suggest that compared with families with two males, those with two females have about 0.53 more children, and those with mixed-sex composition still have about 0.10 more. These estimates are extraordinarily large, since they account for 20 percent and 4 percent, respectively, of the average sibsize (2.67).

Several previous studies have suggested that gender bias is stronger in rural areas (for example, Barcellos, Carvalho, and Lleras-Muney 2014). Columns 4 and 5 of Table 2 show that the son-preferring stopping behaviors are also more evident in rural areas of Taiwan. Family size reduction because of a firstborn son in rural areas is more than 30 percent (=0.288/0.218 – 1) higher than the urban counterpart. Additionally, in urban areas, parents who have a boy and a girl in the first two births have 0.07 more children than those who have two boys (standard error = 0.003). This magnitude in rural areas is more than 65 percent (=0.116/0.070 – 1) higher than the urban counterpart. While Taiwanese families in both rural and urban areas have high demand for sons, it is even higher among rural families than among urban families.

The remainder of Table 2 examines heterogeneity in demand for sons by parental education. Columns 6–9 show that family size reduction due to a firstborn son is more than 35 percent (0.295/0.201 – 1; 0.296/0.219 – 1) stronger if the mother or father has no (vocational or academic) high school diploma (HS–). Similar to the rural–urban comparison, the difference in the demand for multiple sons by parental education levels is also more evident at higher parities, as the bottom panel of the table shows. It should be noted that since parental education is higher in urban areas than in rural areas, the comparison across subpopulations does not isolate the comparative static of interest.

C. Concerning Sex-Selective Abortion

With exceedingly strong demand for multiple sons, Taiwanese families might have adopted sex-testing technologies to select child gender. Endogeneity of child gender would invalidate our empirical strategy that requires sibling gender composition at the first two births to be conditionally exogenous. This subsection examines this concern.

Although prenatal sex testing by ultrasound was introduced in Taiwan during the early 1980s, it was only after 1986 that the technology for sex testing became widely available. Using the same data source as ours, Lin, Liu, and Qian (2014, Figure 1) show that Taiwanese sex ratios at birth start being unbalanced after 1986, and the unbalanced trends are limited to singletons of third and higher parities.¹³ Motivated by their results, we limit our firstborn data to those born before 1985. For these children, the sex ratio of boys to girls at birth is 1.044 ( = 434,729/416,315), and the sex ratio of their next siblings at birth is between 1.053 and 1.070. Both ratios are within the range (1.04–1.08) that demographers consider normal on the basis of historical evidence (Chahnazarian 1988; Johansson and Nygren 1991).

Although the firstborn children in our data were all born before 1985, some have next siblings born after 1985 and exposed to sex-testing technology. Table 3 tests whether or not the firstborn populations whose next siblings were born before and after 1985 are different. A mean difference test in the top row suggests the sex ratio does not differ significantly across the firstborn populations whose next siblings were born earlier or later. In fact, the families of firstborn children whose next siblings were born later are less male-dominated. In contrast, the rest of the table shows significant differences in characteristics between the firstborn populations with older versus younger secondborn siblings. In particular, those with younger secondborn siblings have older and more educated parents and smaller family size. This is because some parents in our firstborn data experienced the expansion of compulsory education from six to nine years during the late 1960s. Compulsory schooling expansion might have postponed the timing of marriage and childbearing.

Furthermore, we use the birth interval between the first two births to detect sex-selective abortion at the second birth. If a male birth is more likely to be selected by son-preferring parents who have a firstborn daughter, the birth spacing could be distorted by the time it takes to abort a female fetus. Regressions of the spacing between the first two births on sibling gender composition and family background factors suggest no evidence of sex-selective abortion at the second birth. As Table 4 shows, the coefficient of the interaction between the indicator for firstborn females (Girl1st) and the indicator for secondborn males (Boy2nd) is small (3.97 days) and statistically insignificant (standard error = 2.81 days), suggesting no evidence of sex-selective abortion at second birth. This estimate is similar to the result when we restrict our sample to firstborns for the pre- or post-1980 birth cohorts. Additionally, the coefficient of Girl1st is significantly negative, indicating that with a firstborn daughter, the next child is born on average only 16 days sooner than if the firstborn had been a son.

All of these estimates suggest that although parents have strong demand for sons, sex-selective abortion is not a concern in our data if we control for the full set of dummies for parental education and year of birth, maternal age at the first birth, and district-level fixed effects.

Additionally, if we restrict the data to mothers who had their first two births before 1985, the estimated decomposed effects of sibling gender on the education of firstborn females are greater because parents with smaller birth spacing after a firstborn girl might have stronger preferences for sons. We report the estimates using the restricted sample in Online Appendix Tables A1–A4. Since birth spacing can be endogenous, our main analysis uses the unrestricted sample.

D. Balancing Tests

Table 5 implements balance tests between samples with a secondborn son versus daughter in samples split by firstborn gender. Columns 1 and 3 show the unconditional mean differences in urban versus rural or parental education by the gender of the secondborn sibling. The estimates show a small but positive association between maternal education and secondborn sons. Using the largest association in the full sample as an example, we find mothers with secondborn sons are 0.3 percentage points more likely to have a vocational high school diploma as their highest qualification than those with secondborn daughters, but this is only less than 2 percent of the sample mean (19 percent; see Table 1).

Columns 2 and 4 additionally include the other covariates, including the full set of indicators for parents’ years of birth, maternal age at the first birth, and the district of the first child’s birthplace. The previously estimated associations now either decrease to nearly zero or become statistically insignificant. Overall, we find little evidence that parental education or the urban dummy differs by secondborn gender if the other covariates are included. Throughout our analysis below, we include the full set of these control variables.

A. Defining the Decomposed Effects at Individual Levels

Consider that the first child’s education Y is determined by the sex of the next sibling D and family size M, conditional on the first child’s gender and observed family backgrounds.¹⁴ Let D be the indicator for a next brother, and M₁ and M₀ denote the potential family size as if the next sibling were male or female, respectively, regardless of the realized sibling gender. Motivated by VanderWeele’s (2013, 2014) causal inference methods, we decompose the total effect (TE) of sibling gender into DE and IE as follows:

(1)

We measure DE by fixing potential sibsize M₁ as if the gender of the next sibling were always male and IE by fixing the gender of the next sibling while comparing the potential sibsize with a different sibling gender. The sum of DE and IE is the total influence of a change in sibling gender on the firstborn.

The “controlled direct effect” CDE is defined by the previous studies under the assumption that sibsize can be fixed:

(2)

When sibsize is fixed and cannot change with sibling gender, families cannot adjust fertility based on the gender composition of the existing children; that is, no families in the estimation can apply the son-preferring stopping rule. We call this measure a naive measure because it assumes parents’ fertility choice M is independent of the firstborn gender. Given the son-preferring stopping rule, a female second birth increases potential sibsize; that is, M₀ tends to exceed M₁. Thus, the naive comparison does not necessarily give a right measure for DE.

B. The Outcome Regression and the Parameters of Interest

Both naive and proposed measures, CDE(M) and DE(M₁), immediately imply that the direct effect of sibling gender on firstborn outcomes is a function of family size.

To allow this possibility, we include an interaction term between family size and sibling gender in the firstborn’s outcome equation. Since we run the outcome equation separately for firstborn males and females, we omit the notation of firstborn gender (and covariates) in this section for ease of exposition:

(3)

where the Greek letters are coefficients, and the error term ϵ_DM can vary with fertility choice M and sibling gender D to allow each firstborn to have idiosyncratic gains from a next brother or a smaller family.¹⁵

Under the assumption of randomized child gender, the following two properties help us to proceed: (i) ϵ_1M and ϵ_0M have the same distribution given the same family size M; (ii) the conditional mean of potential sibsize equals the conditional mean of realized sibsize, given randomized sibling gender. Using these two properties, we derive the sample analogs of the average DE and IE, as functions of the regression coefficients and the conditional means of realized sibsize, by averaging over all possible values of potential sibsize (Online Appendix 1). Again, firstborn gender is omitted for ease of exposition:

(4)

(5)

Their sum is the sample analog of the total effect, ATE. Both ADE and AIE are independent of the gender of the next sibling because both effects measure the impact of a change in the gender of the next sibling. Also, both are independent of family size because we derive both measures by averaging over all possible values of potential sibsize. For comparisons, we also estimate the sample analog of the naive comparison:

(6)

Although the conventional measures for direct effects have omitted the interaction term (by assuming β₃ = 0), we include the interaction to compare traditional methods with ours.

Definitions 4 and 6 immediately imply the following results: (i) Suppose β₃≠0. Then ADE = CDE if and only if E[M|D = 1] = E[M|D = 0] = E[M]. (ii) If β₃ = 0, then ADE = CDE = β₁.

Case (i) considers a general case where β₃ is not necessarily zero. The necessary and sufficient condition for gender neutrality to hold (for example, exclusion of the son-preferring stopping rule; E[M|D] = E[M]) is equivalent to equating the naive comparison CDE with the true average direct effect. As confirmed by our empirical results, the naive comparison indeed produces biased results under the son-preferring stopping rule.

Case (ii) considers a special case where we force β₃ to be zero. This functional form assumption makes the naive comparison seemingly a correct measure for the average direct effect, but purely due to the imposed restriction. Either assuming gender neutrality as in Case (i) or imposing the functional form assumption as in Case (ii) will make the naive comparison appear to be a correct measure for the average direct effect. For the purpose of allowing for more general cases, our estimation does not impose β₃ = 0 and does not assume E[M|D = 1] = E[M|D = 0] = E[M].

Furthermore, the assumption of β₃ = 0 restricts the effect of family size to be independent of sibling gender. However, intensive empirical work has noted that family size effects may vary across firstborn genders (for example, Black, Devereux, and Salvanes 2005, Table 8; Angrist, Lavy, and Schlosser 2010, Table 8). Motivated by the previous literature, our model allows the family size effects to vary with the gender of the next sibling.

C. Addressing Endogeneity of Family Size

It is crucial to distinguish the family size effect from the indirect effect of the next brother. Unlike the indirect effect of a next brother through a reduction in family size, the family size effect is the impact of having one additional sibling (regardless of sibling gender) on firstborn outcomes. Although this study aims to estimate the decomposed effects of sibling gender on firstborn outcomes, we also estimate the family size effect and address the endogeneity of family size because unbiased and consistent estimates of ADE and AIE require unbiased and consistent estimates of (β₁, β₂, β₃).

In the presence of the interaction term in the outcome equation, the family size effect is specific to firstborn gender and the gender of the next sibling. Again, we run regression analysis separately for firstborn males and females so that we can omit the notation of firstborn gender for ease of exposition. The outcome Equation 3 can be rephrased as

(7)

Conditional on firstborn gender, the family size effect is β₂ given a next sister and β₂ + β₃ given a next brother. If the estimated coefficient of the interaction term is significantly positive (β₃ > 0), then any negative impact of one additional sibling on firstborn outcomes can be lessened by the next brother, as confirmed by our empirical result.

Problems of endogeneity arise because fertility choice M can be correlated with unobserved parental preferences, family background, and child characteristics in ϵ_DM. To address this, we apply 2SLS using the incidence of twins at the second birth. The second stage is the outcome Equation 3 or 7. In the first stage, we instrument M with an indicator of whether the second birth is twins (Z = Twin2nd), and we instrument the interaction D × M with an interaction D × Z, as indicated by these two equations:

(8)

(9)

Error terms u and v can be correlated with ϵ_DM. Again for ease of exposition, we suppress the gender of the firstborn and the same set of covariates as in the outcome equation. Since the decomposed effects are both linear in regression coefficients, as Equations 4 and 5 indicate, we calculate their standard errors using conventional methods (by testing for linear restrictions).¹⁶

We predict the direction of omitted variable bias by considering a random-coefficient model,

(10)

where ϵ is independent of D and M, ρ₁ captures parents’ utility gain from having a younger son, and (ρ₂, ρ₃) capture parents’ utility gains from family size. Collecting all error terms yields

which is correlated with M. The OLS estimated result would overstate (understate) the magnitude of the quality–quantity tradeoff if the unobserved factor ϵ_DM is negatively (positively) correlated with fertility M given sibling gender composition. Under the son-preferring fertility-stopping rule, if the next sibling is female (D = 0), then parents’ utility gain from family size ρ₂ is positive, and thus corr(M, ϵ_0M) = corr(M, ϵ + ρ₂ M) = ρ₂ > 0. OLS understates the magnitude of the quality–quantity tradeoff for firstborn outcomes given a next sister.

In contrast, if the next sibling is male (D = 1), then parents’ utility gain from family size (ρ₂ + ρ₃) is negative, and thus corr(M, ϵ_1M) = ρ₂ + ρ₃ < 0 (which implies ρ₃ < 0). OLS overstates the magnitude of the quality–quantity tradeoff for firstborn outcomes given a next brother.

Therefore, the direction of the omitted variable bias for the quality–quantity tradeoff depends on the gender composition of existing children. If the interaction term is also omitted, the bias further expands:

Now we have corr(M, ϵ_0M) = ρ₂ > 0 and corr(M, ϵ_1M) = ρ₂ + ρ₃ + β₃. The direction of the expanded bias also depends on the gender composition of existing children. Given a next sister, 2SLS without the interaction understates the magnitude of the tradeoff. Given a next brother, 2SLS without the interaction can bias upward or downward, depending on the relative magnitudes of β₃ > 0 and ρ₂ + ρ₃ < 0.

In summary, we study how a secondborn brother affects the outcomes of the firstborn, either directly or indirectly through changing sibsize. We instrument sibsize with whether the second birth is a twin birth. Our identification strategy requires the twins instrument and the sex composition of the first two births be both conditionally exogenous. We present empirical evidence justifying these requirements in Sections IV.E and II.C, respectively. To allow fertility choice to be son-preferring, our regression model interacts sibsize with sibling gender, and it instruments this interaction by interacting the twins instrument with sibling gender in the first stage. Omitting the interaction term would lead to biased results, as the results in Section IV suggest.

A. OLS Results

Before presenting the 2SLS results, we report OLS coefficients in the outcome Regression 3 using data that will be utilized for 2SLS estimation. Table 6 shows two sets of results: one includes family composition variables (a next brother D and family size M = Morethan2 or Sibsize) with no controls and the other additionally includes controls. Results in Online Appendix Table A5 using Sibsize as the fertility choice measure show similar patterns. In the shortest regression with sibling gender as the only regressor, the coefficient of a next brother in Columns 1 and 7 represents the unconditional mean difference in firstborn education by gender of the next sibling. In longer regressions with controls, the coefficient of a next brother in Columns 4 and 10 represents the conditional mean difference. Both conditional and unconditional mean differences are smaller than 0.4 percentage point. In particular, the conditional mean difference is small and positive for firstborn females while small and negative for firstborn males. This does not imply the absence of gender bias against girls because a negative ADE and a positive AIE of similar magnitudes might have canceled each other out.

After adding Morethan2 to the regressions with controls, the coefficient of a next brother D is adjusted downward to be (more) negative. This downward adjustment is expected because the correlation between D and Morethan2 is strongly negative due to the son-preferring stopping rule. A next brother induces son-preferring parents to have a smaller family and thus enables them to invest more in the first child. To allow the sibsize effect to vary with sibling gender (or the sibling gender effects to vary with sibsize), we further add the interaction between sibsize and sibling gender in the outcome regression. In the models including the interaction, the coefficients of D and Morethan2 significantly decrease in magnitude with the inclusion of the control variables (for example, parents’ education). This suggests that omitting parents’ education and other family background variables may lead to an overstatement of the two coefficients. Contrary to what we have expected (as explained in Section III), the OLS estimated coefficient of the interaction term is negative, seemingly suggesting that firstborn children’s outcomes are more hurt by a larger family if the next sibling is male. However, the endogeneity issue of fertility choice is intensified by its interaction with gender composition. The OLS estimates with or without interactions are both biased and difficult to compare/interpret.

B. First-Stage Estimates

To address endogeneity of family size and its interaction with sibling gender, we instrument Morethan2 using the occurrence of twins at the second birth (Twin2nd), and we instrument D × Morethan2 using the interaction D × Twin2nd. Table 7 shows a strong first stage, even after adding the interaction term. The F-statistic ranges from 1,372 to 27,123 with the twin instrument Twin2nd only and decreases to 488–12,866 after adding D × Twin2nd as an instrument for the interaction term.

The estimates in Columns 2 and 4 of the top panel suggest that a twin birth significantly increases the probability of having more than two births. Firstborn females have a smaller first-stage estimate than firstborn males because families with a firstborn daughter tend to opt for a larger family, regardless of the occurrence of twins. In particular, when the first two births are both males, the first-stage estimate is largest, at around 60 percentage points (=0.539 + 0.067), as Column 4 shows. In contrast, when the first two births are both females, the first stage is smallest, at around 32 percentage points, as Column 2 indicates. These estimates are precisely estimated, with standard errors no larger than 0.7 percentage points. In Panels 3 and 4, we also report the first-stage estimates for Sibsize and its interaction with a next brother D. The results suggest a twin birth increases the number of siblings by about 0.6–0.7 children (standard error < 0.03) for various compositions of sibling gender.

The first-stage F-statistic for Morethan2 and its interaction with a next brother D, calculated in a manner that takes account of multiple endogenous variables (as in Angrist and Pischke 2009, p. 217–18), is between 2,046 and 12,866. The same F-statistic for Sibsize and its interaction with D is between 488 and 6,300, markedly lower but still outside the range where bias in 2SLS estimates could be a concern. Because the first-stage F-statistic for Morethan2 is considerably larger than that for Sibsize, we focus our discussion on the results using Morethan2.

C. Second-Stage Estimates and Quality–Quantity Tradeoffs

We report the second-stage results for two educational outcomes—high school completion and university admission—at the top of Tables 8 and 9, respectively. We noted in Section III that the coefficient of a next brother could not be interpreted as his direct effect on firstborn outcomes, although it helps estimate the direct rivalry effect (see results in Section IV.D). In what follows, we explain why interactions between sibsize and sibling gender are required for unbiased estimation, discuss how family size effects change with gender composition, and relate our findings to the literature on family size effects.

Under the son-preferring fertility-stopping rule, the omission of interactions between sibsize and sibling gender is equivalent to assuming sibsize and sibling gender affect the first child’s outcomes through two independent channels. This assumption is unrealistic under the stopping rule because parents’ desired family size may change with the gender composition of the existing children. As a result, the magnitude of the quality–quantity tradeoff may also change with sibling gender composition.

As Columns 5 and 10 in both tables show, the 2SLS estimated coefficient of the interaction given a next brother is positive and cancels out a large part of the family size main effect. Firstborn daughters with a next brother receive almost no net family size effect, as suggested by the sum of the family size main effect and the coefficient of the interaction term. In contrast, given a next sister, the family size effect is startlingly strong for firstborn females. If parents go on to have a third child, then firstborn daughters with a next sister face a 10 percentage point reduction in high school completion and a 7 percentage point reduction in university admission. This accounts for nearly 40 percent of the average high school completion rate and the average university admission rate.¹⁷ The pattern of these estimates is robust, irrespective of using Morethan2 or Sibsize as the measure for family size—see Tables 10 and 11. Similar results appear among firstborn sons, although all estimates regarding the effects of family size and interactions are small and insignificant.

To investigate the magnitude of omitted variable bias in estimating the family size effects, we compare the OLS and 2SLS results in Columns 4 and 5 or Columns 9 and 10, both including interactions.¹⁸ We find that the OLS understates the quality–quantity tradeoff given a next sister, but overstates the tradeoff given a next brother. This pattern of biases is particularly evident among firstborn females. Specifically, for firstborn daughters with a next sister, the OLS estimated impact of a third child in the family is only a 1 percentage point reduction in both high school completion and university admission. This accounts for less than 13 percent of the tradeoff suggested by the 2SLS estimates.¹⁹ In contrast, for firstborn daughters with a next brother, OLS estimates suggest a third child in the family reduces firstborn daughters’ educational outcomes by 2 percentage points, significantly greater than the zero effect that the 2SLS estimate would suggest.²⁰ The relationship between the direction of bias and the gender composition of existing children is consistent with what we predicted in Section III.C.

Omitted interaction bias is a special case of omitted variable bias. For the purposes of comparing estimation results using the same set of compliers (that is, those whose parents would respond to a change in the twins instrument) as in the full model, we interact Twin2nd and a next brother D in the first stage while omitting the interaction between Morethan2 and D in the second stage. The 2SLS results in Columns 3 and 8 suggest that omitting interactions in the second stage leads to an understatement of the quality–quantity tradeoffs given a next sister, but overstatement given a next brother.²¹ This is as we predicted in Section III.C for the general directions of omitted variable bias even if interactions are included.

In summary, the second-stage results suggest that under son preference and the son-preferring fertility-stopping rule, the family size effect strongly depends on sibling gender composition, particularly among firstborn daughters. Although not the main focus of this paper, this part of the analysis relates to the literature on the effect of family size on child outcomes or parental behaviors, using variation due to twin births or preferences for a particular gender composition. Rosenzweig and Wolpin (1980) were the first to use twins to identify family size effects, and Angrist and Evans (1998) were the first to use the gender composition of existing children to identify the family size effects on parents’ labor supply. More recently, Black, Devereux, and Salvanes (2005) and Angrist, Lavy, and Schlosser (2010) find no evidence in data from Norway and Israel that an exogenous increase in the number of younger siblings (induced by twins or variation in sibling gender composition) affects adult outcomes. Black, Devereux, and Salvanes (2005) focus on younger Norwegian cohorts and find mixed evidence: no evidence that an exogenous increase in family size induced by gender composition has an impact on IQ scores, but some evidence that an increase in family size induced by twins has negative consequences. Conley and Glauber (2006) and Cáceres-Delpiano (2006) find negative family size effects on attending private school in the United States. However, while the former suggests an adverse impact of family size on grade retention, the latter shows no significant impact.

Other empirical work that exploits an exogenous increase in the number of younger siblings has used data from developing countries. Notable examples are Li, Zhang, and Zhu (2008) and Rosenzweig and Zhang (2009), both using Chinese twins, and Ponczek and Souza (2012) using Brazilian twins. These three studies from developing countries suggest a marked tradeoff between child quality and quantity. Rosenzweig and Wolpin (2000) have noted that in countries such as India where son preference is strong, gender composition might directly affect outcomes due to economies of scale, such as sharing clothes. More generally, gender composition directly affects adult outcomes, including parents’ labor supply and earnings and the relationship between them.²² Thus, child outcomes are also likely affected through channels other than variation in family size. In the following subsection, we show robust evidence of a direct impact of a next brother on the high school completion of firstborn daughters and sons in Taiwan. This suggests that in regions such as Taiwan where the son-preferring stopping rule is prevalent, gender composition is not a valid instrument for family size since it violates the exclusion restriction by directly affecting a child’s education.

D. Main Results: Decomposed Effects of Sibling Gender

The baseline decomposed results appear at the bottom of Tables 8 and 9, summarizing the sample analogs of the total effect, indirect effect and direct effect of a next brother on the education of firstborn daughters and sons (denoted by ATE, AIE, and ADE). As simple statistics suggest that firstborn females are more likely than firstborn males to complete high school or enroll in university, even with markedly strong demand for sons, the ATE of a next brother on firstborn education also seemingly suggests no gender discrimination. For both firstborn females and firstborn males, the ATEs are nearly zero, although some are statistically significant.

However, a zero or positive ATE is likely a result of the son-preferring stopping rule being prevalent. A positive indirect effect of a younger brother—via reduced family size—might have masked the direct rivalry effect on his older sibling. Since the magnitude of the AIE has not been revealed before, we first describe in detail how we estimate AIE using the strategy introduced in Section III. For example, the 2SLS full model in Column 5 of Table 8 shows that AIE = 0.0211, which is derived by multiplying the effect of Morethan2 on the firstborn daughter’s high school completion given a next sister (β₂ = −0.0955) by the effect of a next brother on the probability of having Morethan2 (E[M|D = 1] – E[M|D = 0] = 0.4787 – 0.6991). Similarly, ADE = −0.0192 is derived by the 2SLS estimated coefficient on a next brother (β₁ = −0.0651) being added to the product of the interaction’s coefficient (β₃ = 0.0960) and the probability of Morethan2 when the secondborn is male (E[M|D = 1] = 0.4787). Since both AIE and ADE estimators are linear in the regression coefficients, we use a simple linear restriction to derive the standard errors.

Additionally, we use an alternative family size measure, Sibsize, to estimate the decomposed effects shown in Tables 10 and 11. Because the decomposed results are strikingly similar, irrespective of the choice of family size measure, and because the first stage for Morethan2 is stronger than that for Sibsize, our discussion focuses on the 2SLS results using Morethan2.

The 2SLS estimated AIE suggests that through decreasing potential sibsize of firstborn daughters, a next brother indirectly causes a 2 percentage point increase in the probability of completing high school and a 1.5 percentage point increase in the likelihood of enrolling in university. The results are almost identical using either Morethan2 or Sibsize, as shown in Column 5 of Tables 8 and 10 for high school completion and Tables 9 and 11 for university admission. The magnitude of AIE is not negligible since it accounts for 9 percent (0.0211/0.246) of the high school completion rate and 8 percent (0.0148/0.177) of the university admission rate, both precisely estimated. In contrast, the 2SLS estimates of the AIE on firstborn sons’ high school completion and university admission are only 0.2 percentage points, accounting for 1 percent or less (0.0020/0.239 or 0.0017/0.153) of the sample mean.

The markedly large difference in the magnitude of the AIE between firstborn daughters and sons indicates the presence of gender bias, resulting primarily from two factors. First, firstborn daughters bear a much larger family size effect than firstborn sons do if the next sibling is female. If the next sibling is male, the quality–quantity tradeoffs are nearly zero for firstborn sons and daughters. Second, while a next brother decreases the probability for firstborn sons of having Morethan2 by only 6 percentage points (E [M|D = 1] – E[M|D = 0] = 0.4191 – 0.4831), the reduction is much higher for firstborn daughters, at about 22 percentage points (0.4787 – 0.6991). This result is not surprising because we now allow families to follow the son-preferring stopping rule in the model by interacting sibsize with sibling gender composition.

The large positive AIE on firstborn daughters almost entirely offsets the direct rivalry effect of male siblings, measured by ADE. The 2SLS estimated ADEs in Column 5 of Tables 8 and 9 suggest that a next brother directly reduces the probability for firstborn daughters of completing high school by 1.9 percentage points (standard error = 0.0076) and of enrolling in university by 1.3 percentage points (standard error = 0.0069). Both ADEs account for 7–8 percent (0.0192/0.246 or 0.0125/0.177) of the sample mean. In contrast, the 2SLS estimated ADE for firstborn sons, reported in Column 10, suggests that a next brother has little impact on firstborn sons’ high school completion or university admission. The estimates have standard errors as small as 0.2 percentage points. We note that regardless of firstborn gender, the negative ADE is almost entirely offset by the positive AIE, resulting in a nearly zero total effect.

In the previous literature, the direct rivalry effect of male siblings was measured by the coefficient of sibling gender in regression models without interactions, and sibsize was assumed to be exogenous and independent of the gender composition of older siblings. Our OLS estimated coefficients of a next brother in Columns 1 and 6 indicate a nearly zero direct rivalry effect, regardless of firstborn gender. This coefficient is equal to CDE assuming no interaction term. Even if the interaction term is included, as in Columns 4 and 9, the OLS estimated CDE remains small or statistically insignificant. Overall, the OLS estimates show no sign of sibling rivalry, irrespective of whether the interaction is included.

The 2SLS estimated CDE excluding the interaction also considerably understates ADE, particularly among female firstborns. As Columns 2 and 3 show, without interactions, the 2SLS estimated CDE on firstborn daughters suggests a decrease in high school completion by 0.3–0.6 percentage points and a decrease in university admission by 0.2–0.4 percentage points. These only capture less than one-third of the 2SLS estimated ADE in Column 5 where the interaction term is included. This pattern does not appear in Columns 7 and 8 for firstborn sons.

Looking at Column 5, we can compare the 2SLS estimated ADE and CDE among firstborn females using the same regression model. The former is more than double the latter, and the downward bias of CDE is statistically significant for high school completion but imprecise for university admission. The fundamental difference between these two measures is that ADE is evaluated at the conditional mean of family size as if every firstborn’s next sibling were male. In contrast, CDE is evaluated at the unconditional mean, assuming sibsize does not adjust for the gender composition of existing siblings. The gap between conditional and unconditional family size is larger among firstborn daughters because many parents follow the son-preferring stopping rule. The stronger the demand for sons is, the greater the downward bias of CDE. A firstborn son reduces parental demand for boys, so the downward bias of CDE is smaller among firstborn sons.

Overall, the decomposed effects of sibling gender are larger on firstborn females than on firstborn males. As the estimates of the full 2SLS model suggest, AIE for firstborn females is nine or ten times (0.0148/0.0017 or 0.0211/0.0020) that for firstborn males, and ADE for firstborn women is at least quadruple that for firstborn males. This reveals a strong gender bias against firstborn females’ education, which cannot be uncovered by conventional measures, such as ATE or CDE, or from the coefficient of sibling gender. The decomposition results are robust, regardless of which fertility choice measure (Sibsize or Morethan2) is adopted, as long as we include an interaction term in the 2SLS model.²³

E. Testing for Exogeneity of the Twins Instrument and Robustness Checks

Conditional exogeneity of the twins instrument has been questioned because the birth of twin siblings likely has a direct effect on firstborn children beyond just increasing family size. Although conditional exogeneity of the twins instrument is not testable, this section discusses or addresses three potential concerns: (i) Twins might affect firstborn outcomes because of zero spacing between the next two siblings. (ii) Twins tend to have lower birthweight and/or poor health. (iii) Mothers who have better health or education might be positively selected into having twins.

First, a firstborn child without twin siblings tends to have a lower education if he or she has two siblings born more closely together. Assuming that we can extrapolate this tendency to the case of secondborn twins, in which the interval is zero, the birth of twins not only increases family size but also adversely affects the firstborn through spacing. If such an adverse effect of secondborn twins exists, then our 2SLS result would overstate the magnitudes of both the quantity–quality tradeoff and the average indirect effect.

The potentially negative effect of close spacing between the next two siblings on firstborn outcomes is inherently unmeasurable because most firstborn children have only one sibling. We explore the magnitude of the upward biases by including the logarithmic spacing between the first two births instead. Columns 4 and 8 of Table 12 and Online Appendix Table A7 show that 1 percent longer spacing between the first two births reduces the firstborn’s high school completion or university admission by 0.6–0.9 percentage points. Although this effect is insignificant for the education of firstborn females, inclusion of birth spacing decreases the family size effects on their high school completion and university admission by approximately 2 percentage points (=0.96–0.76) or one-half of the standard error, if the next sibling is also female. The inclusion of birth spacing also decreases the decomposed effects of sibling gender on schooling by approximately 0.4 percentage ( = 0.0154–0.0192; 0.0173–0.0211), about one-half of the standard errors. Because families with firstborn daughters tend to rush to have another child, the impact of sibsize or sibling gender decreases after controlling for birth spacing, as expected. In contrast, the effect of 1 percent longer spacing is significant for the education of firstborn males, as Appendix Table A7 shows. However, the inclusion of birth spacing almost does not change the decomposed effects of sibling gender for firstborn males. Overall, the pattern of our estimation results remains regardless.

Second, weaker health conditions of secondborn twins may induce parents to divert family resources from the twins to the firstborn singleton (if parents have efficiency concerns) or the other way around (if parents have inequality aversion). In either case, estimation results regarding the family size effect and the decomposed effects of sibling gender will be biased. Rosenzweig and Zhang (2009) recommend controlling for the mean birthweight of secondborn twins to address the issue of twins’ endowment deficit. The idea is that by fixing the birthweight of the second birth (in addition to family background), the only channel through which twinning at the second birth can affect the firstborn child’s education is through changing the sibsize. However, this approach is not suitable for studies aiming to detect gender bias because boys are heavier than girls at birth on average. Part of the rivalry effect of a younger brother would be mistaken as a birthweight effect if we included the mean birthweight of the secondborn. In addition, Black, Devereux, and Salvanes (2010) note that inclusion of an initial health condition (such as birthweight or gestational duration) might invalidate conditional exogeneity of Twin2nd because initial health is shaped by other determinants of firstborn education, such as maternal health, the introduction of social programs, and the interplay of genes and environments (Almond, Chay, and Lee 2005; Currie 2009). Appendix Table A8 shows the secondborn twins are lighter at birth in larger families. This suggests the presence of unobserved factors (such as household income or wealth) that are positively correlated with birthweight but negatively correlated with family size. Thus, our main results exclude the endogenous birthweight of secondborn siblings.

Nevertheless, we evaluate this endogeneity issue with initial health by including the conditional mean birthweight percentile of the second birth given gender, instead of the unconditional mean birthweight, to remove the correlation between gender and birth-weight. As Columns 2 and 6 of Table 12 show for firstborn females, the inclusion of the birthweight percentile decreases the estimated decomposed effects and family size effects by at least 40 percent (0.0125/0.0211 – 1 using AIE for high school completion as an example). In contrast, Columns 3 and 7 suggest that inclusion of gestation duration has almost no impact on the estimation results. Since we have no exogenous sources of variation in initial health, our baseline result excludes initial health status.

Third, we find that the family size impact is large among firstborn daughters whose next sibling is also female, but nearly zero or insignificant among firstborn daughters and sons whose next sibling is male. No or low quantity–quality tradeoff can be driven by positive selection of mothers into twinning, as Bhalotra and Clarke (2016) have noted. If more educated/healthier mothers have a higher chance of conceiving twins and prefer a smaller family, then the observed negative impact of family size will be biased downward.

Although our analysis included both mother’s and father’s education levels and birth years and their district of residence when the first child was born, in Table 13 we examine whether the observed correlation between twin births and mother’s education can be captured by other family background covariates, such as father’s education. We present the results using the full sample in the top panel, compared with those using a restricted sample in the bottom panel where the first two births were prior to 1985 (before the enactment of the abortion law, and before the sex ratio of boys to girls for the third birth started rising while that for the first two births remained around the normal range). Columns 1 and 3 show some positive associations between maternal education and secondborn twins, particularly in families with firstborn females. In families with firstborn males in the full sample, the association appears to be absent. After adding father’s education, Columns 2 and 4 suggest that the estimated coefficients of mother’s education levels on twin births become smaller or imprecise. The only exception is the coefficient of mother’s vocational high school diploma among families with firstborn females; it suggests a 0.09 percentage point increase in the probability of having twins at the second birth, about 13 percent ( = 0.0009/0.007) of the sample mean. As for families with firstborn males, mother’s education is less important than father’s education in explaining the occurrence of secondborn twins. All the coefficients of maternal education levels are tiny or negative. In contrast, father’s professional degree is associated with a 0.16 percentage point increase in secondborn twinning, which accounts for a quarter ( = 0.0016/0.0064) of the sample mean. Overall, the evidence for mothers positively selecting into having twins is not as clear in Taiwan as in those countries examined in Bhalotra and Clarke (2016). Our 2SLS results have controlled for both parents’ education levels to minimize the downward bias caused by positive selection.

Finally, to further examine the exclusion restrictions of the twins instrument, we estimate in Table 14 the reduced-form twins effect on the education of children from families whose twins have a smaller impact on fertility. These families are likely to have a lower first stage and to have larger families anyway. The top panel of the table uses the full sample while the lower panel limits our sample to families with 18 months or less between the first two births. For comparison, Table 14 summarizes the first-stage, reduced-form, second-stage, and decomposition results for each sample (using Morethan2 for illustration because of a stronger first stage). Columns 1 and 6 show that the restricted sample has a smaller first stage. However, the reduced-form result for that sample is either close to zero (Column 2) or imprecisely resembles the result using the full sample (Columns 4, 7, and 9).

Thus, twinning has no statistically significant impact on the education of firstborns unless it strongly changes their parents’ fertility choice. This is consistent with the exclusion restrictions required for identification.