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Complementarities in human capital production: Evidence from genetic endowments and birth order

On average, firstborns complete more education than their laterborn siblings. We study whether this effect is amplified by individuals' endowments. Our family-fixed effects approach exploits exogenous variation in birth order and genetic endowments among 14,850 siblings in the UK Biobank. We find that those with higher genetic endowments benefit disproportionately more from being firstborn compared to those with lower genetic endowments, providing a clean example of how nature and nurture interact in producing human capital. Since parental investments are a dominant channel driving birth order effects, our results are consistent with complementarity between endowments and investments in human capital formation.


I. Introduction
It is increasingly accepted that important life outcomes such as educational attainment are influenced by a complex interplay between nature (i.e., genetic variation) and nurture (i.e., environments in which one grows up; Rutter, Moffitt, and Caspi, 2006;Heckman, 2007).
For example, the formation of skills relevant to educational attainment may result from complementarities between (genetic) endowments and parental investments (Ben-Porath, 1967;Becker and Tomes, 1986;. Empirically estimating the separate contributions of genetic endowments and environments, and their possible interaction, is however complicated by the endogenous nature of both. Indeed, environmental characteristics are partially heritable and typically cluster together, e.g., higher educated parents tend to have higher incomes. Hence, disentangling what drives the 'environment-effect' is not straightforward. Similarly, although several studies have found specific genetic variants to be associated with human capital outcomes such as educational attainment , genetic variation is only random conditional on parental genotypes (e.g., Lawlor, Harbord, Sterne, Timpson, & Davey Smith, 2008;Davey Smith & Ebrahim, 2003). Not controlling for the latter implies that the 'genetic-effect' may in fact reflect 'genetic nurture' -that is, the parental genotype can shape the environment in which children grow up, thereby producing a spurious association between the child's genetic variants and their outcomes (e.g., Belsky et al., 2018;Kong et al., 2018).
This paper exploits exogenous variation in both genetic endowments for education and the family environment to analyze the importance of gene-environment interactions for educational attainment. Genetic endowments are measured using a so-called "polygenic score" (PGS), also referred to as a "polygenic index" . A PGS is a highly predictive index that is constructed as the sum of all measured genetic variants, weighted by the strength of their correlation with educational attainment ; . The PGS can (currently) explain up to 13% of the variance in educational attainment. Our measure of the environment is an individual's birth order, which is consistently negatively correlated with educational attainment in developed countries (see e.g. Bagger et al., 2013;Behrman et al., 1986;Black, Devereux, & Salvanes, 2005;Booth & Kee, 2009;De Haan, 2010;De Haan, Plug, & Rosero, 2014;Kantarevic et al., 2006). 1 We overcome endogeneity issues by exploiting within-family variation in both birth order and genetic endowments. Indeed, siblings' birth order is randomly determined within families (e.g., Damian and Roberts, 2015), and genetic variants are randomly assigned across siblings within a family according to Mendel's Law of Independent Assortment.
Hence, genetic variants are unrelated to birth order by construction. 2 With exogenous variation in both genes and environments, this provides a compelling context in which we can fundamentally improve our understanding of the nature-nurture interplay in shaping life outcomes. Do advantageous environments complement genetic advantages?
Answering this question constitutes our first main contribution.
We choose birth order as a measure of the environment not merely for being conveniently uncorrelated with genetic endowments. In fact, birth order has been shown to proxy for parental investments (see below). As such, economic theories of human capital production provide a specific hypothesis of the sign of the interaction term. Becker & Tomes (1986, eq. 4) specify a human capital production function of the form where denotes human capital, represents parental investments, denotes public investments, and refers to genetic and cultural endowments (with index t for time). A critical assumption made to explain intergenerational persistence of earnings and assets is that there are complementarities between endowments and investments (Becker & Tomes, 1986; eq. 5): 2 −1 > 0.
In words, children with higher (genetic) endowments benefit more from parental investments −1 .
The complementarity assumption is not specific to Becker & Tomes (1986). In fact, it is also embedded in so-called "Ben-Porath neutrality" where the stock of human capital raises the productivity of investments in human capital (Ben-Porath 1967;Heckman 1976;Rosen 1976). Moreover, it is the key building block for the concept of dynamic complementarity in skill production , where skills produced at a given age raise the productivity of investments at later ages. Hence, complementarity between the child's endowments and parental investments is a central assumption underlying seminal economic theories of human capital production.
We use an individual's birth order as an exogenous and highly predictive proxy for parental investments because we do not observe parental investments directly, and because realized parental investments are endogenous to the child's endowments (e.g., Becker & Tomes, 1986;Behrman, Pollak, & Taubman, 1982;Breinholt & Conley, 2019;Rosenzweig & Wolpin, 1995;. The theoretical literature on the 'quantity-quality trade-off' (Becker, 1960;Becker & Lewis, 1973;Becker & Tomes, 1976;Galor & Weil, 2000) posits that with each additional child, it is more expensive to maintain the same 'quality' children (i.e., with the same level of education or health), implying that parents invest less in laterborn children. Moreover, with parental preferences for fairness in investments over equality in outcomes (see e.g., Berry, Dizon-Ross, & Jagnani, 2020), parents distribute their resources equally over their children, leading to natural dilutions in time investments for laterborn children compared to their firstborn siblings (Blake 1989;Downey 2001).
Indeed, firstborns have undivided attention until the arrival of the second child (Breining et al. 2020), and the empirical literature suggests that a dominant channel through which birth order affects educational attainment is parental time investments in early life (see e.g., Birdsall, 1991;Black, Grönqvist, & Öckert, 2018;Breining et al., 2020;De Haan, 2010;Del Bono et al., 2016;Monfardini & See, 2012;Pavan, 2016;Price, 2008). Using the American Time Use Survey (ATUS), Price (2008) shows that firstborns receive 20-30 minutes more daily quality time compared to their younger siblings (see also Black et al., 2018;Monfardini & See, 2012), with the gap being largest at early ages. For laterborn children, mothers postpone prenatal care, breastfeed less, are more likely to smoke when not breastfeeding (Lehmann, Nuevo-Chiquero, & Vidal-Fernandez, 2018), and fathers take shorter periods of parental leave (Sundström and Duvander 2002). Hotz and Pantano (2015) additionally show that parents have less stringent parenting strategies for laterborn children. Hence, whilst we do not dismiss other potential channels through which birth order may affect educational attainment, 3 parental investments are a prominent channel through which these effects arise, consistent with the evidence that parental investments are an important input into the child's human capital production (e.g., Del Boca, Flinn, & Wiswall, 2014).
Economic theories of human capital production, and in particular the central assumption of complementarity between endowments and investments, therefore, provides a clear prediction for the sign of the Gene-by-Environment (G×E) interaction term: in a sample of siblings, being firstborn should show a positive interaction with genetic endowments.
Hence, apart from providing a rare context in which economic theory helps formulate hypotheses about fundamental interactions between genetic variation and the environment, the interaction between birth order and genetic endowments is also 3 Eirnaes & Pörtner (2004) distinguish three additional environmental channels through which birth order effects may arise: (i) younger children may benefit from the interaction with their older siblings; (ii) firstborns benefit from a lower maternal age and better maternal immune system (e.g., Behrman, 1988;Black et al., 2016); and (iii) in some societies the oldest son (or older children more generally) are favored as they are the first to become economically independent. Other channels could be that later born children are exposed to more family disruptions (Björklund, Ginther, and Sundström 2021), or experiencing illness at younger ages due to older siblings bringing home viruses (Daysal et al. 2021).
informative about the existence of complementarities in human capital production. This constitutes our second main contribution.
Our empirical analysis exploits data from 14,850 full siblings from the UK Biobank, a population-based sample from the United Kingdom (Fry et al. 2017). To measure participants' genetic endowments, we construct polygenic scores for educational attainment based on the results from our own tailor-made genome-wide association study (GWAS) that uses the UK Biobank sample but excludes all siblings and their relatives. 4 We adopt a family fixed effects approach to exploit within-family variation in genetic endowments and birth order to study the G, E and G×E effects on educational attainment, and we apply Obviously-Related Instrumental Variables estimation (ORIV;  to reduce random measurement error in the polygenic score. We mainly focus on firstborns versus laterborns, since the literature suggests birth order effects are particularly salient at this margin (e.g., Breining et al., 2020).
We confirm earlier findings that firstborns have a higher level of education than laterborns, and that one's genetic endowments are a strong predictor of educational attainment. We also confirm that genetic endowments do not differ systematically across birth order within a family. This finding corroborates that birth order effects must be due to environmental influences (see also Isungset, Freese, Andreassen, & Lyngstad, 2021), such as greater parental time investments in firstborns. Our main finding is that birth order and genetic endowments interact: being firstborn and having higher genetic endowments for education exhibit a positive interaction, meaning that those with a higher polygenic score benefit disproportionally more from being firstborn compared to those with a lower polygenic score. This finding is a clean example of how genetic endowments and the environment interact in producing important life outcomes such as educational attainment. Moreover, our empirical results are consistent with the existence of complementarity between endowments and investments in human capital production: additional parental time investments associated with being firstborn are more 'effective' among those siblings who randomly inherited higher genetic endowments for educational attainment.
Our paper speaks to three main literatures. First, we contribute to an emerging literature on gene-environment interactions (G×E), which addresses how the environment moderates the effect of genetic variants, and vice versa. Previous studies have typically examined interactions between polygenic scores and endogenous environments such as socio-economic status (for studies within the social sciences, see e.g., Barth, Papageorge, & Thom, 2020;Bierut, Biroli, Galama, & Thom, 2018;Ronda et al., 2020), childhood trauma (e.g., Mullins et al., 2016;Peyrot et al., 2014), or a partner's death (e.g., Domingue, Liu, Okbay, & Belsky, 2017). Interpretation of these findings is not straightforward, because individuals with certain genetic predispositions may self-select into different environments (known as gene-environment correlation, or rGE; see Jencks, 1980;Schmitz & Conley, 2017a individuals' educational attainment. 5 We push this literature one step further by not only considering exogenous variation in the environment, but also in genetic endowments by exploiting within-family variation in polygenic scores. 5 Studies with other outcomes that exploit exogenous environments include, e.g., Barban, Cao, & Francesconi (2021), Barcellos, Carvalho, & Turley (2018), , and Schmitz & Conley (2016). For a recent review of the G×E literature in economics and social science, see Dias Pereira et al. (2022).
A second strand of literature that we speak to is the literature on birth order effects. This literature consistently finds that in developed countries, laterborn children have lower educational attainment. Birth order effects have also been found for other outcomes, though sometimes with mixed results, such as early life cognitive skills and intelligence (Black, Devereux, & Salvanesz, 2011;Fenson et al., 1994;Keller, Groesch and Trob, 2015), health (Black, Devereux, and Salvanes 2016;Pruckner et al. 2019), personality and leadership skills (Black, Grönqvist, and Öckert 2018), and delinquency (Breining et al. 2020).
We contribute to this literature by studying heterogeneity in the birth order effect on educational attainment with respect to genetic endowments. The potential interaction between birth order and genetic endowments is not merely an important source of heterogeneity in the treatment effect, but one that -if present -carries over to the next generation, potentially exacerbating intergenerational inequalities (Barclay, Lyngstad, and Conley 2021;Havari and Savegnago 2022).
Finally, our study offers new empirical insights on the human capital production process (e.g., Becker and Tomes, 1986;Todd & Wolpin, 2003). Empirically testing complementarity in human capital formation requires independent variation in initial endowments and subsequent investments , and is therefore extremely challenging . Indeed, the previous literature has almost exclusively relied on early-life outcomes such as birthweight as a measure of endowments (e.g., Datar, Kilburn, & Loughran, 2010;Figlio, Guryan, Karbownik, & Roth, 2014). However, such early life outcomes are affected by prenatal investments Rosenzweig and Schultz 2015), meaning they partially capture parental choices and are therefore endogenous. Furthermore, parents respond to children's endowments (e.g., Adhvaryu & Nyshadham, 2016;Becker & Tomes, 1986;Bharadwaj, Eberhard, & Neilson, 2018;Datar et al., 2010;Frijters, Johnston, Shah, & Shields, 2013;Giannola, 2020;Hsin & Felfe, 2014), with recent studies suggesting that parental investments also respond to the genetic endowments of children (Breinholt & Conley, 2019;Fletcher, Wu, Zhao, & Lu, 2020;Houmark, Ronda, & Rosholm, 2020;Sanzde-Galdeano & Terskaya, 2019). Hence, measures of children's endowments are rarely clean of parental investments, and parental investments are rarely independent of endowments, posing a formidable empirical challenge to accurately identify complementarities in human capital production. 6 We contribute to this literature by exploiting random within-family variation in genetic endowments and birth order induced reductions in parental investments. Indeed, our measure of endowments is randomly assigned within families and fixed at conception. It is therefore clean from parental investments. Furthermore, we proxy for parental investments using individuals' birth order, which is strongly associated with parental investments, but uncorrelated with genetic endowments. Essentially, employing birth order as a proxy for parental investments exploits the natural reduction in the time and money available with the arrival of a laterborn child, and is thus independent of the child's endowments. There could be other channels through which birth order may impact educational attainment, such as through interaction with younger siblings . However, unless these other channels have completely opposite interaction effects with genetic endowments, birth order and genetic endowments provide a promising setting to empirically test a necessary condition for complementarity in human capital production.
Finding support for complementarity between endowments and investments is important for understanding the nature of the human capital production function. The production function of the child's human capital is -next to a parental budget and time constraintan important input into the broader optimization problem where parents decide between 6 A recent set of studies has examined rare cases where exogenous variation exists in both initial endowments as well as later-life investments, with mixed evidence. Some studies find evidence consistent with complementarity , whereas others find weaker evidence or even substitutability between endowments and investments . See Appendix B for a detailed overview.
own consumption and investments in their children. By informing the shape and properties of the production function, our analysis is an important precursor to a structural model of parental investment decisions, estimation of which is beyond the scope of this paper. 7 Evidence in support of complementarity also speaks to whether later-life investments can reduce or eliminate damage originating early in life , and emphasizes the importance of early-life investments being followed-up by later-life investments to reap the full benefits in terms of human capital outcomes (e.g., . More generally, this paper shows how economic theory can inform empirical G×E analyses, and how genetic data can be leveraged to test economic theories. We show that the analysis of G×E within-family, exploiting exogenous G as well as E, provides a way to test for complementarities more generally, which is not restricted to birth order effects, but extends to other (exogenous) parental investments and policy changes (e.g., on studentteacher ratios, minimum school leaving ages, etc.). Our findings further provide one of the first pieces of causal evidence of how genetic variation (here measured by the polygenic score for years of education) and the environment (here measured by birth order) jointly shape and interact in producing important life outcomes such as educational attainment.
While this finding was long anticipated by numerous scholars (e.g., Heckman, 2007;Rutter et al., 2006), finding credible and independent sources of variation in genes and environments is rare given how tightly genetic and environmental influences are entangled (e.g., Koellinger & Harden, 2018). Showing evidence of an interaction between genetic variation and environments is therefore a leap forward in our fundamental understanding of how nature and nurture jointly shape human capital, and provide an antidote against arguments of genetic or environmental determinism.

II. Empirical Strategy
We analyze gene-environment (G×E) interactions between genetic endowments for educational attainment and birth order. The empirical specification is rooted in the human capital production function (1), where we take adult educational attainment as a proxy for human capital by the end of childhood, as in Cunha & Heckman (2008) and .
Following Todd & Wolpin (2003) and Cunha & Heckman (2008), we specify a linear production function, where years of completed education for individual i of family j (denoted by ), is a function of initial endowments, parental investments, and unobserved parental characteristics. Empirically, we measure initial endowments by the polygenic score for educational attainment (denoted by ) 8 , we proxy parental investments by an indicator for being firstborn ( ), and assume parental characteristics are fixed and therefore subsumed into the family fixed effect ( ). We then allow for an interaction term between and , leading to the following specification: where is a set of individual level controls, including gender, month, and year of birth dummies (Black, Devereux, and Salvanes 2005;Handy and Shester 2020). It also includes the vector of the first 40 principal components (PCs) of the genetic relatedness matrix. 9 Finally, is the error term; we use heteroskedasticity-robust standard errors, clustered at the family level. The parameter 2 captures the association between the standardized polygenic score for education and years of schooling, whilst 3 estimates the average Our polygenic score incorporates two sources of measurement error. First, since the GWAS underlying the construction of a polygenic score is based on a finite sample (see Section 3), our estimated polygenic score that is based on the GWAS estimates is a noisy proxy for the true (latent) polygenic score (e.g., Benjamin, Cesarini, Laibson, & Turley, 2020; Van Kippersluis et al., 2020). Second, GWASs typically do not control for parental genotypes.
Trejo & Domingue (2019) show that not controlling for parental genotypes causes the polygenic scores to be measured with error. Both sources of measurement error lead to an attenuation bias in the coefficient of the polygenic score in within-family analyses. 11 10 In section 4.5, we study the interaction effect using more flexible approaches than the linear interaction presented here. 11 The reason why classical measurement error as a result of finite-sample bias leads to attenuation bias is well known. It is more subtle why the exclusion of parental genotype in the discovery GWAS leads to an attenuation bias in within-family studies. The reason is that a polygenic score constructed on basis of a GWAS that did not control for parental genotype will reflect both direct genetic effects arising from the individual's genotype as well as indirect genetic effects arising from the omitted parental genotypes. The latter effects are known as 'genetic nurture' (e.g., Kong et al., 2018). When applied within families, the differences in the polygenic score arising from parental genotype are spurious since parental genotype is the same across siblings. Hence, part of the differences across siblings in the polygenic score reflect genetic nurture (i.e., indirect genetic effects) and can be considered measurement error attenuating the resulting within-family estimates (Trejo & Domingue, 2019).
While we cannot solve the attenuation bias arising from the omission of parental genotype in the GWAS, we follow DiPrete, Burik, & Koellinger (2018) and  in applying Instrumental Variables (IV) to tackle the classic measurement error problem.
More specifically, we split our discovery GWAS sample into two equal halves and construct two polygenic scores based on the two discovery samples. To ensure that these subsamples are unrelated, we randomly select only one individual from each cousin cluster. Even though the two polygenic scores individually have lower predictive power, the measurement error in the two is plausibly orthogonal and so they can be used as instrumental variables for each other. Using these two polygenic scores, we apply Obviously-Related Instrumental Variables (ORIV; . See Appendix C for more details.

III. Data
We use data from the UK Biobank, a population-based cohort with approximately 500,000 individuals aged between 40-69 at the time of interview in 2006-2010 and living within a radius of 40 km from one of the 22 assessment centers in England, Wales, and Scotland (Fry et al. 2017). It contains survey data, biomarker and DNA samples, physical measurements, and linkage to inpatient registers and death records (Sudlow et al. 2015).
Because participation in the UK Biobank is voluntary, it is not a representative sample of the UK population (see Fry et al. (2017) for a detailed analysis).
We apply the following sample selection criteria. The original data include 502,498 consented individuals. We follow the literature and remove those of non-European descent (92,892 observations), as well as twins and multiple births (9,310 observations), and individuals with missing or conflicting information regarding the number of siblings and/or family size (3,801 observations). In doing so, we arrive at a sample of 396,494 individuals. We further restrict this sample to individuals with at least one sibling 12 in the UK Biobank and without missing values on any of the variables included in our analysis (i.e., years of education, birth order, family size, year and month of birth, principal components of the genetic relatedness matrix, gender, and the polygenic score for education). Since the UK Biobank did not specifically target families, this leads to a final sample size of N = 14,850 individuals within 7,281 full-sibling clusters.
We follow the literature (see e.g.,  and convert individuals' qualifications to equivalent years of education using the International Standard Classification of Education (ISCED). 13 The average years of education for the sibling sample is 15 years (see Table 1). We construct individuals' birth order based on their response to a question of how many older siblings they have. If a respondent reports zero older siblings, the birth order is set to one.  Table 1 shows that 39.4% of our sample are firstborns, with an average birth order of 1.91 (where we have censored birth order at 5 for the 245 respondents with birth order beyond 5). Around 37% of our sample is lastborn, and the average family size is 3 (i.e., the average number of siblings is 2).
13 Years of education ranges from 7 to 20, where College or University degree is equivalent to 20 years, National Vocational Qualification (NVQ), Higher National Diploma (HND), or Higher National Certificate (HNC) to 19 years, other professional qualifications to 15 years, having an A or AS levels or similar to 13 years, O levels, (General) Certificate of Secondary Education ((G)CSE) to 10 years, and if none of the above to the lowest level of 7 years.
Our measure of genetic endowment for education is the polygenic score for education. A polygenic score is a weighted sum of genetic variants called Single Nucleotide Polymorphisms (SNPs, see Appendix A for details). The SNP weights are determined by the association between a SNP and years of education  in an independent (discovery) sample: where is the value for the polygenic score for individual i, is the regression coefficient of SNP j ( = 1, … , ) from our own tailor-made GWAS (see below), and is the genotype of individual i for SNP j (coded as 0, 1 or 2, indicating the number of "effect" alleles). We use the LDpred software  to correct for the correlation structure across SNP (i.e., to correct for "linkage disequilibrium"). The polygenic scores are standardized within the sibling sample to have mean 0 and standard deviation 1.
The polygenic score measures the genetic predisposition towards educational attainment within the environmental context and demographic characteristics of the discovery GWAS sample (Domingue, Trejo, Armstrong-Carter, & Tucker-Drob, 2020). It is therefore preferable to select discovery and analysis samples from the same environmental context, especially when analyzing gene-environment interactions. At the same time, the discovery sample should be independent of the analysis sample to avoid overfitting . Navigating this trade-off, we therefore construct the polygenic score by using the weights from our own tailor-made GWAS that uses the UK Biobank sample, but without the siblings (our analysis sample) and their relatives. The GWAS discovery sample comprises 389,419 individuals from the UK Biobank; we use the summary statistics from these analyses to create the polygenic scores on the sample of 14,850 siblings. This tailormade polygenic score alleviates the differences between the discovery and the analysis samples in terms of demographics and environmental context, as well as measurement, i.e., the variables of interest are measured in the same way (Tropf et al. 2017;Elam et al. 2019). Moreover, running our own GWAS enables the construction of two independent polygenic scores, obtained by splitting the discovery GWAS sample into two equal halves, which can be used in ORIV. This approach has been shown to outperform a single polygenic score that is based on meta-analyzing multiple cohorts (Van Kippersluis et al., 2020). 14 IV. Results 4.1. Predictive power of the polygenic score for educational attainment Figure 1 shows that our polygenic score for years of education is approximately normally distributed. For the scatterplot, we divided the polygenic score in 200 bins; the dots represent the average years of education for each bin. The line through the dots is obtained from a local polynomial regression of years of education on our polygenic score.
In line with the literature , the polygenic score is positively correlated with years of education (r = 0.23, p < 0.001). The average difference between those two standard deviations below the mean of the polygenic score, and those two standard deviations above the mean is almost 4 years of completed education, highlighting the substantial predictive power of the polygenic score.
Furthermore, Figure 1 suggests the relationship is approximately linear. Notes: Plotted using 200 bins of the polygenic score. Table 2 shows that the incremental R 2 of the polygenic score (i.e., the additional variance explained by the polygenic score after controlling for gender, month, and year of birth, and the first 40 principal components of the genetic relatedness matrix) is 5.4% in the betweenfamily analysis (i.e., 0.113-0.059=0.054; Columns 1 and 2). In the specifications with family fixed effects (Columns 3 and 4), the incremental (within) R 2 for the polygenic score is reduced to 1%. This reduction in predictive power when moving from between-family to within-family estimates is well-established in the literature (see e.g., Koellinger & Harden, 2018;Kong et al., 2018;Selzam et al., 2019), and reflects the fact that family fixed effects account for the shared family environment and parental genotype, which was not accounted for in the between-family specification. In terms of the effect sizes, we observe that a one standard deviation increase in the polygenic score is associated with an increase of 1.155 years of education. With family fixed effects, the effect size is reduced to 0.646.  Figure 2 shows differences in years of education (residualized with respect to year and month of birth, family size, gender, and 40 first principal components of the genetic relatedness matrix) by birth order (panel A). Not taking into account family fixed effects, we find that years of education for laterborn children is lower (albeit with larger confidence intervals) than that for firstborns. When pooling all laterborns (panel B), the difference between firstborns and laterborns becomes more pronounced. Notes: 95% confidence intervals. Years of education is residualized with respect to year and month of birth, family size, gender, and the first 40 principal components of the genetic related matrix. Table 3 confirms the birth order effects in the specifications with and without family fixed effects. We observe a consistent gap of 0.3-0.4 years of schooling between first-and laterborn children. The direction and magnitude of the effect is robust to using the binary indicator or the categorical variable for birth order. The within-family effect size for 5 th born children does not reach statistical significance due to the relatively small number of observations (see Table 1).  Robust standard errors in parentheses, clustered by family in the within-family analysis; Coefficients for the control variables (year and month of birth, gender and the first 40 principal components of the genetic relatedness matrix)) are not displayed, but available upon request from the authors.

The relationship between birth order and the polygenic score for years of education
To measure causal gene-environment interactions, it is important to verify that birth order is orthogonal to the polygenic score. Figure   Consistent with the graphical evidence,  Robust standard errors in parentheses, clustered by family in the within-family analysis; Coefficients for the control variables (year and month of birth, gender and the first 40 principal components of the genetic relatedness matrix) are not displayed, but available upon request from the authors.
The evidence is however not sufficient to claim that birth order is unrelated to any genetic endowments. To check if there are systematic differences by birth order in other, possibly related, polygenic scores, we analyze the polygenic scores from the Polygenic Index Repository  for all anthropometric, health, health behavior, and personality traits. We find no systematic difference by birth order in any of the 29 available polygenic scores (see Appendix G for details). These results corroborate that there is no systematic association between birth order and genetic endowments, and hence no evidence for gene-environment correlation (rGE). Table 5 presents the gene-environment interaction results in the between-family and within-family analyses. Comparison of Columns 1 and 4 in Table 5 to the estimates presented above (Table 2 and Table 3) shows that the addition of the educational attainment polygenic score does not change the direct effect of birth order on years of schooling. This comparison confirms once again that the polygenic score and birth order are independent. Firstborns enjoy on average 0.40 -0.42 extra years of schooling compared to laterborns. Furthermore, a one standard deviation increase in the polygenic score is estimated to raise years of education by 1.14 years (between-family analysis) and 0.65 years (within-family analysis). Coefficients for the control variables (year and month of birth, gender and the first 40 principal components of the genetic relatedness matrix) are not displayed, but available upon request from the authors. We do not report the R 2 for the ORIV specifications in Column (3) and (6) given the differences in its interpretation and computation for the instrumental variable type of regressions.

Gene-environment interaction: complementarity in human capital production
The between-family design shows a positive interaction between the polygenic score and being firstborn (Column 2), which is borderline statistically significant at conventional thresholds. When we use Obviously-Related Instrumental Variables (ORIV) regression (Column 3), the interaction term becomes larger in magnitude and statistically significant at the 5% threshold, suggesting that measurement error in the polygenic score attenuated the main effect of the polygenic score and the interaction term. In the family fixed effects specification, we find that the interaction effect is significant in both the OLS and the ORIV specifications. The measurement error correction again strengthens the interaction term in terms of magnitude.
As a complementary approach to test the statistical significance of our main results, we also conduct permutation inference by employing 10,000 permutations of the polygenic score and birth order, each time re-estimating the within-family specification (Fisher 1935;Rosenbaum 2002). This approach does not rely on repeated sampling of a hypothetical population and enables gauging how rare the size and significance of our interaction effect is for other permutations of the polygenic score and birth order. Figure 4 provides the distribution of t-statistics for all 10,000 permutations with the t-statistic of the actual estimate shown as a solid grey vertical line. The t-statistic of the estimated interaction term has an exact p-value (i.e., the proportion of t-statistics more extreme than our estimate) of 0.04. The positive and statistically significant interaction term provides strong evidence for geneenvironment interactions in education and is consistent with the existence of complementarity between endowments and investments in human capital production: the effect of being firstborn (associated with more parental investments) is complementary to a higher value for the polygenic score for years of education. In other words: those with a higher polygenic score benefit more from the higher parental investments that are associated with being firstborn. The magnitude of the coefficients suggests that for those with a polygenic score two standard deviations below the mean, there is no advantage of being firstborn. In contrast, for those with a high polygenic score, being firstborn increases one's years of education. For example, firstborns with a polygenic score two standard deviations above the mean completed on average 0.82 (OLS) to 1 (ORIV) more years of education compared to their laterborn siblings with a similar polygenic score. 15

Robustness checks
In this section, we check the robustness of our results against potential non-linearities in the functional form of the polygenic score as well as birth order, against the addition of further control variables, and against potentially endogenous fertility choices. While the linear form adopted in section 4.4 seems justified based on the visual inspection of Figure   1, we explore robustness of our results by allowing for possible non-linearities. Table 6 compares the within-family specification in continuous form (replicated in Column 1 for comparison), with those where we specify the polygenic score in binary form (above and below the mean, Column 2), in quartiles (Column 3) and in squared form (Column 4). We observe positive interaction terms across all specifications. In line with our main findings, the effect of being firstborn is insignificant for those with a lower polygenic score, and the effects are concentrated among those in the top quartile of the polygenic score 15 We also investigated potential gender differences in the interaction of interest. This analysis relies on families with a mixed gender composition to ensure variation in gender within families, reducing the analysis sample considerably. While we do find that gender is an important predictor of educational attainment for the cohorts examined here, with men having 1.46 more years of education than women, the interaction term does not differ significantly by gender. distribution. We also find that the main result in Column 1 is robust to the inclusion of a quadratic version of the polygenic score.  Table 7 reports the sensitivity of the results to an alternative specification of birth order.
Column 1 replicates the main result from Table 5 for comparison. In Column 2, we include dummies for each birth rank with firstborns as the reference category. Hence, we expect the effects to be reversed as compared to Column 1. All point estimates of the main effects and the interaction terms are in line with the birth order literature and consistent with complementarity: on average, second borns have lower educational attainment compared to firstborns and benefit less from having a high polygenic score. The estimates for higher ranks are rather imprecisely estimated due to the relatively small sample sizes for those with a birth rank of three or higher, but the point estimates consistently suggest that laterborns benefit less from having a high polygenic score.
In Table 8, we explore robustness of our results to potentially endogenous fertility decisions. A possible correlation between our measure of endowments and birth order could arise when fertility decisions are based on the genetic endowments of the children, known in the literature as the "child stopping rule" (Black et al., 2005;Pavan, 2016).
Whereas Section 4.3 shows that such a correlation does not exist, here we explicitly control for a possible child-stopping rule by including a dummy variable for being lastborn, which is set to one if an individual's birth order is equal to the total number of children in his/her family. To facilitate an easy comparison, Column 1 in Table 8 replicates the results from Column 5 in Table 5. As can be seen from Column 2, the lastborn dummy is not statistically significant and does not meaningfully affect our results, suggesting that potential endogenous fertility decisions do not change any of our conclusions. In Columns 3 and 4, we report the within-family results based on data from the families with at most three siblings (N = 11,364) and two siblings only (N = 7,918) to check if our results are sensitive to the exclusion of relatively large families with possibly different characteristics. Even though this sample restriction is endogenous, it is reassuring that our point estimates are virtually identical in these restricted samples.   Table 5; Column 2 additionally controls for a dummy indicating whether the individual is the lastborn; Column 3 restricts the sample to families with less than four siblings; Columns 4 restricts the sample to families with two siblings only. Robust standard errors in parentheses, clustered by family; Coefficients for the control variables (year and month of birth, family size, and the first 40 principal components of the genetic relatedness matrix) are not displayed, but available upon request from the authors.

V. Discussion
A large literature shows consistently higher educational attainments for firstborn children.
Using within-family data we move beyond the existing literature by showing that children benefit disproportionally from being firstborn when they have a relatively high polygenic score for educational attainment. More specifically, firstborns with an average polygenic score enjoy 0.415 years (≈ 5 months) of additional schooling compared to their laterborn siblings, on average. However, firstborns with a polygenic score that is one standard deviation above the mean enjoy an additional 0.204 years (≈ 2.5 months) of education, compared to their laterborn siblings with the same genetic endowment. In contrast, for individuals with below-average polygenic scores, being firstborn does not provide an advantage in terms of educational attainment. Since we provide evidence that genetic endowments are orthogonal to birth order, and previous literature suggests that birth order effects on children's education are mainly driven by parental investments, we interpret the positive and significant interaction term as providing support for the existence of the complementarity between endowments and investments in human capital production.
An alternative interpretation of our finding that birth order effects are concentrated among those with higher polygenic scores could be that the additional investments associated with being firstborn are higher for those with higher polygenic scores. That is, the positive interaction effect could also be explained by parents investing more in the firstborn, or altering fertility decisions, when the firstborn child has a higher polygenic score. While we cannot fully rule out this explanation, we believe this explanation is less plausible for three reasons. First, Breinholt & Conley (2019) and Houmark et al. (2020) show that parenting during infancy is not driven by genetic make-up because these endowments are not clearly expressed yet, and parental investment responses to polygenic scores do not arise before age six. This is long after the typical arrival of subsequent children, and so the most precious time of undivided attention for the firstborn is unlikely to be influenced by -at that time unobserved -differences in polygenic scores. 16 Second, whereas it is established that parents do respond to the polygenic scores of children at later ages (e.g., Sanz-De-Galdeano & Terskaya, 2019), we control for this with the main effect of the polygenic score in our specification. Only when -for some reasonparents respond more to the polygenic score when the child is firstborn, our interpretation of the interaction term would be challenged. We cannot rule this out, but deem it less plausible, as the additional investments associated with being firstborn decrease with age (Price, 2008), and become increasingly modest at the ages where genetic endowments are more clearly expressed (Breinholt and Conley 2019). Third, recent evidence exploiting within-family differences in polygenic scores for educational attainment suggests that parents compensate for, rather than reinforce, genetic differences in education between siblings. More specifically, Fletcher et al. (2020) show that the association between the education polygenic score and educational attainment is stronger for siblings with the lower polygenic score. This is consistent with parental preferences for equality among siblings, and compensating parental investments, and does not support the alternative interpretation that parents invest more in the firstborn only when the firstborn has a higher polygenic score. Our findings therefore suggest that the additional investments associated with being firstborn are driven by less restrictive time and budget constraints and are independent of the child's genetic endowment. The appearance of a positive interaction between endowments and being firstborn is therefore consistent with complementarity in human capital production.
Another potential explanation for our findings is that other mechanisms through which birth order effects arise (e.g., parental age at birth, interactions with younger siblings) could interact with genetic endowments. It is reassuring that additionally accounting for parental age at birth and other potential confounders does not affect our results (see Appendix E). Unfortunately, we cannot test alternative explanations regarding sibling interactions directly because the UK Biobank is very limited in measures of parental and sibling interactions. Still, Table 7 suggests that our findings are mainly driven by a distinction between first-and all laterborns, with significant differences appearing prominently between the first and secondborns. If interactions with younger siblings would be driving our results, one would expect a more gradual decrease in the magnitude of the interaction terms with rising birth rank, as some second-and thirdborns also benefit from interactions with younger siblings. Moreover, even if birth order only partially captures parental investment -a premise that should not be controversial given the overwhelming evidence in the literature -then unless these other channels exhibit completely opposite interaction effects, a necessary (but not sufficient) condition for complementarity would be a positive interaction between birth order and genetic endowments. This is indeed what we find.
A number of limitations should be acknowledged. First, our specification may not be the perfect empirical translation of the human capital production function. In particular, we do not measure skills or human capital directly (see e.g., Araujo et al. 2016 for a study that measures skills directly). Instead, we follow Cunha & Heckman (2008) and  who specify adult human capital as a combination of skills accumulated by the end of childhood, and employ a commonly used and convenient proxy: years of education.
Moreover, we do not measure parental investments directly, and use an environmental variable closely related to parental investments: birth order. The upside of using birth order rather than a direct measure of parental investments is that birth order is randomly assigned within families, whereas parental investments are known to be endogenous to offspring endowments. Moreover, whereas birth order cannot distinguish between earlylife and later-life investments, it does capture a persistent difference across siblings rather than a one-time shock in investments that many other papers rely on (see Almond et al., 2018, and Appendix B).
A second limitation is that our measure of genetic endowments is imperfect. In particular, a polygenic score captures only common genetic variations in the human genome, and even within the realm of common variations the measure is subject to measurement error.
While the use of ORIV reduces concerns about classical measurement error, our family fixed effects estimates of the polygenic score are still subject to attenuation bias due to genetic nurture. Since the sign of the bias arising from genetic nurture is known to be negative, our effect size represents in fact a conservative estimate.
The polygenic score should also not be interpreted narrowly as a measure of immutable biological endowments. While within-family analyses allow us to interpret the effect of the polygenic score as a causal effect of genetic variation, it is well-established that the environment may mediate this effect (e.g., Breinholt & Conley, 2019;Houmark, Ronda, & Rosholm, 2020), including the family environment (Fletcher et al., 2020). Hence, a polygenic score measures education-enhancing endowments, and will reflect how on average in the discovery sample environments (including parental investments) respond to differences in genetic endowments. Importantly though, since the measure is fixed at conception and orthogonal to birth order in the within-family analysis, the measure does not reflect parental investments of the child's own parents, nor does it reflect parental genetic (nurture) effects. As a result, the inclusion of environmental responses to genetic variation into the construction of the polygenic score is not a source of concern for our identification strategy, but does imply that our finding of complementarity between genetic endowments and birth order is not an immutable property of the production function for human capital and may be specific to the context studied.
A related limitation regards the external validity of the empirical findings. As mentioned in the data section, there is sample selection into the UK Biobank, with a bias towards healthier and higher-educated individuals (Fry et al. 2017). On top of this, we focus on European-ancestry individuals and the coincidental sampling of siblings even though these were not specifically targeted, further reducing the representativeness of the sample.
Finally, we construct our polygenic score on basis of a tailor-made GWAS, again on basis of the same UK Biobank excluding the siblings and their relatives. While the latter choice helps to maintain the same environments across discovery and prediction sample, it may further increase the likelihood that our results are specific to the UK Biobank. Future research should replicate our findings, but in light of our evidence in support of economic theories of human capital production undergirding our results, we have good reasons to be positive about their replicability in comparable contexts.
For Online Publication Appendix to: "Complementarities in human capital production: Evidence from genetic endowments and birth order" Two unrelated human beings share approximately 99.6% of their DNA, and most genetic differences across humans can be attributed to single nucleotide polymorphisms (SNP) (Auton et al. 2015). A SNP is a locus in the DNA at which two different nucleotides can be observed in the population. Each of the two possible nucleotides is called an allele for that SNP. An individual's genotype is coded as 0, 1, or 2, depending on the number of "effect" alleles present. In the human genome, there are at least 85 million SNPs with a "minor" allele prevalence of at least 1% (Auton et al. 2015).
Genome-Wide Association Studies (GWASs) aim to identify genetic variants that are associated with a particular trait of interest by relating each variant to the trait in a hypothesis-free approach.
Stringent significance thresholds are used to identify variants that are robustly associated with the trait, with other independent samples used for replication. Using the GWAS approach, thousands of genetic discoveries have been made .
Individual SNPs typically explain less than 0.02% of the variance in a behavioral outcome (Chabris et al. 2015;Visscher et al. 2017). It is therefore common to combine multiple SNPs into a polygenic score , constructed as a weighted sum of SNPs. Through increases in GWAS sample sizes, the predictive power of the polygenic score for education has increased from 2-3% , to 6-8% , to 11-13% . In terms of biological pathways, there is evidence that many of the identified genes associate with health, cognitive, and central nervous system traits . Likewise, the majority of the significant SNPs in  and  relate to genomic regions responsible for gene expression in a child's brain during the prenatal period.
Methods. Relatedness. As a first step, we identify siblings and their relatives using the kinship matrix provided by the UK Biobank. The kinship matrix is based on genetically identified relatedness and contains relatives of third degree and closer identified using the KING software (Manichaikul et al. 2010). The UK Biobank does not have information about self-reported relatedness (Bycroft et al. 2018). The degree of relatedness between the pairs of individuals is based on the combination of the kinship coefficient and genetic similarity in terms of the identity by state (IBS0) coefficient. IBS0 measures the fraction of markers for which the related individuals do not share alleles. We follow the KING manual regarding the thresholds for how to determine family relationship (see Table A.1). The identified number of pairs per relationship type differs slightly from that of Bycroft et al. (2017), because some UK Biobank participants withdrew their consent to analyse their data since then. For our analyses, we go one step further by separating those who are related to the siblings up to the 3 rd degree (kinship coefficient ≥ 0.025), i.e., siblings, parents of siblings, cousins of siblings (See Table A.2). In this way, our holdout sample for polygenic score construction and prediction (i.e., the sibling subsample) is unrelated to the GWAS discovery sample which is used to calibrate the SNP weights that are used to construct the polygenic score.  We further quality control the resulting GWAS summary statistics using EasyQC tool (Winkler et al. 2014) and meta-analyse our tailor-made GWAS weights with the summary statistics from . We use these for constructing an alternative polygenic score that is used in the robustness analysis (see footnote Error! Bookmark not defined.). Meta-analysis is conducted using the software package METAL (Willer, Li, and Abecasis 2010).
Polygenic scores. The polygenic scores are constructed while accounting for linkage disequilibrium between SNPs using LDpred , version 1.06, and Python, version 3.6.6.
Linkage disequilibrium pertains to the non-random correlations between SNPs at various loci of a single chromosome. LDpred is a software package based on Python that adjusts the GWAS weights for LD using a Bayesian approach. We follow the steps as outlined in Mills, Barban, & Tropf (2020), including the coordination of the base and target files, computing the LD adjusted weights, and then applying them for polygenic score construction using PLINK (Purcell et al. 2007). We reweight the SNP effects on the basis of LD and the supposed fraction of causal SNPs, which we set to 1, as is standard practice for behavioral traits (Cesarini and Visscher 2017).

B. Empirical evidence on complementarity in human capital production
Testing complementarity between children's endowments and parental investments is challenging, since it requires independent variation in initial endowments and later-life investments .  and  adopt a structural approach, modelling both skills as well as parental investments as low-dimensional latent variables, and find evidence consistent with (dynamic) complementarity. A number of studies have examined whether the effect of specific interventions or investments varies by initial skills.  correct early life health measures for certain prenatal investments, and find that pre-school enrolment is more productive for children with higher levels of this residualized measure of endowments.  use entrance-age in kindergarten as plausibly exogenous variation in initial cognitive skills, and find some evidence for complementarity, although the effect dies out after the first grade.
A recent set of papers have examined rare cases where there exists exogenous variation in both initial endowments as well as later-life investments. For example,  study the interaction between exogenous variation in access to better schools and variation in family backgrounds induced by access to abortion in Romania. Their findings do not suggest a meaningful interaction between initial endowments and later-life investments. Rossin-Slater & Wüst (2020) exploit a nurse home visiting program as an exogenous shock to endowments, and staggered access to high quality preschool childcare in Denmark as an exogenous shock to investment, and find that these interventions are substitutes rather than complements.  exploit a unique combination where a tornado struck an area of Bangladesh that was coincidentally involved in a randomized experiment on vitamin A supplementation. Their findings are consistent with complementarity since children treated with Vitamin A supplements were better protected from the consequences of the earthquake.  exploit local rainfall in the year of birth as exogenous variation in endowments, and randomized cash incentives from Progresa as an exogenous shock to investment. Their main finding is that children from families who received cash transfers were protected better against adverse endowments, consistent with complementarity. Similarly,  also use a combination of adverse weather shocks and conditional cash transfers in Colombia to show that children born under normal weather conditions benefit more from the cash transfers. Finally, Johnson & Jackson (2019) exploit the rollout of Head Start and the implementation of court-ordered school finance reforms (SFRs) that increased spending at public K-12 schools as two exogenous shocks to human capital investment, again finding evidence in favour of complementarity.

C. Obviously-Related Instrumental Variable (ORIV) regression
In this section, we describe Obviously-Related Instrumental Variable (ORIV;  regression. Suppose we would like to predict an outcome variable of interest, Y, using a polygenic score, i.e., estimate the following model: where α is a constant, is the effect of a true polygenic score * and is the error term. We have two estimates of the true polygenic score: 1 = * + 1 and 2 = * + 2 . The covariance between the two measurement error terms 1 , 2 is zero, ( 1 , 2 ) = 0, and they have the same relative variance of the measurement errors 1 , 2 . That is: where one instruments the stack of estimated polygenic scores � 1+ where N is the sample size and 0 is a Nx1 vector with zero's. We include a family-stack fixed effect to conduct the within-family comparisons within a stack of the data. Standard errors are clustered at both the family and individual level following Correia (2017Correia ( , 2019.

D. Early-life parental investments
The only early-life parental investments observed in the UK Biobank are whether the child was breastfed and whether the mother smoked around birth. We also observe the age gap between subsequent siblings. interaction. This shows whether mothers change their behavior depending on whether the child is first-or laterborn and the polygenic score of their children. The results show that the probability of being breastfed (Column 1) and the likelihood of maternal smoking around pregnancy (Column 2) are similar between first-and laterborns. Furthermore, the coefficient of the polygenic score for (the child's) education is not significantly different from zero, and we find no evidence of any differences in maternal investments around pregnancy by the firstborn's polygenic score.
Next, we explore the relationship between the polygenic score and the age gap between siblings.
Column 3 presents the estimates from a regression of the age gap in months between every two consecutive siblings on the polygenic score of the older sibling in the pair. These between-family estimates suggest that birth spacing is one month shorter for every one standard deviation increase in the polygenic score of the older sibling. However, note that these results derive from a between-family comparison, and therefore should not be interpreted causally. For families for whom we observe three consecutive siblings or more (N = 2,542), we can check this result in a within-family specification. The within-family estimates (Column 4) are very noisy because of the small sample size, and the wide confidence intervals do not allow drawing any firm conclusions here. Robust standard errors in parentheses, clustered by family in the within-family analysis of columns 1-2 and 4; Coefficients for the control variables (year and month of birth, gender, family size, and the first 40 principal components of the genetic relatedness matrix) are not displayed, but available upon request from the authors. Sample sizes vary depending on the availability of early life parental investments and the number of siblings included in the analyses.

E. Additional confounders
In Table E.1. we present further analyses to investigate robustness of our results against the inclusion of additional confounders. Column 1 reproduces our main results (Table 5, Column 5). In Column 2 and 3, we employ the correction for missing confounders in gene-environment interaction analyses suggested by Keller (2014) without (column 2) and with (column 3) corrections for measurement error. Specifically, we interact both the dummy for being firstborn and the polygenic score for education with year of birth, month of birth, gender, and first 40 principal components of the genetic relatedness matrix, and include all these interactions as additional control variables in the analysis. The direct effect sizes of being firstborn and the polygenic score for years of education are now relative to the reference categories of the control variables (born in 1937, born in January, being female). The standard error of the interaction term increases due to the much larger number of regressors in both columns. Whereas the magnitude of the interaction term drops somewhat in column 2, the point estimates in both column 2 and 3 do not significantly differ from our baseline specification in column (1), and show a robust interaction effect, at least after correction for measurement error.
We also explore whether our results are robust against the inclusion of parental age as an additional control, as this may be one mechanism through which birth order effects arise . In our sample, we have only 5,601 individuals with information about the age of the mother and 3,085 individuals with information about the age of the father (both measured at time of birth of the child). The inclusion of these variables as control variables in Column 4 and Column 5 respectively of Table E.1. and the subsequent drop in the sample size decreases the statistical power of our tests. Indeed, the interaction term loses statistical significance when controlling for father's age at birth. However, the effect sizes of the interaction term are similar to our main results, and if anything, are larger. 3,085 Notes: Robust standard errors in parentheses, clustered by family; Coefficients for the control variables (year and month of birth, gender and the first 40 principal components of the genetic relatedness matrix) are not displayed, but available upon request from the authors. Column 1 replicates our main results for comparison. Column 2 includes as additional control variables the interactions between firstborn and the polygenic score with year of birth, month of birth, gender and the first 40 principal components. Column (3) provides the ORIV estimation of the Column (2). Column (4) and (5) include controls for maternal age at birth and paternal age at birth, respectively. F. Replication of main results with a different polygenic score for educational attainment Columns 1-3 of Table F.1 replicate the main analyses using the polygenic score based on the metaanalysis of the GWAS summary statistics of our own UK Biobank GWAS described in Section 3 and the GWAS summary statistics of 23andMe . The procedure for constructing the polygenic score is identical to the one used in the main analysis. From Columns 1-2, we can see that the incremental R 2 of the polygenic scores is 1.1% in the within-family specification. The GxE interaction term (Column 3) is slightly smaller than in the main analysis, but not significantly different from our main result. Notes: Robust standard errors in parentheses, clustered by family; Coefficients for the control variables (year and month of birth, gender and the first 40 principal components) are not displayed, but available upon request from the authors; The PGS for years of education in Columns 1-3 is based on a metaanalysis of summary statistics from our own UK Biobank GWAS and 23andMe.
G. Systematic differences in polygenic scores for other traits by birth order.
In this section, we test if there are any systematic differences in polygenic scores for anthropometric, health, and personality traits by birth order. Polygenic scores for these analyses are obtained from the Polygenic Index Repository. These polygenic scores are constructed using PLINK 2 (Chang et al. 2015), where SNPs are corrected for linkage disequilibrium using LDpred  for further technical details, see . Table G.1 reports the results of regressions of the selected single-trait polygenic scores on a dummy for being firstborn and control variables for year and month of birth, gender, the first 40 principal components of the genetic relatedness matrix, and family fixed effects. All associations are very small in magnitude, and only 2 out of 28 individually reach statistical significance at the 5% level.
After a Bonferroni correction for multiple testing (i.e., with a Bonferroni-corrected p-value significance threshold of 0.0018 (=0.05/28)), being firstborn is not significantly associated with any of the selected polygenic scores.