Estimating Intergenerational and Assortative

Processes in Extended Family Data∗

M. Dolores Colladoa Ignacio Ortuno-Ortınb Jan Stuhlerb

August 2019

Abstract

We propose a new approach to quantify intergenerational processes that exploits different degrees

of kinship within the same generation. This “horizontal” approach has several advantages: it can

be applied within the same data source and time period, scales well in administrative sources, is

informative about assortative processes, and yields many more kinship moments than a vertical

approach. This allows us to fit a detailed model that accounts for the transmission of observable and

latent advantages via intergenerational, sibling and assortative processes. Using Swedish registry

and Spanish census data, we find strong persistence in the latent determinants of socioeconomic

status, and a striking degree of assortative mating – to rationalize our kinship data, spouses must

be far more similar in latent than in observable advantages. A standard genetic model cannot fit

the data. Instead, we fit an extended model that allows both for genetic and non-genetic latent

mechanisms. Genes explain about seven percent of the variation in educational attainment, and

assortative mating occurs primarily in non-genetic factors.

∗Previous drafts of this paper circulated with the title “Kinship Correlations and Intergenerational Mobility”. We aregrateful for helpful comments and suggestions from Adrian Adermon, Anders Bjorklund, Lorenzo Cappellari, GregoryClark, Alexia Furnkranz, Hans-Martin von Gaudecker, Marc Goni, Ines Helm, Paul Hufe, Markus Jantti, Mikael Lindahl,Matilde Machado, Martin Nybom, Andreas Peichl, Simon Rabate, David Stromberg and seminar participants at theUniversite du Quebec a Montreal, Nuffield College at Oxford University, the Joint Research Center of the EuropeanUnion (Ispra), Rotterdam University, Stockholm University, the Selten Institute (Cologne), Cemfi (Madrid), Universityof Mannheim, CPB Netherlands, the “Inequality, Social Origins and Intergenerational Mobility” workshop at UniversitaCattolica in Milan, and the “Opportunities, Mobility and Well-being” workshop in Warsaw. We thank the SwedishInstitute for Social Research and the Instituto Cantabro de Estadıstica for data access.(a) Universidad de Alicante; (b) Universidad Carlos III de Madrid

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1 Introduction

Research on intergenerational mobility in socioeconomic status has gained renewed interest in recent

years. In addition to novel evidence on geographic differences in parent-child mobility (e.g. Chetty

et al. 2014) and its relation with income inequality (Corak 2013), researchers have began to provide

evidence on multigenerational mobility across several generations. This evidence suggests that mobility

is perhaps much lower than what most economists used to think (Clark 2014, Lindahl et al. 2015, Barone

and Mocetti 2016, Adermon et al., 2019). In particular, it contradicts a common interpretation of the

available parent-child evidence – that the correlation between individuals in one generation and their

ancestors decreases geometrically as we go back in time, so that after, say, three or four generations

the link is already very weak.1

Instead, recent empirical studies suggest a much higher persistence of socioeconomic status. Using

historical data from a series of countries and periods, Clark and co-authors (e.g., Clark 2014, Clark et al.

2015) show that the average socioeconomic status of surnames regresses at a rather slow rate. Similarly,

in historical data from Florence, the average status of surnames still correlates across generations that

are six centuries apart (Barone and Mocetti, 2016). Other studies link individuals across multiple

generations to directly estimate multigenerational persistence (Lindahl et al. 2015, Braun and Stuhler

2018, Colagrossi et al. 2019), or to study the role of grandparents in the transmission process (Anderson

et al. 2018). These individual-level studies likewise find high multigenerational persistence, if not as

high as studies on the surname level.

One problem with this “vertical” approach are the data requirements. Comparable socioeconomic

information for more than two generations is difficult to obtain. In many countries, there is very little

variation in formal education among older generations, in which a majority had only basic schooling.

Occupational classifications are useful, but going sufficiently back in time we invariably end up with

samples that consist mostly of farmers. And because family links are rarely observed, most studies

use surnames as an imperfect proxy for actual ancestor relations – triggering a lively debate on how

informative surname-level evidence can be about individual-level mobility. Studies based on direct

links avoid this problem, but only track the more immediate ancestors (i.e. grandparents or great-

grandparents). As surname-level studies, they yield only a small set of vertical moments, limiting our

ability to distinguish between competing models of intergenerational transmission (Cavalli-Sforza and

Feldman, 1981).

1Such interpretations are based on the iteration of parent-child regressions, i.e. the assumption that the correlationbetween grandparents and grandchildren outcomes is basically the square of the parent-offspring correlation (e.g., Stuhler,2012). Since parent-offspring correlations in income, education or other socioeconomic outcomes are always moderate,ancestor correlations would decrease fast as we go back in time. As put by Becker and Tomes (1986), “Almost all earningsadvantages and disadvantages of ancestors are wiped out in three generations. Poverty would not [...] persist for severalgenerations.”

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We therefore propose a new approach to quantify intergenerational and assortative processes that

does not require information on distant ancestors. Instead, we use ”horizontal” information, that is,

information about individuals of the same generation, or very close generations, who are relatives of a

certain degree, for example, siblings, cousins, uncle-nephew, and so on. The underlying idea is simple.

Say that we would like to asses the link between grandparents and grandsons, but we do not have data

on the socioeconomic status of grandparents. If instead we have good data for cousins we can infer the

strengths of grandparents-grandsons links from the cousins’ links. Thus, horizontal information can

overcome the lack of vertical information, and be used to study long-run intergenerational mobility in

countries in which there exists no comparable data across multiple generations. Another advantage is

that socio-economic outcomes can be measured within the same data source and at approximately the

same age and time.2

Moreover, horizontal moments are not just a convenient and better-measured substitute for vertical

moments. By tracking affine kinships, such as siblings-in-law, we can identify very distant relatives,

using modern administrative instead of historical data sources. As consanguine (“blood”) kinships,

affine relations are a function of – and therefore informative about – intergenerational and assortative

processes. But the identification of distant consanguine kins requires the identification of distant

ancestors. For example, cousins are defined via their shared grandparents, while the identification of

second-degree cousins requires observation of their great-grandparents, and so on. In contrast, affine

kins can be identified from spousal and parental links irrespective of their degree of separation. In a

first step, we identify a person’s sibling (via their shared parent). In a second step, we identify the

sibling’s spouse (e.g., via their mutual child). Implemented once, these steps identify a sibling-in-law.

Implemented twice, we identify the sibling-in-law of the sibling-in-law, and so on.

The horizontal approach scales particularly well in administrative data sources that cover a large fraction

of a population, such as registry or census data. We consider two such sources, from Sweden and Spain.

The Swedish registers are extensive, include family links over multiple generations, as well as multiple

socioeconomic and other outcomes. Because our sample covers more than one-third of the population,

we can identify very distant in-laws up to five degrees of separations (i.e. individuals separated by

five sibling and five spousal links). In contrast, the Spanish data are limited to a single cross-section

from the region of Cantabria, contain only educational outcomes, and names instead of direct family

links. We exploit Spanish naming conventions to recover parent-child links (and therefore siblings,

first-cousins, uncles-nephews and siblings-in-law) for a large share of our sample. We can identify 141

distinct kinship moments in the Swedish registers, and 65 moments in the Spanish sources. To our

knowledge, these are the most extensive sets of kinship moments that have been compiled so far.3

2An important influence for our approach is the literature on sibling correlations, on which we comment below. Inrelated work, Hallsten (2014) considers the correlation between cousins and second cousins, and discusses the advantageof observing socioeconomic outcomes within the same generation.

3Some extended twin-family studies consider up to 80 different types of relatives. However, most of those kinshipcorrelations involve some type of twin, have very low sample sizes, and capture similarity in behaviors (such as smoking)

3

The horizontal perspective yields therefore many more, more distant, more comparable, and better-

measured kinship moments than the vertical approach. That opens the door for the estimation of more

detailed intergenerational models, and a deeper understanding of intergenerational and assortative

processes. Our model builds on the key implication from the recent multigenerational literature: latent

advantages must play an important role in the transmission process because parent-child correlations

in observables are too low to rationalize the persistence of inequalities across multiple generations.4

However, it deviates in three aspects from this prior literature. First, we allow for both direct and

latent transmission mechanisms instead of considering only observable or latent factors. Second, we

allow for assortative mating in both the observable and the latent dimension. Third, we allow for each

of those mechanisms to vary with the gender of the child and parent.

Intergenerational, Sibling and Assortative Processes

In the first part of the paper, we study the extent to which socioeconomic advantages are being trans-

mitted. Our objective here is to quantify how intergenerational, sibling, and assortative processes

contribute to the overall persistence of inequality across generations, without trying to isolate any par-

ticular causal pathway. However, to the extent that they have specific implications about the pattern

of inequality across kins, our approach can be informative also about causal mechanisms (we return to

this question below).

We calibrate our model using the kinship moments from Sweden and Spain. We use educational

attainment as our baseline outcome, but also consider income and other outcomes. We find strong

persistence in the intergenerational, sibling, and assortative processes. The parent-child correlation of

the latent factor is around 0.6 in our baseline for Sweden, and about 0.8 in the Spanish sources. This

strong transmission of latent advantages suggests that the distribution of educational advantages in

a given generation correlate with the socioeconomic status of their ancestors as much as four or five

generations back in time. In contrast, the observed educational attainment of the parent has a positive,

but only limited, independent association with the educational attainment of their child (consistent

with evidence on the causal effect of parents’ schooling, see e.g. Holmlund et al., 2011).

Striking is the high rate of assortative matching that our data imply. The kinship correlations in

education decay slowly across siblings-in-law, falling by only about 25% with each degree of separation

(e.g., comparing siblings-in-law vs. the cosiblings-in-law’s spouses). To rationalize this slow decay,

spouses must be far more similar to each other in latent factors that determine economic prospects than

they are in observable characteristics. The implied spousal correlation in the latent factor is around

instead of socioeconomic outcomes (see Truett et al. 1994, Eaves et al. 1999, Maes et al. 2018).4That part of the transmission process may be inherently unobservable has long been recognized in the theoretical

literature (e.g., Duncan 1969, Goldberger 1972). The latent variable in our model is close in spirit to Becker and Tomes(1979), who assume that a person’s “endowment” represent a great variety of cultural and genetic attributes.

4

0.75 in the Swedish and about 0.9 in the Spanish sources – far higher than the spousal correlation in

educational attainment or other common measures of socioeconomic status.

We further estimate that siblings share important influences that are not reflected in their educational

attainment. Similar as spouses, siblings must be far more similar to each other in what determines the

socioeconomic success of their descendants than what is reflected in their observable characteristics. The

implied sibling correlations in the latent factor are about 0.7 in both Spanish and Swedish sources, while

the sibling correlation in observables such as years of schooling is below 0.5. Our findings generalize to

other outcomes, such as income data obtained from the Swedish registers. The latent advantages are

more strongly transmitted than income itself, in each of the intergenerational, sibling, and assortative

dimension of our model. However, we also find that those latent factors that influence income are not

as strongly transmitted than those that determine educational attainment.

Because we observe a greater set of empirical moments than previous studies, we can study the robust-

ness of these findings in more detail. We test the out-of-sample performance of the fitted model, and

study if our results remain robust to calibration from different subset of our moments. Our baseline

results remain similar when dropping two thirds of the empirical moments from the Swedish registers.

Moreover, the model calibrated from this restricted set of moments successfully predicts a diverse set

of kinship correlations not included in the calibration, including vertical, horizontal and distant mo-

ments. Our model appears therefore identifiable from information that is far more limited than what

we observe in the Swedish or even Spanish sources.

These results contribute to different strands of the literature on inequality and intergenerational mobil-

ity. First, they relate to those strands that quantify the importance of family background by comparing

close relatives. Bjorklund and Jantti (2019) review the evidence on intergenerational correlations, sib-

ling correlations and equality of opportunity, and note that these approaches point to different rates of

persistence. Sibling correlations account for a higher share of total inequality than parent-child cor-

relations, suggesting that the role of family background is not fully captured by the latter. Similarly,

the equality-of-opportunity approach misses important circumstances that cannot be directly observed

(Niehues and Peichl, 2014), and only a fraction of either measure can be linked to specific causal mech-

anisms (such as the effect of parent on child education). Consistent with these arguments, we propose

a more comprehensive way to account for the importance of family background, by considering many

different kinship types, including distant kinships.

Second, we corroborate recent findings based on multigenerational correlations, which suggest that

inequality is more persistent than previously thought (e.g., Clark 2014, Lindahl et al. 2015, Barone and

Mocetti 2016). Importantly, our approach is very different from prior studies. It is based on direct

family links instead of surnames, and does not require socioeconomic information on distant ancestors.

Our results therefore suggest that high multigenerational persistence is not just a statistical artifact from

the use of name-based grouping estimators, whose validity have been contested (see Chetty et al. 2014,

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Torche and Corvalan 2015, Adermon et al. 2019, Vosters and Nybom 2017, Guell et al. 2018, Solon 2018,

Clark 2018 and Santavirta and Stuhler 2019). Instead, high multigenerational persistence is consistent

with the strong similarity between horizontal kins that we document here. However, our estimates

imply less extreme persistence than name-based studies, and are more in line with multigenerational

evidence from direct family links (Lindahl et al. 2015, Braun and Stuhler 2018, Adermon et al. 2018,

Neidhofer and Stockhausen 2019, Colagrossi et al. 2019).

Compared to the prior literature, our model is more general, and characterizes the multigenerational

process more thoroughly. This also allows us to rationalize sibling, intergenerational and multigen-

erational correlations within a single framework, and to study their relation. Because they account

for unobserved factors that are orthogonal to the observed status of parents, sibling correlations are

a more comprehensive measure of family background than parent-child correlations (see Solon 1999,

Levine and Mazumder 2007, and Jantti and Jenkins, 2014). However, they measure the similarity

of siblings in observables, which may not reflect how similar siblings are in unobservable advantages.

By considering more distant relatives, we can account for such latent advantages, and disentangle the

“non-transferrable” part of family background that is only shared by siblings from the “transferrable”

part that is also partially shared by more distant relatives. We find that siblings share latent advantages

to a far greater extent than they share observable advantages in education or income. Moreover, it

is those latent advantages that primarily determine the prospects of future generations. Our results

therefore suggest that sibling correlations still understate the importance of family background.

Third, our study provides a novel perspective on the role of assortative mating in the intergenerational

process. Because spousal correlations in education and other socioeconomic outcomes are fairly high,

they are an important determinant of income mobility (Ermisch et al., 2006) and inequality (Fernandez

and Rogerson, 2001). For instance, increased educational sorting might have contributed to the rise in

income inequality that many developed countries have experienced in recent decades (Greenwood et al.

2014, Eika et al. 2014). Our results, however, suggest that sorting in observable characteristics such as

education greatly understates the sorting in latent advantages. To rationalize the similarity of kins in

the extended family, spouses must be far more similar to each other than they are in observables. The

implied degree of sorting in latent advantages is striking, with spousal correlations around 0.75 in the

Swedish and 0.9 in the Spanish sources. These results also suggest that shifts in educational sorting

may have little effect on intergenerational mobility, unless they reflect similar shifts in sorting on latent

advantages.

Genetic and Socio-cultural Factors

Our model is general enough to nest specific models, such as the standard genetic model from behavioral

genetics. This allows us to quantify how genetic and non-genetic transmission mechanisms contribute

to the intergenerational and assortative processes in the second part of our paper.

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Many papers estimate the relative importance of nature and nurture by comparing status correlations

for different type of siblings, like monozygotic and dizygotic twins, siblings, half siblings or adoptees

(see Sacerdote 2011 for a literature review).5 In contrast, we do not make use of twins or adoptees,

and instead consider the correlations for many different, including distant kins. More importantly,

our baseline model decomposes the family background into transferrable and non-transferrable rather

than into genetic and environmental components. The obvious disadvantage is that we remain largely

agnostic about what mechanisms the latent factor and the pathways of our model represent. The

principal advantage of our approach, however, is that it provides a more comprehensive account of

intergenerational and assortative processes. Our latent factor is a more comprehensive object than the

“genotype” considered in behavioral genetics, as it also captures non-genetic advantages that matter

for the socioeconomic success of the next generation. That is, by avoiding an apriori stand on the

causal channels via which transmission occurs, we can capture those channels more completely.6

One may however ask if genes are an important component of the advantages encapsulated in the latent

factor of our model. We therefore estimate the relative importance of genetic and non-genetic factors.

We first test if the standard model in quantitative genetics, as nested by our baseline model, can fit the

wide set of kinship correlations observed in the Swedish registers. We show that it performs fairly well

when considering body height, which is known to be strongly influenced by genes (Yang et al., 2015).

In contrast, the standard genetic model cannot fit the kinship correlations in educational attainment

(even the in-sample fit is extremely bad). Our findings are therefore inconsistent with a purely genetic

interpretation. The observation of a wide range of kinship moments is critical for this conclusion. We

show that the standard genetic model can fit a small number of close kins; its inadequacy becomes

apparent only when challenged to fit a more extensive set of kinships.

The observation that a purely genetic model cannot fit the data does however not preclude the view that

genes are an important determinant of socioeconomic status. To quantify their role more thoroughly,

we decompose the latent factor of our model into a genetic and a non-genetic (i.e. socio-cultural)

component. Specifically, we extend our baseline model to a general assortative mating model with two

latent factors, assuming that the genetic factor follows the standard model of genetic inheritance with

assortative mating as used in quantitative genetics (Crow and Felsenstein, 1968). Estimation of this

two-factor model is more challenging than the estimation of our baseline model, but remains feasible

5The standard approach in this literature decomposes the total variance of the output of interest into three additiveterms, representing genetic factors, environmental factors shared by the siblings, and a factor that is idiosyncratic tothe individuals. Among these papers, the most related to our work are Behrman and Taubman (1989) and Bjorklundet al. (2005). Both papers make use of correlations across several sibling types and find the values of the parameters thatbest fit the empirical correlations, in a similar way as we do here. Behrman and Taubman focus on years of schoolingand assume that family environment and genes are uncorrelated. Bjorklund et al. (2005) focus on earnings and considerdifferent possible models, allowing for the possibility that environment and genes are correlated. See Goldberger (1979)for a critique of the approach, and Bingley et al. (2019) for a variant that relies on less restrictive assumptions.

6We also do not have to deal with the complicated problem of the relationship between genes and environment. Bydefinition, the non-transferable component captures all the effects that siblings share and that are not correlated with thetransferable components.

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in the large set of kinship moments from the Swedish registers.

We find that genes (i) explain about 7% in the variation in years of schooling in Sweden, and (ii) are

transmitted to a similar degree from parents to children as the non-genetic latent factor. The former

result is consistent with recent evidence from molecular genetic data, and polygenic scores from genome-

wide association studies (GWAS, e.g. Okbay and et al. 2016, Lee and et al. 2018, Papageorge and Thom

2018).7 The latter result suggests that genetic and non-genetic factors are difficult to distinguish based

on ancestor-child correlations alone.8 We can successfully distinguish them here because they play a

very different role in the horizontal dimension – while spouses are very similar to each other in the

non-genetic latent factor, the correlation in their genotype appears negligible. Remarkably, this finding

is again consistent with recent, direct evidence on spousal correlations in polygenic scores related to

educational achievement (Yengo et al. 2018, Domingue et al. 2014).

Our findings match therefore recent evidence from behavioral genetics, even though our approach is very

different. Moreover, our research design captures not only genetic but also non-genetic mechanisms. It

therefore integrates and contributes to different strands of literature from the natural and social sciences.

Within the latter, our findings provide a rich characterization of intergenerational, multigenerational,

sibling and assortative processes, and how they relate to each other. They corroborate and extend recent

evidence based on multigenerational correlations, but also suggest that they are not very informative

about the underlying transmission mechanisms. Instead, the key element to distinguish genetic and

non-genetic mechanisms turns out to be the observation of distant affine relatives.

We believe that the horizontal approach has enormous potential, in particular when applied in population-

wide register data. However, it is subject to two significant limitations. A fundamental problem for

the estimation of any distributional model is that the sample moments may vary over time, contrary

to the assumption that those moments are in a steady-state equilibrium (Atkinson and Jenkins 1984,

Nybom and Stuhler 2019). Because horizontal kins can be observed at approximately the same time,

our approach is arguably less sensitive to this issue than a multigenerational approach based on distant

ancestors. Still, it could affect our results. We address those concerns by studying the variability of

each kinship correlation over our analysis period, and by not relying on moments that appear less

stable. Another limitation is that we do not model the fertility process. Again this is standard in the

literature, but it might be a concern here because the existence of a distant sibling-in-law depends on

the existence of a sibling – skewing the sample towards larger families with lower socioeconomic status.

To address this issue, we consider only siblings-in-law up to five degrees of separation, among which this

socioeconomic bias is still modest. Future work could improve on our approach by explicitly modeling

selective fertility and off-steady-state dynamics.

7Our estimate of a ”heritability” of only 7%, as well as the values provided by GWAS, are significantly smaller thanthe values obtained in most twin-studies (Branigan et al., 2013). Section 6.2 provides a discussion of this discrepancy.

8This difficulty has been recognized for a long time in the literature on population genetics. For example, Cavalli-Sforzaand Feldman (1981) noticed that ”any variable (other than genotype) that is vertically transmitted and not measured willvery frequently be indistinguishable from genotype” (pp. 287-288).

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The paper proceeds as follows. Section 2 sets out the basic model and develops our method. Section

3 describes the data. Section 4 presents our baseline empirical findings and robustness checks for

Sweden, while Section 5 presents our results for Spain. Section 6 sets out the standard genetic and a

generalized two-factor model, and quantifies genetic and non-genetic transmission mechanisms. Section

7 concludes. We include some additional information about the models and additional results in the

Appendixes.

2 Theory

Our baseline model deviates in three important aspects from the prior literature. First, we allow for

direct (observable) and latent (unobservable) transmission mechanisms. Second, we allow for assortative

mating along two distinct dimensions, and account for both parents explicitly. Third, we consider how

the strength of the transmission mechanisms vary with the gender of the child and the parent. In Section

6.2 we extend this model further to isolate genetic from latent non-genetic transmission channels.

2.1 General Model

Suppose that y is a socioeconomic outcome of interest in our economy, such as income or education.

We henceforth identify y with years of schooling, the baseline outcome in our empirical exercise. All

theoretical implications remain valid when studying other outcomes. We want to study the link of such

variable y between individuals and their ancestors.

Specifically, assume that the outcome y for an individual from generation t is given by

ykt = βkykt−1 + zkt + xkt + ukt (1)

where the superscript k stands for male (k = m) and female (k = f). The first component ykt−1 is the

weighted average socioeconomic status of parents,

ykt−1 = αkyymt−1 + (1− αky)y

ft−1,

where αky ∈ [0, 1]. The parameters βk and αky capture therefore the direct transmission of parental on

child outcomes. As the importance of this channel may vary with the gender of the child (βf 6= βm)

and the gender of the parent (αfy 6= αmy ), we effectively allow it to be distinct for each of the four

parent-child gender combinations.

The latent factor zkt captures the importance of unobservable determinants of child outcomes that are

passed from parents to children (see Section 2.2). As the observable determinant, it depends on the

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weighted average latent status of the parents zkt−1,

zkt = γkzkt−1 + ekt + vkt , (2)

where

zkt−1 = αkzzmt−1 + (1− αkz)z

ft−1 (3)

and αkz ∈ [0, 1]. The parameters γk and αkz capture the strength of indirect transmission channels, i.e.

factors that impact observable outcomes but that are not directly observed themselves. We allow for

distinct transmission pattern across all four parent-child gender combinations. Equations (2) and (3)

do not necessarily map into one particular (e.g., genetic or behavioral) mechanism, but may represent

a great number of underlying mechanisms. Such “reduced-form” representations have been common in

theoretical work (e.g., Becker and Tomes, 1986), and we discuss its interpretation in Section 2.5.

Finally, the model includes three types of shocks. The individual component ukt in the observed outcome

is a white-noise error term. The sibling component xkt is shared by siblings of the same gender, can be

correlated across siblings of different gender, and is uncorrelated with the other variables (in particular

with zt and yt−1). Similarly, the error term vkt in the latent factor is a white-noise error term, and the

sibling component ekt is shared by all siblings of the same gender and can be correlated across siblings of

different genders. Allowing, flexibly, for shared influences among siblings over and above the parental

influence, allows us to extend our analysis in the horizontal dimension (see Section 2.4).

We allow for assortative mating both in the observable and latent socioeconomic status (see Section

2.3).9 In particular, we consider the linear projections of zft−1 and yft−1 on zmt−1 and ymt−1:(zft−1

yft−1

)=

(rmzz rmzyrmyz rmyy

)(zmt−1

ymt−1

)+

(wmt−1

εmt−1

)(4)

where wmt−1 and εmt−1 might be correlated but are uncorrelated with zmt−1 and ymt−1, and the rmsd (s, d = y, z)

coefficients are functions of the following correlations and standard deviations ρzmym , ρzmzf , ρzmyf ,

ρymzf , ρymyf , σzm , σzf , σym and σyf . In Appendix G we provide the formulas for these coefficients, as

well as the corresponding coefficients from the linear projections of zmt−1 and ymt−1 on zft−1 and yft−1, and

show that ρzmym and ρzfyf are functions of the other parameters through two steady-state equations. As

we show in Appendix J, it is not generally possible to write this model with two parents as a one-parent

model without imposing restrictions on either the assortative or the intergenerational process.

9See Behrman and Rosenzweig (2002) for a related model with assortative mating in two dimensions.

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2.2 Direct and Latent Transmission Channels

The model incorporates both direct (via observables) and indirect (via latent variables) transmission

channels. That part of the transmission process may be inherently unobservable has long been recog-

nized in the literature. Goldberger (1972) describes how latent factors such as “ambition” played a

central role in the early sociological work (such as Duncan, 1969). The latent variable in our model

is closer in spirit to Becker and Tomes (1979), who assume that a person’s “endowment” represent a

great variety of cultural and genetic attributes. The distinction between direct and latent transmission

mechanisms has also been influential in a literature on cultural transmission based on models from

population genetics (e.g., Rice et al. 1978, Cavalli-Sforza and Feldman 1981).

Empirical work in economics has however focused on intergenerational correlations in observable char-

acteristics, or particular causal channels, such as the effect of parental education on child education

(see reviews by Solon, 1999, and Black and Devereux, 2011). Latent transmission channels have seen

renewed interest due to recent work on multigenerational transmission, which documents that inequal-

ities appear more persistent than indicated by traditional parent-child correlations. Studies based on

historical records indicate that inequalities between surnames persist over very long periods (Clark and

Cummins 2012, Clark 2014, Collado et al. 2014, Barone and Mocetti 2016), and estimates from direct

family links across multiple generations also point to high persistence (see Lindahl et al. 2015, Dribe

and Helgertz 2016, Braun and Stuhler 2018, Adermon et al. 2018, Long and Ferrie 2018, Neidhofer and

Stockhausen 2019, Colagrossi et al. 2019).

The existence of latent transmission mechanisms would rationalize these findings (Clark and Cummins

2012, Stuhler 2012).10 If the latent variable is comparatively persistent across generations (γk > βk),

but explains only part of the inequality in socioeconomic outcomes (σ2zk< σ2

yk), then the parent-child

correlation may greatly understate the actual transmission of advantages or how this transmission

varies across groups, areas, and time. Intuitively, the observable socioeconomic status is only an imper-

fect proxy for status or prospects, a type of “measurement error” that may attenuate the traditional

parent-child correlations. This observation is related to the insight that sibling correlations are a more

comprehensive measure of the importance of family background than parent-child correlations (see

Jantti and Jenkins 2014).

Moreover, multigenerational data can help to identify the parameters of a transmission model, includ-

ing the inherently unobserved component. This potential has been noted already by Becker and Tomes

10Other potential rationalizations include the idea that intergenerational transmission occurs via multiple channels ofvarying rates of persistence (see Clark 2014, Stuhler, 2012), or that it does not follow a Markov process (Mare, 2011).A recent literature on “grandparent effects” aims to quantify the causal contribution of grandparents on their children.Intuitively, this literature agrees with the diagnosis (a missing component in the transmission process) but considers amore specific solution (considering a missing person, i.e. grandparent) than the approach that we follow here (consideringa latent factor). Anderson et al. (2018), Breen (2018), and Lundberg (2019) review and critically assess this alternativeinterpretation.

11

(1979) and Goldberger (1989), and has been exploited in the recent literature. Clark and Cummins

(2012) use surname averages across two or more generations to estimate the underlying rate of persis-

tence in a simple latent factor model. Braun and Stuhler (2018) illustrate that this model can also be

identified from direct family linkages across three or more generations. However, the model considered

in these studies is very restrictive: socioeconomic status is transmitted exclusively via the latent factor,

without any role for direct transmission (i.e. βm = βf = 0 in our model) or shared influences among

siblings (i.e. σ2x = σ2

e = 0). We aim to account for latent transmission mechanisms without imposing

such restrictions, and to quantify the importance of both direct and indirect transmission channels.

2.3 Assortative Mating

Most intergenerational studies consider a simplified one-parent family structure, in which the assorta-

tive mating process enters only implicitly. But the assortative process is fundamental to understand

the recent multigenerational evidence, as high rates of persistence across generations requires strong

assortative mating (Diaz-Vidal and Clark 2015, Clark 2017, Braun and Stuhler 2018). The intuition for

this argument follows from equations (2) and (3), which suggest that the father’s and mother’s latent

status zmt−1 and zft−1 can only be both strongly correlated with their child’s status zkt if they are also

strongly correlated with each other (0� ρzmzf ).

Spousal correlations in socioeconomic outcomes such as income or education are typically between 0.4

and 0.6 in developed countries (e.g., Fernandez and Rogerson 2001, Ermisch et al. 2006, Greenwood et al.

2014). But as we show below, there needs to be far greater sorting between spouses to rationalize the

socioeconomic similarity between distant kins. Spousal correlations in observable characteristics appear

too low to explain multigenerational dependence, and the pattern of dependence across horizontal kins

that we present in this paper.

We rationalize this discrepancy by allowing spouses to be similar not only in observable but also

unobservable characteristics that determine the socioeconomic status of their offspring. In particular,

spouses may be more similar to each other in latent (zkt−1 in our model) than observable outcomes (ykt−1).

Intuitively, spousal correlations in socioeconomic outcomes may reflect only a superficial similarity, and

not the effective degree of assortative mating in more fundamental characteristics that determine the

success of future generations. Our data provide an opportunity to test this hypothesis.11

The assortative process is, therefore, a key component to rationalize the pattern of dependence in our

kinship data. Conversely, this dependence will be informative about formerly unknown aspects of the

11Ermisch et al. (2006) shows the potential of this approach, illustrating how the degree of assortative mating in one-dimensional matching model with a latent human capital variable can be identified from data on children, their parents,and their parents in-law. Similarly, Ruby et al. (2018) exploit the observation of sibling-in-law to study the role ofassortative mating in the transmission of human longevity. As these papers, we model assortative mating as a reduced-form relationship. Bidner and Knowles (2018) develop an equilibrium model of assortative mating with latent variables,which can be used to study the implications of redistributive and other policies.

12

assortative process. We therefore model this process in more detail than the previous literature. We

allow for assortative mating along two dimensions, with the spousal correlation in the socioeconomic

outcome (ρymyf ) potentially differing from the spousal correlation in the latent status (ρzmzf ).

2.4 Horizontal Kinship

Our model is comparatively general and therefore too complex to be identified from inter- or multigen-

erational moments alone, which has been the traditional approach. Thus, we switch from this “vertical”

to a “horizontal” approach, considering information about relatives in the same generation (such as

siblings or cousins). Because siblings may share influences over and above the parental influence, we

model these influences in a flexible way. We allow for shared sibling components in both the observable

outcome y and the latent advantage z, and for the distribution of those components to vary across

gender combinations. Importantly, by modeling both assortative and sibling processes we can also

consider siblings-in-law in our analysis.

The horizontal approach offers important advantages in terms of data quality, feasibility and scope.

First, studies of multigenerational persistence have to compare socio-economic outcomes from distant

generations. As income is rarely observed in historical sources, this problem typically boils down

to making occupational outcomes more comparable over time – a difficult task, which has received

considerable attention in sociology and, more recently, economics (e.g., Long and Ferrie 2013b, Modalsli

2017). This problem is much diminished when considering horizontal kins, for whom socio-economic

outcomes can be measured at approximately the same age and time. We also avoid problems related

to recall bias or the comparison of data from different sources.

Second, the horizontal approach remains feasible in settings in which there is only limited information

on ancestors available. This benefit has long been recognized in the extensive literature on sibling

correlations (Solon et al. 1991, Levine and Mazumder 2007, Bjorklund et al. 2009, Jantti and Jenkins

2014). Sibling correlations can be estimated even when parental outcomes are not or not well observed,

and therefore require less data than intergenerational measures. Similarly, our horizontal approach

remains feasible in settings in which the intergenerational dimension of the data is restrictive, such as

the Spanish sources discussed below.

Third, and most importantly, the incorporation of horizontal kins yields a much greater set of empirical

moments than a purely vertical approach. As better data is becoming available, the literature has

started to document the pattern of socioeconomic inequalities beyond the nuclear family. For example,

Hallsten (2014) studies kinship correlations between first and second cousins in Sweden, while Adermon

et al. (2019) provide evidence across a broad range of kinship relations, including horizontal kins. Such

evidence is valuable not only from a descriptive perspective, but also opens the door for the identification

13

Figure 1: The Identification of Siblings-in-law

GP1 GP2 GP3 GP4

… a x b y c z …

ax by cz

Notes: Hypothetical family trees across child, parent, and grandparent (GP) generation. Consanguine (affine) relationships in black(orange).

of more detailed intergenerational models – providing a deeper understanding of transmission processes

within the nuclear family and the assortative process.

We can distinguish several dozen consanguine (“blood”) kinships in our data, but extend our analysis

also to affine (“in-law”) kinships such as siblings-in-law. While they may not descend from a common

ancestor, affine kins have similar informational value as consanguine kins – both are a function of,

and therefore informative about, intergenerational and assortative processes (see Appendix G). The

informational value of non-genetic relatives has also been illustrated in recent work in population

genetics. For example, the observation of considerable correlations between in-law relatives has led to

substantially lower estimates of the heritability of human longevity (Ruby et al., 2018).

The key advantage of affine as compared to consanguine relations is that they can be traced over ex-

ceptionally long “distances”, in particular in modern administrative data sources. Figure 1 illustrates

the logic in hypothetical family trees across a child, parent, and grandparent (GP) generation. The

identification of more distant consanguine kins necessitates the observation of more distant ancestors.

For example, the identification of cousins (e.g., ax-by) requires the observation of their shared grand-

parents (GP2), while the identification of second-degree cousins would require the identification of their

great-grandparents. As a consequence, distant family links are never directly observed, and existing

evidence on mobility in the very long run is instead based on their probabilistic approximation via

surnames (e.g., Clark 2014, Barone and Mocetti, 2016).

In contrast, affine relationships are defined only via spousal and parental links – irrespective of their

degree of separation. In a first step, we identify a person’s spouse (e.g., a-x ) via their shared descendant

(ax ). In a second step, we identify the spouse’s sibling (b) via their shared parent (GP2 ). Implemented

once, these steps identify a pair of siblings-in-law (a-b). Implemented twice, we identify the sibling-in-

14

law of the sibling-in-law (a-c), and so on. In population-wide data in which every spouse and sibling

is observed, one can reiterate these linkages – and therefore the number of empirical moments – ad

infinitum.

In this study, we observe more than one-third of the Swedish population, and sizable samples for 141

distinct kinship types up to fifth-order affinity relations. We systematically consider all kinship types,

such as siblings-in-law (e.g., a-b), co-siblings-in-law involving the spouse of the sibling-in-law (a-y) or

the sibling of the sibling-in-law (x-c), as well as vertical relations such as the uncle in-law (ax-y) and

cousins-in-law (ax-cz ). We also consider all possible gender combinations within each kinship type,

such as brother-brother, sister-sister, brother-sister for sibling correlations. The observation of distant

affine relations will play a key role, as in distant kins it becomes directly discernible if a model can fit

the pattern of economic inequalities across generations.

The horizontal perspective yields, therefore, many more, and much more distant family relations than

the vertical perspective. As a result, the model from Section 2.1 is heavily over-identified. Over-

identification helps us to pin down the parameters of the model and to test its fit more systematically

than what has been possible in the prior literature. Specifically, we test if our results remain robust

to considering different kinship types, if our model can fit kinship moments out-of-sample, and if

alternative models can provide a similarly good fit to the data.

2.5 Interpretation

Our model can be interpreted as a reduced-form representation of the causal effect of family background,

decomposed into its intergenerational and assortative, observable and latent dimensions. In our baseline

model, we do not impose any specific interpretation of its parameters. Our primary objective is instead

a statistical one – to capture the transmission of economic inequalities across a wide range of kins,

by formulating a model that is sufficiently flexible in the intergenerational and assortative dimensions.

Most existing work focuses instead on simpler descriptive measures (such as intergenerational or sibling

correlations) or specific causal channels (such as the causal effect of parental on child education).

Compared to more targeted studies, our approach has distinct advantages and disadvantages. The

obvious disadvantage is that we remain largely agnostic about what mechanisms the pathways of our

model represent, in particular with respect to the latent factor zkt .12

12Many economic models remain agnostic in some aspects, while targeting specific mechanisms in others. For example,Becker and Tomes (1979) focus on the effect of parental investments on child outcomes, but also consider the role of“endowments” that have a similarly broad interpretation as the latent factor in our model. Two recent studies extendstandard economic models to account more explicitly for genetic factors. Rustichini et al. (2018) incorporate a polygenicscore from genome-wide association studies into the Becker and Tomes framework, replacing the standard scalar represen-tation of skills with a detailed model of skill transmission that incorporates genetic factors. Bidner and Knowles (2018)develop a model of equilibrium sorting in an unobserved heritable characteristic, such as genotype.

15

The principal advantage of our approach, however, is that it provides a more comprehensive account of

intergenerational transmission.13 For example, our latent factor zkt is a more comprehensive object than

the “genotype” considered in behavioral genetics, as it also captures non-genetic advantages that matter

for the socioeconomic success of the next generation. That is, by avoiding an apriori stand on the causal

channels via which advantages are transferred, we capture those channels more comprehensively.14

Similar to Ruby et al. (2018), we quantify the “transferable variance” of a socioeconomic outcome

variable, which includes the heritability due to genetic factors but also the variance due to inherited

socio-cultural factors, as well as the covariance between the two.

One limitation of our approach is that we can quantify the extent to which latent advantages are

transmitted between generations, but not necessarily the extent to which they determine status within

any given generation. We can estimate how strongly the latent factor zkt determine a particular so-

cioeconomic outcome ykt , but zkt may also affect other dimensions of socioeconomic status that are not

considered by our model. As such, it is not clear if the observed measure ykt or the latent status zktrepresents a better measure of a person’s experienced socioeconomic status. A recent strand of the

literature provides evidence on this question by aggregating multiple measures of parent and/or child

socioeconomic status, and is complementary to our approach (Vosters and Nybom 2017, Adermon et al.,

2019, Blundell and Risa 2018).

While our primary objectives are statistical, our approach can be informative about causal mechanisms,

to the extent that those mechanisms have specific statistical implications about the pattern of inequality

across kins. Most importantly, our baseline model is sufficiently general to nest a standard genetic model

of transmission. We can, therefore, test how the standard genetic model fits the data compared to our

more general model (Section 6.1), or attempt to decompose the overall transferability of socioeconomic

advantages into its genetic and socio-cultural components (Section 6.2).

3 Data and Calibration

We describe our data sources and estimation procedures in this section. We calibrate the model in

Section 2.1 for two different countries, Sweden and Spain. The comparison will contribute to the

13The extent to which advantages are transmitted from one generation to the next has long been a central question inthe literature, and the evidence on this question has changed drastically over recent decades. A more careful treatment ofmeasurement error nearly tripled estimates of income persistence in the United States (Solon 1999, Mazumder 2005) ascompared to earlier studies (Becker and Tomes 1986). However, some authors argue that such estimates still drasticallyunderstate the transmission of advantages (e.g., Clark, 2014). This controversial hypothesis can be studied separatelyfrom the question which particular pathways contribute to the overall rate of persistence.

14Our approach shares therefore aspects with the literature on siblings correlations. Because they account for all factorsshared by siblings, including those that are orthogonal to the observed status of parents, sibling correlations are a morecomprehensive measure of family background than intergenerational correlations (Jantti and Jenkins, 2014). However,sibling correlations still capture only those advantages that are directly observable in the child generation. In contrast,our approach captures also advantages that are not visible in the child generation, but that affect a family’s prospects infuture generations (i.e., the sibling correlation in latent advantages).

16

Table 1: Descriptive Statistics in Swedish Registers

Variable Generation Cohorts Observations Mean Std. Dev. Mean Std. Dev.Education Child 1966-76 1,026,084 12.35 2.16 12.81 2.12

Parent 1920-62 2,502,869 10.70 3.01 10.60 2.88Income Child 1958-68 1,062,847 12.30 0.55 11.97 0.50

Parent 1923-53 1,811,895 12.37 0.52 11.82 0.80Height Child 1974-80 269,475 179.73 6.55

Parent 1951-66 378,686 178.99 6.37

Men WomenTable: Descriptive Statistics in Swedish Registers

Notes: Education is measured as years of education. Income is measured as the logarithm of ten-year average incomes at age 30-39for children and age 45-54 for parents. Height is observed for men only.

literature on mobility differences across countries, which contains little evidence on Spain (see Black

and Devereux, 2011). More importantly for our purposes, we study how different components of the

transmission process contribute to those observed differences in mobility.

The comparison between Swedish and Spanish sources is interesting also from a methodological per-

spective, to demonstrate the feasibility of our approach in different settings. The Swedish registers are

very extensive, include family links over multiple generations, and cover a large share of the population.

In contrast, the Spanish data are limited to a single cross-section, contain only educational outcomes

and names, and lack direct family links to define kinship.

Our baseline outcome is educational attainment, which is observed for both countries. In the Swedish

registers we consider two additional outcomes, income and height. We study variation in the overall

rate of persistence, but also if certain components – such as the role of assortative mating in observable

or latent factors – are more important for some outcomes than others. Because external evidence exists

on some of these components, the comparison serves as a sanity check of our approach.

3.1 Swedish Multigenerational Registers

Our sample is based on a random 35 percent draw of the Swedish population born between 1932

and 1967, as well as their biological parents, siblings and children. Family links are biological links,

with a man and woman considered to be spouses if they have a child together. We match individual

characteristics from bi-decennial censuses (starting from 1960), official registers, and military enlistment

tests. Table 1 shows basic descriptive statistics on our main outcome variables:

Education. Educational registers were compiled in 1970, 1990, and about every third year thereafter

up to 2007. We consider the highest schooling level recorded across these years, and translate it into

years of education, with seven years for the old compulsory school being the minimum, and 20 years

17

for a doctoral degree the maximum.15 Educational information in 1970 is available only for those born

1911 and later. It may be missing also if parents had died or emigrated before 1970, but the share of

affected observations is small. As the data are collected from official registers, they are not subject to

standard non-response problems. We measure education up to cohorts born in the mid-1970s, after

which the information on educational attainment becomes less reliable at the top of the attainment

distribution.

Income. We measure long-run income by averaging over multiple annual incomes, which are observed

for the years 1968-2007. We consider total (pre-tax) income, which is the sum of an individual’s labor

(and labor-related) earnings, early-age pensions, and net income from business and capital realizations,

and express all incomes in 2005 prices. Incomes for parents are necessarily measured at a later age

than incomes for their offspring, which may bias intergenerational estimates. To reduce this bias we

construct log ten-year average incomes measured at age 30-39 for children and age 45-54 for parents.16

These averages can be observed from the 1923 cohort in the parent generation up to the 1968 cohort in

the child generation. To probe the influence of measurement error, we estimate our model also based

on 5-year average and annual incomes.

Height. We observe height from military enlistment data, for male individuals born between 1951 and

1980. Military enlistment took place at age 18 or 19 and was at the time universal for all men. Height

was recorded as part of the medical examination. Because we observe height only for birth cohorts

spanning three decades, we can consider father-child and other vertical correlations only for fathers who

were sufficiently young at the birth of their child (we impose a minimum age of 18 years). However,

selectivity with respect to parental age appears less problematic for height than for other outcomes. For

example, the father-son correlation in education increases in parental age at birth, while the father-son

correlation in height remains fairly stable.

Standardizations. Kins can be born in different cohorts and their outcomes being measured in different

years. To abstract from this source of variation, we de-mean all outcomes by birth cohort and gender.

Moreover, income averages are censored at the 1st and 99th percentiles (again by birth cohort and

gender) to reduce the influence of outliers. In a robustness test we standardized also the variance of

each outcome (i.e., z-scores), but this transformation had only negligible effects on our results. These

standardizations are performed in the full sample, before selecting pairs of observation for each kinship

moment.

Cohort Selection. The selection of sub-samples for each kinship moment is non-trivial. Multiple sources

of selection need to be taken into account, separately for each outcome, and separately for horizontal and

15In the 1970 Census, we impute 7 years for the (old) primary school, 9 years for (new) compulsory schooling, 9 years forpost-primary school (realskola), 11 years for short high school, 12 years for long high school, 14 years for short university,16 years for long university, and 20 years for a PhD. Schooling levels are recorded in more details in later registers.

16Nybom and Stuhler (2017) study the magnitude of attenuation and life-cycle biases from the approximation of long-run income with short average incomes in the same data source. The magnitude of these biases are small in our chosenage range.

18

vertical kinship types. We first select cohorts for which the outcome is reliably observed, as described

above. We then assess which kinship types can be reliably identified within those cohorts. For example,

the identification of siblings requires observation of their parents, while for the identification of cousins

we need to observe grandparents, and so on. Our aim here is to avoid selectivity with respect to the age

difference between kins, as kinship correlations vary systematically along this dimension. We abstain

from kinship types that depend on the identification of great-grandparents for this reason. We therefore

consider only three generations to identify kins, and only two generations to measure outcomes. We

describe these issues in more details in Appendix A, and provide evidence on the stability of the kinship

moments over cohorts in Appendix B.

Distant chains and duplicate entries. Larger families are necessarily overrepresented in the more distant

moments, but our results are not very sensitive to weighting by family size (see Section 3.3). A second

issue is that the chain of sibling and spousal connections that links distant in-laws may contain duplicate

entries. For example, an individual may be his own second-degree brother-in-law if two families are

connected via more than one spousal link, or a person may feature multiple times because he or she

has children with different partners. In principle, the observation of duplicate entries are a reflection

of the assortative process, and should be retained in the analysis. For example, inequality might be

more persistent if in-law relations “circle” within closed groups defined by geography, ethnicity, or

other characteristics. However, because we observe only a random subset of the Swedish population,

duplicate entries occur at a higher rate in our sample than in the full population. We therefore drop

all chains with duplicate entries, which has only a small effect on the most distant kinship correlations.

3.2 Spanish Census Data

The 2001 population census for Spain, which is available nationwide, does not allow to identify families

unless they are living in the same household. However, for the Spanish region of Cantabria we obtained

information on the full name of each person, and we can use this information to identify parents and

children. The Census also reports, among other variables, the gender, age and educational level of all

individuals living in the region (526, 339 persons). We define the t-generation as all persons born in

Cantabria between 1956 and 1976 (71, 479 males and 68, 830 females) and the (t − 1)-generation as

their parents.

Matching. Surnames in Spain are passed from parents to children according to the following rule: A

newborn person, regardless of gender, receives two surnames that are kept for life. The first surname

is the father’s first surname and the second the mother’s first surname. This naming convention allows

us to identify fathers and mothers. For each person i in generation t we define the set of potential

parents as all the couples born before 1956 such that the husband first surname coincides with person

i first surname and the wife first surname coincides with person i second surname. Then, we say

19

Table 2: Descriptive Statistics in Spanish Census

Mean Std. Dev Mean Std. Dev Mean Std. Dev Mean Std. DevAge 33.61 5.91 35.42 6.16 33.70 5.92 35.50 6.15Years of schooling 10.53 3.71 9.71 3.64 10.99 3.71 10.11 3.69Observations 45,61925,860

WomenMenTable: Descriptive Statistics in Spanish Census

Matched Unmatched Matched Unmatched

44,22024,610

that we identify the parents if there is only one couple in the set of potential parents and the age

difference between both parents and the child is at least 16 years. We identify the parents for 25, 860

males and 24, 610 females, which is approximately 36.2% and 35.8% of the male and female population,

respectively.

To assess how well our strategy to identify parents and children works, we exploit the fact that we can

directly identify parents and children when they live together (without using surnames). We use this

information to estimate the percentage of incorrect matchings derived from our identification strategy.

We identify 51,923 parent-child pairs using the surnames, with 23,694 of these children co-residing with

their (real) parents and 28,229 living in different households. For the sub-sample of children co-residing

with their parents, the percentage of identification mistakes is 6.1%. We exclude these 1,453 pairs from

our sample and the final sample size is 50,470. If the percentage of incorrect identifications for the

sub-sample of parents-child not living together were also 6.1% we would expect 1,722 mistakes (3.4%)

in the total sample.

Once we have identified parents and children, siblings are immediately identified, and when children

are married we also identify siblings-in-law. Finally, we assume that siblings in the parents’ generation

are identified when there are at most four individuals in the over 25 population sharing the same two

surnames. Once siblings in the parent generation are identified, uncles and nephews, and cousins are

immediately identified. Again it is important to estimate how well our strategy to identify siblings in the

parents’ generation works. We cannot directly detect identification mistakes in the parent generation,

but can test the reliability of our approach in the child generation. Specifically, we repeat the exercise to

detect identification mistakes in the sample of co-residing children as described above, but restrict that

sample to children with surnames held by between two and four individuals in the over 25 population.

As expected, the percentage of incorrect identifications is now lower, 2.5%.

Education. We use the information on each individual’s educational attainment and convert it to years

of schooling following Calero et al. (2007).17 We de-mean years of schooling using gender-birth-cohort

17We assign 2 years of education to those who did not complete primary education, 5 years to primary education, 8to compulsory education, 10 to vocational training, 12 to secondary education, 15 to sort university degrees, 17 to longuniversity degrees other than engineering and medicine, 18 for engineers and medical doctors and 19 for a Ph.D. All ourresults are robust to other reasonable ways to assign years of education as, for example, assigning 0 years of education tothose who did not complete primary education, 4 years to primary education, 9 to vocational training and 11 to secondary

20

averages. Table 2 shows some basic descriptive statistics. The matched sample is almost two years

younger than the unmatched one. The reason is that the older a person is, the more likely the parents

are not living together or one of them has died. Since the matched sample is younger it is also more

educated (0.8 more years of schooling than the unmatched sample).

3.3 Estimation and Calibration

We compute each kinship correlation based on the sample restrictions described above. Because the

number of family members varies across families we need to decide how to weight large compared to

small families. A family or “cluster” is defined by the most recent common ancestor (such as the

common grandparents shared by cousins) for biological kins, or by the linking spouse for in-laws. We

considered four different sets of weights, ranging from uniform to weights that are proportional to the

number of kinship pairs per family (see Solon et al. 2000). The sample correlations are not very sensitive

to this choice, even though the number of kinship pairs per family varies strongly for distant kins (e.g.

the number of cousins varies more strongly than the number of siblings).18 We therefore picked an

intermediate scheme, weighting each family by the square root of their number of distinct pairs.

Since we can directly estimate σym and σyf from the data, we have 20 unknown parameters that we

write as the vector v,

v = {βm, γm, σzm , σxm , βf , γf , σzf , σxf , ρxmxf , ρzmzf , ρymyf , ρzmyf , ρymzf ,

αmy , αmz , α

fy , α

fz , σem , σef , ρemef },

and therefore we need at least 20 correlations between relatives of different kinship. We calibrate the

parameters in v by solving the following minimization problem,19

Minv∈F∑i∈C

pi(ρi − ρi)2, (5)

where ρi are the theoretical correlations, ρi the empirical correlations, pi the weight given to each term,

F is the set of feasible values for the unknown parameters, and C denotes the set of correlations. In

education.18The correlation between the set of sample moments estimated under the two most extreme weighting schemes is

greater than 0.99 (0.98) in the Swedish (Spanish) sample.19We cannot estimate the parameters by GMM as in Abowd and Card (1989) because the units of analysis, that are

families, are not well defined. Moreover, most individuals will belong to different families and therefore the sample unitswill not be independent (see Appendix C). We have used Mathematica 11.3 to solve the minimization problem. The codeis in the [Online Appendix]. We have used the Simulated Annealing algorithm, which is a stochastic function minimizer.In most exercises we have used a minimum of 10,000 random starting points from the set of feasible values F . In most ofour main exercises, and in particular in our benchmark case, we reach the same minimum for most of the starting points,so that we are confident that we have found a global minimum. We also tried other algorithms for constrained globaloptimization (Nelder-Mead, Differential Evolution and Random Search) and never found a different global minimum.

21

our benchmark case the set C will contain 105 different kinship correlations. In most cases we give the

same weight to all the terms so that pi = 1 for all i ∈ C, but the results are very similar if we weight

each moment by the number of families used to calculate the sample correlation.

4 Kinship Correlations in Sweden

In this section, we report our baseline results for Sweden, considering years of schooling as our dependent

variable. Table 3 provides a partial list of the kinship types that we can distinguish in our data, as

well as the number of moments within each group. Considering siblings-in-law up to the five degrees of

separation, we observe 141 distinct moments. The formulas for each of these correlations as a function

of the model parameters are presented in Appendix G.

4.1 Sample Moments

Table 4 reports the sample correlation in years of schooling for each the 141 kinship moments. The

moments are sorted by kinship type, from closely related to more distant kins. Columns (1) and (2)

report the number of pairs and sample correlations. The pairs are weighted inversely by the square root

of family size, as described in Section 3.3. The sample correlations in years of schooling span between

one half for close kins, such as spouses or parents, to only a fraction of that for the most distant kinship

types. Owing to the large number of observations, all correlations are precisely estimated. As far as

they overlap, they appear consistent with estimates from the previous literature.20

4.2 Calibrated Moments

In our baseline calibration we include siblings-in-law up to three degrees of separation, but do not include

cousins or higher-order siblings-in-law. With these restrictions, our baseline calibration is based on 105

distinct kinships, grouped into fourteen different kinship types. We calibrate the model as described in

Section 3, and report the predicted moments as well as the percentage deviation between the observed

and predicted moments in columns (3) and (4) of Table 4. Moments that were not included in the

calibration are printed in italics.

Figure 2 illustrates the in-sample fit graphically, by plotting both sample moments (orange) and pre-

dicted moments from the calibrated model (blue dots). The model explains the data well, for both

20For example, Bjorklund et al. (2009) report that the correlation in years of schooling between brothers in Sweden isslightly below 0.5 (cf. 0.44 in our sample). Hallsten (2014) estimates that the corresponding correlation for cousins isabout 0.15, which fits our model predictions but is below our own sample correlation. We return to this contrast below.

22

Table 3: List of Kinships

kinship kinship type # correlationsa–x spouses direct, horizontal 1x–b siblings direct, horizontal 3ax–by cousins direct, horizontal 10ax–a child-parent direct, vertical 4ax–b child-uncle/aunt direct, vertical 8a–b siblings in-law (degree 1) affinity, horizontal 4a–y spouse of sib-in-law (dg 1) affinity, horizontal 3x–c sibling of sib-in-law (dg 1) affinity, horizontal 4a–c siblings in-law (degree 2) affinity, horizontal 8a–z spouse of sib-in-law (dg 2) affinity, horizontal 4x–d sibling of sib-in-law (dg 2) affinity, horizontal 10a–d siblings in-law (degree 3) affinity, horizontal 16… … affinity, horizontal …ax–y child-sibling in law (dg 1) affinity, vertical 8… … " …

ß

Table: List of Kinships

Notes: The number of distinct moments within each kinship type is determined by the potential set of gender combinations (suchas brother-brother, sister-sister, and brother-sister). See Figure 1 for definition of kinship types.

vertical and horizontal moments, and for both direct (“blood”) and affinity (“in-law”) kinships. The

mean absolute error across all moments used in the calibration is 1.9 percent, and across all moments 7.2

percent. In percentage terms, the out-of-sample fit is worst for cousins and extremely distant in-laws.

We return to those issues below.

These results suggest that it is possible to fit the pattern of inequality across very different kinship

types, and for both narrow and distant relatives using a parsimonious model with a limited set of

transmission mechanisms. As we will illustrate in Section 4.7, simpler models that do not allow for

latent advantages in intergenerational and assortative processes would not successfully fit the data.

4.3 Intergenerational Transmission

Table 5 summarizes our baseline findings. Panel A reports the calibrated parameters for the intergen-

eration or “vertical” components of our model. As motivated in Section 2.2, we distinguish between

the direct transmission of advantages that are reflected in our outcome of interest (i.e. educational at-

tainment), and the transmission of latent advantages that are not necessarily reflected in that outcome,

but that nevertheless contribute to the socioeconomic success of future descendants.

The direct transmission channels captured by the parameter βk represent the causal effect of parental

education, but also other advantages that are closely correlated with years of schooling (Bjorklund and

Salvanes, 2011). With βm = 0.14 and βf = 0.13, this channel contributes very little to the overall

23

Table 4: Sample and Predicted Moments in Swedish Registers

# ## name number sample predicted percent # ## name number sample predicted percentof pairs correlation correlation error of pairs correlation correlation error

(1) (2) (3) (4) (1) (2) (3) (4)I 1 Husband-Wife 413,062 0.491 0.489 -0.3 …II 2 Brother 387,028 0.432 0.431 -0.3 XII 72 MFMS 299,602 0.138 0.135 -2.2

3 Sister 431,698 0.416 0.417 0.3 73 FMMS 273,809 0.126 0.124 -1.4 4 Brother-Sister 800,127 0.375 0.377 0.5 XIII 74 M-MMMS 160,726 0.102 0.098 -3.5

III 5 Father-Son 396,304 0.380 0.381 0.2 75 M-MMFS 174,261 0.103 0.106 3.4 6 Father-Daughter 376,255 0.321 0.321 0.1 76 M-MFMS 158,401 0.107 0.109 1.9 7 Mother-Son 422,374 0.366 0.367 0.2 77 M-MFFS 160,105 0.106 0.106 0.3 8 Mother-Daughter 400,337 0.347 0.349 0.5 78 M-FMMS 147,949 0.102 0.103 0.9

IV 9 Brother in-law (HS) 602,262 0.302 0.296 -2.1 79 M-FMFS 156,876 0.103 0.111 7.9 10 Brother-Sister in-law (WB) 578,269 0.296 0.304 2.7 80 M-FFMS 133,588 0.104 0.103 -1.0 11 Brother-Sister in-law (HS) 650,127 0.298 0.307 3.0 81 M-FFFS 131,756 0.101 0.101 -0.5 12 Sister in-law (WB) 596,540 0.278 0.277 -0.2 82 F-MMMS 152,751 0.087 0.086 -0.1

V 13 Nephew-Uncle (BF) 280,067 0.254 0.249 -1.7 83 F-MMFS 165,828 0.091 0.093 3.1 14 Niece-Uncle (BF) 266,289 0.218 0.220 1.2 84 F-MFMS 151,100 0.094 0.095 1.7 15 Nephew-Uncle (BM) 312,019 0.241 0.238 -1.2 85 F-MFFS 153,065 0.089 0.093 4.9 16 Niece-Uncle (BM) 295,580 0.209 0.210 0.5 86 F-FMMS 140,585 0.093 0.092 -1.1 17 Nephew-Aunt (SF) 285,618 0.234 0.229 -2.1 87 F-FMFS 150,162 0.097 0.099 2.9 18 Niece-Aunt (SF) 270,325 0.217 0.203 -6.7 88 F-FFMS 126,129 0.093 0.092 -1.2 19 Nephew-Aunt (SM) 333,141 0.251 0.245 -2.3 89 F-FFFS 124,968 0.085 0.090 5.3 20 Niece-Aunt (SM) 316,625 0.234 0.218 -7.0 XIV 90 M-MMM-M 84,025 0.094 0.082 -13.4

VI 21 Brother in-law (WWS) 252,232 0.252 0.246 -2.2 91 M-MMF-M 100,261 0.101 0.086 -15.2 22 Sister in-law (HHB) 226,795 0.229 0.232 1.1 92 M-MFM-M 93,237 0.105 0.090 -13.9 23 Brother-Sister in-law (HWBS) 464,081 0.222 0.227 1.9 93 M-FMM-M 80,486 0.097 0.085 -12.0

VII 24 Nephew-Aunt in-law (BF) 231,767 0.192 0.192 -0.3 94 M-MMM-F 79,690 0.087 0.073 -16.7 25 Niece-Aunt in-law (BF) 221,287 0.172 0.171 -0.8 95 M-MMF-F 95,733 0.094 0.076 -19.9 26 Nephew-Aunt in-law (BM) 254,534 0.187 0.183 -2.2 96 M-MFM-F 89,364 0.093 0.080 -13.6 27 Niece-Aunt in-law (BM) 241,873 0.164 0.161 -2.0 97 M-MFF-F 95,020 0.095 0.075 -20.4 28 Nephew-Uncle in-law (SF) 227,403 0.190 0.188 -1.5 98 M-FMM-F 76,514 0.095 0.076 -20.0 29 Niece-Uncle in-law (SF) 215,068 0.163 0.167 2.2 99 M-FMF-F 89,054 0.088 0.079 -9.8 30 Nephew-Uncle in-law (SM) 264,524 0.197 0.198 0.8 100 M-FFM-F 77,332 0.094 0.076 -19.0 31 Niece-Uncle in-law (SM) 251,782 0.171 0.175 2.1 101 M-FFF-F 80,067 0.082 0.072 -12.9

VIII 32 Male Cousins (B) 70,137 0.208 0.159 -23.8 102 F-MMM-F 76,344 0.080 0.064 -20.3 33 Male Cousins (S) 82,049 0.215 0.160 -25.4 103 F-MMF-F 91,080 0.090 0.066 -26.6 34 Male Cousins (BS) 156,747 0.202 0.152 -24.8 104 F-MFM-F 84,736 0.092 0.070 -23.4 35 Female Cousins (B) 63,032 0.169 0.126 -25.5 105 F-FMM-F 72,410 0.082 0.068 -17.4 36 Female Cousins (S) 73,649 0.197 0.124 -37.0 XV 106 XMMMM 288,374 0.103 0.103 -0.2 37 Female Cousins (BS) 140,522 0.177 0.118 -33.2 107 XMMMF 312,703 0.102 0.105 3.5 38 Male-Female Cousins (B) 144,100 0.179 0.141 -20.9 108 XMMFM 311,795 0.111 0.113 1.4 39 Male-Female Cousins (S) 170,577 0.196 0.141 -28.2 109 XMMFF 162,928 0.099 0.104 5.4 40 Male-Female Cousins (BS) 148,691 0.179 0.133 -25.5 110 XMFMM 308,163 0.116 0.115 -1.5 41 Male-Female Cousins (SB) 148,631 0.184 0.135 -26.5 111 XMFMF 166,250 0.121 0.117 -2.6

IX 42 XMMM 461,883 0.185 0.191 3.5 112 XMFFM 304,684 0.114 0.114 0.3 43 XMMF 500,448 0.192 0.196 1.8 113 XFMMM 278,416 0.117 0.111 -4.9 44 XMFM 481,006 0.207 0.212 2.6 114 XFMFM 149,478 0.131 0.122 -7.0 45 XFMM 447,263 0.208 0.207 -0.6 115 XFFMM 143,733 0.115 0.112 -2.2

X 46 MMM 362,409 0.156 0.157 0.8 XVI 116 MMMM 230,313 0.087 0.085 -2.5 47 MMF 393,579 0.156 0.161 3.4 117 MMMF 251,223 0.087 0.087 0.2 48 MFM 375,442 0.179 0.173 -3.4 118 MMFM 248,811 0.097 0.093 -4.0 49 MFF 391,389 0.158 0.160 1.2 119 MMFF 259,925 0.083 0.086 3.0 50 FMM 353,470 0.160 0.161 0.8 120 MFMM 245,814 0.104 0.094 -9.7 51 FMF 378,720 0.164 0.165 0.5 121 MFMF 265,220 0.101 0.096 -4.5 52 FFM 341,316 0.157 0.160 2.1 122 MFFM 241,998 0.099 0.093 -6.0 53 FFF 351,350 0.148 0.148 0.0 123 MFFF 248,449 0.084 0.086 1.7

XI 54 M-MMM 202,632 0.119 0.127 6.7 124 FMMM 224,873 0.091 0.087 -5.1 55 M-MMF 219,007 0.129 0.130 1.1 125 FMMF 246,186 0.092 0.089 -3.9 56 M-MFM 192,819 0.134 0.140 4.6 126 FMFM 237,791 0.100 0.095 -5.1 57 M-MFF 199,811 0.129 0.129 -0.3 127 FMFF 247,495 0.088 0.088 0.0 58 M-FMM 183,670 0.125 0.133 6.2 128 FFMM 223,661 0.086 0.087 0.3 59 M-FMF 196,631 0.132 0.136 3.6 129 FFMF 240,328 0.094 0.089 -5.8 60 M-FFM 160,857 0.132 0.132 0.6 130 FFFM 213,155 0.091 0.086 -5.5 61 M-FFF 164,528 0.122 0.122 -0.2 131 FFFF 220,553 0.084 0.079 -4.9 62 F-MMM 192,818 0.104 0.112 7.1 XVII 132 MMMMS 176,790 0.071 0.066 -8.3 63 F-MMF 208,008 0.114 0.114 0.2 133 MMMFS 199,041 0.075 0.071 -6.0 64 F-MFM 183,929 0.116 0.123 6.1 XVIII 134 MMMMM 153,057 0.047 0.046 -3.2 65 F-MFF 191,177 0.112 0.113 1.2 135 FFFFF 144,976 0.054 0.043 -20.1 66 F-FMM 175,507 0.110 0.119 8.0 XIX 136 MMMMMS 117,473 0.047 0.035 -25.5 67 F-FMF 187,178 0.121 0.122 0.5 137 MMMMFS 135,096 0.042 0.038 -9.9 68 F-FFM 151,606 0.112 0.118 5.5 XX 138 MMMMMM 106,844 0.031 0.025 -21.9 69 F-FFF 155,658 0.113 0.109 -3.7 139 FFFFFF 100,871 0.043 0.023 -46.5

XII 70 MMMS 278,938 0.122 0.122 -0.5 XXI 140 MMMMMMS 82,523 0.032 0.019 -40.0 71 MMFS 310,160 0.132 0.132 -0.6 141 MMMMMFS 96,840 0.027 0.021 -24.4

Table: Estimated and Calibrated Moments in Swedish RegistersKinship type Data Calibration Kinship type Data Calibration

Notes: HS=Husband of the sister, WB=Wife of the brother, BF=Brother of the father, BM=Brother of the mother, SF=Sister ofthe father, SM=Sister of the mother, WWS=Wives are sisters, HHB=Husbands are brothers, HWBS=Husband and wife are brotherand sister, B=Fathers are brothers, S=Mothers are sisters.

24

Table 5: Calibrated Parameters in Swedish Registers

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.144 0.129 0.664 0.565σ2

ym σ2yf σ2

zm σ2zf σ2

um σ2uf

4.648 4.465 2.070 1.560 1.978 2.329αym αyf αzm αzf

0.390 0.020 0.658 0.773Parent-child correlations in z:

Father-Son Father-Dau Mother-Son Mother-Dau0.586 0.600 0.527 0.508

Ancestor correlations in y and z:Father-Son Grandf-… GGrandf-… GGGrandf-Son

in y 0.381 0.209 0.121 0.071 in z 0.586 0.343 0.201 0.118

Panel B: Sibling ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef

0.178 0.246 0.069 0.657 0.711 0.625Variance Shares: in y 3.8% 5.5% 1.5% 14.1% 15.9% 13.7% in z - - - 31.7% 45.6% 34.8%Sibling correlations in z:

Brothers Sisters Brother-Sister0.678 0.824 0.711

Panel C: Assortative ProcessesParameters:

rmzz rm

zy rmyz rm

yy σ2ωm σ2

ϵm

0.663 -0.008 0.696 0.143 0.673 2.919rf

zz rfzy rf

yz rfyy

0.747 0.112 0.662 0.249Spousal correlations in y and z:

ρymyf ρzmzf ρymzf ρzmyf

0.489 0.754 0.540 0.580Panel D: Variance Decomposition

% y z x Cov(y,z)Male 0.013 0.445 0.038 0.038Female 0.016 0.349 0.055 0.030

Table: Calibrated Parameters in Swedish Registers

25

Figure 2: Baseline Fit in Swedish Registers

I II III IV V VI VII IX X XI XII XIII XV XVI1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

Notes: See Table 4 for the corresponding list of kinship moments.

transmission of status from one generation to the next. Accordingly, only about one percent of the

variation in years of schooling of the offspring is explained by parental education itself (Panel D of

Table 5).21 This finding holds for all possible combinations of parent and child gender.

Instead, the transmission of advantages occurs predominantly via the latent factor, as measured by the

parameter γk. We find γm = 0.66 and γf = 0.57, implying that the set of unobserved advantages that

this factor represents are more strongly transmitted from parents to their children than educational

attainment itself. Moreover, this latent factor explains a large share of the variance in child education,

about 45 percent for men and 35 percent for women.

These results are consistent with the finding from recent multigenerational studies, which document

that kinship correlations decline more slowly across generations than a simple iteration of the parent-

child correlation would suggest (see for example Clark 2014, Lindahl et al. 2015, or Braun and Stuhler

2018, and Hallsten 2014 for a survey of the earlier literature). For illustration, we compute the implied

autocorrelation of the child and his or her ancestors on the male side (Panel A in Table 5). Because

education is only an imperfect proxy for an individual’s unobserved characteristics, the observed cor-

21Accordingly, the parameters αmy and αf

y measuring the relative weight of mothers and fathers in this direct effect arenot very consequential either.

26

relation in parent-child status understates the true persistence of socio-economic advantages across

generations.22

4.4 Siblings and Horizontal Transmission

In order to incorporate the horizontal family dimension, our model allows for common shocks among

siblings (that are allowed to vary with gender), in both observable and latent factors. We find that

siblings share influences in both dimensions, as summarized in Panel B of Table 5. The shared sibling

component in the observable outcome (captured by xkt ) explains about five percent of the overall

variation in years of schooling, while the shared component in the latent factor (captured by ekt )

explains about 15 percent of the variation in years of schooling (and between 30 and 45 percent of the

variation in the latent factor itself).

Siblings share, therefore, important influences over and above what can be accounted for by parental

characteristics. This finding is consistent with the literature on sibling correlations, which has shown

that siblings share many additional influences that are orthogonal to the observed socioeconomic status

of parents (see Jantti and Jenkins 2014, and Section 2.4). Our results provide a richer characterization

of this process. In particular, they suggest that siblings share advantages to a much greater extent than

what would be reflected in their education.

Intuitively, while the correlation in educational attainment between siblings is high, siblings (and

spouses) must be substantially more similar to rationalize why kinship correlations decay so slowly

across siblings-in-law. For Sweden, the implied correlation in the latent status between siblings is

around 0.7 or 0.8 (depending on their gender, see Panel B of Table 5). These correlations are more

than 50 percent higher than the observed sibling correlation in years of education (Table 4), and also

much higher than sibling correlations in other socio-economic outcomes. Our findings therefore imply

that sibling correlations still understate the importance of family background, even though they are a

more comprehensive measure than the conventional intergenerational associations.

We observe sibling correlations to be lower for mixed than for brother or sister pairs (after controlling for

the mean education by gender and cohort), a pattern that has been previously observed in the literature.

Our results, however, suggest that male and female siblings share latent influences to a similar degree

(cf. σ2em , σ2

efand σemef in Table 5). Instead, the effect of family background varies with child gender

conditional on their latent status – mixed siblings share only about one third of direct influences in

education compared to same-gender pairs (cf. σ2xmt

, σ2xft

and σxmt ,x

ft). A potential interpretation is

22The correlation between grandfathers and their grandchildren as predicted by our model is very similar (0.210 vs.0.216) to the corresponding correlation as directly measured in a Swedish data set in Lindahl et al. (2015). This similarityis notable, as this moment was not targeted by our calibration, and we do not make use of any higher-order autocorrelationsacross three or more generations.

27

that siblings share fundamental factors that determine the prospects of their descendants irrespectively

of gender, but educational choices conditional on those factors do depend on gender – for example,

because the returns to education or other market conditions may be different for women than men.

4.5 Assortative Mating

Spousal correlations in years of schooling are around one half in our data (see Table 4), in line with prior

evidence from Sweden and other countries (Raaum et al., 2007). But while the similarity of spouses in

observable characteristics are well quantified, our model aims to account also for assortative mating in

unobservable factors that matter for child outcomes.

The calibrated parameter values for both the observed and latent dimensions of assortative mating are

reported in Panel C of Table 5. In the projection of zft−1 and yft−1 on zmt−1 and ymt−1 ,

E

(zft−1

yft−1

|zmt−1

ymt−1

)=

(0.663 −0.008

0.696 0.143

)(zmt−1

ymt−1

), (6)

the latent status of the mother is predominantly explained by the latent status of the father. With

a coefficient just below zero, the educational attainment of the father has no additional predictive

power. However, father’s education does have some predictive power for the educational attainment

of the mother, over and above what can be explained by the father’s latent factor. The corresponding

projection matrix for males is similar, with the educational attainment of mothers having slightly more

predictive power for the characteristics of their spouses.

These results suggest that the spousal correlation in years of schooling is predominantly a by-product

of sorting in latent characteristics. Moreover, it is those latent and more fundamental factors that

determine the educational attainment of the next generation, not parental educational attainment per

se (see Section 4.3). The key parameter for understanding the transmission of socioeconomic inequalities

across generations is thus the degree of assortative mating in latent factors. By exploiting variation

in the magnitude of kinship correlations between close and more distant kins, our approach offers an

estimate of that parameter.

The spousal correlations implied by the parametrization of our model are reported in the last block

in Panel C of Table 5. The first entry is the calibrated spousal correlation in educational attainment,

which at ρymyf = 0.49 is very similar to its sample counterpart. In contrast, the implied spousal

correlation in the latent factor z is substantially higher at ρzmzf = 0.75. This estimate is also far higher

than spousal correlations in observed outcomes estimated for countries with strong assortative mating,

such as the United States (Raaum et al., 2007).

28

Our estimates, therefore, suggest that spouses are substantially more similar in those factors that

determine the economic success of their children than they are in educational outcomes. Intuitively,

while spousal correlations in educational attainment are high, they are still far too low to rationalize

the slow decay of kinship correlations between distant relatives observed in our data. Spouses must

therefore be much more similar to each other than what is reflected in observables. This finding is

consistent also with recent evidence on the distribution of surnames (Diaz-Vidal and Clark 2015), and

the pattern of intergenerational transmission across multiple generations (Braun and Stuhler 2018).

These results add an interesting perspective on the relation between assortative mating and inequality.

A recent strand of the literature asks if increased educational sorting may explain the rise in economic

inequality that many developed countries have experienced since the early 1980s (Greenwood et al.

2014, Eika et al. 2014). But while contributing to inequalities between households, our results suggest

that trends in educational sorting may not affect the transmission of inequalities across generations

much – unless they reflect similar assortative trends in the latent factor. Because the latter explains

less than half of the variation of the former (see Panel C of Table 5), even large shifts in educational

sorting may have little implication for assortative mating in those factors that primarily affect child

outcomes. An interesting question for future research would therefore be if the sorting in educational

and latent advantages has shifted similarly over time.

4.6 Fit and Robustness

By incorporating horizontal kins, we observe a greater set of empirical moments than prior studies in the

literature. Apart from identifying a more detailed model, this also allows us to study the robustness of

our results in more detail. In particular, we study their sensitivity to the choice of empirical moments,

and the performance of the calibrated model in terms of predicting kinship correlations that were used

(in-sample fit) or not used (out-of-sample fit) in its calibration.

In a first robustness exercise, we exclude 32 moments that involve siblings-in-law of second degree,

specifically those involving the child of the sibling-in-law (group 12 in Table 5) or his or her spouse

(group 13). While reducing the number of moments from 105 to 73, calibration based on this reduced

set of moments implies nearly the same kinship correlations and model parameters as our baseline

calibration based on the full set (results available upon request). Moreover, the model also predicts the

excluded moments well.

In a second exercise, we drop nearly two thirds of our empirical moments, keeping only kinships that

we also observe in the Spanish data (described in the next section). Specifically, we drop all moments

in moment group 10 and higher, reducing the set of empirical moments by two thirds, from 105 to

35 moments. Moreover, because we drop all distant kins, this exercise tests our model’s ability to

29

extrapolate distant kinship correlations from the more narrow ones. The exercise constitutes therefore

a particularly challenging robustness test of our results for Sweden. Moreover, it illustrates if our

complete model with 20 parameters can be identified from a restricted set of moments that is even

more limited than the set of moments in our Spanish sample.

We find that our results remain quite stable even when using this restricted set of moments. Figure 3

plots the out-of-sample fit of the calibrated model for the 70 excluded moments. The model calibrated

from the restricted set of moments successfully predicts a diverse set of kinship correlations not included

in the calibration, including vertical, horizontal and distant moments. Our baseline results are therefore

robust to the choice of moments. Moreover, the model appears identifiable from information that is far

more limited than what we observe in the Swedish data.

However, our model does not provide a good fit for two kinship types, cousins and very distant in-laws.

Our baseline calibration understates the correlation between cousins by on average 27 percent, and the

correlations for in-laws of four degrees of separation (i.e. separated by four siblings and four spouses) or

higher by an even larger amount (although the deviations are small in absolute value). This gap closes

only partially if we directly target those moments in the calibration. After examination, the apparent

understatement of the kinship correlation between cousins may reflect a measurement problem, while

the understatement of the most distant kinship correlations might point to a limitation of our model.

A fundamental problem for the estimation of any distributional model is that the moments that we wish

to explain may vary over time, contrary to the identifying assumption that all moments are in a steady-

state equilibrium (see Atkinson and Jenkins 1984, Nybom and Stuhler 2014). Because the outcome

for horizontal kins can be measured at approximately the same time, our approach is arguably less

sensitive to this assumption than recent multigenerational studies, which exploit vertical information

across three or more generations. Still, various sources of selection bias – with respect to measurement

of the outcome variable, the age of parents, or identification of kins via their common ancestor (see

Section 3.1) – limit our ability to freely choose a cohort range for each kinship type.

This limitation is most severe for cousins. Because the identification of cousins requires the identification

of grandparents, our sample for cousins has to be based on more recent cohorts than the other moments

(see Appendix A). This would be unproblematic if the kinship correlations were stable over cohorts,

but the sample correlations for cousins appear less stable than the other moments.23 When including

earlier birth cohorts we estimate a substantially smaller cousin correlation that is similar to the estimate

reported by Hallsten (2014). His estimate of the cousin correlation in years of education is 0.15, which

would be very similar to the predicted cousin correlations from our theoretical model.

23To study this question, we estimated the time trends in our kinship correlations based on a two-step procedure. In afirst step, we estimated each kinship correlation separately by birth year. In the second step, we estimated a liner timetrend over the first-stage coefficients (separately for each of the 144 kinships). Appendix B provides examples, the fullevidence is available upon request from the authors. See also Bjorklund et al. (2009), who find that brother correlationsin years of schooling have been approximately constant in Sweden during our study period.

30

Figure 3: Out-of-Sample Fit in Swedish Registers

X XI XII XIII XV XVI

46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed0.00

0.05

0.10

0.15

0.20

0.25

However, including earlier birth cohorts can be problematic because their grandparents can only be

identified if their parents were born after 1932 (and thus comparatively young). Because the cousin

correlations also vary with parental age-at-birth, we chose not to include them in the estimation.

Our evidence regarding cousins remains therefore inconclusive: the mismatch between the sample and

predicted cousin correlations may reflect a data limitation and selection problem, or may point to

shared influences between cousins (see Lundberg, 2019). Data covering an even wider range of cohorts

(e.g., future versions of the Swedish registers) would be informative about this question. While we do

not use cousin correlations in our baseline calibration, including them has only negligible effects on our

parameter estimates.

In contrast, our model’s tendency to understate the most distant kinship correlations (cf. in-laws of

4th or 5th degrees of separation in Table 5) cannot be due to their variability over time. As shown in

Appendix B, the time trends in the siblings-in-law – which can be estimated for a wide range of cohorts,

because the identification of in-laws does not require the identification of grandparents – appear quite

stable. Instead, we may understate the similarity of extremely distant family members because we allow

for only a single latent factor in our model. Traditionally, the latent factor is thought to summarize

the influence of many transmission channels, such as cultural, genetic, or social determinants (e.g.,

Becker and Tomes 1986). However, some of those determinants may have higher rates of persistence

than others, and those with the strongest persistence will explain a greater share of of the kinship

correlation among distant kins.24 A model with a single latent factor may therefore still understate

the similarity of very distant kins. Consideration of a more general model that includes multiple latent

factors with different rates of persistence could therefore be an interesting avenue for future research.

24For example, the determinants of socio-economic status that relate to race (such as discrimination) or geography (suchas local economic structures) may be quite persistent if interracial marriage is rare or geographic mobility low. Stuhler(2012) argues that this “multiplicity” of transmission mechanisms is another explanation why multigenerational correla-tions do not decay as quickly as a model with a single factor would suggest. Models with persistent group membershipcan be seen as the extreme case of this argument. For example, Becker and Tomes (1986) include group dummies torationalize why racial gaps in the United States decay much more slowly than their model would otherwise predict.

31

Finally, we test the robustness our results to variation in the observed empirical correlations. Because

they are estimated from very large samples, sampling variation is not our primary concern.25 Instead,

we are concerned about a conceptual mismatch between the population that we sample from and its

theoretical counterpart. One source for such mismatch is variation in kinship correlations over time

(as detailed above), but deviations may also arise because of other factors not explicitly modeled (such

as fertility pattern, non-linearities, and so on). To check if such deviations could have a significant

effect on our findings, we test how changes in the empirical moments map into variation in the model

parameters, and vice versa.

In a first exercise, we perturb each of the 20 parameters of our baseline calibration, by multiplying it with

a random variable that is uniformly distributed between 0.95 and 1.05. Implementing this perturbation

one million times, we compare the distribution of the kinship correlations based on these perturbations

with the kinship correlation as predicted by the baseline calibration. The model predictions are centered

and smoothly distributed around the kinship correlations measured in the actual data (results available

upon request). In Appendix C we report a second exercise, in which we independently perturb each

of the 105 empirical correlations used in the benchmark specification. We repeat this exercise 10,000

times and compute descriptive statistics for each parameter. The general picture is very similar in

all the simulations, and the standard deviations and interquartile ranges are small. This observation

suggests that small variation in the underlying empirical moments would only have limited effects on

our findings.

4.7 Restricted Models

Because our model is comparatively general, one may ask if some of its components – i.e. the direct

and indirect transmission processes in the intergenerational, assortative, and sibling dimension – could

be removed without significantly reducing the model’s ability to explain the horizontal and vertical

(intergenerational) pattern of socioeconomic inequality. Figure 4 provides evidence on this question.

As a reference point, we first shut down the direct transmission mechanism, calibrating the model with

the restriction βm = βf = 0. This restricted model explains the data nearly as well as our benchmark

model, as illustrated in Figure 5a. The observation that the direct transmission mechanism is not

necessary is consistent with the observation that it explains less than two percent in the variation of

educational attainment in our benchmark model.

In contrast, it is crucial to allow for the transmission of latent characteristics. The fit of the restricted

model with γm = γf = 0 is very poor, as shown in Figure 5b. With this restriction, the intergenerational

transmission has to be captured by direct mechanisms instead (βm = 0.61 and βf = 0.54). While some

25Sampling variation is negligible even for the most distant moments in our baseline calibration.

32

Figure 4: Restricted Models

I II III IV V VI VII IX X XI XII XIII XV XVI

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

(a) No direct transmission (β = 0)

I II III IV V VI VII IX X XI XII XIII XV XVI

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

(b) No latent transmission (γ = 0)

I II III IV V VI VII IX X XI XII XIII XV XVI

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

(c) No shared sibling component

I II III IV V VI VII IX X XI XII XIII XV XVI

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

(d) Assortative mating only in observables

type of kinship correlations are understated (such as the spousal correlation, moment group I), others

are greatly overstated (such as the parent-child and sibling-in-law correlations, groups III and IV). This

misspecification cancels out for some of the more distant siblings-in-law (e.g., groups XV and XVI).

We next study how important it is to allow for siblings to share more influences than what is captured

by the direct and latent transmission channels β and γ. As shown in Figure 5c, the fit of a restricted

model in which all sibling components have variance zero (σ2xm = σ2

xf= σ2

em = σ2ef

= 0) is not as good

as the benchmark model. The sibling correlations (group II) are heavily understated, which illustrates

that siblings share distinct influences over and above the average rate of intergenerational transmission

in direct and latent factors (as captured by β and γ). The correlations for the distant siblings-in-law

(groups XV and XVI) are also understated, suggesting that the implications of shared sibling influences

on more distant relatives cannot be “imitated” by other parts of the model.

Finally, to study the role of the assortative process we calibrate the model under the assumption that

assortative mating occurs exclusively in the observed outcome, as has been a standard modeling choice

in the literature. With this more restrictive assortative process, the model does not explain the data

33

well, as shown in Figure 5d. The spousal correlation is overstated by 45 percent (0.71, outside of

the plot area), and the sibling (group II) and sibling-in-law (group IV) correlations are substantially

understated. In contrast, the distant in-law correlations are overstated, with the prediction error

increasing in relative terms with distance. Our results therefore suggest that the assortative mating in

latent advantages is crucial to match the pattern of socioeconomic inequality across distant kins.26

In sum, these results suggest that all components of our model are important, except of the direct

transmission mechanism captured by β. We nevertheless retain this channel in our model because this

channel has received attention in the economic literature; it is therefore desirable to not exclude it

apriori. Moreover, the channel might be more important for other outcomes, as we will test below.

The latent factor is the key part of the model, both in the intergenerational and assortative dimension.

Based on our results, it seems doubtful that models that do not account for the role of unobservable

characteristics could successfully fit the pattern of socioeconomic inequality across kins. We base this

finding on the observation of many and distant kinship moments. We show below that it is much harder

to discriminate between transmission models with the narrow set of moments that have typically been

used in the literature.

4.8 Other Outcomes: Income

Education is the key mediator for the transmission of socioeconomic advantages in standard sociological

and economic models (see Goldthorpe 2014). However, the transmission of educational advantages may

follow distinct patterns that may or may not generalize to other socioeconomic advantages. To study

this question, we also calibrate our model based on income as a potentially more direct measure of

socioeconomic origins and destinations.27 We summarize the results here, and provide a more detailed

account in Appendix D.

Because the Swedish registers track income profiles over nearly six decades, we can construct high-

quality measures of income for both the parent and child generation (see details in Section 3.1). Our

primary measure is the logarithm of ten-year averages of annual total pre-tax income, centered around

age 35 for children and age 45 for parents. We observe 141 distinct correlations and use the same 105

moments as in our baseline calibration for education. As for education, the calibrated model explains

26We also calibrated a model in which we allowed for one-dimensional assortative mating in the latent factor, but notin observables. The results are similar to our benchmark (results available upon request). This result suggests that ifresearchers wish to consider a simple one-dimensional model of assortative mating the latent factor should be its keycomponent, i.e. by interpreting the observed spousal correlation in education within an errors-in-variables model. Anexample for this approach is Ermisch et al. (2006).

27Income has been the primary measure of socioeconomic status in the economic literature, while education or occupationhave been more central in sociological research. This division is becoming less sharp, however. Recent sociological workemphasizes the informative content of income (Kim et al. 2018). In economics, occupational and educational measureshave become common in comparative, historical and multigenerational studies.

34

the data well, providing a close fit to both vertical and horizontal moments, and for both consanguine

and affine relationships (see Appendix Table D.1 and Appendix Figure D.1). Appendix Table D.2

reports the calibrated parameters of the transmission model, separately by intergenerational (Panel A),

sibling (Panel B), and assortative processes (Panel C).

The transmission of income follows a qualitatively similar pattern as the transmission of years of school-

ing. The latent determinants of income are more strongly transmitted than income itself, across all

three dimensions (intergenerational, sibling and assortative processes). As for education, spouses are

far more similar in latent than in observed income. At about 0.80, its implied spousal correlation is

similar to the corresponding estimate for education. Siblings share similar latent advantages irrespec-

tively of whether the socioeconomic status is measured in terms of education or income. However, the

implied father-son correlation in the latent factor is only about 0.35. That estimate is twice as large

as the sample correlation in observed income, but substantially below the corresponding estimate for

education. Our results therefore suggest that those factors that determine educational attainment are

more strongly transmitted from one generation to the next than those that determine income. These

findings are discussed further in Appendix K.

5 Kinship Correlations in Spain

We next calibrate our model for Spain, for which the intergenerational evidence has so far been quite

limited (see for example Black and Devereux 2011). Because the available data sources do not report

income for both parents and children, estimates of income mobility are based on a two-sample instru-

mental variable approach (Cervini-Pla 2015). The cross-country comparisons of educational mobility

by Hertz et al. (2008) or Blanden (2011) do not contain evidence for Spain either.28 We circumvent

these data limitations by using surnames to identify kins in the complete-count 2001 Census from the

Spanish region of Cantabria.29

In contrast to the administrative registers from Sweden, the Census data are limited to a single cross-

section, contain only educational outcomes and names, and lack direct family links to define kinship

– allowing us to illustrate the feasibility of our approach in settings with scarce data. As described

in Section 3.2, we exploit that children in Spain inherit surnames from both their parents to recover

their family links. This naming convention allows us to track both maternal and paternal lines, and at

28Blanden (2011) demonstrates that the ranking of countries in terms of educational mobility and income mobility isquite similar, with a pair-wise correlation between the two type of measures of around 0.7. While our evidence pertainsto education, it is therefore likely to be informative about the transmission of economic advantages more generally.

29Other studies have developed novel name-based estimators to study intergenerational mobility in Spain. For example,Collado et al. (2014) study multigenerational mobility in the 19th and the 20th century in Cantabria, while Guell et al.(2015) study intergenerational mobility in more recent data from Catalonia. In contrast to these studies, we use namesto identify direct family relationships and to estimate individual-level processes.

35

around 36%, the match rates are far higher than the rates that have been achieved in Census data from

other countries. As a consequence, we can compare kinship correlations between Spain and Sweden,

across a wide range of kinship types. More importantly, we can distinguish how latent intergenerational,

sibling, and assortative processes contribute to differences in intergenerational mobility rates between

the two countries.

5.1 Sample and Calibrated Moments

Table 6 reports the sample correlation in years of schooling for each kinship, sorted from closely related

to more distant kins. Columns (1) and (2) report the number of pairs and sample correlations. The

pairs are weighted inversely by the square root of family size, as described in Section 3. We observe

65 distinct moments that can be classified into groups from very close kins (such as spouses, group

I) to relatively distant kins (such as second-degree siblings-in-law, group X). Two kinship types that

were not observed in the Swedish data are child-parents in-law (group III-b in Table 6) and the Spouse

of nephew/niece-uncle/aunt (group VII-b), as its child generation was too young for this definition to

be meaningful. The sample sizes are much smaller than in the Swedish sources. They are however

still large enough for precise measurements of kinship types involving siblings and parents, and become

noisy only for the more distant types.

The kinship correlations in educational attainment tend to be slightly larger in our Spanish data than

the corresponding moments for Sweden. For example, the brother correlation in years of schooling in

Spain is 0.46, compared to 0.43 for Sweden. The gap is smaller for in-law and vertical kinships, and

inverses for those moments that involve females in the parent generation (such as the mother-son or

aunt-nephew relationships).

In our calibration we include all groups (including III-b and VII-b). We therefore use 65 distinct

moments from 12 different kinship types. We calibrate the model as described in Section 3, and

report the predicted moments as well as the percentage deviation between the observed and predicted

moments in columns (3) and (4) of Table 6. Overall, the calibrated model explains the data well.

Figure 5 illustrates the fit graphically, by plotting the sample moments and predicted moments from

the calibrated model. For comparability, we plot the same 105 moments that are also plotted for

our Swedish data in Figure 2. The mean absolute prediction error across all included moments is

9.9 percent. The prediction errors are therefore larger than in the Swedish data, consistent with the

presence of sampling error from the smaller sample sizes.

The model however appears to fit the pattern of inequality transmission in Spain across vertical and

horizontal kins, and across both direct and in-law relationships. Given the more restricted set of mo-

ments and number of observations, this conclusion is not as well supported as for the case of Sweden.

36

Table 6: Sample and Predicted Moments in Spanish Census

# ## name number sample predicted percent # ## name number sample predicted percentof pairs correlation correlation error of pairs correlation correlation error

(1) (2) (3) (4) (1) (2) (3) (4)I 1 Husband-Wife 24,819 0.543 0.569 4.7 …II 2 Brother 11,109 0.464 0.464 0.0 30 Nephew-Uncle in-law (SM) 3,334 0.199 0.205 3.0

3 Sister 10,316 0.420 0.425 1.2 31 Niece-Uncle in-law (SM) 3,067 0.168 0.183 8.5 4 Brother-Sister 21,017 0.410 0.414 0.9 VII-b 24-b W-Nephew-Aunt in-law (BF) 1,738 0.220 0.197 -10.1

III 5 Son-Father 25,860 0.385 0.369 -4.2 25-b H-Niece-Aunt in-law (BF) 1,930 0.192 0.191 -0.2 6 Daughter-Father 24,610 0.335 0.321 -4.3 26-b W-Nephew-Aunt in-law (BM) 1,737 0.213 0.198 -6.9 7 Son-Mother 25,860 0.323 0.310 -4.2 27-b H-Niece-Aunt in-law (BM) 1,873 0.224 0.210 -6.5 8 Daughter-Mother 24,610 0.300 0.284 -5.3 28-b W-Nephew-Uncle in-law (SF) 1,537 0.161 0.146 -9.1

III-b 5-b Son-Father in-law 13,191 0.262 0.276 5.1 29-b H-Niece-Uncle in-law (SF) 1,746 0.112 0.140 24.3 6-b Daughter-Father in-law 11,628 0.280 0.265 -5.4 30-b W-Nephew-Uncle in-law (SM) 1,559 0.196 0.175 -10.9 7-b Son-Mother in-law 13,191 0.245 0.225 -8.0 31-b H-Niece-Uncle in-law (SM) 1,648 0.176 0.185 4.8 8-b Daughter-Mother in-law 11,628 0.236 0.219 -7.1 VIII 32 Male Cousins (B) 2,053 0.202 0.200 -1.0

IV 9 Brother in-law (HS) 12,260 0.281 0.304 8.2 33 Male Cousins (S) 1,779 0.200 0.214 7.0 10 Brother-Sister in-law (WB) 11,184 0.300 0.309 3.0 34 Male Cousins (BS) 3,752 0.209 0.189 -9.4 11 Brother-Sister in-law (HS) 12,339 0.296 0.292 -1.2 35 Female Cousins (B) 1,747 0.145 0.140 -3.5 12 Sister in-law (WB) 10,743 0.287 0.255 -10.9 36 Female Cousins (S) 1,523 0.182 0.167 -8.1

V 13 Nephew-Uncle (BF) 3,787 0.237 0.258 8.6 37 Female Cousins (BS) 3,368 0.188 0.140 -25.4 14 Niece-Uncle (BF) 3,487 0.201 0.219 9.0 38 Male-Female Cousins (B) 3,817 0.187 0.167 -10.5 15 Nephew-Uncle (BM) 3,602 0.241 0.257 6.7 39 Male-Female Cousins (S) 3,364 0.192 0.189 -1.7 16 Niece-Uncle (BM) 3,337 0.229 0.229 0.2 40 Male-Female Cousins (BS) 3,604 0.191 0.167 -12.6 17 Nephew-Aunt (SF) 3,452 0.151 0.191 26.8 41 Male-Female Cousins (SB) 3,625 0.172 0.159 -7.9 18 Niece-Aunt (SF) 3,253 0.140 0.164 16.9 IX 42 XMMM 3,045 0.159 0.158 -0.8 19 Nephew-Aunt (SM) 3,334 0.221 0.227 2.9 43 XMMF 2,924 0.167 0.147 -11.8 20 Niece-Aunt (SM) 3,067 0.181 0.204 12.3 44 XMFM 3,089 0.176 0.194 10.3

VI 21 Brother in-law (WWS) 4,156 0.272 0.227 -16.5 45 XFMM 3,132 0.245 0.212 -13.3 22 Sister in-law (HHB) 3,296 0.249 0.221 -11.3 X 46 MMM 1,966 0.113 0.146 29.4 23 Brother-Sister in-law (HWBS) 7,061 0.234 0.207 -11.6 47 MMF 1,950 0.072 0.133 84.2

VII 24 Nephew-Aunt in-law (BF) 3,787 0.175 0.201 14.9 48 MFM 2,009 0.181 0.166 -8.6 25 Niece-Aunt in-law (BF) 3,487 0.153 0.169 10.8 49 MFF 1,881 0.121 0.121 0.1 26 Nephew-Aunt in-law (BM) 3,602 0.172 0.195 13.5 50 FMM 1,854 0.142 0.165 16.4 27 Niece-Aunt in-law (BM) 3,337 0.163 0.173 6.1 51 FMF 1,807 0.134 0.149 11.2 28 Nephew-Uncle in-law (SF) 3,452 0.192 0.178 -7.6 52 FFM 1,792 0.185 0.155 -16.1 29 Niece-Uncle in-law (SF) 3,253 0.136 0.151 11.0 53 FFF 1,710 0.124 0.112 -9.7

Table: Estimated and Calibrated Moments in Spanish CensusKinship type Data Calibration Kinship type Data Calibration

Nevertheless, two observations suggest that our results are robust. First, we demonstrated in the

Swedish sample that the full model (including the different channels via which inequalities are trans-

mitted) could be calibrated from a set of moments that was much more restrictive than the set available

for Spain (see Section 4.6). Second, our findings for Spain appear not too sensitive to changes in the

underlying set of moments or their weights.30

5.2 Intergenerational Transmission

Panel A of Table 7 reports the calibrated parameters for the intergenerational or “vertical” transmission

in Spain. We again find that the direct transmission channels captured by the parameter βk contribute

very little to the transmission of educational inequalities. At βm = 0.03 and βf = 0.11 the estimates

30Because educational attainment for females has been increasing rapidly in our observation period, its distribution isquite different in the parent and child generation. The main results remain however robust to excluding some momentsthat involve females in the parent generation, such as the mother in-law or aunt. Our results are also robust to droppingdistant kinships types for which sample sizes become very small, or to weight moments by the square root of the samplesize (results available upon request).

37

Figure 5: Fit in Spanish Census

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

Notes: See Table 6 for the corresponding list of kinship moments.

are close to the corresponding estimates for Sweden. Only about one percent of the variation in years

of schooling is directly explained by parental education (panel D of Table 7 ).

As for Sweden, the transmission of advantages occurs predominantly via the latent factor. At γm = 0.92

and γf = 0.84, the rate by which this latent factor is transmitted from parents to children is substantially

higher in Spain than in Sweden. As a consequence, the implied correlation in the latent status between

parents and children is also much higher (about 30 percent higher). The share of variance explained by

the latent factor is similar as in Sweden for men, but much lower for women.31 Our results suggest that

educational correlations decay more slowly across generations in Spain than in Sweden. For example,

the implied correlation in educational attainment between children and their great-great-grandfathers

is more than twice as large in Spain (0.16 vs. 0.07, Panel A of Tables 5 and 7).

In sum, our results have three major implications. First, the pattern of inequality transmission in Spain

and Sweden are qualitatively similar, with the transmission of advantages occurring predominantly via

latent variables, and only a minor role for parental education itself. Second, the stronger transmission

31One potential explanation for this pattern could be that secular trends in females’ educational attainment were strongerin Spain, such that less of the total variation in educational attainment is explained by individual- and family-specificfactors.

38

Table 7: Calibrated Parameters in Spanish Census

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.027 0.111 0.915 0.842σ2

ym σ2yf σ2

zm σ2zf σ2

um σ2uf

13.579 13.213 6.519 2.779 5.162 7.003αym αyf αzm αzf0.742 0.855 0.587 0.127

Parent-child correlations in z:Father-Son Father-Dau Mother-Son Mother-Dau0.760 0.827 0.732 0.883

Ancestor correlations in y and z:Father-Son Grandf-… GGrandf-… GGGrandf-…

in y 0.369 0.271 0.205 0.156 in z

Panel B: Sibling ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef

1.650 2.644 2.089 0.558 0.001 0.018Variance Shares: in y 12.1% 20.0% 15.6% 4.1% 0.0% 0.1% in z - - - 8.6% 0.0% 0.4%Sibling correlations in z:

Brothers Sisters Brother-Sister0.674 0.784 0.667

Panel C: Assortative ProcessesParameters:

rmzz rm

zy rmyz rm

yy σ2ωm σ2

ϵm

0.731 -0.139 0.418 0.357 0.381 8.369rf

zz rfzy rf

yz rfyy

1.291 0.083 0.576 0.441Spousal correlations in y and z:

ρymyf ρzmzf ρymzf ρzmyf

0.569 0.903 0.483 0.549Panel D: Variance Decomposition

% y z x Cov(y,z)Male 0.001 0.480 0.121 0.009Female 0.010 0.210 0.200 0.009

Table: Calibrated Parameters in Spanish Census

39

of educational inequalities in Spain is explained by stronger transmission in the latent factor. That is,

the difference in the intergenerational process between Spain and Sweden is not superficial, but due to

fundamental differences in the extent of status transmission. Third, standard measures understate the

difference in intergenerational transmission between Sweden and Spain. While parent-child and sibling

correlations are only slightly larger in Spain, the gap is greater for more distant relatives.

5.3 Siblings and Horizontal Transmission

Panel B of Table 5 summarizes our findings that pertain to siblings, which quantify what siblings share

over and above the average rate of intergenerational transmission discussed in the previous section. As

in the Swedish data, siblings must be far more similar to each other than what is captured by sibling

correlations in years of education. For Spain, the implied correlation in the latent status between siblings

are around 0.7 or higher, about 50 percent larger than the sibling correlation in years of education. The

similarity in siblings in observable and latent characteristics captured by xk and ek explains between 15

percent (brothers and mixed pairs) and 20 percent (sisters) of the variation in educational attainment.

The degree to which siblings are subject to common influences is therefore similar in Spain as in Sweden.

However, in Spain, most of this similarity is explained by the shared sibling influences in the observable

outcome (xk), while the shared sibling components in the latent factor (ekt ) are less important. The

variance shares explained by the shard influences in observables is about three times larger than the

corresponding variance shares for Sweden. One potential explanation for this finding are location-

specific shocks and trends. Because siblings grow up in the same location, structural changes in the

local provision of schooling would tend to be reflected in this component. Spain may have experienced

more such changes in our analysis period, which would then contribute to the similarity of siblings in

educational attainment. Because we lack long-distance horizontal relationships in the Spanish data, it

is however difficult to distinguish the two types of shared sibling components.

5.4 Assortative Mating

The calibrated parameter values for both the observed and latent dimensions of assortative mating in

Spain are summarized in Panel C of Table 5. Spousal correlations in years of schooling are around

0.54 in our Spanish sources, about 10 percent higher than the corresponding moment in the Swedish

registers (see table 4). The expectation of zft−1 and yft−1 conditional on zmt−1 and ymt−1 is estimated as

E

(zft−1

yft−1

|zmt−1

ymt−1

)=

(0.731 −0.139

0.418 0.357

)(zmt−1

ymt−1

), (7)

40

As was the case for Sweden, the latent status of the mother is predominantly explained by the latent

status of the father, while his educational attainment has no additional predictive power. However, the

father’s education has a substantial association with maternal education, over and above what can be

explained by the father’s latent factor. The corresponding projection matrix for females is similar.

The spousal correlations implied by these parameters are reported in the last block of Panel C. The

first entry is simply the calibrated spousal correlation in educational attainment, which at ρymyf = 0.57

is similar to its sample counterpart. In contrast, the implied spousal correlation in the latent factor is

substantially higher, which at ρzmzf = 0.90 is also more than 10 percent higher than the corresponding

estimate for Sweden. Our results, therefore, suggest stronger assortative mating in Spain compared

to Sweden – not only in educational attainment, but also in the latent determinants of socioeconomic

status. Spouses in Spain appear extremely similar in those factors that ultimately determine the

educational attainment of their descendants.

6 Genetic and Non-Genetic Transmission

Our objective so far was to quantify how strongly advantages are transmitted from one generation

to the next, and to distinguish how intergenerational, sibling and assortative processes contribute to

socioeconomic inequality. To capture their overall strength we interpreted those processes broadly, and

remained agnostic about the specific causal mechanisms that they represent. However, our approach

can be used to distinguish between different causal mechanisms, to the extent that they have specific

statistical implications about the pattern of inequality transmission across kinship types.

We first illustrate that our model nests the standard model of genetic transmission (Section 6.1),

and test if this genetic model can explain the transmission of educational attainment or body height

– as compared to the more general model, which allows for additional, non-genetic pathways. We

demonstrate that a purely genetic model cannot fit the distribution of educational advantages across

kins. Importantly, the failure of the standard genetic model becomes visible only when a sufficiently

large number of kinship moments becomes available.

We then generalize our baseline model to separate the contribution from genetic and non-genetic latent

factors. We show that the extensive array of kinship data underlying our study does in principle allow

for separate identification of multiple, distinct latent processes, and calibrate this more general model

in educational advantages in Sweden (Section 6.2). We find that genetic and non-genetic transmission

mechanisms have similar statistical implications in the vertical dimension, but very distinct implications

in the horizontal dimension across siblings-in-law. The horizontal approach therefore allows us to

quantify each channel separately.

41

6.1 The Standard Genetic Model

In this section we describe the standard model in quantitative genetics to evaluate the correlation be-

tween relatives. The fundamental idea here is that the observed outcome or “phenotype” is determined

by genetic and environmental factors. Each individual receives half of its genetic contribution from the

father and the other half from the mother. In the simplest form the model assumes that genetic and

environmental factors contribute additively to the phenotype, i.e. do not interact and are uncorrelated.

Assortative mating is based on similarity in the phenotype and the population is at the steady state.

This standard quantitative genetic model is a special case of our general model. Specifically, in Ap-

pendix K we obtain formulas for the correlations between relatives that coincide with the ones in, for

example, Crow and Felsenstein (1968).

The genetic model is nested in our general model by imposing three sets of restrictions. First, assume

that the parental outcome (phenotype) does not have any direct association with the child outcome

(imposing the restrictions βk = 0, k = f,m). Second, because the latent factor represents genes it is

transmitted from parents to children as

zkt =zmt−1 + zft−1

2+ vkt (8)

where vkt is uncorrelated across relatives and to zmt−1 and zft−1 (imposing γk = 1, αkz = 0.5, and σ2ek

= 0,

k = f,m).32 Finally, assortative mating occurs only in the outcome (phenotype) y. As a consequence,

ρzmyf , ρymzf and ρzmzf are functions of ρymyf and some of the other parameters of the model. This

genetic model has only 5 parameters, which account for the share of variance in the phenotype explained

by the genotype (σ2zm = σ2

zf= σ2

z), the environmental effects shared by siblings (σ2xm , σ2

xfand σxmxf ),

and the assortative mating in the phenotype (ρymyf ). Heritability is defined as the share of the variance

in the outcome variable that is due to genetic variance.

The Standard Genetic Model and Education

We calibrate this standard genetic model for years of schooling in the Swedish registers, using the same

105 kinships that we used for calibration of our baseline model (Section 4). Figure 6a illustrates the

in-sample fit graphically, to be compared against the fit of our baseline model (Figure 2). The standard

genetic model fits substantially worse, overstating kinship correlations within the nuclear family while

understating those for more distant relatives. Its mean prediction error is 59.7 percent, compared to

only 4.5 percent for our baseline model.

32One might wonder why equation (8) contains an error term, given that children inherit their genes exclusively fromtheir parents. The explanation is that a genotype is defined by a combination of genes/alleles, and different childrenreceive different combinations (with the exception of monozygotic twins). This result is derived formally in AppendixSection K.

42

Figure 6: Model Fit (Standard Genetic Model, Education)

(a) Using 105 Moments

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

(b) Using 15 Moments

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

Notes: See Table 4 for the corresponding list of sample moments.

Our findings are therefore inconsistent with a purely genetic interpretation. We base this rejection

on a simple statistical observation – the genetic model is not able to provide a close in-sample fit to

the distribution of educational advantages across kins. However, this observation is not necessarily

inconsistent with the view that genes are one of the decisive factors in the transmission from parents to

their children. Instead, we reject the standard genetic model from quantitative genetics, in particular

the assumption that assortative mating occurs only on the phenotype. Spouses must be far more

similar in fundamental determinants of socioeconomic success than what can be discerned from spousal

correlations in socioeconomic outcomes themselves.

This finding is based on the observation of distant siblings-in-law – intuitively, failure of the assor-

tative assumptions contained in the standard genetic model becomes evident when trying fit kinship

correlations that are a function of multiple assortative matches. In contrast, the genetic model appears

to perform well when fitting a restricted set of kinship correlations that includes only close kins. To

illustrate this point, we recalibrate the standard genetic model using only sibling correlations (group

II in Table 4), parent-child correlations (group III), and nephew/niece/uncle/aunt correlations (group

V). As shown in Figure 6b, the genetic model provides an excellent in-sample fit for those 15 close

kinships. However, it provides an extremely bad fit to all other kinship moments. To test the validity

of transmission models based on their statistical properties, researchers therefore need to observe a suf-

ficiently wide range of sufficiently distant kinship correlations.33 We argue that applied to large-scale

administrative registers, a horizontal approach can provide the necessary moments.

33Most economic research to date has been based on even narrower set of kinship correlations than the 15 momentsused in this example. Prior work may therefore not have had enough “power” to distinguish different transmission modelsbased on their statistical properties. Ruby et al. (2018) make a similar point with respect to estimation of the heritabilityof human longevity.

43

The Standard Genetic Model and Height

That the standard genetic model cannot fit the kinship pattern in educational advantages is not a

mechanical consequence of the model having comparatively few parameters. To demonstrate this, we

also calibrate the genetic model for body height. Because genes are known to be the primary source of

variation in height, a genetic model should in principle provide a reasonably good fit in this case. In

Appendix F we show that this is the case. The heritability, i.e. the share of variation in height explained

by the latent genetic factor, is about 73 percent (see Panel D of Table F.1). This estimate is slightly

below estimates of the heritability of height from quantitative genetics. However, we also find that the

baseline model still provides a better fit than the genetic model. Given the low spousal correlation in

phenotype height, the standard genetic model cannot explain why the kinship correlations in height

remain non-negligible for distant siblings-in-law.

6.2 General Assortative Mating Model with Two Latent Factors

The latent variable z is a ”black box” that might be composed of variables of a very different nature.

Although the objective of our study is not to decipher the elements that make up z, it may be useful

to try to quantify which part of the latent factor z is composed of genetic factors and which is based

on cultural or environmental factors. Thus, we now decompose the latent factor zkt into a genetic

factor, zG,kt , and a socio-cultural or ”cultural” factor, zC,kt . The factor zC,kt captures all the cultural and

environmental factors that are transmitted but which are not genetic. We assume that the outcome y

for an individual from generation t and gender k is given by

ykt = βkykt−1 + zG,kt + zC,kt + xkt + ukt (9)

We model the genetic factor following the standard model of genetic inheritance with assortative mating

used in Quantitative Genetics (Crow and Felsenstein, 1968). The genetic factor of the child, zG,kt ,

depends on the father zG,mt−1 as well as on the mother zG,ft−1

zG,kt =zG,mt−1 + zG,ft−1

2+ vG,kt (10)

where vG,kt is a white-noise error term. The variances of zG,mt and zG,ft are the same. Notice that we

do not need to impose the condition that is often assumed in quantitative genetics that the “environ-

ments” of parents and offspring are independent since the latent variable zC,kt captures such shared

environment.34 The cultural factor, zC,kt , depends on the father zC,mt−1 and the mother zC,ft−1, as zkt in our

baseline model. The remaining elements are also define analogously.

34We also allow for a possible correlation between the environments of siblings. In Quantitative Genetics it is commonto assume that the covariance of the genetic contribution and the environment is zero. In our case, if we see zC,k

t as part

44

We assume there is assortative mating both in years of schooling and in the cultural factor, but there

is no direct assortative mating in the genetic factor.35 This assumption means that the coefficient of

zG,mt−1 in the linear projections of zG,ft−1 , zC,ft−1 and yft−1 on zG,mt−1 , z

C,mt−1 and ymt−1 is zero. The model initially

depends on the 15 correlations between the elements of (zG,mt−1 , zC,mt−1 , y

mt−1, z

G,ft−1 , z

C,ft−1, y

ft−1). However, in

the online appendix, we show that, in the steady-state, there are only four free correlations, ρzC,mzC,f ,

ρzC,my,f , ρymzC,f and ρymy,f , the remaining ones are functions of these four and the other parameters

of the model. Then, the two-factor model has only one more parameter than the benchmark, that is

the variance of the genetic factor σ2zG

that is the same for both genders. In the online appendix, we

provide formulas for the correlations between relatives as a function of the parameters of the model.

We now calibrate the parameters using years of schooling for Sweden and the 105 correlations of our

baseline specification.36 The estimated parameters are presented in Table 8. The general picture is very

similar to the benchmark specification. The indirect transmission channel through the cultural factor

is much more important than the direct transmission channel. The shocks to the cultural factor shared

by siblings are very significant and, therefore, the sibling correlation in the cultural factor is much

bigger than in the observed outcome. Assortative mating takes place mainly in the cultural factor.

Educational attainment is correlated between spouses because it is correlated with their respective

“cultural” latent factor.

An interesting part of this exercise is to asses the relative contribution of the genetic and non-genetic

factors to the variance in years of schooling. We find that the cultural factor explains 38% of the

variance of the observed outcome for males and 31% for females, whereas the ”heritability”, i.e. the

share of the variance explained by genes, is only 7%. This estimated heritability is in line with recent

evidence from genome-wide association studies (GWAS) based on direct genetic information. For

instance, Lee and et al. (2018), using a sample of more than a million individuals, find that a polygenic

score based on genes explains between 11− 13% of the variance in years of schooling. Papageorge and

of the environment, such covariance does not need to be zero; however, in our calibration, it turns out to be basically zero.An additional difference with the standard genetic model is that we only consider an “additive” genetic component andwe leave out the “dominance” contribution. Thus, we estimate the so-called weak heritability. In any case, this dominancecontribution, which is shared by siblings, if it were of positive magnitude would be picked up by our variable x. What wedo leave out of this model are the possible non-additive gene-environment interactions. This is an important restrictionthat is shared by most classical heritability statistics. See Feldman and Ramachandran (2018).

35Bidner and Knowles (2018) develop a theoretical model to analyze intergenerational transmission of innate ability,which is a latent variable that can be seen as genes, in which assortative mating plays a fundamental role. However,they assume no gender differences so that the distributions of all the variables and transmission mechanisms are the samefor men and women. Rice et al. (1978) and Cloninger et al. (1979) used Wright’s work (Wright, 1921) on path analysisto develop a series of theoretical model very much related to our model here. In particular, they allow for assortativemating, gender differences and genetic and cultural transmission. Those model were applied in the empirical literatureon extended twin-family in which a large number of kinship correlations are used (Truett et al. 1994, Eaves et al. 1999).The main differences with our model are that they do not consider both types of assortative mating, do not allow for adirect effect of the output of the parent on the output of the child, and most of the correlations involved some type oftwin. Moreover, they study personality and social attitudes instead of socioeconomic outcomes, based on samples thatare much smaller than the ones we consider here.

36The fit is very good, which is not surprising, since the fit was already very good for our benchmark model. As in thebenchmark case, the out-of-sample predictions are also good.

45

Table 8: Calibrated Parameters in Swedish Registers (Education, Two Factor Model)

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.113 0.098 0.691 0.586σ2

ym σ2yf σ2

zcm σ2zcf σ2

zgm

4.648 4.465 1.778 1.375 0.312αym αyf αzm αzf

0.447 0.000 0.574 0.657Within-person correlations in y and z:

ρyzc ρyzg ρzczg males 0.670 0.301 0.032 females 0.599 0.301 0.032Parent-child correlations in z:

Father-Son Father-Dau Mother-Son Mother-Dau in zc 0.578 0.579 0.537 0.509 in zg 0.512 0.512 0.512 0.512 in zc+zg 0.584 0.584 0.549 0.527Ancestor correlations in y and z:

Father-Son Grandf-… GGrandf-… GGGrandf-… in y 0.379 0.210 0.122 0.071 in zc+zg 0.584 0.342 0.200 0.117

Panel B: Sibling ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef

0.118 0.174 0.000 0.679 0.729 0.651Variance Shares: in y 2.5% 3.9% 0.0% 14.6% 16.3% 14.3% in z - - - 38.2% 53.0% 41.6%Sibling correlations in z:

Brothers Sisters Brother-Sister in zc 0.750 0.886 0.778 in zg 0.512 0.512 0.512

Panel C: Assortative ProcessesSpousal correlations in y and z:

ρymyf ρymzcf ρymzgf ρzcmyf ρzcmzcf ρzcmzgf

0.491 0.518 0.096 0.548 0.703 0.079ρzgmyf ρzgmzcf ρzgmzgf

0.080 0.047 0.024Panel D: Variance Decomposition

% y z zg xMale 0.009 0.383 0.067 0.025Female 0.010 0.308 0.070 0.039

Table: Calibrated Parameters in Swedish Registers (Two Factor Education)

46

Thom (2018) apply the same polygenic scores in the Health and Retirement Study and find values

between 4.5−9.7%.37 These estimates are potentially subject to an upward bias from the correlation of

polygenic scores with environmental factors.38 To address this problem, researchers use within-family

designs that exploit random variation of actual around the expected genetic relatedness.39 Estimates

with family fixed effects tend to be smaller, but much of the relationship between polygenic scores and

educational attainment remains (Papageorge and Thom, 2018).

Estimates from GWA studies are much lower than those typically found in the behavioral genetics

literature using correlations between different types of twins or adoptees (Bjorklund et al., 2006),

which estimates that genes explain at least between 30 and 40% of the variance in years of schooling

(Branigan et al., 2013). This difference between the traditional twin-based estimates and the more

direct evidence from GWA studies is sometimes called “missing heritability” (see Golan et al., 2014).

Some authors consider that those high values are a consequence of the variance analysis used and the

lack of a correct recognition of a non-genetic transmission of many cultural traits in twin studies.40

Thus, Feldman and Ramachandran (2018) claim that the results obtained from those twins studies are

not the standards relative to which other variance analyses should be compared.

In particular, the twin approach is sensitive to assumptions on the degree of assortative mating, which

cannot be well estimated in traditional data with only close kins. Ruby et al. (2018) illustrate this point

by using large-scale ancestry information from the Ancestry repository to estimate the heritability

of human longevity.41 While their estimates based on close genetic relatives agree with the prior

literature, estimates based on more distant relatives suggest that assortative mating in longevity is

37Previous GWA studies based on smaller samples found lower estimates (e.g., Okbay and et al., 2016), but the predictivepower of PGS increases with sample size. Cesarini and Visscher (2017) argue that recent studies based on millions ofobservations (such as Lee and et al., 2018) come however close to the theoretical upper bound of the approach.

38Estimates from GWA studies capture only additive effects of certain single-nucleotide polymorphisms, and mighttherefore represent a lower bound of the genetic contribution (Selzam et al. 2017). On the other hand, they may bebiased upward by the correlation of polygenic scores with environmental factors (Domingue et al., 2018). The distinctionbetween environmental and genetic factors may not be sharp. For example, the observation that the non-transmittedalleles of parents predict child outcomes over and above the polygenic score constructed from transmitted alleles suggeststhat environmental factors may itself be partly determined by the genetic endowments of the parents (“genetic nurture”,Kong et al. 2018).

39Whether a sibling receives a piece of DNA from one parent or another resembles the outcome of a coin toss (randomMendelian segregation of DNA). Siblings share half of their DNA on average, but the actual share varies around thisaverage. Within-family variation in genetic relatedness therefore resembles a “natural experiment” and should be closeto independent of pre-existing environmental factors. However, the number of sibling pairs is often too small to yieldprecise heritability estimates. Young et al. (2018) propose a new approach that exploits variation in how much more orless related any pair of individuals is than would be expected from the relatedness of their parents. The method requiresgenetic information of the parents and a sufficiently large sample has so far been only available for Iceland. They findthat genes explain about 17% of the variation in educational attainment, but with a standard error of 9.4.

40Twin studies are subject to a number of conceptual issues, including that (i) gene-environment interactions are notcaptured by the linear model used, (ii) the overall effect of genes may not add up linearly (e.g., dominance of alleles orepistasis, dominance of genes), (iii) the environment may be different for twins or adoptees than for other sibling types(Goldberger, 1979), and (iv) there might be less variability in the phenotype among families with twins or adoptees.

41Kaplanis et al. (2018) use a similar approach to Ruby et al. (2018) to estimate the heritability of longevity using datafrom publicly-available online repositories. They consider 16 different types of kinships but do not include siblings-in-law,and do not allow for assortative mating.

47

much stronger, and the heritability of longevity therefore much weaker. Ruby et al. exploit that the

phenotype correlation between siblings-in-law must be mostly due to assortative mating. We note

that by tracking distant relatives in large-scale administrative register data, the same strategy can be

applied to socioeconomic outcomes. The large coverage of the administrative sources potentially allows

researchers to consider extremely distant siblings-in-law.42

Our results are therefore consistent with evidence on the heritability of educational attainment from

recent GWA studies based on direct genetic information. The observation that we find smaller values

of such heritability than the literature on different types of siblings and twins seems in turn consistent

with recent “big data” studies on the heritability of longevity based on publicly available pedigree

data. By exploiting administrative data sources, we can overcome two important limitations that these

two recent kinds of literature face. First, we can track distant siblings-in-law and therefore consider

many more, and more distant empirical moments. This allows us to consider a more general model

and subject it to more challenging out-of-sample tests. Second, administrative data contain many more

variables, including socioeconomic outcomes, which are of more direct interest in the social sciences.

While the genetic factor explains only a small share of the variance in years of schooling, this does not

imply that the correlation between the genetic factor and years of schooling is small. We find that this

correlation is 30% for both men and women. It is also interesting to note that we found a close to zero

correlation (0.03) between the latent genetic factor and the latent cultural factor. If we see the cultural

factor as part of the socioeconomic environment provided by parents, our findings are in accordance

with Papageorge and Thom (2018), where it is proved that the relationship between polygenic scores

and the socioeconomic status of the family is also very small. Another interesting result that provides

some reassurance in our method is the degree of genetic assortative mating that we find, 2.5%, which

is similar to the estimates that recent GWAS studies find (Domingue et al. 2014, Yengo et al. 2018).

Splitting the unobservable factor into a genetic and a not-genetic component has no consequences for

the implied long-run correlations. The correlations with grandparents, great-grand-parents and great-

great-grandparents in years of schooling are basically identical to those obtained from the benchmark

model. Moreover, if we define the total latent factor as zkt = zG,kt + zC,kt , the long-run correlations in

this aggregate factor are also identical to the implied long-run correlations in the latent factor from

our baseline specification. This observation relates to the argument that the observation of distant

vertical correlations would not be informative about the relative contribution of genetic and non-genetic

pathways to status persistence – our estimates suggest that their transmission patterns are statistically

too similar to be distinguishable by a vertical perspective alone.

42For reasons explained above, we restrict our attention to siblings-in-law of five degrees of separation. Ruby et al.(2018) consider siblings-in-law of one degree of separation (corresponding to our kinship group IV) and two types ofco-siblings-in-law “law-sib-law” (our group VI) and “sib-law-sib” (group IX). A comparison between these moments willbe informative about the relative similarity of spouses as compared to siblings. However, the observation of siblings-in-lawof higher degrees of separation are useful for testing the sensitivity of results, or to evaluate the model fit out-of-sample(see Section 4.6).

48

7 Conclusions

We have proposed a new approach to estimate long-run intergenerational socioeconomic mobility. While

sharing conceptual similarities with prior work on the correlation between relatives in quantitative

genetics and economics (e.g., Behrman and Taubman, 1989), it deviates in two crucial aspects. First,

our objectives are different. We are primarily interested in the overall extent to which advantages are

transmitted from one generation to the next, less in the nature-nurture debate at the heart of prior work.

We thus do not need to deal with the complicated relation between genes and environment (Goldberger,

1979), and instead capture both, providing a more comprehensive account of intergenerational and

assortative processes.

Second, we exploit large-scale administrative registers to track “horizontal” relatives, such as distant

siblings-in-law. This horizontal approach has many advantages compared to the conventional “vertical”

approach. It remains feasible in settings in which distant ancestors are not available, and socioeconomic

outcomes can be measured within a single data source, at approximately the same age and time. The

horizontal also yields many more kinship moments than the vertical approach. In the Swedish registers,

we obtained precise information on 141 distinct kinship moments. In a second application, we obtained

65 moments from a single cross-sectional census from Spain.

The set of kinship moments considered in this study is thus far more extensive than what has been

available in past research (for example, Behrman and Taubman considered only eight kinships). As a

consequence, we can distinguish transmission mechanisms in more detail than what has been possible

in previous work. In particular, we allow for latent transmission mechanisms in each component of the

transmission model – in intergenerational, sibling and assortative processes. This choice is motivated

by recent multigenerational studies, which show that socioeconomic inequalities are more persistent

across multiple generations than what can be rationalized by parent-child transmission in observable

characteristics (e.g. Clark 2014, Lindahl et al. 2015, Braun and Stuhler 2018).

We find strong persistence not only in the intergenerational, but also in the sibling and assortative

processes. Conventional measures based on observable characteristics greatly understate the impor-

tance of family background, and the extent to which latent advantages correlate between kins. This

applies even to sibling correlations, which are known to be a more comprehensive measure of family

background than intergenerational correlations (Jantti and Jenkins, 2014). We however find that they

still understate – by about 50 percent – the similarity of siblings in latent advantages. This similarity

matters in so far as it is those latent advantages that determine the socioeconomic prospects of their

descendants.

Most striking is the degree of assortative matching that our data imply. Applied in large-scale admin-

istrative data, our approach allows us to track distant siblings-in-law. Because they are separated by

49

many spousal links, it becomes quite apparent if a model cannot fit the assortative process in latent

or observable characteristics. Our data suggests an extremely high degree of assortative matching,

with correlations in the latent advantages estimated to be around 0.75 in Sweden and as high as 0.9

in Spain. These findings imply that there is far greater sorting between spouses than previous studies

have indicated.

The tracking of distant siblings-in-law also allowed us to contribute to the nature vs. nurture debate, in

the final part of our paper. By observing both consanguine (“blood”) and affine (“in-law”) relatives, we

can distinguish the role of genetic and non-genetic transmission mechanisms. We show that a standard

genetic model cannot fit the kinship pattern in educational advantages. The observation of a large set

of kinship moments is key for this conclusion. The genetic model appears to explain the transmission

process well when considering a small set of primarily vertical moments. However, it fails to explain

the remarkable similarity of distant siblings-in-law and other distant relatives that we document in this

study.

Genetic and non-genetic pathways are difficult to distinguish because they generate a similar statistical

pattern in the intergenerational dimension. In contrast, their assortative pattern turns out to be

very distinct. The horizontal approach can, therefore, be used to decompose genetic from non-genetic

mechanisms. Calibrating an extended two-factor model, we show that genetic and non-genetic latent

factors are transmitted at a similar rate from parents to children, with about 7% of the variation in

years of schooling being explained by variation in genotype. However, the similarity of spouses is nearly

exclusively due to non-genetic factors. Remarkably, our quantitative results match recent findings from

behavioral genetics, even though our approach is very different.

We hope our study illustrates the enormous potential of tracking distant consanguine and affine kins

in large-scale administrative data. Using this approach, we believe it might be possible to provide a

comprehensive and unified account of intergenerational and assortative processes that links evidence

from population genetics with intergenerational, sibling and multigenerational evidence from the social

sciences. This study is a first step in providing such unified account, and we hope that future work will

extend on our approach. In particular, it would be useful to treat fertility outcomes more explicitly,

to allow for multiple latent factors with different rates of transmission, and to rely less on steady-state

assumptions than we do here.

50

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Empirical Appendix

A Cohort Distribution

Figure A.1: Distribution of Birth Cohorts (Sweden, Education)

0.0

1.0

2.0

3.0

4D

ensi

ty

1920 1940 1960 1980birth year

(a) Brothers(2)

0.0

2.0

4.0

6.0

8.1

Den

sity

1920 1940 1960 1980birth year

(b) Male Cousins(32)

0.0

5.1

Den

sity

1920 1940 1960 1980birth year

(c) Father-Son(5)

0.0

1.0

2.0

3.0

4D

ensi

ty

1920 1940 1960 1980birth year

(d) Brother-in-law(9)

0.0

1.0

2.0

3.0

4.0

5D

ensi

ty

1920 1940 1960 1980birth year

(e) Brother-in-law, 2nd-degree(46)

0.0

1.0

2.0

3.0

4.0

5D

ensi

ty

1920 1940 1960 1980birth year

(f) Brother-in-law, 3rd-degree(116)

Notes: The figure plots the histogram of sampled birth cohorts for six example moments. (kinship ID in parentheses).

Multiple selection issues need to be taken into account when selecting sub-samples for each kinship

moment. We first select cohorts for which the outcome is reliably observed, as described in Section

3.1. We then assess which kinship types can be reliably identified within those cohorts. For example,

the identification of siblings requires observation of their parents, while for the identification of cousins

we need to observe grandparents, and so on. In principle, our data contain family links up to four

generations. However, we are more likely to observe great-grandparents if individuals became parent

at a comparatively young age, which could introduce a selection bias in our estimates (see also Hallsten

2014). We therefore abstain from kinship types that depend on the identification of great-grandparents,

such as second-degree cousins. We further minimize selectivity concerning the age difference between

kins. For example, parent-child correlations tend to be lower for young parents, for whom the age

difference to their children is small. We avoid this problem by considering a range of child cohorts that

60

is sufficiently narrow, such that their parents and their outcomes are observed irrespective of parental

age-at-birth. Horizontal moments, such as sibling and spousal correlations, vary less systematically with

the difference in age. However, the issue could become more severe for the very distant in-law moments,

because the age difference between kins tends to increase with the degree of separation – e.g., siblings

tend to be more closely spaced than cousins. We avoid this problem by considering a broad range of

cohort for all horizontal moments. A final issue is that intergenerational and assortative processes may

vary across birth cohorts, which represents a fundamental problem for the analysis of any distributional

model (see Section 4.6). We address this issue in two ways. First, we select similar cohort ranges for

different kinship types, as far as possible given the other constraints mentioned above. Second, we

examine explicitly if kinship correlations change over birth cohorts. As we describe in Appendix B, the

kinship correlations in education remain quite stable in our analysis period. Most problematic are time

trends in the cousin correlation, because the requirement to identify grandparents forces us to consider

more recent birth cohorts (such that grandparents are observed irrespective of parental age-at-birth).

We return to this issue below.

Figure A.1 plots the distribution of birth cohorts for six example moments. Sub-figures (a)-(c) show

the histograms for brothers, male cousins, and father-son pairs. The sample of siblings is restricted

to individuals born between 1932 (the first cohort for which parents and therefore siblings are linked)

and 1976 (the last cohort before right-censoring in years of schooling becomes apparent). We further

restrict the sample to those who had a child until the end of our sampling range in 2003, to increase

comparability with other kinship moments that are subject to this restriction by definition. We impose

the same sampling restrictions for all other “horizontal” moments. The exceptions are the cousins

(sub-figure b). Because the observation of cousins requires the observation of their grandparents,

we are forced to sample cousins from a more recent and narrow range of cohorts. Given that the

grandparent generation can be reliably observed only for parents born after 1932, we consider cousins

born between 1966 and 1976 (see Hallsten 2014 for a more detailed discussion of sampling limitations

in the identification of cousins in the Swedish registers). Most parents of these cohorts have been born

after 1932, while sampling earlier cohorts would introduce selectivity with respect to parental age at

birth. For comparability, we sample the same cohort range for all other vertical kins, such as the father-

son correlation (sub-figure c). Sub-figures (d)-(f) show the corresponding histograms for brothers-in-law

of first, second and third degree of separation. The cohort distribution of the brothers-in-law (blue)

is shifted to the right relative to the index person (green) because the considered brother-in-law is the

husband of the sister, and husbands are on average older than their wives. Comparison between (d)-(f)

illustrates that this age gap increases with each degree of separation (i.e., with each linkage of female

spouses). To avoid selectivity issues from the interaction between such systematic and kinship-specific

age shifts and the fixed sampling windows, it is essential to consider a wide range of birth cohorts, as

we do here.

61

B Cohort Trends

Figure B.1: Mobility Trends over Cohorts (Sweden, Education)

0.1

.2.3

.4.5

corr

elat

ion

1930 1940 1950 1960 1970 1980cohort

observed linear trend

(a) Brothers(2)

0.1

.2.3

.4.5

corr

elat

ion

1966 1968 1970 1972 1974 1976cohort

observed linear trend

(b) Male Cousins(32)

0.1

.2.3

.4.5

corr

elat

ion

1966 1968 1970 1972 1974 1976cohort

observed linear trend

(c) Father-Son(5)

0.1

.2.3

.4.5

corr

elat

ion

1920 1940 1960 1980cohort

observed linear trend

(d) Brother-in-law(9)

0.1

.2.3

.4.5

corr

elat

ion

1920 1940 1960 1980cohort

observed linear trend

(e) Brother-in-law, 2nd-degree(46)

0.1

.2.3

.4.5

corr

elat

ion

1920 1940 1960 1980cohort

observed linear trend

(f) Brother-in-law, 3rd-degree(116)

Notes: The figure plots the estimated kinship correlation separately for each birth cohort for six example moments (kinship ID inparentheses). The size of each circle is proportional to the number of observations. The birth cohorts as indicated on the x-axis aredefined for one side of each kinship pair (e.g. for the son in sub-figure b) and remain unrestricted for the paired observation (e.g.the father).

Kinship moments are defined over different cohorts, and may, therefore, be sensitive to variation in the

outcome distribution or mobility trends across cohorts. Because we can measure outcomes within the

same data source and in the same periods, our approach should be less sensitive to this problem than

the traditional multigenerational approach. We have nevertheless studied the non-stationarity of the

outcome and kinship distributions in some detail, and rescaled the outcome variables to minimize its

consequences. Each outcome variable is demeaned by birth cohort and gender. Further standardizations

of the variance (by constructing z-scores for each cohort and gender) had only negligible effects on our

results. Shifts in the marginal distribution of educational outcomes across cohorts are therefore not

our primary concern. A more severe issue is that kinship correlations may not be stable across birth

cohorts. To study those trends, we estimated each of our 141 kinship moments separately for each

birth cohort. We find that most kinship correlations are quite stable across cohorts. For illustration,

Figure B.1 plots the kinship correlation in years of schooling for the same six example moments that

we also considered in Appendix A.1. The kinship correlations for brother, male cousin and father-son

62

pairs (subfigures a-c) are quite stable, as are the kinship correlations for brothers-in-law of different

degree of separation (sub-figures d-f). The cohort variation tends to be a bit more pronounced for

moments involving females, but always remain small within our sampling window. We conclude that

the kinship correlations in education are sufficiently stable within the generation and range of cohorts

in which they can be directly estimated. However, kinship moments may vary more substantially

between generations. In particular, we estimate the cousin correlations in a narrow band of cohorts

born between 1966 and 1976, and while the cousin correlations trend only slightly upward within

this range they might of course vary more substantially for cohorts born in the 1940s or 50s. We

discuss this difficulty of fitting cousin correlations in the main text. The mobility trends are also more

pronounced for the kinship correlations in income, in particular for pairings that involve female kins. As

we acknowledge in the manuscript, such non-stationarity is a fundamental problem for the estimation of

any distributional model. While our “horizontal” approach is arguably less affected than the “vertical”

approach, accounting for non-stationarity remains a central objective for future work.

C Robustness Checks

We cannot estimate the parameters by GMM following Abowd and Card (1989) because the units of

analysis (families) are not well defined. Moreover, most individuals will belong to different families and

therefore the sample units will not be independent, as illustrated in the following example. Using the

notation in the Figure 1, consider family a composed by a, x (spouse of a), ax (son of a and x), b

(sibling of x and sibling-in-law of a), y (spouse of b and spouse of the sibling-in-law of a), by (nephew

of x and nephew-in-law of a), and so on up to siblings-in-law of degree 5 of a. Now consider family d

composed by d (who also belongs to family a because d is the sibling-in-law of degree 3 of a), w (the

spouse of d and also part of family a), dw (son of d and w who is not a member of family a since

we are not considering nephews-in-law of degree 2), the siblings-in-law up to degree 2 of d (who are

also members of family a), and the more distant in-laws of d that are not part of family a since we

only consider siblings-in -law up to degree 5. As this example shows there is an overlap of some family

members across families and therefore, families are not independent.

63

Table C.1: Sensitivity of Main Parameters to Perturbations of Empirical Moments

Main parametersOriginal Mean St.dev Min P05 P25 Median P75 P95 Max IQR

σ2zm 2.070 1.965 0.197 1.175 1.585 1.860 1.997 2.101 2.235 2.468 0.240σ2zf

1.560 1.459 0.187 1.024 1.141 1.313 1.481 1.602 1.738 2.039 0.289

ρzmzf 0.754 0.726 0.049 0.561 0.638 0.691 0.734 0.763 0.794 0.844 0.071ρzmyf 0.580 0.570 0.017 0.474 0.535 0.563 0.574 0.582 0.590 0.601 0.019

ρymzf 0.540 0.527 0.024 0.480 0.487 0.508 0.530 0.546 0.564 0.594 0.038

ρymyf 0.489 0.489 0.007 0.476 0.478 0.483 0.489 0.495 0.500 0.502 0.012

γm 0.664 0.663 0.015 0.604 0.637 0.653 0.663 0.674 0.688 0.711 0.021γf 0.565 0.564 0.015 0.510 0.538 0.553 0.564 0.574 0.589 0.618 0.022βm 0.144 0.162 0.034 0.069 0.113 0.138 0.157 0.181 0.228 0.289 0.044βf 0.129 0.146 0.029 0.085 0.110 0.125 0.138 0.163 0.203 0.270 0.038σ2xm 0.178 0.175 0.038 0.019 0.113 0.147 0.176 0.204 0.233 0.288 0.057σ2xm 0.246 0.249 0.034 0.152 0.195 0.222 0.249 0.274 0.303 0.352 0.052σxmxf 0.069 0.065 0.031 0.000 0.016 0.042 0.065 0.089 0.115 0.163 0.047σ2em 0.657 0.675 0.035 0.573 0.625 0.650 0.670 0.694 0.742 0.858 0.044σ2ef

0.711 0.720 0.023 0.637 0.686 0.705 0.718 0.734 0.763 0.843 0.029

σemef 0.625 0.642 0.030 0.564 0.602 0.621 0.637 0.657 0.698 0.788 0.036αzm 0.658 0.641 0.053 0.440 0.550 0.607 0.643 0.680 0.724 0.825 0.073αym 0.390 0.407 0.085 0.003 0.251 0.355 0.418 0.469 0.527 0.691 0.114αzf 0.773 0.748 0.054 0.518 0.647 0.715 0.754 0.787 0.826 0.891 0.072αyf 0.020 0.082 0.102 0.000 0.000 0.000 0.011 0.164 0.277 0.407 0.164

Since we cannot estimate the parameters by GMM, we cannot obtain standard errors. Instead, we

perform a simulation exercise to check how robust our results are. We independently perturb each of

the 105 empirical correlations used in the benchmark specification, and obtain the calibrated parameters

for this artificial economy.43 We repeat this exercise 10,000 times and compute descriptive statistics for

each parameter. The results are presented in Table C.1. The mean and the median are very close to

the calibrated parameters from the original correlations and both standard deviations and interquartile

ranges are small. The general picture is very similar in all the simulations. The direct transmission

channels captured by βm and βf play a minor role. The transmission of advantages occurs instead

predominantly via the latent factor, with γm ranging from 0.60 to 0.71 and γf from 0.51 to 0.62, both

much larger than the corresponding β. Assortative mating occurs mainly in the latent factor with the

spouses’ correlation in the latent factor ranging from 0.56 to 0.84.

Next, for each of the simulations we compute the implied correlation of the male (female) child and

his or her ancestors on the male (female) line, both in the observable outcome and in the underlying

43Specifically, we multiply each correlation by a random draw from a uniform distribution in the interval [0.975,1.025].We have also performed a simulation exercise sampling the correlations from a 99% confidence interval based on asymptoticstandard errors of the correlations. The results are similar and available upon request.

64

Table C.2: Sensitivity of Long-run Predictions to Perturbations of Empirical Moments

Long-run correlationsy (male) y (female) z (male) z (female)

Parent 0.381 0.349 0.586 0.508Mean 0.381 0.349 0.574 0.498Std. dev. 0.005 0.005 0.022 0.018Grand-Parent 0.209 0.163 0.343 0.263Mean 0.204 0.161 0.331 0.255Std. dev. 0.010 0.004 0.024 0.016Great-Grand-Parent 0.121 0.082 0.201 0.137Mean 0.116 0.080 0.191 0.131Std. dev. 0.010 0.004 0.020 0.012GGreat-Grand-Parent 0.071 0.043 0.118 0.071Mean 0.067 0.041 0.110 0.067Std. dev. 0.008 0.003 0.015 0.008

Table C.3: Sensitivity of Sibling Correlations to Perturbations of Empirical Moments

Sibling correlationsCorrY CorrZ

Brothers 0.431 0.678Mean 0.431 0.700Std. dev. 0.006 0.040Sisters 0.417 0.824Mean 0.417 0.866Std. dev. 0.006 0.079Brother-sister 0.377 0.711Mean 0.377 0.741Std. dev. 0.005 0.050

latent factor. The results are presented in Table C.2. The average long-run correlations are similar

to those implied by the original correlations and the standard deviations are very small. The kinship

correlations decline more slowly with the distance between kins than a simple iteration of the parent-

child correlation would suggest. In Table C.3 we report means and standard deviations of the sibling

correlation in the observed and latent factors implied by our simulations. For all gender combinations,

the average sibling correlations are very similar to the one based on the original correlations, and we

always find that the correlation in the latent status is much stronger than in the observed outcome.

65

D The Baseline Model and Income

Figure D.1: Sample and Predicted Moments (Income)

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Notes: See Table D.1 for the corresponding list of kinship moments.

We consider income as a potentially more direct measure of socioeconomic origins and destinations.44

Because the Swedish registers track income profiles over nearly six decades, we can construct high-

quality measures of income for both the parent and child generation. As detailed in Section 3.1 we

measure income as the logarithm of ten-year averages of annual income centered total pre-tax around

age 35 for children and age 45 for parents.45 As for education, we observe 141 distinct correlations,

classified into 21 kinship types. Columns (1) and (2) of Appendix Table D.1 report the sample size and

sample correlations for a subset of those 141 moments.46 The correlations are systematically lower for

44Income has been the primary measure of socio-economic status in the economic literature, while education or oc-cupation play a more important role in sociological research. This division is becoming less sharp, however. Recentsociological work emphasizes the informative content of income (Kim et al. 2018). In economics, occupational and edu-cational measures have become common in comparative or historical (e.g. Hertz, 2007, or Long and Ferrie, 2013a) andmultigenerational studies (e.g. Stuhler, 2018).

45For robustness we also considered shorter averages (five-year and annual), and Spearman rank correlations insteadof log-linear Pearson correlations. The results vary somewhat with the income definition (ten-year, five-year and annualincomes, rank or log incomes), but the broad pattern remains stable. Results for alternative income definitions are availableupon request.

46Our estimates for intergenerational and sibling correlations in income are broadly in line with prior evidence forSweden. Estimates based on longer income spans or adjusted for measurement error are larger, see Bjorklund et al. (2009)

66

mixed or female than for male pairs. This pattern is consistent with the observation that women were

less likely to participate in the labor force, in particular for the parent generation in our sample.47 We

base our calibration on the same 105 moments that we use also in our baseline calibration for Sweden,

which includes siblings-in-law up to three degrees of separation (group XVI). We plot those moments

in Figure D.1, and report all moments in Table D.1. As for education, the calibrated model explains

the data well, providing a close fit to both vertical and horizontal moments, and for both direct and

affinity relationships. The model also replicates the asymmetric transmission pattern across genders.

As for education, we understate the correlation for cousins and very distant siblings-in-law.

Table D.2 reports the parameters of the calibrated model for income, separately by intergenerational

(Panel A), sibling (Panel B), and assortative processes (Panel C). The findings vary only mildly with

the chosen income definition (ten-year, five-year and annual incomes, rank or log incomes), and are

qualitatively similar to our benchmark calibration based on years of schooling. The latent advantages

are more strongly transmitted than income itself, in each of the intergenerational, sibling and assor-

tative processes. However, it is not as strongly transmitted as for the case of education. The latent

determinants of income appear more persistent in the horizontal dimension. The sibling correlation in

the latent factor varies around between 0.6 and 0.8, and the shared sibling components in the observable

and the latent factor explain a large share of the similarity between siblings (Panel B of Table D.2).

The shared sibling component in the observable is far more important than the direct transmission of

income from parents to children, which explains only two percent of the variation in income for males

and less for females (see Panel D of Table D.2). The spousal correlation in log income is as low as 0.12,

consistent with endogenous labor supply decisions at the household level. However, the implied spousal

correlation in the latent factor is 0.8. (Panel C of Table D.2). This estimate is similar as for the case

of years of schooling. We, therefore, find again that spouses are much more similar in the determinants

of future socioeconomic success than they are in observable characteristics.

In sum, these estimates confirm that our qualitative findings extend to socioeconomic outcomes other

than educational attainment. However, the strength of different transmission processes does vary

with the outcome under study. One interpretation is that different factors influence different aspects

of socioeconomic status, and that some of those factors have higher persistence than others. The

recent multigenerational literature does not provide much evidence on income, because income is either

not observed, or not very informative in historical sources. Our “horizontal” approach does not face

such constraints and can, therefore, be used to study transmission across a broader set of outcomes.

Our evidence suggests that those factors that determine educational attainment are more strongly

transmitted from one generation to the next than those factors that influence earnings.

or Nybom and Stuhler (2017).47We do not account explicitly for labor supply decisions. However, our model allows for gender differences in all its

components, and for the transmission of other advantages apart from income in intergenerational and assortative processes.It is an empirical question if that model provides sufficient flexibility to fit kinship correlations for a comparatively complexoutcome such as income.

67

Table D.1: Estimated and Calibrated Moments in Swedish Registers (Income)

# ## name number sample predicted percent # ## name number sample predicted percentof pairs correlation correlation error of pairs correlation correlation error

(1) (2) (3) (4) (1) (2) (3) (4)I 1 Husband-Wife 412,735 0.122 0.121 -1.1 …II 2 Brother 200,803 0.238 0.241 1.2 XII 72 MFMS 154,974 0.041 0.045 11.0

3 Sister 217,722 0.144 0.145 0.7 73 FMMS 119,219 0.033 0.030 -10.1 4 Brother-Sister 408,331 0.122 0.123 0.0 XIII 74 M-MMMS 83,216 0.020 0.023 15.9

III 5 Father-Son 388,190 0.227 0.228 0.4 75 M-MMFS 108,087 0.035 0.034 -0.7 6 Father-Daughter 366,590 0.147 0.147 0.0 76 M-MFMS 83,133 0.021 0.025 19.6 7 Mother-Son 446,039 0.082 0.082 0.7 77 M-MFFS 94,984 0.037 0.034 -6.4 8 Mother-Daughter 422,142 0.096 0.096 0.8 78 M-FMMS 62,612 0.014 0.025 69.8

IV 9 Brother in-law (HS) 339,682 0.143 0.138 -3.4 79 M-FMFS 78,556 0.035 0.037 5.1 10 Brother-Sister in-law (WB) 296,051 0.090 0.099 10.9 80 M-FFMS 56,750 0.027 0.025 -7.6 11 Brother-Sister in-law (HS) 362,169 0.089 0.094 5.6 81 M-FFFS 62,983 0.043 0.033 -22.3 12 Sister in-law (WB) 301,464 0.079 0.063 -19.9 82 F-MMMS 78,128 0.013 0.017 30.5

V 13 Nephew-Uncle (BF) 149,817 0.106 0.093 -12.7 83 F-MMFS 102,222 0.029 0.026 -8.5 14 Niece-Uncle (BF) 142,096 0.074 0.071 -3.5 84 F-MFMS 79,019 0.018 0.020 6.0 15 Nephew-Uncle (BM) 201,393 0.090 0.078 -13.8 85 F-MFFS 90,190 0.022 0.026 17.7 16 Niece-Uncle (BM) 190,299 0.060 0.060 0.2 86 F-FMMS 59,524 0.023 0.020 -13.3 17 Nephew-Aunt (SF) 152,065 0.048 0.054 11.8 87 F-FMFS 75,079 0.024 0.030 22.1 18 Niece-Aunt (SF) 142,912 0.047 0.042 -10.0 88 F-FFMS 53,001 0.021 0.020 -7.3 19 Nephew-Aunt (SM) 217,131 0.049 0.054 9.9 89 F-FFFS 59,009 0.019 0.027 36.5 20 Niece-Aunt (SM) 205,748 0.054 0.044 -18.7 XIV 90 M-MMM-M 41,300 0.017 0.019 8.5

VI 21 Brother in-law (WWS) 156,164 0.119 0.118 -0.9 91 M-MMF-M 61,036 0.025 0.019 -22.5 22 Sister in-law (HHB) 112,873 0.072 0.057 -20.9 92 M-MFM-M 45,581 0.027 0.021 -22.0 23 Brother-Sister in-law (HWBS)251,377 0.074 0.078 4.2 93 M-FMM-M 31,435 0.033 0.020 -39.4

VII 24 Nephew-Aunt in-law (BF) 120,226 0.030 0.047 55.8 94 M-MMM-F 38,113 0.022 0.015 -32.9 25 Niece-Aunt in-law (BF) 114,422 0.034 0.038 11.6 95 M-MMF-F 58,304 0.022 0.015 -31.5 26 Nephew-Aunt in-law (BM) 158,116 0.034 0.044 29.9 96 M-MFM-F 43,614 0.030 0.017 -44.6 27 Niece-Aunt in-law (BM) 149,645 0.030 0.034 10.1 97 M-MFF-F 57,298 0.028 0.015 -46.4 28 Nephew-Uncle in-law (SF) 124,725 0.070 0.064 -8.1 98 M-FMM-F 29,951 0.017 0.016 -3.2 29 Niece-Uncle in-law (SF) 117,191 0.051 0.051 0.5 99 M-FMF-F 43,353 0.007 0.016 131.3 30 Nephew-Uncle in-law (SM) 179,624 0.068 0.066 -2.2 100 M-FFM-F 30,120 0.026 0.016 -38.9 31 Niece-Uncle in-law (SM) 170,499 0.052 0.051 -2.7 101 M-FFF-F 38,840 0.022 0.014 -34.0

VIII 32 Male Cousins (B) 36,543 0.077 0.040 -47.4 102 F-MMM-F 36,226 0.010 0.011 19.0 33 Male Cousins (S) 62,125 0.072 0.037 -48.6 103 F-MMF-F 54,841 0.019 0.011 -38.8 34 Male Cousins (BS) 93,329 0.067 0.036 -46.1 104 F-MFM-F 41,223 0.024 0.013 -46.9 35 Female Cousins (B) 32,984 0.036 0.025 -30.3 105 F-FMM-F 28,318 0.014 0.013 -7.4 36 Female Cousins (S) 55,675 0.045 0.022 -50.6 XV 106 XMMMM 133,431 0.024 0.030 24.3 37 Female Cousins (BS) 82,310 0.034 0.022 -34.0 107 XMMMF 143,881 0.027 0.021 -20.2 38 Male-Female Cousins (B) 70,834 0.044 0.032 -28.0 108 XMMFM 156,686 0.032 0.034 7.3 39 Male-Female Cousins (S) 122,876 0.049 0.029 -41.3 109 XMMFF 80,788 0.020 0.021 3.4 40 Male-Female Cousins (BS) 88,467 0.044 0.028 -36.3 110 XMFMM 136,695 0.038 0.034 -11.0 41 Male-Female Cousins (SB) 87,582 0.036 0.029 -19.1 111 XMFMF 73,466 0.026 0.024 -9.8

IX 42 XMMM 223,084 0.051 0.059 13.8 112 XMFFM 140,818 0.036 0.034 -5.8 43 XMMF 238,392 0.042 0.041 -3.1 113 XFMMM 129,638 0.050 0.049 -2.1 44 XMFM 232,232 0.059 0.066 11.5 114 XFMFM 76,528 0.057 0.055 -3.8 45 XFMM 217,877 0.095 0.094 -0.9 115 XFFMM 63,988 0.050 0.049 -2.0

X 46 MMM 190,450 0.063 0.072 14.5 XVI 116 MMMM 115,877 0.036 0.037 4.5 47 MMF 204,255 0.046 0.050 8.6 117 MMMF 125,068 0.029 0.026 -9.6 48 MFM 207,353 0.078 0.080 3.0 118 MMFM 139,577 0.047 0.042 -12.1 49 MFF 213,663 0.048 0.050 3.9 119 MMFF 143,320 0.028 0.026 -7.3 50 FMM 163,588 0.046 0.053 16.0 120 MFMM 123,894 0.048 0.042 -13.9 51 FMF 173,923 0.039 0.037 -6.3 121 MFMF 133,139 0.032 0.029 -8.2 52 FFM 157,713 0.055 0.053 -3.1 122 MFFM 131,608 0.047 0.042 -12.5 53 FFF 160,923 0.033 0.033 -0.8 123 MFFF 134,253 0.029 0.026 -10.4

XI 54 M-MMM 113,118 0.037 0.040 9.5 124 FMMM 97,448 0.027 0.027 3.3 55 M-MMF 121,696 0.029 0.028 -3.8 125 FMMF 105,868 0.025 0.019 -21.9 56 M-MFM 108,145 0.045 0.045 -0.3 126 FMFM 113,190 0.031 0.031 -1.2 57 M-MFF 111,238 0.023 0.028 20.7 127 FMFF 115,798 0.022 0.019 -13.3 58 M-FMM 84,217 0.040 0.044 10.1 128 FFMM 92,992 0.027 0.027 1.7 59 M-FMF 89,595 0.027 0.031 11.6 129 FFMF 99,601 0.020 0.019 -4.5 60 M-FFM 73,134 0.046 0.044 -4.7 130 FFFM 92,574 0.033 0.027 -17.3 61 M-FFF 74,708 0.021 0.027 26.8 131 FFFF 95,272 0.026 0.017 -34.4 62 F-MMM 106,389 0.026 0.031 17.7 XVII 132 MMMMS 82,420 0.022 0.021 -4.4 63 F-MMF 114,915 0.020 0.022 11.1 133 MMMFS ##### 0.034 0.032 -7.5 64 F-MFM 102,747 0.032 0.035 9.2 XVIII 134 MMMMM 75,822 0.026 0.019 -26.3 65 F-MFF 105,813 0.025 0.022 -13.5 135 FFFFF 61,230 0.018 0.009 -51.0 66 F-FMM 79,920 0.034 0.035 1.2 XIX 136 MMMMMS 53,670 0.025 0.011 -56.9 67 F-FMF 85,898 0.026 0.024 -5.1 137 MMMMFS 79,620 0.026 0.016 -37.9 68 F-FFM 68,471 0.038 0.035 -8.6 XX 138 MMMMMM 53,098 0.015 0.010 -31.9 69 F-FFF 69,793 0.019 0.022 11.5 139 FFFFFF 42,278 0.003 0.005 39.8

XII 70 MMMS 137,086 0.040 0.040 1.4 XXI 140 MMMMMMS 37,586 0.009 0.006 -40.1 71 MMFS 185,967 0.058 0.061 5.7 141 MMMMMFS 57,789 0.023 0.009 -62.3

Table: Estimated and Calibrated Moments in Swedish Registers: Log IncomeKinship type Data Calibration Kinship type Data Calibration

Notes: HS=Husband of the sister, WB=Wife of the brother, BF=Brother of the father, BM=Brother of the mother, SF=Sister ofthe father, SM=Sister of the mother, WWS=Wives are sisters, HHB=Husbands are brothers, HWBS=Husband and wife are brotherand sister, B=Fathers are brothers, S=Mothers are sisters. 68

Table D.2: Calibrated Parameters in Swedish Registers (Income)

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.144 0.103 0.618 0.429σ2

ym σ2yf σ2

zm σ2zf σ2

um σ2uf

0.304 0.248 0.057 0.027 0.219 0.201αym αyf αzm αzf1.000 0.643 0.054 0.210

Parent-child correlations in z:Father-Son Father-Dau Mother-Son Mother-Dau0.354 0.400 0.430 0.443

Ancestor correlations in y and z:Father-Son Grandf-… GGrandf-… GGGrandf-…

in y 0.228 0.065 0.021 0.007 in z

Panel B: Sibling ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef0.014 0.014 0.000 0.034 0.011 0.018

Variance Shares: in y 4.8% 5.8% 0.0% 11.4% 4.3% 6.5% in z - - - 60.8% 39.6% 45.9%Sibling correlations in z:

Brothers Sisters Brother-Sister0.794 0.599 0.651

Panel C: Assortative ProcessesParameters:

rmzz rm

zy rmyz rm

yy σ2ωm σ2

ϵm0.517 0.029 0.566 -0.003 0.010 0.230rf

zz rfzy rf

yz rfyy

1.158 -0.006 1.535 -0.044Spousal correlations in y and z:

ρymyf ρzmzf ρymzf ρzmyf0.121 0.795 0.444 0.269

Panel D: Variance Decomposition% y z x Cov(y,z)Male 0.021 0.187 0.048 0.012Female 0.007 0.109 0.058 0.007

Table: Calibrated Parameters in Swedish Registers (Income)

69

E The Baseline Model and Height

Figure E.1: Sample and Predicted Moments (Height)

(a) Baseline model

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

0.6

(b) Genetic model

I II III IV V VI VII IX X XI XII XIII XV XVI

1 2-4 5-8 9-12 13-20 21-23 24-31 42-45 46-53 54-69 70-73 74-89 106-115 116-131

Prediction Observed

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Notes: See Table E.1 for the corresponding list of kinship moments.

In this section we calibrate the standard genetic model for body height, which we observe from military

enlistment tests that were universal for the male Swedish population (see Section 3.1). Height is

an interesting reference point because it is known to be primarily determined by genes, at least in

populations that are not exposed to famine or undernutrition.48 It is subject to only comparatively

weak assortative processes, which was one of the reasons why Francis Galton considered height in his

famous work on linear regression (Galton, 1886). Because the transmission of body height is better

understood, the plausibility of our findings is easier to evaluate for height than for socioeconomic

outcomes. Moreover, we observe height only for males, and it is an interesting question to what extent

female outcomes need to be observed to identify our model, including the assortative and gender-specific

processes.

Table E.1 reports the sample correlations. We observe 39 (male) kinship correlations.49 As in our

baseline case, we include siblings-in-law of three degrees of separation for the calibration (i.e., excluding

kinship groups XX and XXI). To these 37 moments, we add the correlation in height between spouses

and between sisters from external sources.50 The predicted moments from the calibrated model are

reported in column (3) of Table E.1, while Figure E.1a plots the 105 kinship moments that are included

48The proportion of the total variation in body height in a population that is due to genetic variation is estimated tobe around 0.8 (Silventoinen 2003). Accordingly, the correlation in body height is much higher in biological than fosterfamilies, and can be as high as 0.99 for monozygotic twins.

49Because the military enlistments tests cover only birth cohorts born between 1950 and 1980, we observe fewer obser-vations than for the other outcomes. The moments are however precisely estimated, and in line with prior evidence. Forexample, the father-son correlation in height in our sample is 0.48, the same value as reported by Gronqvist et al. (2017).

50Price and Vandenberg (1980) reports a spousal correlation in height of 0.27 for Swedish couples. To calibrate thecorrelation in height for sisters, we assume that the gap to the corresponding correlation for brothers is as large as inNorwegian sources reported in Tambs et al. (1992).

70

Table E.1: Estimated and Calibrated Moments in Swedish Registers (Height)

# ## name number sample # ## name number sampleof pairs correlation general genetic of pairs correlation general genetic

(1) (2) (3) (4) (1) (2) (3) (4)I 1 HUSB-WIF External 0.270 0.270 0.287 …II 2 BROTHERS 16,437 0.545 0.540 0.549 XI 56 M-MFM 37,029 0.053 0.054 0.049

3 SISTERS External 0.535 0.535 0.540 58 M-FMM 43,068 0.047 0.057 0.0384 BROTH-SIS 0.486 0.477 60 M-FFM 29,437 0.059 0.057 0.045

III 5 FATH-SON 46,441 0.483 0.483 0.470 XII 71 MMFS 39,288 0.023 0.032 0.0066 FATH-DAUGH 0.468 0.470 XIII 75 M-MMFS 29,311 0.017 0.030 0.0137 MOTH-SON 0.586 0.470 77 M-MFFS 21,488 0.022 0.030 0.0128 MOTH-DAUGH 0.603 0.470 79 M-FMFS 26,207 0.037 0.039 0.012

IV 9 BL-HS 135,006 0.111 0.111 0.137 81 M-FFFS 16,828 0.033 0.033 0.01110 BSL-WB 0.201 0.158 XIV 90 M-MMM-M 24,274 0.025 0.034 0.02411 BSL-HS 0.146 0.155 91 M-MMF-M 27,032 0.043 0.032 0.02612 SL-WB 0.249 0.137 92 M-MFM-M 27,089 0.040 0.040 0.025

V 13 NEU-BF 52,618 0.260 0.274 0.279 93 M-FMM-M 25,681 0.036 0.043 0.023… XV 113 XFMMM 50,330 0.041 0.034 0.01015 NEU-BM 66,270 0.289 0.295 0.288 114 XFMFM 25,234 0.042 0.040 0.013

VI 21 BL-WWS 44,034 0.067 0.060 0.045 115 XFFMM 29,208 0.031 0.034 0.010VII 28 NEHASF 28,901 0.077 0.075 0.083 XVI 116 SIL-MMMM 30,183 0.025 0.023 0.003

30 NEHASM 39,634 0.073 0.075 0.089 118 SIL-MMFM 30,538 0.026 0.026 0.003VIII 32 MC-B 21,153 0.160 0.152 0.163 120 SIL-MFMM 36,879 0.032 0.026 0.003

33 MC-S 21,937 0.188 0.188 0.186 122 SIL-MFFM 33,478 0.025 0.026 0.00334 MC-BS 45,689 0.167 0.153 0.175 XII 133 MMMFS 19,846 0.016 0.018 0.001

IX 45 XFMM 99,219 0.074 0.079 0.075 XIII 134 MMMMM 16,980 0.020 0.013 0.000X 46 SIL-MMM 59,544 0.039 0.042 0.019 XIX 137 MMMMFS 11,286 0.009 0.010 0.000

48 SIL-MFM 63,798 0.053 0.048 0.024 XX 138 MMMMMM 10,309 0.015 0.007 0.000XI 54 M-MMM 46,771 0.048 0.047 0.040 XXI 141 MMMMMFS 6,873 0.044 0.006 0.000

predicted correlation predicted correlation

Table: Estimated and Calibrated Moments in Swedish Registers: HeightKinship type Data Calibration Kinship type Data Calibration

Notes: Kinship correlations from Swedish registers. Moments printed in italics were not used in the calibration.

in our baseline calibration for educational attainment. As for education and income, the calibrated

model explains our data well, providing a close fit to both vertical and horizontal moments, and for

both direct and affinity relationships. However, in the absence of direct observations, we appear unable

to fit the correlations for female kins – the calibrated model predicts much larger correlations for distant

kinships that involve female than those that involve male.

Table E.2 reports the estimated parameters. The transmission process for height can be well approx-

imated by direct transmission channels – observed height explains well the height of descendants and

other family members. Despite its strong intergenerational transmission (βm = 0.934 and βf = 0.844),

the father-child and mother-child correlations in height remain modest because the spousal correlation

in height (ρymyf = 0.27) is much smaller than the corresponding spousal correlations in socioeconomic

outcomes.51 Siblings also appear to share some additional environmental influences over and above

what can be explained by the heigh of their parents – the shared sibling component explains about 5

percent of the variation in height.

51The implied spousal correlations in the latent factor is large, but less relevant because the latent factor explains hardlyany of the variation in the outcome.

71

Table E.2: Calibrated Parameters in Swedish Registers (Height)

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.934 0.844 0.781 0.018σ2

ym σ2yf σ2

zm σ2zf σ2

um σ2uf

1.000 1.000 0.070 0.001 0.435 0.465αym αyf αzm αzf

0.341 0.389 1.000 0.290Parent-child correlations in z:

Father-Son Father-Dau Mother-Son Mother-Dau0.781 0.054 0.590 0.046

Ancestor correlations in y and z:Father-Son Grandf-… GGrandf-… GGGrandf-…

in y 0.483 0.233 0.113 0.055 in z

Panel B: Siblings ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef

0.053 0.070 0.022 0.002 0.001 0.004Variance Shares: in y 5.3% 7.0% 2.2% 0.2% 0.1% 0.4% in z - - - 3.5% 54.4% 46.1%Sibling correlations in z:

Brothers Sisters Brother-Sister0.645 0.547 0.503

Panel C: Assortative ProcessesParameters:

rmzz rm

zy rmyz rm

yy σ2ωm σ2

ϵm

0.090 -0.006 -2.131 0.264 0.000 0.611rf

zz rfzy rf

yz rfyy

6.386 -0.153 -6.046 0.273Spousal correlations in y and z:

ρymyf ρzmzf ρymzf ρzmyf

0.270 0.756 -0.187 -0.565Panel D: Variance Decomposition

% y z x Cov(y,z)Male 0.534 0.070 0.053 -0.072Female 0.419 0.001 0.070 -0.001

Table: Calibrated Parameters in Swedish Registers (Height)

72

F The Standard Genetic Model and Height

Genes are known to be the primary source of variation in height (Silventoinen, 2003). We therefore study

if the standard genetic model described in Section 6.1 can provide a comparatively good fit, despite

having much fewer parameters than our baseline model. We use the same 39 kinship correlations as for

calibration of the baseline model, as described in Appendix E. The sample and predicted moments are

reported in Table E.1. As illustrated in Figure E.1b, the genetic model provides a worse fit to in-sample

moments.52 However, it fits close relatives such as spouses and siblings quite well and generates out-of-

sample predictions that appear more stable within kinship types than the widely varying predictions

of the general model. Table F.1 summarizes the results.53 The latent factor explains about 73 percent

of the variation in height (see Panel D). This estimate is slightly below estimates of the heritability of

height from quantitative genetics, which tend to be around 80 percent (Silventoinen, 2003). In principle,

our approach can capture the transmission of latent factors other than genes, and can therefore be

interpreted as an upper bound for heritability.54 That our estimates are close to those from the genetic

literature suggests therefore that the parent-child correlation in height is nearly exclusively due to

genes, and not other factors. The remaining variation in height is predominantly explained by factors

that are not shared by siblings (Panels A and B). Finally, the spousal correlation in phenotype height is

greater than the spousal correlation in genotype height (Panel C). This result follows directly from the

assumption that assortative mating occurs exclusively on the phenotype. In contrast, our baseline model

allows for spouses to be more similar than indicated by their observed phenotypes. As a consequence,

the genetic and general model yield substantially different predictions for the distant kinship types in

our data (cf. ancestor correlations reported in Tables E.2 and F.1).

The results are therefore ambiguous. On the one hand, our approach yields parameter estimates that

match well with estimates of the heritability of body height from quantitative genetics. On the other

hand, the standard genetic model cannot explain why body height remains correlated between the

most distant family members in our data. The genetic model fits a narrow set of kinship moments,

but not the full set observed in this study. The potential culprit is the assumption that assortative

mating occurs only in the phenotype. If spouses match on factors other than phenotype height, and

those other factors have an independent association with genotype height, then the standard genetic

model understates the correlation between relatives – even if the intergenerational transmission process

itself is exclusively due to genetic factors. The advantage of our approach is that such failures become

visible. Because we observe distant relatives, erroneous assumptions in the assortative process become

noticeable even if they have only negligible implications for the close kinship correlations that have

been studied in the previous literature.

52The mean prediction error up to second-degree siblings-in-law is only 3.7 percent for the general model, but 13.6percent for the genetic model.

53In contrast to the standard model described above, we allow σ2ek to be non-zero in the calibration (e.g. to account for

the presence of monozygotic twins in our data). This added flexibility has only a negligible effect on the results.54See Ruby et al. (2018) for a similar upper-bound argument for the case of human longevity.

73

Table F.1: Calibrated Parameters in Swedish Registers (Height, Genetic Model)

Panel A: Intergenerational ProcessesParameters:

βm βf γm γf

0.000 0.000 1.000 1.000σ2

ym σ2yf σ2

zm σ2zf σ2

um σ2uf

1.000 1.000 0.731 0.731 0.163 0.237αym αyf αzm αzf

0.000 0.000 0.500 0.500Parent-child correlations in z:

Father-Son Father-Dau Mother-Son Mother-Dau0.605 0.605 0.605 0.605

Ancestor correlations in y and z:Father-Son Grandf-… GGrandf-… GGGrandf-…

in y 0.470 0.285 0.172 0.104 in z

Panel B: Sibling ProcessesParameters:

σ2xm σ2

xf σxmxf σ2em σ2

ef σemef

0.107 0.032 0.000 0.000 0.066 0.035Variance Shares: in y 10.7% 3.2% 0.0% 0.0% 6.6% 3.5% in z - - - 0.0% 9.0% 4.8%Sibling correlations in z:

Brothers Sisters Brother-Sister0.605 0.695 0.653

Panel C: Assortative ProcessesParameters:

rmzz rm

zy rmyz rm

yy σ2ωm σ2

ϵm

0.000 0.210 0.000 0.287 0.687 0.917rf

zz rfzy rf

yz rfyy

0.000 0.210 0.000 0.287Spousal correlations in y and z:

ρymyf ρzmzf ρymzf ρzmyf

0.287 0.210 0.246 0.246Panel D: Variance Decomposition

% y z x Cov(y,z)Male 0.000 0.731 0.107 0.000Female 0.000 0.731 0.032 0.000

Table: Calibrated Parameters in Swedish Registers (Height, Genetic)

74

Theoretical Appendix

G General Assortative Mating Model

We assume that the value of the outcome y for an individual from generation t is given by

ykt = βkykt−1 + zkt + xkt + ukt (G.1)

where the superscript k = f stands for males and k = m for females. We assume that

ykt−1 = αkyymt−1 + (1− αky)y

ft−1

and the socioeconomic status of the child, zkt , depends on the father zmt−1 as well as on the mother zft−1

zkt = γkzkt−1 + ekt + vktzkt−1 = αkzz

mt−1 + (1− αkz)z

ft−1

(G.2)

Regarding the shocks, we assume that xkt and ekt are shared by all siblings of the same gender, can be correlated

across siblings of different gender and are uncorrelated to each other and with the other variables (in particular

with zkt−1 and ylt−1, l = m, f). Finally ukt and vkt are white-noise errors.

G.1 Assortative mating process

We assume there is assortative mating both in years of schooling and in socioeconomic status (see Behrman and

Rosenzweig, 2002, for a related model with assortative mating in two dimensions). In particular we consider the

linear projections of zft−1 and yft−1 on zmt−1 and ymt−1:(zft−1

yft−1

)=

(rmzz rmzyrmyz rmyy

)(zmt−1

ymt−1

)+

(wmt−1

εmt−1

)

where wmt−1 and εmt−1 might be correlated but are uncorrelated with zmt−1 and ymt−1.

The coefficients of the linear projections depend on ρzmym , ρzmzf , ρzmyf , ρymzf and ρymyf , as well as on the

standard deviations of zkt−1 and ykt−1, k = m, f :

rmzz =1

(1− ρ2zmym)

σzf

σzm(ρzmzf − ρzmymρymzf )

rmzy =1

(1− ρ2zmym)

σzf

σym(ρymzf − ρzmymρzmzf )

75

rmyz =1

(1− ρ2zmym)

σyf

σzm(ρzmyf − ρzmymρymyf )

rmyy =1

(1− ρ2zmym)

σyf

σym(ρymyf − ρzmymρzmyf )

The variance matrix of (wmt−1, εmt−1) is given by

V ar

(wmt−1

εmt−1

)= V ar

(zft−1

yft−1

)−

(rmzz rmzyrmyz rmyy

)V ar

(zmt−1

ymt−1

)(rmzz rmzyrmyz rmyy

)′

We use these matching functions to write years of schooling, ykt , and social status, zkt , as a function of father’s

years of schooling, ymt−1, and social status zmt−1. We can write (G.2) as

zkt = γk(αkzz

mt−1 + (1− αkz)z

ft−1

)+ ekt + vkt

= γk(αkzz

mt−1 + (1− αkz)

(rmzzz

mt−1 + rmzyy

mt−1 + wmt−1

))+ ekt + vkt

= Gkzmzmt−1 +Gkymy

mt−1 + gkmω

mt−1 + ekt + vkt

where

Gkzm = γk(αkz + (1− αkz)rmzz)

Gkym = γk(1− αkz)rmzygkm = γk(1− αkz)

and (G.1) as

ykt = βk(αkyy

mt−1 + (1− αky)y

ft−1

)+ zkt + xkt + ukt

= βk(αkyy

mt−1 + (1− αky)

(rmyzz

mt−1 + rmyyy

mt−1 + εmt−1

))+ zkt + xkt + ukt

= βk(αkyy

mt−1 + (1− αky)

(rmyzz

mt−1 + rmyyy

mt−1 + εmt−1

))+Gkzmz

mt−1 +Gkymy

mt−1 + gkmω

mt−1 + ekt + vkt + xkt + ukt

ykt = Bkymy

mt−1 +Bk

zmzmt−1 + bkmε

mt−1 + gkmω

mt−1 + ekt + vkt + xkt + ukt

where

Bkym = βk

(αky + (1− αky)rmyy

)+Gkym

Bkzm = βk(1− αky)rmyz +Gkzm

bkm = βk(1− αky)

76

All these expressions will be used to compute correlations between relatives that are related through their fathers.

However, when we consider relatives that are related through their mothers, we need to find expressions for yktand zkt as functions of mother’s years of schooling, yft−1, and social status zft−1.We then also consider the linear

projections of zmt−1 and ymt−1 on zft−1 and yft−1:(zmt−1

ymt−1

)=

(rfzz rfzy

rfyz rfyy

)(zft−1

yft−1

)+

(wft−1

εft−1

)

where wft−1 and εft−1 might be correlated but are uncorrelated with zft−1 and yft−1.

The coefficients of the linear projections depend on ρzfyf , ρzmzf , ρzmyf , ρymzf and ρymyf , as well as on the

standard deviations of zkt−1 and ykt−1, k = m, f :

rfzz =1

(1− ρ2zfyf

)

σzm

σzf(ρzmzf − ρzfyfρzmyf )

rfzy =1

(1− ρ2zfyf

)

σzm

σyf(ρzmyf − ρzfyfρzmzf )

rfyz =1

(1− ρ2zfyf

)

σym

σzf(ρymzf − ρzfyfρymyf )

rfyy =1

(1− ρ2zfyf

)

σym

σyf(ρymyf − ρzfyfρymzf )

Using these linear projections, we can write (G.2) as

zkt = Gkzfzft−1 +Gkyfy

ft−1 + gkfω

ft−1 + ekt + vkt

where

Gkzf = γk(αkzrfzz + (1− αkz))

Gkyf = γkαkzrfzy

gkf = γkαkz

and (G.1) as

ykt = Bkyfy

ft−1 +Bk

zfzft−1 + bkfε

ft−1 + gkfω

ft−1 + ekt + vkt + xkt + ukt

77

where

Bkyf = βk

(αkyr

fyy + (1− αky)

)+Gkyf

Bkzf = βkαkyr

fyz +Gkzf

bkf = βkαky

G.2 Steady state assumption

We assume that the second order moments of all variables are time invariant. This steady state assumption

implies that ρzmym and ρzfyf depend on the remaining parameters of the model as shown below.

Cov(ymt , zmt ) = Cov(βmymt−1 + zmt , z

mt ) = Cov(βm(αmy y

mt−1 + (1− αmy )yft−1), γm(αmz z

mt−1 + (1− αmz )zft−1)) + σ2

zm

= βmαmy γmαmz Cov(ymt−1, z

mt−1) + βmαmy γ

m(1− αmz )Cov(ymt−1, zft−1)

+ βm(1− αmy )γmαmz Cov(yft−1, zmt−1) + βm(1− αmy )γm(1− αmz )Cov(yft−1, z

ft−1) + σ2

zm

Dividing by σzm and σym , we have

ρzmym = βmαmy γmαmz ρzmym + βmαmy γ

m(1− αmz )σzf

σzmρymzf

+ βm(1− αmy )γmαmzσyf

σymρzmyf + βm(1− αmy )γm(1− αmz )

σzf

σzm

σyf

σymρzfyf +

σzm

σym

and rearranging

(1− βmαmy γmαmz )ρzmym − βm(1− αmy )γm(1− αmz )σzf

σzm

σyf

σymρzfyf (G.3)

=σzm

σym+ βmαmy γ

m(1− αmz )σzf

σzmρymzf + βm(1− αmy )γmαmz

σyf

σymρzmyf

analogously

−βf (1− αfy )γf (1− αfz )σzm

σzf

σym

σyfρzmym + (1− βfαfyγfαfz )ρzfyf (G.4)

=σzf

σyf+ βfαfyγ

f (1− αfz )σzm

σzfρzmyf + βf (1− αfy )γfαfz

σym

σyfρymzf

78

In matrix form (1− βmαmy γmαmz ) −βm(1− αmy )γm(1− αmz )σzf

σzm

σyf

σym

−βf (1− αfy )γf (1− αfz )σzmσzf

σym

σyf

(1− βfαfyγfαfz )

( ρzmym

ρzfyf

)

=

σzmσym

+ βmαmy γm(1− αmz )

σzf

σzmρymzf + βm(1− αmy )γmαmz

σyf

σymρzmyf

σzf

σyf

+ βfαfyγf (1− αfz )σzmσzfρzmyf + βf (1− αfy )γfαfz

σym

σyfρymzf

and ρzmym and ρzfyf depend on ρymzf , ρzmyf and some other parameters of the model.

We then have that the model has 20 parameters γk, βk, αkz , αky , σ

2zk, σ2

xk, σ2

ek, k = m, f , and σxmxf , σemef , ρzmzf , ρymzf , ρzmyf

and ρymyf

G.3 Covariances

G.3.1 Main covariances

We first compute the main covariances (husband-wife, parent-child and siblings). Then, the covariances for other

relatives are obtained recursively.

Husband and wife

Cov(ymt−1, yft−1) = Cov(ymt−1, r

myzz

mt−1 + rmyyy

mt−1 + εmt−1) = rmyzCov(ymt−1, z

mt−1) + rmyyσ

2ym

= ρzmymσzmσym + rmyyσ2ym

Parent–child

Let k = m, f be the gender of the child and n = m, f the gender of the parent

Cov(zkt , znt−1) = Cov(Gkyny

nt−1 +Gkznz

nt−1, z

nt−1)

= GkynCov(ynt−1, znt−1) +Gkznσ

2zn

Cov(zkt , ynt−1) = Cov(Gkyny

nt−1 +Gkznz

nt−1, y

nt−1)

= Gkynσ2yn +GkznCov(ynt−1, z

nt−1)

Cov(ykt , znt−1) = Cov(Bk

ynynt−1 +Bk

znznt−1, z

nt−1)

= BkynCov(ynt−1, z

nt−1) +Bk

znσ2zn

79

Cov(ykt , ynt−1) = Cov(Bk

ynynt−1 +Bk

znznt−1, y

nt−1)

= Bkynσ

2yn +Bk

znCov(ynt−1, znt−1)

Siblings

We denote by k, l = m, f the genders of the siblings. We can compute the covariances projecting on the father

(n = m) or on the mother (n = f)

Cov(zk,it , zl,jt ) = Cov(Gkynynt−1 +Gkznz

nt−1 + gknω

nt−1 + ek,it , Glyny

nt−1 +Glznz

nt−1 + glnω

nt−1 + el,jt−1)

= GkynGlynσ

2yn+GkznG

lznσ

2zn+

(GkynG

lzn+GkznG

lyn

)Cov(ynt−1, z

nt−1)

+gknglnσ

2wn+σekel

Cov(zk,it , yl,jt ) = Cov(Gkynynt−1 +Gkznz

nt−1 + gknω

nt−1 + ek,it−1, B

lyny

nt−1 +Bl

znznt−1 + blnε

nt−1 + glnω

nt−1 + el,jt−1)

= GkynBlynσ

2yn+GkznB

lznσ

2zn+

(GkynB

lzn+GkznB

lyn

)Cov(ynt−1, z

nt−1)

+gknglnσ

2wn+σekel+g

knblnCov(εnt−1, ω

nt−1)

Cov(yk,it , zl,jt ) = Cov(Bkyny

nt−1 +Bk

znznt−1 + bknε

nt−1 + gknω

nt−1 + ek,it−1 + xkt−1, G

lyny

nt−1 +Glznz

nt−1 + glnω

nt−1 + el,jt−1)

= BkynG

lynσ

2yn+Bk

znGlznσ

2zn+

(BkynG

lzn+Bk

znGlyn

)Cov(ynt−1, z

nt−1)

+gknglnσ

2wn+σekel+b

kng

lnCov(εnt−1, ω

nt−1)

Cov(yk,it , yl,jt ) = Cov(Bkyny

nt−1+Bk

znznt−1+bknε

nt−1+gknω

nt−1+ek,it−1+xkt−1, B

lyny

nt−1+Bl

znznt−1+blnε

nt−1+glnω

nt−1+el,jt−1+xlt−1)

= BkynB

lynσ

2yn+Bk

znBlznσ

2zn+

(BkynB

lzn+Bk

znBlyn

)Cov(ynt−1, z

nt−1) + bknb

lnσ

2εn

+gknglnσ

2wn+σekel+

(bkng

ln+gknb

ln

)Cov(εnt−1, ω

nt−1) + Cov(xkt−1, x

lt−1)

G.3.2 Other covariances

Before we obtain the remaining covariances for different degrees of kinship we compute the linear projections of

zi,kt−1 and yi,kt−1 on zj,lt−1 and yj,lt−1, k, l = m, f, where i and j are siblings

(zi,kt−1

yi,kt−1

)=

(rk,lzz rk,lzy

rk,lyz rk,lyy

)(zj,lt−1

yj,lt−1

)+

(wk,lt−1

εk,lt−1

)

where wk,lt−1 and εk,lt−1 might be correlated but are uncorrelated with zj,lt−1 and yj,lt−1. We have that

rk,lzz =1

σ2zlσ2yl− σ2

zlyl

(σ2ylσzi,kzj,l − σzlylσzi,kyj,l

)

rk,lzy =1

σ2zlσ2yl− σ2

zlyl

(σ2zlσzi,kyj,l − σzlylσzi,kzj,l

)rk,lyz =

1

σ2zlσ2yl− σ2

zlyl

(σ2ylσyi,kzj,l − σzlylσyi,kyj,l

)

80

rk,lyy =1

σ2zlσ2yl− σ2

zlyl

(σ2zlσyi,kyj,l − σzlylσyi,kzj,l

)Let’s now consider several couples ”a” and ”x”, ”b” and ”y”, ”c” and ”z”, ”d” and ”w”, etc., such that ”x” and

”b” are siblings, ”y” and ”c” are siblings, etc. Let ”ax” be a child of ”a” and ”x”, ”by” a child of ”b” and ”y”,

etc., and ”a′x′” the spouse of ”ax”.

Generation t− 1 aspouse− x

sibling− b

spouse− y

sibling− c

spouse− z

sibling− d

spouse− w

↓child

↓child

↓child

↓child

Generation t axspouse− a′x′ by cz dw

Consanguine relatives (”blood”)

Vertical covariances

Uncle/aunt (siblings of the parents)

We have to compute the covariances between ”ax” and ”b”. Let a∗ = m, f be the gender of ”ax” and l = m, f

the gender of the ”b”. We project ax on x (his/her father or mother) who has gender n′

Cov(zax,n∗

t , zb,lt−1) = Cov(Gn∗zn′z

x,n′

t−1 +Gn∗yn′y

x,n′

t−1 , zb,lt−1) = Gn

∗zn′Cov(zx,n

′

t−1 , zb,lt−1) +Gn

∗yn′Cov(yx,n

′

t−1 , zb,lt−1)

Cov(zax,n∗

t , yb,lt−1) = Cov(Gn∗zn′z

x,n′

t−1 +Gn∗yn′y

x,n′

t−1 , yb,lt−1) = Gn

∗zn′Cov(zx,n

′

t−1 , yb,lt−1) +Gn

∗yn′Cov(yx,n

′

t−1 , yb,lt−1)

Cov(yax,n∗

t , zb,lt−1) = Cov(Bn∗zn′z

x,n′

t−1 +Bn∗yn′y

x,n′

t−1 , zb,lt−1) = Bn∗

zn′Cov(zx,n′

t−1 , zb,lt−1) +Bn∗

yn′Cov(yx,n′

t−1 , zb,lt−1)

Cov(yax,n∗

t , yb,lt−1) = Cov(Bn∗zn′z

x,n′

t−1 +Bn∗yn′y

x,n′

t−1 , yb,lt−1) = Bn∗

zn′Cov(zx,n′

t−1 , yb,lt−1) +Bn∗

yn′Cov(yx,n′

t−1 , yb,lt−1)

where ”x” and ”b” are siblings.

Horizontal covariances

Cousins

We have to compute the covariances between ”ax” and ”by”. Let a∗ = m, f be the gender of ”ax” and l∗ = m, f

the gender of the ”ay”. We project by on b (his/her father or mother) who has gender l

Cov(zax,n∗

t , zby,l∗

t ) = Cov(zax,n∗

t , Gl∗zlz

b,lt−1 +Gl

∗yly

b,lt−1) = Gl

∗zlCov(zax,n

∗

t , zb,lt−1) +Gl∗ylCov(zax,n

∗

t , yb,lt−1)

Cov(zax,n∗

t , yby,l∗

t ) = Cov(zax,n∗

t , Bl∗zlz

b,lt−1 +Bl∗

ylyb,lt−1) = Bl∗

zlCov(zax,n∗

t , zb,lt−1) +Bl∗ylCov(zax,n

∗

t , yb,lt−1)

Cov(yax,n∗

t , zby,l∗

t ) = Cov(yax,n∗

t , Gl∗zlz

b,lt−1 +Gl

∗yly

b,lt−1) = Gl

∗zlCov(yax,n

∗

t , zb,lt−1) +Gl∗ylCov(yax,n

∗

t , yb,lt−1)

Cov(yax,n∗

t , yby,l∗

t ) = Cov(yax,n∗

t , Bl∗zlz

b,lt−1 +Bl∗

ylyb,lt−1) = Bl∗

zlCov(yax,n∗

t , zb,lt−1) +Bl∗ylCov(yax,n

∗

t , yb,lt−1)

81

where ”b” is the uncle/aunt of ”ax”.

Affinity relatives (”in-law”)

Vertical covariances

Parents in-law

We have to compute the covariances between ”a′x′” and ”a”. Let a∗′

= m, f be the gender of ”a′x′” and n = m, f

the gender of the ”a”. We project ”a′x′” on his/her spouse ”ax”

Cov(za′x′,a∗

′

t , za,nt−1) = Cov(rn∗zz z

ax,n∗

t−1 + rn∗zy y

ax,n∗

t−1 , za,nt−1) = rn∗zzCov(zax,n

∗

t , za,nt−1) + rn∗zyCov(yax,n

∗

t−1 , za,nt−1)

Cov(za′x′,a∗

′

t , ya,nt−1) = Cov(rn∗zz z

ax,n∗

t−1 + rn∗zy y

ax,n∗

t−1 , ya,nt−1) = rn∗zzCov(zax,n

∗

t , ya,nt−1) + rn∗zyCov(yax,n

∗

t−1 , ya,nt−1)

Cov(ya′x′,a∗

′

t , za,nt−1) = Cov(rn∗yz z

ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , za,nt−1) = rn∗yzCov(zax,n

∗

t , za,nt−1) + rn∗yyCov(yax,n

∗

t−1 , za,nt−1)

Cov(ya′x′,a∗

′

t , ya,nt−1) = Cov(rn∗yz z

ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , ya,nt−1) = rn∗yzCov(zax,n

∗

t , ya,nt−1) + rn∗yyCov(yax,n

∗

t−1 , ya,nt−1)

where ”a” is the father/mother of ”ax”.

Spouse of the uncle/aunt (spouses of the siblings of the parents)

We have to compute the covariances between ”ax” and ”y”. Let a∗ = m, f be the gender of ”ax” and l′ = m, f

the gender of the ”y”. We project y on his/her spouse b

Cov(zax,n∗

t , zy,l′

t−1) = Cov(zax,n∗

t , rlzzzb,lt−1 + rlzyy

b,lt−1) = rlzzCov(zax,n

∗

t , zb,lt−1) + rlzyCov(zax,n∗

t , yb,lt−1)

Cov(zax,n∗

t , yy,l′

t−1) = Cov(zax,n∗

t , rlyzzb,lt−1 + rlyyy

b,lt−1) = rlyzCov(zax,n

∗

t , zb,lt−1) + rlyyCov(zax,n∗

t , yb,lt−1)

Cov(yax,n∗

t , zy,l′

t−1) = Cov(yax,n∗

t , rlzzzb,lt−1 + rlzyy

b,lt−1) = rlzzCov(yax,n

∗

t , zb,lt−1) + rlzyCov(yax,n∗

t , yb,lt−1)

Cov(yax,n∗

t , yy,l′

t−1) = Cov(yax,n∗

t , rlyzzb,lt−1 + rlyyy

b,lt−1) = rlyzCov(yax,n

∗

t , zb,lt−1) + rlyyCov(yax,n∗

t , yb,lt−1)

where ”b” is uncle/aunt of ”ax”.

Uncles/Aunts of the spouse

We have to compute the covariances between ”a′x′” and ”b”. Let a∗′

= m, f be the gender of ”a′x′” and l = m, f

the gender of the ”b”. We project ”a′x′” on his/her spouse ”ax”

Cov(za′x′,a∗

′

t , zb,lt−1) = Cov(rn∗zz z

ax,n∗

t−1 + rn∗zy y

ax,n∗

t−1 , zb,lt−1) = rn∗zzCov(zax,n

∗

t , zb,lt−1) + rn∗zyCov(yax,n

∗

t−1 , zb,lt−1)

Cov(za′x′,a∗

′

t , yb,lt−1) = Cov(rn∗zz z

ax,n∗

t−1 + rn∗zy y

ax,n∗

t−1 , yb,lt−1) = rn∗zzCov(zax,n

∗

t , yb,lt−1) + rn∗zyCov(yax,n

∗

t−1 , yb,lt−1)

Cov(ya′x′,a∗

′

t , zb,lt−1) = Cov(rn∗yz z

ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , zb,lt−1) = rn∗yzCov(zax,n

∗

t , zb,lt−1) + rn∗yyCov(yax,n

∗

t−1 , zb,lt−1)

82

Cov(ya′x′,a∗

′

t , yb,lt−1) = Cov(rn∗yz z

ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , yb,lt−1) = rn∗yzCov(zax,n

∗

t , ya,nt−1) + rn∗yyCov(yax,n

∗

t−1 , yb,lt−1)

where ”b” is the uncle/aunt of ”ax”.

Siblings of the siblings in law of the parents

We have to compute the covariances between ”ax” and ”c”. Let n∗ = m, f be the gender of ”ax” and k = m, f

the gender of the ”c”. We project c on his/her sibling y

Cov(zax,n∗

t , zc,kt−1) = Cov(zax,n∗

t , rk,l′

zz zy,l′

t−1 + rk,l′

zy yy,l′

t−1) = rk,l′

zz Cov(zax,n∗

t , zy,l′

t−1) + rk,l′

zy Cov(zax,n∗

t , yy,l′

t−1)

Cov(zax,n∗

t , yc,kt−1) = Cov(zax,n∗

t , rk,l′

yz zy,l′

t−1 + rk,l′

yy yy,l′

t−1) = rk,l′

yz Cov(zax,n∗

t , zy,l′

t−1) + rk,l′

yy Cov(zax,n∗

t , yy,l′

t−1)

Cov(yax,n∗

t , zc,kt−1) = Cov(yax,n∗

t , rk,l′

zz zy,l′

t−1 + rk,l′

zy yy,l′

t−1) = rk,l′

zz Cov(yax,n∗

t , zy,l′

t−1) + rk,l′

zy Cov(yax,n∗

t , yy,l′

t−1)

Cov(yax,n∗

t , yc,kt−1) = Cov(yax,n∗

t , rk,l′

yz zy,l′

t−1 + rk,l′

yy yy,l′

t−1) = rk,l′

yz Cov(yax,n∗

t , zy,l′

t−1) + rk,l′

yy Cov(yax,n∗

t , yy,l′

t−1)

where ”y” is the spouse of the uncle/aunt of ”ax”. We can analogously compute the covariance between ”ax”

and ”z”, ”d”, etc.

Horizontal covariances

Siblings in law

We have to compute the covariances between ”a” and ”b”. Let n = m, f be the gender of ”a” and l = m, f the

gender of the ”b”. We project a on his/her spouse x

Cov(za,nt−1, zb,lt−1) = Cov(rn

′zzz

x,n′

t−1 + rn′zyy

x,n′

t−1 , zb,lt−1) = rn

′zzCov(zx,n

′

t−1 , zb,lt−1) + rn

′zyCov(yx,n

′

t−1 , zb,lt−1)

Cov(za,nt−1, yb,lt−1) = Cov(rn

′zzz

x,n′

t−1 + rn′zyy

x,n′

t−1 , yb,lt−1) = rn

′zzCov(zx,n

′

t−1 , yb,lt−1) + rn

′zyCov(yx,n

′

t−1 , yb,lt−1)

Cov(ya,nt−1, zb,lt−1) = Cov(rn

′yzz

x,n′

t−1 + rn′yyy

x,n′

t−1 , zb,lt−1) = rn

′yzCov(zx,n

′

t−1 , zb,lt−1) + rn

′yyCov(yx,n

′

t−1 , zb,lt−1)

Cov(ya,nt−1, yb,lt−1) = Cov(rn

′yzz

x,n′

t−1 + rn′yyy

x,n′

t−1 , yb,lt−1) = rn

′yzCov(zx,n

′

t−1 , yb,lt−1) + rn

′yyCov(yx,n

′

t−1 , yb,lt−1)

where ”x” and ”b” are siblings. Notice that since ”x” is the spouse of ”a”, n′ = f when n = m and viceversa.

Spouse of the siblings in law

We have to compute the covariances between ”a” and ”y”. Let n = m, f be the gender of ”a” and l′ = m, f the

gender of the ”y”. We project y on his/her spouse b

Cov(za,nt−1, zy,l′

t−1) = Cov(za,nt−1, rlzzz

b,lt−1 + rlzyy

b,lt−1) = rlzzCov(za,nt−1, z

b,lt−1) + rlzyCov(za,nt−1, y

b,lt−1)

Cov(za,nt−1, yy,l′

t−1) = Cov(za,nt−1, rlyzz

b,lt−1 + rlyyy

b,lt−1) = rlyzCov(za,nt−1, z

b,lt−1) + rlyyCov(za,nt−1, y

b,lt−1)

Cov(ya,nt−1, zy,l′

t−1) = Cov(ya,nt−1, rlzzz

b,lt−1 + rlzyy

b,lt−1) = rlzzCov(ya,nt−1, z

b,lt−1) + rlzyCov(ya,nt−1, y

b,lt−1)

Cov(ya,nt−1, yy,l′

t−1) = Cov(ya,nt−1, rlyzz

b,lt−1 + rlyyy

b,lt−1) = rlyzCov(ya,nt−1, z

b,lt−1) + rlyyCov(ya,nt−1, y

b,lt−1)

83

where ”a” and ”b” are siblings in law. Notice that since ”b” is the spouse of ”y”, l = f when l′ = m and viceversa.

Sibling of the sibling in law

We have to compute the covariances between ”x” and ”c”. Let n′ = m, f be the gender of ”x” and k = m, f the

gender of the ”c”. We project x on his/her sibling b who has gender l

Cov(zx,n′

t−1 , zc,kt−1) = Cov(rn

′,lzz z

b,lt−1 + rn

′,lzy y

b,lt−1, z

c,kt−1) = rn

′,lzz Cov(zb,lt−1, z

c,kt−1) + rn

′,lzy Cov(yb,lt−1, z

c,kt−1)

Cov(zx,n′

t−1 , yc,kt−1) = Cov(rn

′,lzz z

b,lt−1 + rn

′,lzy y

b,lt−1, y

c,kt−1) = rn

′,lzz Cov(zb,lt−1, y

c,kt−1) + rn

′,lzy Cov(yb,lt−1, y

c,kt−1)

Cov(yx,n′

t−1 , zc,kt−1) = Cov(rn

′,lyz z

b,lt−1 + rn

′,lyy y

b,lt−1, z

c,kt−1) = rn

′,lyz Cov(zb,lt−1, z

c,kt−1) + rn

′,lyy Cov(yb,lt−1, z

c,kt−1)

Cov(yx,n′

t−1 , yc,kt−1) = Cov(rn

′,lyz z

b,lt−1 + rn

′,lyy y

b,lt−1, y

c,kt−1) = rn

′,lyz Cov(zb,lt−1, y

c,kt−1) + rn

′,lyy Cov(yb,lt−1, y

c,kt−1)

where ”b” and ”c” are siblings in law.

Sibling in law of degree 2

We have to compute the covariances between ”a” and ”c”. Let n = m, f be the gender of ”a” and l = m, f the

gender of the ”c”. We project a on his/her spouse x

Cov(za,nt−1, zc,lt−1) = Cov(rn

′zzz

x,n′

t−1 + rn′zyy

x,n′

t−1 , zc,lt−1) = rn

′zzCov(zx,n

′

t−1 , zc,lt−1) + rn

′zyCov(yx,n

′

t−1 , zc,lt−1)

Cov(za,nt−1, yc,lt−1) = Cov(rn

′zzz

x,n′

t−1 + rn′zyy

x,n′

t−1 , yc,lt−1) = rn

′zzCov(zx,n

′

t−1 , yc,lt−1) + rn

′zyCov(yx,n

′

t−1 , yc,lt−1)

Cov(ya,nt−1, zc,lt−1) = Cov(rn

′yzz

x,n′

t−1 + rn′yyy

x,n′

t−1 , zc,lt−1) = rn

′yzCov(zx,n

′

t−1 , zc,lt−1) + rn

′yyCov(yx,n

′

t−1 , zc,lt−1)

Cov(ya,nt−1, yc,lt−1) = Cov(rn

′yzz

x,n′

t−1 + rn′yyy

x,n′

t−1 , yc,lt−1) = rn

′yzCov(zx,n

′

t−1 , yc,lt−1) + rn

′yyCov(yx,n

′

t−1 , yc,lt−1)

where ”x” is the sibling of the sibling in law of ”b”. Notice that since ”x” is the spouse of ”a”, n′ = f when

n = m and viceversa. We can compute siblings in law of any degree analogously.

Spouse of the sibling in law of degree 2

We have to compute the covariances between ”a” and ”z”. Let n = m, f be the gender of ”a” and k′ = m, f the

gender of the ”z”. We project z on his/her spouse c

Cov(za,nt−1, zz,k′

t−1) = Cov(za,nt−1, rkzzz

c,kt−1 + rkzyy

c,kt−1) = rkzzCov(za,nt−1, z

c,kt−1) + rkzyCov(za,nt−1, y

c,kt−1)

Cov(za,nt−1, yz,k′

t−1) = Cov(za,nt−1, rkyzz

c,kt−1 + rkyyy

c,kt−1) = rkyzCov(za,nt−1, z

c,kt−1) + rkyyCov(za,nt−1, y

c,kt−1)

Cov(ya,nt−1, zz,k′

t−1) = Cov(ya,nt−1, rkzzz

c,kt−1 + rkzyy

c,kt−1) = rkzzCov(ya,nt−1, z

c,kt−1) + rkzyCov(ya,nt−1, y

c,kt−1)

Cov(ya,nt−1, yz,k′

t−1) = Cov(ya,nt−1, rkyzz

c,kt−1 + rkyyy

c,kt−1) = rkyzCov(ya,nt−1, z

c,kt−1) + rkyyCov(ya,nt−1, y

c,kt−1)

where ”a” and ”c” are siblings in law of degree 2. Notice that since ”c” is the spouse of ”z”, k = f when k′ = m

and viceversa. We can compute spouses of siblings in law of any degree analogously.

84

Sibling of the sibling in law of degree 2

We have to compute the covariances between ”x” and ”d”. Let n′ = m, f be the gender of ”x” and j = m, f the

gender of the ”d”. We project x on his/her sibling b who has gender l

Cov(zx,n′

t−1 , zd,jt−1) = Cov(rn

′,lzz z

b,lt−1 + rn

′,lzy y

b,lt−1, z

d,jt−1) = rn

′,lzz Cov(zb,lt−1, z

d,jt−1) + rn

′,lzy Cov(yb,lt−1, z

d,jt−1)

Cov(zx,n′

t−1 , yd,jt−1) = Cov(rn

′,lzz z

b,lt−1 + rn

′,lzy y

b,lt−1, y

d,jt−1) = rn

′,lzz Cov(zb,lt−1, y

d,jt−1) + rn

′,lzy Cov(yb,lt−1, y

d,jt−1)

Cov(yx,n′

t−1 , zd,jt−1) = Cov(rn

′,lyz z

b,lt−1 + rn

′,lyy y

b,lt−1, z

d,jt−1) = rn

′,lyz Cov(zb,lt−1, z

d,jt−1) + rn

′,lyy Cov(yb,lt−1, z

d,jt−1)

Cov(yx,n′

t−1 , yd,jt−1) = Cov(rn

′,lyz z

b,lt−1 + rn

′,lyy y

b,lt−1, y

d,jt−1) = rn

′,lyz Cov(zb,lt−1, y

d,jt−1) + rn

′,lyy Cov(yb,lt−1, y

d,jt−1)

where ”b” and ”d” are siblings in law of degree 2. We can compute siblings of siblings in law of any degree

analogously.

Cousins in law

We have to compute the covariances between ”ax” and ”cz”. Let n∗ = m, f be the gender of ”ax” and k∗ = m, f

the gender of the ”cz”. We project cz on c (his/her father or mother) who has gender k

Cov(zax,n∗

t , zcz,k∗

t ) = Cov(zax,n∗

t , Gk∗zkz

c,kt−1 +Gk

∗yky

c,kt−1) = Gk

∗zkCov(zax,n

∗

t , zc,kt−1) +Gk∗ykCov(zax,n

∗

t , yc,kt−1)

Cov(zax,n∗

t , ycz,k∗

t ) = Cov(zax,n∗

t , Bk∗zkz

c,kt−1 +Bk∗

ykyc,kt−1) = Bk∗

zkCov(zax,n∗

t , zc,kt−1) +Bk∗ykCov(zax,n

∗

t , yc,kt−1)

Cov(yax,n∗

t , zcz,k∗

t ) = Cov(yax,n∗

t , Gk∗zkz

c,kt−1 +Gk

∗yky

c,kt−1) = Gk

∗zkCov(yax,n

∗

t , zc,kt−1) +Gk∗ykCov(yax,n

∗

t , yc,kt−1)

Cov(yax,n∗

t , ycz,k∗

t ) = Cov(yax,n∗

t , Bk∗zkz

c,kt−1 +Bk∗

ykyc,kt−1) = Bk∗

zkCov(yax,n∗

t , zc,kt−1) +Bk∗ykCov(yax,n

∗

t , yc,kt−1)

where ”c” is the sibling in law of the uncle/aunt of ”ax”. We can compute cousins in law of any degree analogously.

H Assortative Mating Only in z

If we assume that there is assortative mating only in z, that is rmzy = rmyy = rfzy = rfyy = 0, we have that

rmzy =1

(1− ρ2zmym)

σzf

σym(ρymzf − ρzmymρzmzf ) = 0⇒ ρymzf = ρzmymρzmzf (H.1)

rmyy =1

(1− ρ2zmym)

σyf

σym(ρymyf − ρzmymρzmyf ) = 0⇒ ρymyf = ρzmymρzmyf (H.2)

rfzy =1

(1− ρ2zfyf

)

σzm

σyf(ρzmyf − ρzfyfρzmzf ) = 0⇒ ρzmyf = ρzfyfρzmzf (H.3)

rfyy =1

(1− ρ2zfyf

)

σym

σyf(ρymyf − ρzfyfρymzf ) = 0⇒ ρymyf = ρzfyfρymzf

85

and the other coefficients of the linear projections

rmzz =σzf

σzmρzmzf

rmyz =σyf

σzmρzmyf =

Using (H.3)

σyf

σzmρzfyfρzmzf

rfzz =σzm

σzfρzmzf

rfyz =σym

σzfρymzf =

Using (H.1)

σym

σzfρzmymρzmzf

only depend on ρzmzf , ρzmym and ρzfyf .

Moreover, substituting the expressions for ρymzf and ρzmyf in (H.1) and (H.3), in the steady state equations for

ρzmym and ρzfyf ((G.3) and (G.4)), we have that

(1− βmαmy γmαmz )ρzmym − βm(1− αmy )γm(1− αmz )σzf

σzm

σyf

σymρzfyf

=σzm

σym+ βmαmy γ

m(1− αmz )ρzmymρzmzfσzf

σzm+ βm(1− αmy )γmαmz ρzfyfρzmzf

σyf

σym

−βf (1− αfy )γf (1− αfz )σzm

σzf

σym

σyfρzmym + (1− βfαfyγfαfz )ρzfyf

=σzf

σyf+ βfαfyγ

f (1− αfz )ρzfyfρzmzfσzm

σzf+ βf (1− αfy )γfαfzρzmymρzmzf

σym

σyf

which can be written as:

1− βmαmy γm(αmz + (1− αmz )ρzmzf

σzf

σzm

)−βm(1− αmy )γm

σyf

σym

((1− αmz )

σzf

σzm+ αmz ρzmzf

)−βf (1− αfy )γf

σym

σyf

((1− αfz )σzmσ

zf+ αfzρzmzf

)1− βfαfyγf

(αfz + (1− αfz )ρzmzf

σzmσzf

) ( ρzmym

ρzfyf

)

=

σzmσymσzf

σyf

and therefore the coefficients of the linear projections can be written as a function of ρzmzf (and the γs, βs,

etc.). The model then has 17 parameters instead of 20.

I Assortative Mating Only in y

86

If we assume that there is assortative mating only in y, that is rmzz = rmyz = rfzz = rfyz = 0, we have that

rmzz =1

(1− ρ2zmym)

σzf

σzm(ρzmzf − ρzmymρymzf ) = 0⇒ ρzmzf = ρzmymρymzf

rmyz =1

(1− ρ2zmym)

σyf

σzm(ρzmyf − ρzmymρymyf ) = 0⇒ ρzmyf = ρzmymρymyf

rfzz =1

(1− ρ2zfyf

)

σzm

σzf(ρzmzf − ρzfyfρzmyf ) = 0⇒ ρzmzf = ρzfyfρzmyf

rfyz =1

(1− ρ2zfyf

)

σym

σzf(ρymzf − ρzfyfρymyf ) = 0⇒ ρymzf = ρzfyfρymyf

and the coefficients of the linear projections

rmzy =σzf

σymρymzf =

σzf

σymρzfyfρymyf

rmyy =σyf

σymρymyf

rfzy =σzm

σyfρzmyf =

σzm

σyfρzmymρymyf

rfyy =σym

σyfρymyf

only depend on ρymyf , ρzmym and ρzfyf .

Moreover, substituting ρymzf and ρzmyf in the steady state equations for ρzmym and ρzfyf ((G.3) and (G.4))

(1− βmαmy γmαmz )ρzmym − βm(1− αmy )γm(1− αmz )σzf

σzm

σyf

σymρzfyf

=σzm

σym+ βmαmy γ

m(1− αmz )ρzfyfρymyfσzf

σzm+ βm(1− αmy )γmαmz ρzmymρymyf

σyf

σym

−βf (1− αfy )γf (1− αfz )σzm

σzf

σym

σyfρzmym + (1− βfαfyγfαfz )ρzfyf

=σzf

σyf+ βfαfyγ

f (1− αfz )ρzmymρymyfσzm

σzf+ βf (1− αfy )γfαfzρzfyfρymyf

σym

σyf

which can be written as:

87

1− βmαmz γm(αmy + (1− αmy )ρymyf

σyf

σym

)−βm(1− αmz )γm

σzf

σzm

((1− αmy )

σyf

σym+ αmy ρymyf

)−βf (1− αfz )γf σzmσ

zf

((1− αfy )

σym

σyf

+ αfyρymyf)

1− βfαfzγf(αfy + (1− αfy )ρymyf

σym

σyf

) ( ρzmym

ρzfyf

)

=

σzmσymσzf

σyf

and therefore the coefficients of the linear projections can be written as a function of ρymyf (and the γs, βs,

etc.). The model then has 17 parameters instead of 20.

J The Reduced Form Model

We now consider a reduced form model where the outcome y for an individual from generation t only depends

on his father, and is given by

yt = βymt−1 + zt + xt + ut (J.1)

Moreover, the socioeconomic status of the child, zt, only depends on the father zmt−1

zt = γzmt−1 + et + vt (J.2)

Substituting (J.2) in (J.1)

yt = βymt−1 + γzmt−1 + et + vt + xt + ut (J.3)

Regarding the shocks, as in the general model, we assume that xt and et are shared by all siblings and are

uncorrelated to each other and with the other variables (in particular with zmt−1 and ymt−1). Finally ut and vt are

white-noise errors.

We can now compare (J.2) and (J.3) with the expression for zkt and ykt as a function of the father obtained in

Section 1. For the general model we have that

zkt = Gkzmzmt−1 +Gkymy

mt−1 + gkmω

mt−1 + ekt + vkt (J.4)

ykt = Bkymy

mt−1 +Bk

zmzmt−1 + bkmε

mt−1 + gkmω

mt−1 + ekt + vkt + xkt + ukt (J.5)

where

Gkzm = γk(αkz + (1− αkz)rmzz)

Gkym = γk(1− αkz)rmzygkm = γk(1− αkz)

88

Bkym = βk

(αky + (1− αky)rmyy

)+Gkym

Bkzm = βk(1− αky)rmyz +Gkzm

bkm = βk(1− αky)

It is then trivial to see that there are two key differences between the two models:

1. The errors in (J.2) are assumed to be orthogonal to ymt−1 and zmt−1, whereas in (J.4), zkt depends on ymt−1 and

therefore, unless Gkym = 0, if we project ymt−1 on zmt−1, the new error will be correlated to ymt−1.

2. In the reduced form model, zmt−1 has the same coefficient in (J.2) and (J.1), whereas in the general model,

the coefficient of zmt−1 is different in (J.4) and (J.5).

Then, we have that the general model can be written as a reduced form model if and only if

1. Gkym = 0⇐⇒ γk = 0, or αkz = 1 or rmzy = 0

2. Bkzm = Gkzm ⇐⇒ βk = 0, or αky = 1 or rmyz = 0

We then have that the general model can be written as a reduced form model

1. In the trivial case when just the father matters (αkz = αky = 1, or αkz = 1 and βk = 0, or αky = 1 and γk = 0).

2. When βk = 0 and ymt−1 does not influence zft−1 once the effect of zmt−1 has been netted out (rmzy = 0).

3. When γk = 0 and zmt−1 does not influence yft−1 once the effect of ymt−1 has been netted out (rmyz = 0).

4. When ymt−1 does not influence zft−1 once the effect of zmt−1 has been netted out (rmzy = 0), and zmt−1 does not

influence yft−1 once the effect of ymt−1 has been netted out (rmyz = 0).

Notice that Case 2 corresponds to a latent factor model with assortative mating only in z and Case 3 to a direct

effect model with assortative mating only in y.

K The Genetic Model

The genetic model is nested in our general model by imposing the following restrictions:

• There is no a direct effect of parents outcome on children outcome (βk = 0, k = f,m)

• The latent factor is genetic and therefore it is transmitted from parents to children as

zkt =zmt−1 + zft−1

2+ vkt

where vkt is uncorrelated across relatives and to zmt−1 and zft−1 (γk = 1 and σ2ek

= 0, k = f,m)

• The share of the variance explained by the latent factor is equal across genders (σ2zk

= σ2z , k = f,m)

89

• There is assortative mating only in the observed outcome y (ρzmyf , ρymzf and ρzmzf are functions of ρymyf

and some of the other parameters of the model (see Section 3)).

The genetic model has only 5 parameters: σ2z , σ

2xm , σ

2xf, σxmxf , ρymyf .

K.1 Genetic transmission

Suppose that each person has only one gene with two alleles, A and B. One of the allele is inherited from the

father and the other one from the mother. Let XA and XB be the random variables representing the potential

values each allele may take and suppose that the outcome of interest Y depends on Z = XA +XB

Y = Z + U

where U is mean independent of Z.

We have to compute

E(Zt |Zmt−1 = zm, Zft−1 = zf

)The distribution of the child Zt conditional on the parents Xm

A,t−1 = xmA , XmB,t−1 = xmB , Xf

A,t−1 = xfA, XfB,t−1 = xfB

is multinomial with the following probability mass function

Zt =

xmA + xfA, with probability 1

4

xmA + xfB, with probability 14

xmB + xfA, with probability 14

xmB + xfB, with probability 14

Then, the distribution of Zt conditional on XmA,t−1 = xmA , Zmt−1 = zm, Xf

A,t−1 = xfA, Zft−1 = zf is also multinomial,

and the probability mass function is

Zt =

xmA + xfA, with probability 1

4

xmA + zf − xfA, with probability 14

zm − xmA + xfA, with probability 14

zm − xmA + zf − xfA, with probability 14

Then,

E(Zt |Xm

A,t−1, Zmt−1, X

fA,t−1, Z

ft−1

)=

1

4

(XmA,t−1 +Xf

A,t−1

)+

1

4

(XmA,t−1 + Zft−1 −X

fA,t−1

)+

1

4

(Zmt−1 −Xm

A,t−1 +XfA,t−1

) 1

4

(Zmt−1 −Xm

A,t−1 + Zft−1 −XfA,t−1

)=

1

4Zft−1 +

1

4Zmt−1 +

1

4

(Zmt−1 + Zft−1

)=

1

2

(Zmt−1 + Zft−1

)90

Since E(Zt |Xm

A,t−1, Zmt−1, X

fA,t−1, Z

ft−1

)does not depend on Xm

A,t−1 and XfA,t−1, using the law of iterated

expectations

E(Zt |Zmt−1, Z

ft−1

)=

1

2

(Zmt−1 + Zft−1

)If we now consider that each person has n genes, each with two alleles, Ai and Bi. For each gene, one of the

allele is inherited from the father and the other one from the mother. Let XAi and XBi be the random variables

representing the potential values each allele may take and suppose that the outcome of interest Y depends on

Z =∑n

i=1 (XAi +XBi) =∑n

i=1 Zi , where Zi = XAi + XBi . For each gene i, given XmAi,t−1 = xmAi

, Zmi,t−1 = zmi ,

XfAi,t−1 = xfAi

, there are 4 possible realizations of the child alleles of gene i, and therefore 4n possible genomes.

Then, the distribution of the child Zt conditional on the parents XmA1,t−1 = xmA1

, Zm1,t−1 = zm1 , XfA1,t−1 = xfA1

,

Zf1,t−1 = zf1 , ...,XmAn,t−1 = xmAn

, Zmn,t−1 = zmn , XfAn,t−1 = xfAn

, Zfn,t−1 = zfn is also multinomial, and the probability

mass function is

Zt =

(xmA1

+ xfA1

)+(xmA2

+ xfA2

)+ ..+

(xmAn

+ xfAn

), with probability 1

4n(xmA1

+ zf1 − xfA1

)+(xmA2

+ xfA2

)+ ..+

(xmAn

+ xfAn

), with probability 1

4n(zm1 − xmA1

+ xfA1

)+(xmA2

+ xfA2

)+ ..+

(xmAn

+ xfAn

), with probability 1

4n(zm1 − xmA1

+ zf1 − xfA1

)+(xmA2

+ xfA2

)+ ..+

(xmAn

+ xfAn

), with probability 1

4n

...(zm1 − xmA1

+ zf1 − xfA1

)+(zm2 − xmA2

+ zf2 − xfA2

)+ ..+

(zmn − xmAn

+ zfn − xfAn

), with probability 1

4n

As in the model with just one gene, all the xkAicancel when we compute the conditional mean

E(Zt |Xm

A1,t−1, Zm1,t−1, X

fA1,t−1, Z

f1,t−1, ..., X

mAn,t−1, Z

mn,t−1, X

fAn,t−1, Z

fn,t−1

)=

1

2Zm1,t−1 + ...+

1

2Zmn,t−1 +

1

2Zf1,t−1 + ...+

1

2Zfn,t−1 =

1

2

(Zmt−1 + Zft−1

)Then, since the conditional expectation above only depends on Zmt−1 and Zft−1, using the law of iterated expec-

tations

E(Zt |Zmt−1, Z

ft−1

)=

1

2

(Zmt−1 + Zft−1

)Lets now consider two siblings i and j. We have to compute

E(ZitZjt |Zmt−1 = zm, Zft−1 = zf

)The distribution of ZitZjt conditional on the parents Xm

A,t−1 = xmA , XmB,t−1 = xmB , Xf

A,t−1 = xfA, XfB,t−1 = xfB is

91

multinomial with the following probability mass function

ZitZjt =

(xmA + xfA

)2, with probability 1

16(xmA + xfA

)(xmA + xfB

), with probability 1

8(xmA + xfA

)(xmB + xfA

), with probability 1

8(xmA + xfA

)(xmB + xfB

), with probability 1

8(xmA + xfB

)2, with probability 1

16(xmA + xfB

)(xmB + xfA

), with probability 1

8(xmA + xfB

)(xmB + xfB

), with probability 1

8(xmB + xfA

)2, with probability 1

16(xmB + xfA

)(xmB + xfB

), with probability 1

8(xmB + xfB

)2, with probability 1

16

Then, the distribution of ZitZjt conditional on XmA,t−1 = xmA , Zmt−1 = zm, Xf

A,t−1 = xfA, Zft−1 = zf is also

multinomial, and the probability mass function is

ZitZjt =

(xmA + xfA

)2, with probability 1

16(xmA + xfA

)(xmA + zf − xfA

), with probability 1

8(xmA + xfA

)(zm − xmA + xfA

), with probability 1

8(xmA + xfA

)(zm − xmA + zf − xfA

), with probability 1

8(xmA + zf − xfA

)2, with probability 1

16(xmA + zf − xfA

)(zm − xmA + xfA

), with probability 1

8(xmA + zf − xfA

)(zm − xmA + zf − xfA

), with probability 1

8(zm − xmA + xfA

)2, with probability 1

16(zm − xmA + xfA

)(zm − xmA + zf − xfA

), with probability 1

8(zm − xmA + zf − xfA

)2, with probability 1

16

92

Then,

E(ZitZjt |Xm

A,t−1, Zmt−1, X

fA,t−1, Z

ft−1

)=

1

16

(XmA +Xf

A

)((XmA +Xf

A

)+(XmA + Zf −Xf

A

)+(Zm −Xm

A +XfA

)+(Zm −Xm

A + Zf −XfA

))+

1

16

(XmA + Zf −Xf

A

)((XmA +Xf

A

)+(XmA + Zf −Xf

A

)+(Zm −Xm

A +XfA

)+(Zm −Xm

A + Zf −XfA

))+

1

16

(Zm −Xm

A +XfA

)((XmA +Xf

A

)+(XmA + Zf −Xf

A

)+(Zm −Xm

A +XfA

)+(Zm −Xm

A + Zf −XfA

))+

1

16

(Zm −Xm

A + Zf −XfA

)((XmA +Xf

A

)+(XmA + Zf −Xf

A

)+(Zm −Xm

A +XfA

)+(Zm −Xm

A + Zf −XfA

))=

1

8

(XmA +Xf

A

)(Zf + Zm

)+

1

8

(XmA + Zf −Xf

A

)(Zf + Zm

)+

1

8

(Zm −Xm

A +XfA

)(Zf + Zm

)+

1

8

(Zm −Xm

A + Zf −XfA

)(Zf + Zm

)=

1

4

(Zmt−1 + Zft−1

)2

Since E(ZitZjt |Xm

A,t−1, Zmt−1, X

fA,t−1, Z

ft−1

)does not depend on Xm

A,t−1 and XfA,t−1, using the law of iterated

expectations

E(ZitZjt|Zmt−1, Z

ft−1

)=

1

4

(Zmt−1 + Zft−1

)2

We then have that

E(ZitZjt|Zmt−1, Z

ft−1

)= E

(Zit|Zmt−1, Z

ft−1

)E(Zjt|Zmt−1, Z

ft−1

)and Zit and Zjt are uncorrelated conditional on Zmt−1, Z

ft−1. Then, we can write

Zit =1

2

(Zmt−1 + Zft−1

)+ eit

where eit and ejt are uncorrelated across siblings.

93

L General Assortative Mating Model with Two Unobservable Factors

We assume that the value of the output y for an individual from generation t is given by

ykt = βkykt−1 + zG,kt + zC,kt + xkt + ukt (L.1)

where the superscript k stands for males (k = m) and for females (k = f). We assume that

ykt−1 = αkyymt−1 + (1− αky)y

ft−1

zG,kt and zC,kt are two unobservable factors. The genetic factor of the child, zG,kt , depends on the father zG,mt−1 as

well as on the mother zG,ft−1

zG,kt =zG,mt−1 + zG,ft−1

2+ vG,kt (L.2)

The cultural factor of the child, zC,kt , also depends on the father zC,mt−1 as well as on the mother zC,ft−1

zC,kt = γkzkt−1 + eC,kt + vC,kt

zkt−1 = αkzzC,mt−1 + (1− αkz)z

C,ft−1

(L.3)

Regarding the shocks in model (L.1), we assume that xkt , and eC,kt are shared by all siblings of the same gender,

can be correlated across siblings of different gender and are uncorrelated with the other variables (in particular

with zG,kt , zC,kt and yt−1). Finally ukt , vG,kt and vC,kt are individual’s white-noise error terms.

L.1 Assortative mating process

We assume there is assortative mating both in years of schooling and in the cultural factor (see Berhman and

Rosenzweig 2002 for a related model with assortative mating in two dimensions). We then have 15 correlations:

ρzG,mzC,m , ρzG,mym , ρzC,mym , ρzG,f zC,f , ρzG,fyf , ρzC,fyf , ρzG,mzG,f , ρzG,mzC,f , ρzG,my,f , ρzC,mzG,f , ρzC,mzC,f , ρzC,my,f ,

ρymzG,f , ρymzC,f , ρymy,f In particular we consider the linear projections of zG,ft−1 , zC,ft−1 and yft−1 on zG,mt−1 , z

C,mt−1 and

ymt−1:

zG,ft−1 = rmzGzGzG,mt−1 + rmzGzCz

C,mt−1 + rmzGyy

mt−1 + wG,mt−1

zC,ft−1 = rmzCzGzG,mt−1 + rmzCzCz

C,mt−1 + rmzCyy

mt−1 + wC,mt−1

yft−1 = rmyzGzG,mt−1 + rmyzCz

C,mt−1 + rmyyy

mt−1 + εmt−1

Since we assume there is no assortative mating in the genetic factor, we have that rmzGzG

= rmzCzG

= rmyzG

= 0,

and

ρzG,mzG,f =

(ρzG,mymρzC,mym − ρzG,mzC,m

)ρzC,mzG,f +

(ρzG,mzC,mρzC,mym − ρzG,mym

)ρymzG,f

ρ2zC,mym

− 1(L.4)

94

ρzG,mzC,f =

(ρzG,mymρzC,mym − ρzG,mzC,m

)ρzC,mzC,f +

(ρzG,mzC,mρzC,mym − ρzG,mym

)ρymzC,f

ρ2zC,mym

− 1

ρzG,myf =

(ρzG,mymρzC,mym − ρzG,mzC,m

)ρzC,myf +

(ρzG,mzC,mρzC,mym − ρzG,mym

)ρymyf

ρ2zC,mym

− 1

ρzG,mzG,f =

(ρzG,mym

ρzC,mym

−ρzG,mzC,m

)((ρzG,f yf

ρzC,f yf

−ρzG,f zC,f

)ρzC,mzC,f +

(ρzG,f zC,f ρzC,f yf

−ρzG,f yf

)ρzC,myf

)(ρ2zC,mym

−1)(ρ2zC,f yf

−1)

+

(ρzG,mzC,mρzC,mym

−ρzG,mym

)((ρzG,f yf

ρzC,f yf

−ρzG,f zC,f

)ρymzC,f +

(ρzG,f zC,f ρzC,f yf

−ρzG,f yf

)ρymyf

)(ρ2zC,mym

−1)(ρ2zC,f yf

−1)

which reduces the number of free correlations to 12.

The remaining coefficients are:

rmzGzC =1

(1− ρ2zC,mym

)

σzG,f

σzC,m

(ρzC,mzG,f − ρzC,mymρymzG,f )

rmzGzC =1

(1− ρ2zC,mym

)

σzG,f

σzC,m

(ρzC,mzG,f − ρzC,mymρymzG,f )

rmzGy =1

(1− ρ2zC,mym

)

σzG,f

σym(ρymzG,f − ρzC,mymρzC,mzG,f )

rmzCzC =1

(1− ρ2zC,mym

)

σzC,f

σzC,m

(ρzC,mzC,f − ρzC,mymρymzC,f )

rmzCy =1

(1− ρ2zC,mym

)

σzC,f

σym(ρymzC,f − ρzC,mymρzC,mzC,f )

rmyzC =1

(1− ρ2zC,mym

)

σyf

σzC,m

(ρzC,myf − ρzC,mymρymyf )

rmyy =1

(1− ρ2zC,mym

)

σyf

σym(ρymyf − ρzC,mymρzC,myf )

95

Notice that the variance matrix of (wG,mt−1 , wC,mt−1 , ε

mt−1) is given by

V ar

wG,mt−1

wC,mt−1

εmt−1

= V ar

zG,ft−1

zC,ft−1

yft−1

−

0 rmzGzC

rmzGy

0 rmzCzC

rmzCy

0 rmyzC

rmyy

V ar

zG,mt−1

zC,mt−1

ymt−1

0 rmzGzC

rmzGy

0 rmzCzC

rmzCy

0 rmyzC

rmyy

′

=

σ2zG,f σzG,f zC,f σzG,fyf

σzG,f zC,f σ2zC,f σzC,fyf

σzG,fyf σzC,fyf σ2yf

−

0 rm

zGzCrmzGy

0 rmzCzC

rmzCy

0 rmyzC

rmyy

σ2

zG,m σzG,mzC,m σzG,mym

σzG,mzC,m σ2zC,m σzC,mym

σzG,mym σzC,mym σ2ym

0 0 0

rmzGzC

rmzCzC

rmyzC

rmzGy

rmzCy

rmyy

=

σ2zG,f σzG,f zC,f σzG,fyf

σzG,f zC,f σ2zC,f σzC,fyf

σzG,fyf σzC,fyf σ2yf

−

rmzGzC

σzG,mzC,m + rmzGy

σzG,mym rmzGzC

σ2zC,m + rm

zGyσzC,mym rm

zGzCσzC,mym + rm

zGyσ2ym

rmzCzC

σzG,mzC,m + rmzCy

σzG,mym rmzCzC

σ2zC,m + rm

zCyσzC,mym rm

zCzCσzC,mym + rm

zCyσ2ym

rmyzC

σzG,mzC,m + rmyyσzG,mym rmyzC

σ2zC,m + rmyyσzC,mym rm

yzCσzC,mym + rmyyσ

2ym

∗

0 0 0

rmzGzC

rmzCzC

rmyzC

rmzGy

rmzCy

rmyy

We then have that

σ2wG,m = σ2

zG,f − (rmzGzC )2σ2zC,m − 2rmzGzCr

mzGyσzC,mym − (rmzGy)

2σ2ym

σ2wC,m = σ2

zC,f − (rmzCzC )2σ2zC,m − 2rmzCzCr

mzCyσzC,mym − (rmzCy)

2σ2ym

σ2εm = σ2

yf − (rmyzC )2σ2zC,m − 2rmyzCr

myyσzC,mym − (rmyy)

2σ2ym

and

Cov(wG,mt−1 , wC,mt−1 ) = σzG,f zC,f − rmzGzCr

mzCzCσ

2zC,m − (rmzGyr

mzCzC + rmzGzCr

mzCy)σzC,mym − rmzGyr

mzCyσ

2ym

Cov(wG,mt−1 , ε,mt−1) = σzG,fyf − rmzGzCr

myzCσ

2zC,m − (rmzGyr

myzC + rmzGzCr

myy)σzC,mym − rmzGyr

myyσ

2ym

Cov(wC,mt−1 , ε,mt−1) = σzC,fyf − rmzCzCr

myzCσ

2zC,m − (rmzCyr

myzC + rmzCzCr

myy)σzC,mym − rmzCyr

myyσ

2ym

We use these matching functions to write the genetic factor, zG,kt , the cultural factor, zC,kt , and years of schooling,

ykt , as a function of father’s genetic factor, zG,mt−1 , cultural factor, zC,mt−1 , and years of schooling, ymt−1. We write

96

(L.2) as

zG,kt =zG,mt−1 + zG,ft−1

2+ eG,kt

=1

2

(zG,mt−1 + rmzGzCz

C,mt−1 + rmzGyy

mt−1 + wG,mt−1

)+ vG,kt

= GkzgmzG,mt−1 +Gkzmz

C,mt−1 +Gkymy

mt−1 + gkmw

G,mt−1 + vG,kt

where

Gkzgm =1

2

Gkzm =1

2rmzGzC

Gkym =1

2rmzGy

gkm =1

2

(L.3) as

zC,kt = γk(αkzz

C,mt−1 + (1− αkz)z

C,ft−1

)+ eC,kt + vC,kt

= γk(αkzz

C,mt−1 + (1− αkz)

(rmzCzCz

C,mt−1 + rmzCyy

mt−1 + wC,mt−1

))+ eC,kt + vC,kt

= CkzmzC,mt−1 + Ckymy

mt−1 + ckmω

C,mt−1 + eC,kt + vC,kt

where

Ckzm = γk(αkz + (1− αkz)rmzCzC )

Ckym = γk(1− αkz)rmzCyckm = γk(1− αkz)

and (L.1) as

ykt = βk(αkyy

mt−1 + (1− αky)y

ft−1

)+ zG,kt + zC,kt + xkt + ukt

ykt = βk(αkyy

mt−1 + (1− αky)y

ft−1

)+ zG,kt + zC,kt + xkt + ukt

= βk(αkyy

mt−1 + (1− αky)

(rmyzCz

C,mt−1 + rmyyy

mt−1 + εmt−1

))1

2zG,mt−1 +

1

2rmzGzCz

C,mt−1 +

1

2rmzGyy

mt−1 +

1

2wG,mt−1 + vG,kt

CkzmzC,mt−1 + Ckymy

mt−1 + ckmω

C,mt−1 + eC,kt + vC,kt + xkt + ukt

97

ykt = Bkzgmz

G,mt−1 +Bk

zmzC,mt−1 +Bk

ymymt−1 + gkmw

G,mt−1 + ckmω

C,mt−1 + bkmε

mt−1

+eG,kt + eC,kt + xkt + vG,kt + vC,kt + ukt

where

Bkzgm =

1

2

Bkzm = βk(1− αky)rmyzC +

1

2rmzGzC + Ckzm

Bkym = βk

(αky + (1− αky)rmyy

)+

1

2rmzGy + Ckym

bkm = βk(1− αky)

All these expressions will be used to compute correlations between relatives that are related through their fathers.

Analogously, we can compute the linear projections of zG,mt−1 , zC,mt−1 and ymt−1 on zG,ft−1 , z

C,ft−1 and yft−1. The assumption

of no assortative mating in the genetic factor implies rfzGzG

= rfzCzG

= rfyzG

= 0

ρzG,mzG,f =

(ρzG,fyfρzC,fyf − ρzG,f zC,f

)ρzG,mzC,f +

(ρzG,f zC,fρzC,fyf − ρzG,fyf

)ρzG,myf

ρ2zC,fyf

− 1(L.5)

ρzC,mzG,f =

(ρzG,fyfρzC,fyf − ρzG,f zC,f

)ρzC,mzC,f +

(ρzG,f zC,fρzC,fyf − ρzG,fyf

)ρzC,myf

ρ2zC,fyf

− 1

ρymzG,f =

(ρzG,fyfρzC,fyf − ρzG,f zC,f

)ρymzC,f +

(ρzG,f zC,fρzC,fyf − ρzG,fyf

)ρymyf

ρ2zC,fyf

− 1

The definitions of ρzG,mzG,f in (L.4) and (L.5) turn to be identical and therefore the the number of free correlations

in reduced to 10. We can then write the genetic factor, zG,kt , the cultural factor, zC,kt , and years of schooling, ykt ,

as a function of mother’s genetic factor, zG,ft−1 , cultural factor, zC,ft−1, and years of schooling, yft−1.These expressions

will be used to compute correlations between relatives that are related through their mothers.

L.2 Steady state assumption

We assume that the second order moments of all variables are time invariant. This steady state assumption

implies that ρzC,mzG,m , ρzC,mym , ρzG,mym , ρzC,f zG,f , ρzC,fyf , and ρzG,fyf depend on the remaining parameters of

the model as shown below.

98

We first compute ρzG,mzC,m

Cov(zC,mt , zG,mt ) = Cov

(γmαmz z

C,mt−1 + γm(1− αmz )zC,ft−1,

1

2zG,mt−1 +

1

2zG,ft−1

)= γmαmz

1

2Cov

(zC,mt−1 , z

G,mt−1

)+ γmαmz

1

2Cov

(zC,mt−1 , z

G,ft−1

)+γm(1− αmz )

1

2Cov

(zC,ft−1, z

G,mt−1

)+ γm(1− αmz )

1

2Cov

(zC,ft−1, z

G,ft−1

)and dividing by σzC,m and σzG,m and rearranging we have(

1− γmαmz1

2

)ρzG,mzC,m − γm(1− αmz )

1

2

σzC,f

σzC,m

σzG,f

σzG,m

ρzC,f zG,f

= γmαmz1

2

σzG,f

σzG,m

ρzC,mzG,f + γm(1− αmz )1

2

σzC,f

σzC,m

ρzG,mzC,f

analogously we obtain another equation from the steady state assumption on ρzG,f zC,f(1− 1

2γf(

1− αfz))

ρzG,f zC,f −1

2γfαfz

σzC,m

σzC,f

σzG,m

σzG,f

ρzG,mzC,m

=1

2γf(

1− αfz) σzG,m

σzG,f

ρzG,mzC,f +1

2γfαfz

σzC,m

σzC,f

ρzC,mzG,f

We now compute ρzG,mym

Cov(ymt , zG,mt ) = Cov

(βmymt−1 + zG,mt + zC,mt , zG,mt

)= Cov

(βmαmy y

mt−1 + βm(1− αmy )yft−1,

1

2zG,mt−1 +

1

2zG,ft−1

)+σ2

zG,m + Cov(zC,mt , zG,mt

)= βmαmy

1

2Cov

(ymt−1, z

G,mt−1

)+ βmαmy

1

2Cov

(ymt−1, z

G,ft−1

)+ βm(1− αmy )

1

2Cov

(yft−1, z

G,mt−1

)+βm(1− αmy )

1

2Cov

(yft−1, z

G,ft−1

)+ σ2

zG,m + Cov(zC,mt , zG,mt

)and dividing by σym and σzG,m and rearranging we have(

1− 1

2βmαmy

)ρzG,mym −

1

2βm(1− αmy )

σyf

σym

σzG,f

σzG,m

ρzG,fyf

=σzC,m

σymρzG,mzC,m +

1

2βmαmy

σzG,f

σzG,m

ρymzG,f +1

2βm(1− αmy )

σyf

σymρzG,myf +

σzG,m

σym

99

analogously we obtain another equation from the steady state assumption on ρzG,fy,f(1− 1

2βf(

1− αfy))

ρzG,fyf −1

2βfαfy

σym

σyf

σzG,m

σzG,f

ρzG,mym

=σzC,f

σyfρzG,f zC,f +

1

2βf(

1− αfy) σzG,m

σzG,f

ρyf zG,m +1

2βfαfy

σym

σyfρzG,fym +

σzG,f

σyf

We finally compute ρzC,mym

Cov(ymt , zC,mt ) = Cov(βmymt−1 + zG,mt + zC,mt , zC,mt )

= Cov(βmαmy y

mt−1 + βm(1− αmy )yft−1, γ

mαkzzC,mt−1 + γm(1− αkz)z

C,ft−1

)+Cov

(zC,mt , zG,mt

)+ σ2

zC,m

= βmαmy γmαmz Cov

(ymt−1, z

C,mt−1

)+ βmαmy γ

m(1− αkz)Cov(ymt−1, z

C,ft−1

)+βm(1− αmy )γmαkzCov

(yft−1, z

C,mt−1

)+βm(1− αmy )γm(1− αkz)Cov

(yft−1, z

C,ft−1

)+ Cov

(zC,mt , zG,mt

)+ σ2

zC,m

and dividing by σym and σzC,m and rearranging we have

(1− βmαmy γmαmz

)ρzC,mym − βm(1− αmy )γm(1− αmz )

σzC,f

σzC,m

σyf

σymρzC,fyf

=σzG,m

σymρzG,mzC,m + βmαmy γ

m(1− αmz )σzC,f

σzC,m

ρymzC,f + βm(1− αmy )γmαmzσyf

σymρzC,myf +

σzC,m

σym

analogously we obtain another equation from the steady state assumption on ρzC,fy,f

(1− βf

(1− αfy

)γfαfz

)ρzC,fyf − βfαfyγfαfz

σzC,m

σzC,f

σym

σyfρzC,mym

=σzG,f

σyfρzG,f zC,f + βf

(1− αfy

)γfαfz

σzC,m

σzC,f

ρyf zC,m + βfαfyγf(

1− αfz) σymσyf

ρzC,fym +σzC,f

σyf

The six equations for the steady state reduce the number of free correlations to 4: ρzC,mzC,f , ρzC,myf , ρymzC,f , and

ρymyf . Then, this model has 21 parameters: βk, γk, σzG,k , σzC,k , σ2xk, σ2

eC,k , αky , α

kz , k = m, f, σxmxf , σeC,meC,f , ρzC,mzC,f ,

ρzC,myf , ρymzC,f , and ρymyf , 1 parameter more than the one factor model.

L.3 Covariances

L.3.1 Main covariances

We first compute the main covariances (husband-wife, parent-child and siblings). Then, the covariances for other

relatives are obtained recursively.

100

Husband and wife

Cov(ymt−1, yft−1) = Cov(ymt−1, r

myzGz

G,mt−1 + rmyzCz

C,mt−1 + rmyyy

mt−1 + εmt−1) = rmyzGCov(ymt−1, z

G,mt−1 )

+rmyzCCov(ymt−1, zC,mt−1 ) + rmyyσ

2ym

= ρzC,mymσzC,mσym + rmyyσ2ym

Parent–child

Let k = m, f be the gender of the child and n = m, f the gender of the parent

Cov(zG,kt , zG,nt−1) = Cov(GkzgnzG,nt−1 +Gkznz

C,nt−1 +Gkyny

nt−1, z

G,nt−1)

= Gkzgnσ2zG,n +GkznCov(zC,nt−1, z

G,nt−1) +GkynCov(ynt−1, z

G,nt−1)

Cov(zG,kt , zC,nt−1) = Cov(GkzgnzG,nt−1 +Gkznz

C,nt−1 +Gkyny

nt−1, z

C,nt−1)

= GkzgnCov(zG,nt−1 , zC,nt−1) +Gkznσ

2zC,n +GkynCov(ynt−1, z

C,nt−1)

Cov(zG,kt , ynt−1) = Cov(GkzgnzG,nt−1 +Gkznz

C,nt−1 +Gkyny

nt−1, y

nt−1)

= GkzgnCov(zG,nt−1 , ynt−1) +GkznCov(zC,nt−1, y

nt−1) +Gkynσ

2yn

Cov(zC,kt , zG,nt−1) = Cov(CkznzC,nt−1 + Ckyny

nt−1, z

G,nt−1)

= CkznCov(zC,nt−1, zG,nt−1) + CkynCov(ynt−1, z

G,nt−1)

Cov(zC,kt , zC,nt−1) = Cov(CkznzC,nt−1 + Ckyny

nt−1, z

C,nt−1)

= Ckznσ2zC,n + CkynCov(ynt−1, z

C,nt−1)

Cov(zC,kt , ynt−1) = Cov(CkznzC,nt−1 + Ckyny

nt−1, y

nt−1)

= CkznCov(zC,nt−1, ynt−1) + Ckynσ

2yn

Cov(ykt , zG,nt−1) = Cov(Bk

zgnzG,nt−1 +Bk

znzC,nt−1 +Bk

ynynt−1, z

G,nt−1)

= Bkzgnσ

2zG,n +Bk

znCov(zC,nt−1, zG,nt−1) +Bk

ynCov(ynt−1, zG,nt−1)

101

Cov(ykt , zC,nt−1) = Cov(Bk

zgnzG,nt−1 +Bk

znzC,nt−1 +Bk

ynynt−1, z

C,nt−1)

= BkzgnCov(zG,nt−1 , z

C,nt−1) +Bk

znσ2zC,n +Bk

ynCov(ynt−1, zC,nt−1)

Cov(ykt , ynt−1) = Cov(Bk

zgnzG,nt−1 +Bk

znzC,nt−1 +Bk

ynynt−1, y

nt−1)

= BkzgnCov(zG,nt−1 , y

nt−1) +Bk

znCov(zC,nt−1, ynt−1) +Bk

ynσ2yn

Siblings

We denote by k, l = m, f the genders of the siblings. We can compute the covariances projecting on the father

(n = m) or on the mother (n = f)

Cov(zG,k,it , zG,l,jt ) = GkzgnGlzgnσ

2

zG,n +GkznGlznσ

2zC,n +GkynG

lynσ

2

yn+(GkzgnG

lzn +GkznG

lzgn

)Cov(zG,nt−1, z

C,nt−1)

+(GkzgnG

lyn +GkynG

lzgn

)Cov(zG,nt−1, y

nt−1) +

(GkznG

lyn +GkynG

lzn

)Cov(zC,nt−1, y

nt−1) + gkng

lnσ

2wG,n

Cov(zG,k,it , zC,l,jt ) = GkznClznσ

2zC,n +GkynC

lynσ

2

yn+GkzgnC

lznCov(z

G,n

t−1, zC,nt−1) +GkzgnC

lynCov(zG,nt−1 , y

nt−1)

+(GkznC

lyn +GkynC

lzn

)Cov(zC,nt−1, y

nt−1) + gknc

lnCov(wG,nt−1 , ω

C,nt−1)

Cov(zG,k,it , yl,jt ) = GkzgnBlzgnσ

2

zG,n +GkznBlznσ

2zC,n +GkynB

lynσ

2

yn+(GkzgnB

lzn +GkznB

lzgn

)Cov(zG,nt−1, z

C,nt−1)

+(GkzgnB

lyn +GkynB

lzgn

)Cov(zG,nt−1 , y

nt−1) +

(GkznB

lyn +GkynB

lzn

)Cov(zC,nt−1, y

nt−1)

+gknglnσ

2wG,n + gknc

lnCov(wG,nt−1 , ω

C,nt−1) + gknb

lnCov(wG,nt−1 , ε

nt−1)

Cov(zC,k,it , zG,l,jt ) = CkznGlznσ

2zC,n + CkynG

lynσ

2

yn+ CkznG

lzgnCov(z

G,n

t−1, zC,nt−1) + CkynG

lzgnCov(zG,nt−1 , y

nt−1)

+(CkznG

lyn + CkynG

lzn

)(zC,nt−1, y

nt−1) + ckng

lnCov(wG,nt−1 , ω

C,nt−1)

Cov(zC,k,it , zC,l,jt ) = ckznClznσ

2

zC,n + ckynClynσ

2

yn+(CkznC

lyn + CkynC

lzn

)cov(zC,nt−1, y

nt−1) + cknc

lnσ

2

wC,n + σec,kec,l

Cov(zC,k,it , yl,jt ) = CkznBlznσ

2zC,n + CkynB

lynσ

2

yn+ CkznB

lzgnCov(z

G,n

t−1, zC,nt−1) + CkynB

lzgnCov(z

G,n

t−1, ynt−1)

+(CkznB

lyn + CkynB

lzn

)cov(zC,nt−1, y

nt−1) + ckng

lnCov(wC,nt−1, ω

G,nt−1) + cknc

lnσ

2

wC,n+cknblnCov(wC,nt−1, ε

nt−1) + σec,kec,l

Cov(yk,it , zG,l,jt ) = BkzgnG

lzgnσ

2

zG,n +BkznG

lznσ

2zC,n +Bk

ynGlynσ

2

yn+(BkzgnG

lzn +Bk

znGlzgn

)Cov(zG,nt−1, z

C,nt−1)

+(BkzgnG

lyn +Bk

ynGlzgn

)Cov(zG,nt−1, y

nt−1) +

(BkznG

lyn +Bk

ynGlzn

)Cov(zC,nt−1, y

nt−1)

+gkmglnσ

2wG,n + ckng

lnCov(wC,mt−1 , ω

G,nt−1) + bkmg

lnCov(εmt−1, ω

G,nt−1)

Cov(yk,it , zC,l,j

t ) = BkznC

lznσ

2zC,n +Bk

ynClynσ

2

yn+Bk

zgnClznCov(z

G,n

t−1, zC,nt−1) +Bk

zgnClynCov(z

G,n

t−1, ynt−1)

+(BkznC

lyn +Bk

ynClzn

)Cov(zC,nt−1, y

nt−1) + gknc

lnCov(wG,nt−1 , ω

C,nt−1)

+cknclnσ

2wC,n+bknc

lnCov(εmt−1, ω

C,nt−1) + σec,kec,l

102

Cov(yk,it , yl,j

t ) = BkzgnB

lzgnσ

2

zG,n +BkznB

lznσ

2zC,n +Bk

ynBlynσ

2

yn+(BkzgnB

lzn +Bk

znBlzgn

)Cov(zG,nt−1, z

C,nt−1)

+(BkzgnB

lyn +Bk

ynBlzgn

)cov(zG,nt−1, y

nt−1) +

(BkznB

lyn +Bk

ynBlzn

)cov(zC,nt−1, y

nt−1) + bknb

lnσ

2εn + gkng

lnσ

2wG,n

+cknclnσ

2wC,n +

(gknc

ln + ckng

ln

)Cov(ωG,nt−1 , ω

C,nt−1)+

(bkng

ln + gknb

ln

)Cov(ωG,nt−1 , ε

mt−1)

+(bknc

ln + cknb

ln

)Cov(ωc,nt−1, ε

mt−1) + σeC,keC,l + σxkxl

Other covariances

Before we obtain the remaining covariances for different degrees of kinship we compute the linear projections of

zi,G,kt−1 , zi,C,kt−1 and yi,kt−1 on zj,lt−1, zj,C,lt−1 and yj,lt−1, k, l = m, f, where i and j are siblings. We have that

zi,G,kt−1 = rk,lzGzG

zj,G,lt−1 + rk,lzGzC

zj,C,lt−1 + rk,lzGy

yj,lt−1 + wG,k,lt−1

zi.C,kt−1 = rk,lzCzG

zj,G,lt−1 + rk,lzCzC

zj,C,lt−1 + rk,lzCy

yj,lt−1 + wC,k,lt−1

yi.kt−1 = rk,lyzG

zj,G,lt−1 + rk,lyzC

zj,C,lt−1 + rk,lyy yj,lt−1 + εk,lt−1

In matrix form zi,G,kt−1

zi,C,kt−1

yi,kt−1

=

rk,lzGzG

rk,lzGzC

rk,lzGy

rk,lzCzG

rk,lzCzC

rk,lzCy

rk,lyzG

rk,lyzC

rk,lyy

zj,G,lt−1

zj,C,lt−1

yj,lt−1

+

wG,k,lt−1

wC,k,lt−1

εk,lt−1

where wG,k,lt−1 , wC,k,lt−1 and εk,lt−1 might be correlated but are uncorrelated with zj,G,lt−1 , z

j,C,lt−1 and yj,lt−1. We have that

rk,lzGzG

rk,lzGzC

rk,lzGy

rk,lzCzG

rk,lzCzC

rk,lzCy

rk,lyzG

rk,lyzC

rk,lyy

′

=

σ2zG,l σzG,lzC,l σzG,lyl

σzG,lzC,l σ2zC,l σzC,lyl

σzG,lyl σzC,lyl σ2yl

−1 σzG,j,lzG,i,k σzG,j,lzC,i,k σzG,j,lyi,k

σzC,j,lzG,i,k σzC,j,lzC,i,k σzC,j,lyi,k

σyj,lzG,i,k σyj,lzC,i,k σyj,lyi,k

Let’s now consider several couples ”a” and ”x”, ”b” and ”y”, ”c” and ”z”, ”d” and ”w”, etc., such that ”x” and

”b” are siblings, ”y” and ”c” are siblings, etc. Let ”ax” be a child of ”a” and ”x”, ”by” a child of ”b” and ”y”,

etc., and ”a′x′” the spouse of ”ax”.

Generation t− 1 aspouse− x

sibling− b

spouse− y

sibling− c

spouse− z

sibling− d

spouse− w

↓child

↓child

↓child

↓child

Generation t axspouse− a′x′ by cz dw

Consanguine relatives (”blood”)

Vertical covariances

103

Uncle/aunt (siblings of the parents)

We have to compute the covariances between ”ax” and ”b”. Let a∗ = m, f be the gender of ”ax” and l = m, f

the gender of the ”b”. We project ax on x (his/her father or mother) who has gender n′

Cov(zG,ax,n∗

t , zG,b,lt−1 ) = Cov(Gn∗zgn′z

G,x,n′

t−1 +Gn∗zn′z

C,x,n′

t−1 +Gn∗yn′y

x,n′

t−1 , zG,b,lt−1 ) =

Gn∗zgn′Cov(zG,x,n

′

t−1 , zG,b,lt−1 ) +Gn∗zn′Cov(zC,x,n

′

t−1 , zG,b,lt−1 ) +Gn∗yn′Cov(yx,n

′

t−1 , zG,b,lt−1 )

Cov(zG,ax,n∗

t , zC,b,lt−1 ) = Cov(Gn∗zgn′z

G,x,n′

t−1 +Gn∗zn′z

C,x,n′

t−1 +Gn∗yn′y

x,n′

t−1 , zC,b,lt−1 ) =

Gn∗zgn′Cov(zG,x,n

′

t−1 , zC,b,lt−1 ) +Gn∗zn′Cov(zC,x,n

′

t−1 , zC,b,lt−1 ) +Gn∗yn′Cov(yx,n

′

t−1 , zC,b,lt−1 )

Cov(zG,ax,n∗

t , yb,lt−1) = Cov(Gn∗zgn′z

G,x,n′

t−1 +Gn∗zn′z

C,x,n′

t−1 +Gn∗yn′y

x,n′

t−1 , yb,lt−1) =

Gn∗zgn′Cov(zG,x,n

′

t−1 , yb,lt−1) +Gn∗zn′Cov(zC,x,n

′

t−1 , yb,lt−1) +Gn∗yn′Cov(yx,n

′

t−1 , yb,lt−1)

Cov(zC,ax,n∗

t , zG,b,lt−1 ) = Cov(Cn∗

zgn′zG,x,n′

t−1 + Cn∗

zn′zC,x,n′

t−1 + Cn∗

yn′yx,n′

t−1 , zG,b,lt−1 ) =

Cn∗

zgn′Cov(zG,x,n′

t−1 , zG,b,lt−1 ) + Cn∗

zn′Cov(zC,x,n′

t−1 , zG,b,lt−1 ) + Cn∗

yn′Cov(yx,n′

t−1 , zG,b,lt−1 )

Cov(zC,ax,n∗

t , zC,b,lt−1 ) = Cov(Cn∗

zgn′zG,x,n′

t−1 + Cn∗

zn′zC,x,n′

t−1 + Cn∗

yn′yx,n′

t−1 , zC,b,lt−1 ) =

Cn∗

zgn′Cov(zG,x,n′

t−1 , zC,b,lt−1 ) + Cn∗

zn′Cov(zC,x,n′

t−1 , zC,b,lt−1 ) + Cn∗

yn′Cov(yx,n′

t−1 , zC,b,lt−1 )

Cov(zC,ax,n∗

t , yb,lt−1) = Cov(Cn∗

zgn′zG,x,n′

t−1 + Cn∗

zn′zC,x,n′

t−1 + Cn∗

yn′yx,n′

t−1 , yb,lt−1) =

Cn∗

zgn′Cov(zG,x,n′

t−1 , yb,lt−1) + Cn∗

zn′Cov(zC,x,n′

t−1 , yb,lt−1) + Cn∗

yn′Cov(yx,n′

t−1 , yb,lt−1)

Cov(yax,n∗

t , zG,b,lt−1 ) = Cov(Bn∗zgn′z

G,x,n′

t−1 +Bn∗zn′z

C,x,n′

t−1 +Bn∗yn′y

x,n′

t−1 , zG,b,lt−1 ) =

Bn∗zgn′Cov(zG,x,n

′

t−1 , zG,b,lt−1 ) +Bn∗zn′Cov(zC,x,n

′

t−1 , zG,b,lt−1 ) +Bn∗yn′Cov(yx,n

′

t−1 , zG,b,lt−1 )

Cov(yax,n∗

t , zC,b,lt−1 ) = Cov(Bn∗zgn′z

G,x,n′

t−1 +Bn∗zn′z

C,x,n′

t−1 +Bn∗yn′y

x,n′

t−1 , zC,b,lt−1 ) =

Bn∗zgn′Cov(zG,x,n

′

t−1 , zC,b,lt−1 ) +Bn∗zn′Cov(zC,x,n

′

t−1 , zC,b,lt−1 ) +Bn∗yn′Cov(yx,n

′

t−1 , zC,b,lt−1 )

Cov(yax,n∗

t , yb,lt−1) = Cov(Bn∗zgn′z

G,x,n′

t−1 +Bn∗zn′z

C,x,n′

t−1 +Bn∗yn′y

x,n′

t−1 , yb,lt−1) =

Bn∗zgn′Cov(zG,x,n

′

t−1 , yb,lt−1) +Bn∗zn′Cov(zC,x,n

′

t−1 , yb,lt−1) +Bn∗yn′Cov(yx,n

′

t−1 , yb,lt−1)

where ”x” and ”b” are siblings.

104

Horizontal covariances

Cousins

We have to compute the covariances between ”ax” and ”by”. Let a∗ = m, f be the gender of ”ax” and l∗ = m, f

the gender of the ”ay”. We project by on b (his/her father or mother) who has gender l

Cov(zG,ax,n∗

t , zG,by,l∗

t ) = Cov(zG,ax,n∗

t , Gl∗zglz

G,b,lt−1 +Gl

∗zlz

C,b,lt−1 +Gl

∗yly

b,lt−1)

= Gl∗zglCov(zG,ax,n

∗

t , zG,b,lt−1 ) +Gl∗zlCov(zG,ax,n

∗

t , zC,b,lt−1 ) +Gl∗ylCov(zG,ax,n

∗

t , yb,lt−1)

Cov(zG,ax,n∗

t , zC,by,l∗

t ) = Cov(zG,ax,n∗

t , C l∗zglz

G,b,lt−1 + C l

∗zlz

C,b,lt−1 + C l

∗yly

b,lt−1)

= C l∗zglCov(zG,ax,n

∗

t , zG,b,lt−1 ) + C l∗zlCov(zG,ax,n

∗

t , zC,b,lt−1 ) + C l∗ylCov(zG,ax,n

∗

t , yb,lt−1)

Cov(zG,ax,n∗

t , yby,l∗

t ) = Cov(zG,ax,n∗

t , Bl∗zglz

G,b,lt−1 +Bl∗

zlzC,b,lt−1 +Bl∗

ylyb,lt−1)

= Bl∗zglCov(zG,ax,n

∗

t , zG,b,lt−1 ) +Bl∗zlCov(zG,ax,n

∗

t , zC,b,lt−1 ) +Bl∗ylCov(zG,ax,n

∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , zG,by,l∗

t ) = Cov(zC,ax,n∗

t , Gl∗zglz

G,b,lt−1 +Gl

∗zlz

C,b,lt−1 +Gl

∗yly

b,lt−1)

= Gl∗zglCov(zC,ax,n

∗

t , zG,b,lt−1 ) +Gl∗zlCov(zC,ax,n

∗

t , zC,b,lt−1 ) +Gl∗ylCov(zC,ax,n

∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , zC,by,l∗

t ) = Cov(zC,ax,n∗

t , C l∗zglz

G,b,lt−1 + C l

∗zlz

C,b,lt−1 + C l

∗yly

b,lt−1)

= C l∗zglCov(zC,ax,n

∗

t , zG,b,lt−1 ) + C l∗zlCov(zC,ax,n

∗

t , zC,b,lt−1 ) + C l∗ylCov(zC,ax,n

∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , yby,l∗

t ) = Cov(zC,ax,n∗

t , Bl∗zglz

G,b,lt−1 +Bl∗

zlzC,b,lt−1 +Bl∗

ylyb,lt−1)

= Bl∗zglCov(zC,ax,n

∗

t , zG,b,lt−1 ) +Bl∗zlCov(zC,ax,n

∗

t , zC,b,lt−1 ) +Bl∗ylCov(zC,ax,n

∗

t , yb,lt−1)

Cov(yax,n∗

t , zG,by,l∗

t ) = Cov(yax,n∗

t , Gl∗zglz

G,b,lt−1 +Gl

∗zlz

C,b,lt−1 +Gl

∗yly

b,lt−1)

= Gl∗zglCov(yax,n

∗

t , zG,b,lt−1 ) +Gl∗zlCov(yax,n

∗

t , zC,b,lt−1 ) +Gl∗ylCov(yax,n

∗

t , yb,lt−1)

Cov(zG,ax,n∗

t , zC,by,l∗

t ) = Cov(yax,n∗

t , C l∗zglz

G,b,lt−1 + C l

∗zlz

C,b,lt−1 + C l

∗yly

b,lt−1)

= C l∗zglCov(yax,n

∗

t , zG,b,lt−1 ) + C l∗zlCov(yax,n

∗

t , zC,b,lt−1 ) + C l∗ylCov(yax,n

∗

t , yb,lt−1)

Cov(yax,n∗

t , yby,l∗

t ) = Cov(yax,n∗

t , Bl∗zglz

G,b,lt−1 +Bl∗

zlzC,b,lt−1 +Bl∗

ylyb,lt−1)

= Bl∗zglCov(yax,n

∗

t , zG,b,lt−1 ) +Bl∗zlCov(yax,n

∗

t , zC,b,lt−1 ) +Bl∗ylCov(yax,n

∗

t , yb,lt−1)

105

where ”b” is the uncle/aunt of ”ax”.

Affinity relatives (”in-law”)

Vertical covariances

Parents in-law

We have to compute the covariances between ”a′x′” and ”a”. Let a∗′

= m, f be the gender of ”a′x′” and n = m, f

the gender of the ”a”. We project ”a′x′” on his/her spouse ”ax”

Cov(zG,a′x′,a∗

′

t , zG,a,nt−1 ) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , zG,a,nt−1 )

= rn∗

zGzGCov(zG,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗

zGzCCov(zC,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗

zGyCov(yax,n∗

t−1 , zG,a,nt−1 )

Cov(zG,a′x′,a∗

′

t , zC,a,nt−1 ) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , zC,a,nt−1 )

= rn∗

zGzGCov(zG,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗

zGzCCov(zC,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗

zGyCov(yax,n∗

t−1 , zC,a,nt−1 )

Cov(zG,a′x′,a∗

′

t , ya,nt−1) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , ya,nt−1)

= rn∗

zGzGCov(zG,ax,n∗

t−1 , ya,nt−1) + rn∗

zGzCCov(zC,ax,n∗

t−1 , ya,nt−1) + rn∗

zGyCov(yax,n∗

t−1 , ya,nt−1)

Cov(zC,a′x′,a∗

′

t , zG,a,nt−1 ) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , zG,a,nt−1 )

= rn∗

zCzGCov(zG,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗

zCzCCov(zC,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗

zCyCov(yax,n∗

t−1 , zG,a,nt−1 )

Cov(zC,a′x′,a∗

′

t , zC,a,nt−1 ) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , zC,a,nt−1 )

= rn∗

zCzGCov(zG,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗

zCzCCov(zC,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗

zCyCov(yax,n∗

t−1 , zC,a,nt−1 )

Cov(zC,a′x′,a∗

′

t , ya,nt−1) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , ya,nt−1)

= rn∗

zCzGCov(zG,ax,n∗

t−1 , ya,nt−1) + rn∗

zCzCCov(zC,ax,n∗

t−1 , ya,nt−1) + rn∗

zCyCov(yax,n∗

t−1 , ya,nt−1)

Cov(ya′x′,a∗

′

t , zG,a,nt−1 ) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , zG,a,nt−1 )

= rn∗

yzGCov(zG,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗

yzCCov(zC,ax,n∗

t−1 , zG,a,nt−1 ) + rn∗yyCov(yax,n

∗

t−1 , zG,a,nt−1 )

106

Cov(ya′x′,a∗

′

t , zC,a,nt−1 ) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , zC,a,nt−1 )

= rn∗

yzGCov(zG,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗

yzCCov(zC,ax,n∗

t−1 , zC,a,nt−1 ) + rn∗yyCov(yax,n

∗

t−1 , zC,a,nt−1 )

Cov(ya′x′,a∗

′

t , ya,nt−1) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , ya,nt−1)

= rn∗

yzGCov(zG,ax,n∗

t−1 , ya,nt−1) + rn∗

yzCCov(zC,ax,n∗

t−1 , ya,nt−1) + rn∗yyCov(yax,n

∗

t−1 , ya,nt−1)

where ”a” is the father/mother of ”ax”.

Spouse of the uncle/aunt (spouses of the siblings of the parents)

We have to compute the covariances between ”ax” and ”y”. Let a∗ = m, f be the gender of ”ax” and l′ = m, f

the gender of the ”y”. We project y on his/her spouse b

Cov(zG,ax,n∗

t , zG,y,l′

t−1 ) = Cov(zG,ax,n∗

t , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(zG,ax,n∗

t , zG,b,lt−1 ) + rlzGzCCov(zG,ax,n∗

t , zC,b,lt−1 ) + rlzGyCov(zG,ax,n∗

t , yb,lt−1)

Cov(zG,ax,n∗

t , zC,y,l′

t−1 ) = Cov(zG,ax,n∗

t , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(zG,ax,n∗

t , zG,b,lt−1 ) + rlzCzCCov(zG,ax,n∗

t , zC,b,lt−1 ) + rlzCyCov(zG,ax,n∗

t , yb,lt−1)

Cov(zG,ax,n∗

t , yy,l′

t−1) = Cov(zG,ax,n∗

t , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(zG,ax,n∗

t , zG,b,lt−1 ) + rlyzCCov(zG,ax,n∗

t , zC,b,lt−1 ) + rlyyCov(zG,ax,n∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , zG,y,l′

t−1 ) = Cov(zC,ax,n∗

t , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(zC,ax,n∗

t , zG,b,lt−1 ) + rlzGzCCov(zC,ax,n∗

t , zC,b,lt−1 ) + rlzGyCov(zC,ax,n∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , zC,y,l′

t−1 ) = Cov(zC,ax,n∗

t , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(zC,ax,n∗

t , zG,b,lt−1 ) + rlzCzCCov(zC,ax,n∗

t , zC,b,lt−1 ) + rlzCyCov(zC,ax,n∗

t , yb,lt−1)

Cov(zC,ax,n∗

t , yy,l′

t−1) = Cov(zC,ax,n∗

t , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(zC,ax,n∗

t , zG,b,lt−1 ) + rlyzCCov(zC,ax,n∗

t , zC,b,lt−1 ) + rlyyCov(zC,ax,n∗

t , yb,lt−1)

Cov(yax,n∗

t , zG,y,l′

t−1 ) = Cov(yax,n∗

t , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(yax,n∗

t , zG,b,lt−1 ) + rlzGzCCov(yax,n∗

t , zC,b,lt−1 ) + rlzGyCov(yax,n∗

t , yb,lt−1)

107

Cov(yax,n∗

t , zC,y,l′

t−1 ) = Cov(yax,n∗

t , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(yax,n∗

t , zG,b,lt−1 ) + rlzCzCCov(yax,n∗

t , zC,b,lt−1 ) + rlzCyCov(yax,n∗

t , yb,lt−1)

Cov(yax,n∗

t , yy,l′

t−1) = Cov(yax,n∗

t , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(yax,n∗

t , zG,b,lt−1 ) + rlyzCCov(yax,n∗

t , zC,b,lt−1 ) + rlyyCov(yax,n∗

t , yb,lt−1)

where ”b” is uncle/aunt of ”ax”.

Uncles/Aunts of the spouse

We have to compute the covariances between ”a′x′” and ”b”. Let a∗′

= m, f be the gender of ”a′x′” and l = m, f

the gender of the ”b”. We project ”a′x′” on his/her spouse ”ax”

Cov(zG,a′x′,a∗

′

t , zG,b,lt−1 ) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , zG,b,lt−1 )

= rn∗

zGzGCov(zG,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗

zGzCCov(zC,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗

zGyCov(yax,n∗

t−1 , zG,b,lt−1 )

Cov(zG,a′x′,a∗

′

t , zC,b,lt−1 ) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , zC,b,lt−1 )

= rn∗

zGzGCov(zG,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗

zGzCCov(zC,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗

zGyCov(yax,n∗

t−1 , zC,b,lt−1 )

Cov(zG,a′x′,a∗

′

t , yb,lt−1) = Cov(rn∗

zGzGzG,ax,n∗

t−1 + rn∗

zGzCzC,ax,n∗

t−1 + rn∗

zGyyax,n∗

t−1 , yb,lt−1)

= rn∗

zGzGCov(zG,ax,n∗

t−1 , yb,lt−1) + rn∗

zGzCCov(zC,ax,n∗

t−1 , yb,lt−1) + rn∗

zGyCov(yax,n∗

t−1 , yb,lt−1)

Cov(zC,a′x′,a∗

′

t , zG,b,lt−1 ) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , zG,b,lt−1 )

= rn∗

zCzGCov(zG,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗

zCzCCov(zC,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗

zCyCov(yax,n∗

t−1 , zG,b,lt−1 )

Cov(zC,a′x′,a∗

′

t , zC,b,lt−1 ) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , zC,b,lt−1 )

= rn∗

zCzGCov(zG,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗

zCzCCov(zC,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗

zCyCov(yax,n∗

t−1 , zC,b,lt−1 )

Cov(zC,a′x′,a∗

′

t , yb,lt−1) = Cov(rn∗

zCzGzG,ax,n∗

t−1 + rn∗

zCzCzC,ax,n∗

t−1 + rn∗

zCyyax,n∗

t−1 , yb,lt−1)

= rn∗

zCzGCov(zG,ax,n∗

t−1 , yb,lt−1) + rn∗

zCzCCov(zC,ax,n∗

t−1 , yb,lt−1) + rn∗

zCyCov(yax,n∗

t−1 , yb,lt−1)

108

Cov(ya′x′,a∗

′

t , zG,b,lt−1 ) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , zG,b,lt−1 )

= rn∗

yzGCov(zG,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗

yzCCov(zC,ax,n∗

t−1 , zG,b,lt−1 ) + rn∗yyCov(yax,n

∗

t−1 , zG,b,lt−1 )

Cov(ya′x′,a∗

′

t , zC,b,lt−1 ) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , zC,b,lt−1 )

= rn∗

yzGCov(zG,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗

yzCCov(zC,ax,n∗

t−1 , zC,b,lt−1 ) + rn∗yyCov(yax,n

∗

t−1 , zC,b,lt−1 )

Cov(ya′x′,a∗

′

t , yb,lt−1) = Cov(rn∗

yzGzG,ax,n∗

t−1 + rn∗

yzCzC,ax,n∗

t−1 + rn∗yyy

ax,n∗

t−1 , yb,lt−1)

= rn∗

yzGCov(zG,ax,n∗

t−1 , yb,lt−1) + rn∗

yzCCov(zC,ax,n∗

t−1 , yb,lt−1) + rn∗yyCov(yax,n

∗

t−1 , yb,lt−1)

where ”b” is the uncle/aunt of ”ax”.

Siblings of the siblings in law of the parents

We have to compute the covariances between ”ax” and ”c”. Let a∗ = m, f be the gender of ”ax” and k = m, f

the gender of the ”c”. We project c on his/her sibling y

Cov(zG,ax,n∗

t , zG,c,kt−1 ) = Cov(zG,ax,n∗

t , rk,l′

zGzGzG,y,l

′

t−1 + rk,l′

zGzCzC,y,l

′

t−1 + rk,l′

zGyyy,l

′

t−1)

= rk,l′

zGzGCov(zG,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zGzCCov(zG,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zGyCov(zG,ax,n

∗

t , yy,l′

t−1)

Cov(zG,ax,n∗

t , zC,c,kt−1 ) = Cov(zG,ax,n∗

t , rk,l′

zCzGzG,y,l

′

t−1 + rk,l′

zCzCzC,y,l

′

t−1 + rk,l′

zCyyy,l

′

t−1)

= rk,l′

zCzGCov(zG,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zCzCCov(zG,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zCyCov(zG,ax,n

∗

t , yy,l′

t−1)

Cov(zG,ax,n∗

t , yc,kt−1) = Cov(zG,ax,n∗

t , rk,l′

yzGzG,y,l

′

t−1 + rk,l′

yzCzC,y,l

′

t−1 + rk,l′

yy yy,l′

t−1)

= rk,l′

yzGCov(zG,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

yzCCov(zG,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

yy Cov(zG,ax,n∗

t , yy,l′

t−1)

Cov(zC,ax,n∗

t , zG,c,kt−1 ) = Cov(zC,ax,n∗

t , rk,l′

zGzGzG,y,l

′

t−1 + rk,l′

zGzCzC,y,l

′

t−1 + rk,l′

zGyyy,l

′

t−1)

= rk,l′

zGzGCov(zC,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zGzCCov(zC,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zGyCov(zC,ax,n

∗

t , yy,l′

t−1)

Cov(zC,ax,n∗

t , zC,c,kt−1 ) = Cov(zC,ax,n∗

t , rk,l′

zCzGzG,y,l

′

t−1 + rk,l′

zCzCzC,y,l

′

t−1 + rk,l′

zCyyy,l

′

t−1)

= rk,l′

zCzGCov(zC,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zCzCCov(zC,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zCyCov(zC,ax,n

∗

t , yy,l′

t−1)

109

Cov(zC,ax,n∗

t , yc,kt−1) = Cov(zC,ax,n∗

t , rk,l′

yzGzG,y,l

′

t−1 + rk,l′

yzCzC,y,l

′

t−1 + rk,l′

yy yy,l′

t−1)

= rk,l′

yzGCov(zC,ax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

yzCCov(zC,ax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

yy Cov(zC,ax,n∗

t , yy,l′

t−1)

Cov(yax,n∗

t , zG,c,kt−1 ) = Cov(yax,n∗

t , rk,l′

zGzGzG,y,l

′

t−1 + rk,l′

zGzCzC,y,l

′

t−1 + rk,l′

zGyyy,l

′

t−1)

= rk,l′

zGzGCov(yax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zGzCCov(yax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zGyCov(yax,n

∗

t , yy,l′

t−1)

Cov(yax,n∗

t , zC,c,kt−1 ) = Cov(yax,n∗

t , rk,l′

zCzGzG,y,l

′

t−1 + rk,l′

zCzCzC,y,l

′

t−1 + rk,l′

zCyyy,l

′

t−1)

= rk,l′

zCzGCov(yax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

zCzCCov(yax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

zCyCov(yax,n

∗

t , yy,l′

t−1)

Cov(yax,n∗

t , yc,kt−1) = Cov(yax,n∗

t , rk,l′

yzGzG,y,l

′

t−1 + rk,l′

yzCzC,y,l

′

t−1 + rk,l′

yy yy,l′

t−1)

= rk,l′

yzGCov(yax,n

∗

t , zG,y,l′

t−1 ) + rk,l′

yzCCov(yax,n

∗

t , zC,y,l′

t−1 ) + rk,l′

yy Cov(yax,n∗

t , yy,l′

t−1)

where ”y” is the spouse of the uncle/aunt of ”ax”. We can analogously compute the covariance between ”ax”

and ”z”, ”d”, etc.

Horizontal covariances

Siblings in law

We have to compute the covariances between ”a” and ”b”. Let n = m, f be the gender of ”a” and l = m, f the

gender of the ”b”. We project a on his/her spouse x

Cov(zG,a,nt−1 , zG,b,lt−1 ) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , zG,b,lt−1 )

= rn′

zGzGCov(zG,x,n′

t−1 , zG,b,lt−1 ) + rn′

zGzCCov(zC,x,n′

t−1 , zG,b,lt−1 ) + rn′

zGyCov(yx,n′

t−1 , zG,b,lt−1 )

Cov(zG,a,nt−1 , zC,b,lt−1 ) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , zC,b,lt−1 )

= rn′

zGzGCov(zG,x,n′

t−1 , zC,b,lt−1 ) + rn′

zGzCCov(zC,x,n′

t−1 , zC,b,lt−1 ) + rn′

zGyCov(yx,n′

t−1 , zC,b,lt−1 )

Cov(zG,a,nt−1 , yb,lt−1) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , yb,lt−1)

= rn′

zGzGCov(zG,x,n′

t−1 , yb,lt−1) + rn′

zGzCCov(zC,x,n′

t−1 , yb,lt−1) + rn′

zGyCov(yx,n′

t−1 , yb,lt−1)

Cov(zC,a,nt−1 , zG,b,lt−1 ) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , zG,b,lt−1 )

= rn′

zCzGCov(zG,x,n′

t−1 , zG,b,lt−1 ) + rn′

zCzCCov(zC,x,n′

t−1 , zG,b,lt−1 ) + rn′

zCyCov(yx,n′

t−1 , zG,b,lt−1 )

110

Cov(zC,a,nt−1 , zC,b,lt−1 ) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , zC,b,lt−1 )

= rn′

zCzGCov(zG,x,n′

t−1 , zC,b,lt−1 ) + rn′

zCzCCov(zC,x,n′

t−1 , zC,b,lt−1 ) + rn′

zCyCov(yx,n′

t−1 , zC,b,lt−1 )

Cov(zC,a,nt−1 , yb,lt−1) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , yb,lt−1)

= rn′

zCzGCov(zG,x,n′

t−1 , yb,lt−1) + rn′

zCzCCov(zC,x,n′

t−1 , yb,lt−1) + rn′

zCyCov(yx,n′

t−1 , yb,lt−1)

Cov(ya,nt−1, zG,b,lt−1 ) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , zG,b,lt−1 )

= rn′

yzGCov(zG,x,n′

t−1 , zG,b,lt−1 ) + rn′

yzCCov(zC,x,n′

t−1 , zG,b,lt−1 ) + rn′yyCov(yx,n

′

t−1 , zG,b,lt−1 )

Cov(ya,nt−1, zC,b,lt−1 ) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , zC,b,lt−1 )

= rn′

yzGCov(zG,x,n′

t−1 , zC,b,lt−1 ) + rn′

yzCCov(zC,x,n′

t−1 , zC,b,lt−1 ) + rn′yyCov(yx,n

′

t−1 , zC,b,lt−1 )

Cov(ya,nt−1, yb,lt−1) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , yb,lt−1)

= rn′

yzGCov(zG,x,n′

t−1 , yb,lt−1) + rn′

yzCCov(zC,x,n′

t−1 , yb,lt−1) + rn′yyCov(yx,n

′

t−1 , yb,lt−1)

where ”x” and ”b” are siblings. Notice that since ”x” is the spouse of ”a”, n′ = f when n = m and viceversa.

Spouse of the siblings in law

We have to compute the covariances between ”a” and ”y”. Let n = m, f be the gender of ”a” and l′ = m, f the

gender of the ”y”. We project y on his/her spouse b

Cov(zG,a,nt , zG,y,l′

t−1 ) = Cov(zG,a,nt , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(zG,a,nt , zG,b,lt−1 ) + rlzGzCCov(zG,a,nt , zC,b,lt−1 ) + rlzGyCov(zG,a,nt , yb,lt−1)

Cov(zG,a,nt , zC,y,l′

t−1 ) = Cov(zG,a,nt , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(zG,a,nt , zG,b,lt−1 ) + rlzCzCCov(zG,a,nt , zC,b,lt−1 ) + rlzCyCov(zG,a,nt , yb,lt−1)

Cov(zG,a,nt , yy,l′

t−1) = Cov(zG,a,nt , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(zG,a,nt , zG,b,lt−1 ) + rlyzCCov(zG,a,nt , zC,b,lt−1 ) + rlyyCov(zG,a,nt , yb,lt−1)

Cov(zC,a,nt , zG,y,l′

t−1 ) = Cov(zC,a,nt , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(zC,a,nt , zG,b,lt−1 ) + rlzGzCCov(zC,a,nt , zC,b,lt−1 ) + rlzGyCov(zC,a,nt , yb,lt−1)

111

Cov(zC,a,nt , zC,y,l′

t−1 ) = Cov(zC,a,nt , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(zC,a,nt , zG,b,lt−1 ) + rlzCzCCov(zC,a,nt , zC,b,lt−1 ) + rlzCyCov(zC,a,nt , yb,lt−1)

Cov(zC,a,nt , yy,l′

t−1) = Cov(zC,a,nt , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(zC,a,nt , zG,b,lt−1 ) + rlyzCCov(zC,a,nt , zC,b,lt−1 ) + rlyyCov(zC,a,nt , yb,lt−1)

Cov(ya,nt , zG,y,l′

t−1 ) = Cov(ya,nt , rlzGzGzG,b,lt−1 + rlzGzCz

C,b,lt−1 + rlzGyy

b,lt−1)

= rlzGzGCov(ya,nt , zG,b,lt−1 ) + rlzGzCCov(ya,nt , zC,b,lt−1 ) + rlzGyCov(ya,nt , yb,lt−1)

Cov(ya,nt , zC,y,l′

t−1 ) = Cov(ya,nt , rlzCzGzG,b,lt−1 + rlzCzCz

C,b,lt−1 + rlzCyy

b,lt−1)

= rlzCzGCov(ya,nt , zG,b,lt−1 ) + rlzCzCCov(ya,nt , zC,b,lt−1 ) + rlzCyCov(ya,nt , yb,lt−1)

Cov(ya,nt , yy,l′

t−1) = Cov(ya,nt , rlyzGzG,b,lt−1 + rlyzCz

C,b,lt−1 + rlyyy

b,lt−1)

= rlyzGCov(ya,nt , zG,b,lt−1 ) + rlyzCCov(ya,nt , zC,b,lt−1 ) + rlyyCov(ya,nt , yb,lt−1)

where ”a” and ”b” are siblings in law. Notice that since ”b” is the spouse of ”y”, l = f when l′ = m and viceversa.

Sibling of the sibling in law

We have to compute the covariances between ”x” and ”c”. Let n′ = m, f be the gender of ”x” and k = m, f the

gender of the ”c”. We project x on his/her sibling b who has gender l

Cov(zG,x,n′

t−1 , zG,c,kt−1 ) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, zG,c,kt−1 )

= rn′,lzGzG

Cov(zG,b,lt−1 , zG,c,kt−1 ) + rn

′,lzGzC

Cov(zC,b,lt−1 , zG,c,kt−1 ) + rn

′,lzGy

Cov(yb,lt−1, zG,c,kt−1 )

Cov(zG,x,n′

t−1 , zC,c,kt−1 ) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, zC,c,kt−1 )

= rn′,lzGzG

Cov(zG,b,lt−1 , zC,c,kt−1 ) + rn

′,lzGzC

Cov(zC,b,lt−1 , zC,c,kt−1 ) + rn

′,lzGy

Cov(yb,lt−1, zC,c,kt−1 )

Cov(zG,x,n′

t−1 , yc,kt−1) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, yc,kt−1)

= rn′,lzGzG

Cov(zG,b,lt−1 , yc,kt−1) + rn

′,lzGzC

Cov(zC,b,lt−1 , yc,kt−1) + rn

′,lzGy

Cov(yb,lt−1, yc,kt−1)

Cov(zC,x,n′

t−1 , zG,c,kt−1 ) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, zG,c,kt−1 )

= rn′,lzCzG

Cov(zG,b,lt−1 , zG,c,kt−1 ) + rn

′,lzCzC

Cov(zC,b,lt−1 , zG,c,kt−1 ) + rn

′,lzCy

Cov(yb,lt−1, zG,c,kt−1 )

112

Cov(zC,x,n′

t−1 , zC,c,kt−1 ) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, zC,c,kt−1 )

= rn′,lzCzG

Cov(zG,b,lt−1 , zC,c,kt−1 ) + rn

′,lzCzC

Cov(zC,b,lt−1 , zC,c,kt−1 ) + rn

′,lzCy

Cov(yb,lt−1, zC,c,kt−1 )

Cov(zC,x,n′

t−1 , yc,kt−1) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, yc,kt−1)

= rn′,lzCzG

Cov(zG,b,lt−1 , yc,kt−1) + rn

′,lzCzC

Cov(zC,b,lt−1 , yc,kt−1) + rn

′,lzCy

Cov(yb,lt−1, yc,kt−1)

Cov(yx,n′

t−1 , zG,c,kt−1 ) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, z

G,c,kt−1 )

= rn′,lyzG

Cov(zG,b,lt−1 , zG,c,kt−1 ) + rn

′,lyzC

Cov(zC,b,lt−1 , zG,c,kt−1 ) + rn

′,lyy Cov(yb,lt−1, z

G,c,kt−1 )

Cov(yx,n′

t−1 , zC,c,kt−1 ) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, z

C,c,kt−1 )

= rn′,lyzG

Cov(zG,b,lt−1 , zC,c,kt−1 ) + rn

′,lyzC

Cov(zC,b,lt−1 , zC,c,kt−1 ) + rn

′,lyy Cov(yb,lt−1, z

C,c,kt−1 )

Cov(yx,n′

t−1 , yc,kt−1) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, y

c,kt−1)

= rn′,lyzG

Cov(zG,b,lt−1 , yc,kt−1) + rn

′,lyzC

Cov(zC,b,lt−1 , yc,kt−1) + rn

′,lyy Cov(yb,lt−1, y

c,kt−1)

where ”b” and ”c” are siblings in law.

Sibling in law of degree 2

We have to compute the covariances between ”a” and ”c”. Let n = m, f be the gender of ”a” and k = m, f the

gender of the ”c”. We project a on his/her spouse x

Cov(zG,a,nt−1 , zG,c,kt−1 ) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , zG,c,kt−1 )

= rn′

zGzGCov(zG,x,n′

t−1 , zG,c,kt−1 ) + rn′

zGzCCov(zC,x,n′

t−1 , zG,c,kt−1 ) + rn′

zGyCov(yx,n′

t−1 , zG,c,kt−1 )

Cov(zG,a,nt−1 , zC,c,kt−1 ) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , zC,c,kt−1 )

= rn′

zGzGCov(zG,x,n′

t−1 , zC,c,kt−1 ) + rn′

zGzCCov(zC,x,n′

t−1 , zC,c,kt−1 ) + rn′

zGyCov(yx,n′

t−1 , zC,c,kt−1 )

Cov(zG,a,nt−1 , yc,kt−1) = Cov(rn′

zGzGzG,x,n′

t−1 + rn′

zGzCzC,x,n′

t−1 + rn′

zGyyx,n′

t−1 , yc,kt−1)

= rn′

zGzGCov(zG,x,n′

t−1 , yc,kt−1) + rn′

zGzCCov(zC,x,n′

t−1 , yc,kt−1) + rn′

zGyCov(yx,n′

t−1 , yc,kt−1)

113

Cov(zC,a,nt−1 , zG,c,kt−1 ) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , zG,c,kt−1 )

= rn′

zCzGCov(zG,x,n′

t−1 , zG,c,kt−1 ) + rn′

zCzCCov(zC,x,n′

t−1 , zG,c,kt−1 ) + rn′

zCyCov(yx,n′

t−1 , zG,c,kt−1 )

Cov(zC,a,nt−1 , zC,c,kt−1 ) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , zC,c,kt−1 )

= rn′

zCzGCov(zG,x,n′

t−1 , zC,c,kt−1 ) + rn′

zCzCCov(zC,x,n′

t−1 , zC,c,kt−1 ) + rn′

zCyCov(yx,n′

t−1 , zC,c,kt−1 )

Cov(zC,a,nt−1 , yc,kt−1) = Cov(rn′

zCzGzG,x,n′

t−1 + rn′

zCzCzC,x,n′

t−1 + rn′

zCyyx,n′

t−1 , yc,kt−1)

= rn′

zCzGCov(zG,x,n′

t−1 , yc,kt−1) + rn′

zCzCCov(zC,x,n′

t−1 , yc,kt−1) + rn′

zCyCov(yx,n′

t−1 , yc,kt−1)

Cov(ya,nt−1, zG,c,kt−1 ) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , zG,c,kt−1 )

= rn′

yzGCov(zG,x,n′

t−1 , zG,c,kt−1 ) + rn′

yzCCov(zC,x,n′

t−1 , zG,c,kt−1 ) + rn′yyCov(yx,n

′

t−1 , zG,c,kt−1 )

Cov(ya,nt−1, zC,c,kt−1 ) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , zC,c,kt−1 )

= rn′

yzGCov(zG,x,n′

t−1 , zC,c,kt−1 ) + rn′

yzCCov(zC,x,n′

t−1 , zC,c,kt−1 ) + rn′yyCov(yx,n

′

t−1 , zC,c,kt−1 )

Cov(ya,nt−1, yc,kt−1) = Cov(rn

′

yzGzG,x,n′

t−1 + rn′

yzCzC,x,n′

t−1 + rn′yyy

x,n′

t−1 , yc,kt−1)

= rn′

yzGCov(zG,x,n′

t−1 , yc,kt−1) + rn′

yzCCov(zC,x,n′

t−1 , yc,kt−1) + rn′yyCov(yx,n

′

t−1 , yc,kt−1)

where ”x” is the sibling of the sibling in law of ”c”. Notice that since ”x” is the spouse of ”a”, n′ = f when

n = m and viceversa. We can compute siblings in law of any degree analogously.

Spouse of the sibling in law of degree 2

We have to compute the covariances between ”a” and ”z”. Let n = m, f be the gender of ”a” and k′ = m, f the

gender of the ”z”. We project z on his/her spouse c

Cov(zG,a,nt , zG,z,k′

t−1 ) = Cov(zG,a,nt , rkzGzGzG,c,kt−1 + rkzGzCz

C,c,kt−1 + rkzGyy

c,kt−1)

= rkzGzGCov(zG,a,nt , zG,c,kt−1 ) + rkzGzCCov(zG,a,nt , zC,c,kt−1 ) + rkzGyCov(zG,a,nt , yc,kt−1)

Cov(zG,a,nt , zC,z,k′

t−1 ) = Cov(zG,a,nt , rkzCzGzG,c,kt−1 + rkzCzCz

C,c,kt−1 + rkzCyy

c,kt−1)

= rkzCzGCov(zG,a,nt , zG,c,kt−1 ) + rkzCzCCov(zG,a,nt , zC,c,kt−1 ) + rkzCyCov(zG,a,nt , yc,kt−1)

114

Cov(zG,a,nt , yz,k′

t−1) = Cov(zG,a,nt , rkyzGzG,c,kt−1 + rkyzCz

C,c,kt−1 + rkyyy

c,kt−1)

= rkyzGCov(zG,a,nt , zG,c,kt−1 ) + rkyzCCov(zG,a,nt , zC,c,kt−1 ) + rkyyCov(zG,a,nt , yc,kt−1)

Cov(zC,a,nt , zG,z,k′

t−1 ) = Cov(zC,a,nt , rkzGzGzG,c,kt−1 + rkzGzCz

C,c,kt−1 + rkzGyy

c,kt−1)

= rkzGzGCov(zC,a,nt , zG,c,kt−1 ) + rkzGzCCov(zC,a,nt , zC,c,kt−1 ) + rkzGyCov(zC,a,nt , yc,kt−1)

Cov(zC,a,nt , zC,z,k′

t−1 ) = Cov(zC,a,nt , rkzCzGzG,c,kt−1 + rkzCzCz

C,c,kt−1 + rkzCyy

c,kt−1)

= rkzCzGCov(zC,a,nt , zG,c,kt−1 ) + rkzCzCCov(zC,a,nt , zC,c,kt−1 ) + rkzCyCov(zC,a,nt , yc,kt−1)

Cov(zC,a,nt , yz,k′

t−1) = Cov(zC,a,nt , rkyzGzG,c,kt−1 + rkyzCz

C,c,kt−1 + rkyyy

c,kt−1)

= rkyzGCov(zC,a,nt , zG,c,kt−1 ) + rkyzCCov(zC,a,nt , zC,c,kt−1 ) + rkyyCov(zC,a,nt , yc,kt−1)

Cov(ya,nt , zG,z,k′

t−1 ) = Cov(ya,nt , rkzGzGzG,c,kt−1 + rkzGzCz

C,c,kt−1 + rkzGyy

c,kt−1)

= rkzGzGCov(ya,nt , zG,c,kt−1 ) + rkzGzCCov(ya,nt , zC,c,kt−1 ) + rkzGyCov(ya,nt , yc,kt−1)

Cov(ya,nt , zC,z,k′

t−1 ) = Cov(ya,nt , rkzCzGzG,c,kt−1 + rkzCzCz

C,c,kt−1 + rkzCyy

c,kt−1)

= rkzCzGCov(ya,nt , zG,c,kt−1 ) + rkzCzCCov(ya,nt , zC,c,kt−1 ) + rkzCyCov(ya,nt , yc,kt−1)

Cov(ya,nt , yz,k′

t−1) = Cov(ya,nt , rkyzGzG,c,kt−1 + rkyzCz

C,c,kt−1 + rkyyy

c,kt−1)

= rkyzGCov(ya,nt , zG,c,kt−1 ) + rkyzCCov(ya,nt , zC,c,kt−1 ) + rkyyCov(ya,nt , yc,kt−1)

where ”a” and ”c” are siblings in law of degree 2. Notice that since ”c” is the spouse of ”z”, k = f when k′ = m

and viceversa. We can compute spouses of siblings in law of any degree analogously.

Sibling of the sibling in law of degree 2

We have to compute the covariances between ”x” and ”d”. Let n′ = m, f be the gender of ”x” and j = m, f the

gender of the ”d”. We project x on his/her sibling b who has gender l

Cov(zG,x,n′

t−1 , zG,d,jt−1 ) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, zG,d,jt−1 )

= rn′,lzGzG

Cov(zG,b,lt−1 , zG,d,jt−1 ) + rn

′,lzGzC

Cov(zC,b,lt−1 , zG,d,jt−1 ) + rn

′,lzGy

Cov(yb,lt−1, zG,d,jt−1 )

115

Cov(zG,x,n′

t−1 , zC,d,jt−1 ) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, zC,d,jt−1 )

= rn′,lzGzG

Cov(zG,b,lt−1 , zC,d,jt−1 ) + rn

′,lzGzC

Cov(zC,b,lt−1 , zC,d,jt−1 ) + rn

′,lzGy

Cov(yb,lt−1, zC,d,jt−1 )

Cov(zG,x,n′

t−1 , yd,jt−1) = Cov(rn′,lzGzG

zG,b,lt−1 + rn′,lzGzC

zC,b,lt−1 + rn′,lzGy

yb,lt−1, yd,jt−1)

= rn′,lzGzG

Cov(zG,b,lt−1 , yd,jt−1) + rn

′,lzGzC

Cov(zC,b,lt−1 , yd,jt−1) + rn

′,lzGy

Cov(yb,lt−1, yd,jt−1)

Cov(zC,x,n′

t−1 , zG,d,jt−1 ) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, zG,d,jt−1 )

= rn′,lzCzG

Cov(zG,b,lt−1 , zG,d,jt−1 ) + rn

′,lzCzC

Cov(zC,b,lt−1 , zG,d,jt−1 ) + rn

′,lzCy

Cov(yb,lt−1, zG,d,jt−1 )

Cov(zC,x,n′

t−1 , zC,d,jt−1 ) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, zC,d,jt−1 )

= rn′,lzCzG

Cov(zG,b,lt−1 , zC,d,jt−1 ) + rn

′,lzCzC

Cov(zC,b,lt−1 , zC,d,jt−1 ) + rn

′,lzCy

Cov(yb,lt−1, zC,d,jt−1 )

Cov(zC,x,n′

t−1 , yd,jt−1) = Cov(rn′,lzCzG

zG,b,lt−1 + rn′,lzCzC

zC,b,lt−1 + rn′,lzCy

yb,lt−1, yd,jt−1)

= rn′,lzCzG

Cov(zG,b,lt−1 , yd,jt−1) + rn

′,lzCzC

Cov(zC,b,lt−1 , yd,jt−1) + rn

′,lzCy

Cov(yb,lt−1, yd,jt−1)

Cov(yx,n′

t−1 , zG,d,jt−1 ) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, z

G,d,jt−1 )

= rn′,lyzG

Cov(zG,b,lt−1 , zG,d,jt−1 ) + rn

′,lyzC

Cov(zC,b,lt−1 , zG,d,jt−1 ) + rn

′,lyy Cov(yb,lt−1, z

G,d,jt−1 )

Cov(yx,n′

t−1 , zC,d,jt−1 ) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, z

C,d,jt−1 )

= rn′,lyzG

Cov(zG,b,lt−1 , zC,d,jt−1 ) + rn

′,lyzC

Cov(zC,b,lt−1 , zC,d,jt−1 ) + rn

′,lyy Cov(yb,lt−1, z

C,d,jt−1 )

Cov(yx,n′

t−1 , yd,jt−1) = Cov(rn

′,lyzG

zG,b,lt−1 + rn′,lyzC

zC,b,lt−1 + rn′,lyy y

b,lt−1, y

d,jt−1)

= rn′,lyzG

Cov(zG,b,lt−1 , yd,jt−1) + rn

′,lyzC

Cov(zC,b,lt−1 , yd,jt−1) + rn

′,lyy Cov(yb,lt−1, y

d,jt−1)

where ”b” and ”d” are siblings in law of degree 2. We can compute siblings of siblings in law of any degree

analogously.

Cousins in law

We have to compute the covariances between ”ax” and ”cz”. Let a∗ = m, f be the gender of ”ax” and k∗ = m, f

116

the gender of the ”cz”. We project cz on c (his/her father or mother) who has gender k

Cov(zG,ax,n∗

t , zG,cz,k∗

t ) = Cov(zG,ax,n∗

t , Gk∗zgkz

G,b,kt−1 +Gk

∗zkz

C,b,kt−1 +Gk

∗yky

b,kt−1)

= Gk∗zgkCov(zG,ax,n

∗

t , zG,b,kt−1 ) +Gk∗zkCov(zG,ax,n

∗

t , zC,b,kt−1 ) +Gk∗ykCov(zG,ax,n

∗

t , yb,kt−1)

Cov(zG,ax,n∗

t , zC,cz,k∗

t ) = Cov(zG,ax,n∗

t , Ck∗zgkz

G,b,kt−1 + Ck

∗zk z

C,b,kt−1 + Ck

∗yky

b,kt−1)

= Ck∗zgkCov(zG,ax,n

∗

t , zG,b,kt−1 ) + Ck∗zkCov(zG,ax,n

∗

t , zC,b,kt−1 ) + Ck∗ykCov(zG,ax,n

∗

t , yb,kt−1)

Cov(zG,ax,n∗

t , ycz,k∗

t ) = Cov(zG,ax,n∗

t , Bk∗zgkz

G,b,kt−1 +Bk∗

zkzC,b,kt−1 +Bk∗

ykyb,kt−1)

= Bk∗zgkCov(zG,ax,n

∗

t , zG,b,kt−1 ) +Bk∗zkCov(zG,ax,n

∗

t , zC,b,kt−1 ) +Bk∗ykCov(zG,ax,n

∗

t , yb,kt−1)

Cov(zC,ax,n∗

t , zG,cz,k∗

t ) = Cov(zC,ax,n∗

t , Gk∗zgkz

G,b,kt−1 +Gk

∗zkz

C,b,kt−1 +Gk

∗yky

b,kt−1)

= Gk∗zgkCov(zC,ax,n

∗

t , zG,b,kt−1 ) +Gk∗zkCov(zC,ax,n

∗

t , zC,b,kt−1 ) +Gk∗ykCov(zC,ax,n

∗

t , yb,kt−1)

Cov(zC,ax,n∗

t , zC,cz,k∗

t ) = Cov(zC,ax,n∗

t , Ck∗zgkz

G,b,kt−1 + Ck

∗zk z

C,b,kt−1 + Ck

∗yky

b,kt−1)

= Ck∗zgkCov(zC,ax,n

∗

t , zG,b,kt−1 ) + Ck∗zkCov(zC,ax,n

∗

t , zC,b,kt−1 ) + Ck∗ykCov(zC,ax,n

∗

t , yb,kt−1)

Cov(zC,ax,n∗

t , ycz,k∗

t ) = Cov(zC,ax,n∗

t , Bk∗zgkz

G,b,kt−1 +Bk∗

zkzC,b,kt−1 +Bk∗

ykyb,kt−1)

= Bk∗zgkCov(zC,ax,n

∗

t , zG,b,kt−1 ) +Bk∗zkCov(zC,ax,n

∗

t , zC,b,kt−1 ) +Bk∗ykCov(zC,ax,n

∗

t , yb,kt−1)

Cov(yax,n∗

t , zG,cz,k∗

t ) = Cov(yax,n∗

t , Gk∗zgkz

G,b,kt−1 +Gk

∗zkz

C,b,kt−1 +Gk

∗yky

b,kt−1)

= Gk∗zgkCov(yax,n

∗

t , zG,b,kt−1 ) +Gk∗zkCov(yax,n

∗

t , zC,b,kt−1 ) +Gk∗ykCov(yax,n

∗

t , yb,kt−1)

Cov(zG,ax,n∗

t , zC,cz,k∗

t ) = Cov(yax,n∗

t , Ck∗zgkz

G,b,kt−1 + Ck

∗zk z

C,b,kt−1 + Ck

∗yky

b,kt−1)

= Ck∗zgkCov(yax,n

∗

t , zG,b,kt−1 ) + Ck∗zkCov(yax,n

∗

t , zC,b,kt−1 ) + Ck∗ykCov(yax,n

∗

t , yb,kt−1)

Cov(yax,n∗

t , ycz,k∗

t ) = Cov(yax,n∗

t , Bk∗zgkz

G,b,kt−1 +Bk∗

zkzC,b,kt−1 +Bk∗

ykyb,kt−1)

= Bk∗zgkCov(yax,n

∗

t , zG,b,kt−1 ) +Bk∗zkCov(yax,n

∗

t , zC,b,kt−1 ) +Bk∗ykCov(yax,n

∗

t , yb,kt−1)

where ”c” is the sibling in law of the uncle/aunt of ”ax”. We can compute cousins in law of any degree analogous.

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