Discussion Paper No. 1011

INTERGENERATIONAL TRANSMISSION
OF EDUCATION IN JAPAN:NONPARAMETRIC BOUNDS ANALYSISWITH MULTIPLE TREATMENTS

Nobuyoshi Kikuchi

September 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Intergenerational Transmission of Education in Japan:
Nonparametric Bounds Analysis with Multiple Treatments∗
Nobuyoshi Kikuchi†
September 19, 2017
Abstract
This paper investigates the intergenerational effects of education in Japan using a nonparametric bounds approach. The educational levels of parents are considered key factors in explaining
children’s educational success. Nevertheless, the literature has not reached consensus on the causal
effects of parents’ education on their child’s schooling. This is because both parents’ and the child’s
schooling depend on unobserved heterogeneity. Moreover, the strong positive correlation of themother’s and father’s schooling makes it difficult to separate the effects of each parent’s schooling,making it unclear how to control spousal schooling in the analysis. Therefore, this paper estimates
a set of semi-ordered vectors of both parents’ schooling as an application of the nonparametricbounds method with multiple treatments. It thus derives bounds depending on relatively weaksemi-monotonicity assumptions on treatment response, selection, and instrumental variables. A
combination of these assumptions provides informative bounds on the average treatment effect ofboth parents’ education on their child’s schooling. The main results show that the tightest lower
bounds suggest the positive causal effects of parents’ schooling, but the tightest upper bounds onthe effects are lower than the point estimates that rely on the assumptions of an exogenous selection
for parents’ schooling. These results suggest that simple regressions overestimate the true causal
effect of parents’ education.
JEL Classification: I24, I26, J62, C14
Keywords: Intergenerational transmission, returns to education, nonparametric bounds, multiple
treatments∗ This paper is based on part of my dissertation at the University of Tokyo and on an awarded paper at the 16th Labor

Economics Conference at Nihon University. I would like to thank my advisor, Hidehiko Ichimura, as well as committee
members Hideo Akabayashi, Yuji Genda, Hideo Owan, and Yasuyuki Sawada for their valuable advice and comments.
I am also grateful to Takeshi Aida, Daiji Kawaguchi, Hirokazu Matsuyama, Tsunao Okumura, as well as the seminar
participants at the University of Tokyo, and the participants at the 16th Labor Economics Conference at Nihon University,and the 2013 Japanese Economic Association Autumn Meeting at Kanagawa University for their helpful comments andsuggestions on earlier versions of this paper. Any remaining errors are my own. This research was financially supportedby the Grant-in-Aid for Japan Society for the Promotion of Science Fellows (DC2), Grant Number: 12J10408, and theGrant-in-Aid for Research Activity Start-up No. 16H06925 from the Japan Society for the Promotion of Science.
† Institute of Social and Economic Research, Osaka University. Address: 6-1, Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. Tel: +81-6-6879-8582, E-mail: kikuchi@iser.osaka-u.ac.jp.

1

Introduction

Parent’s educational level is key to explaining children’s educational success. If more educatedparents provide better nurturing environments and invest in their children (e.g., Guryan, Hurst, andKearney, 2008; Carneiro, Meghir, and Parey, 2013; Del Bono et al., 2016), an improvement of theeducation levels in one generation might also affect subsequent generations. This causal chain implies
potential policy interventions to address inequality transmitted across generations by encouraging
education. Therefore, numerous studies in social science, including economics, have paid attention
to the significance of the intergenerational transmission of education and empirically found a positive
correlation between the parents’ and their child’s schooling.1
Unfortunately, the literature faces at least two important difficulties in quantifying the causal
impacts of parents’ education on their child’s schooling. First, the schooling choice depends on
unobserved heterogeneity, such as preferences, genetic endowments, or other personal attributes
acquired during earlier life stages. If such unobservables determine both parents’ and their child’s
schooling, the positive intergenerational correlation does not necessarily reflect a causal relation. To
estimate the causal impact, the literature depends on identification strategies that use identical twins,
adopted children, and instrumental variables (IV). However, as Holmlund, Lindahl, and Plug (2011)
point out, different strategies lead to different results, and thus the literature does not reach a consensus
on the magnitude of the causal effect.
Second, a strong positive correlation between the father’s and mother’s schooling makes it difficult
to interpret the effect of each parent separately. Therefore, it is unclear whether spouses’ education
should be controlled in the analysis, and if so, how. Most previous studies focus on the effect of a single
treatment of each parent. The estimated effect might not be the causal effect of interest, because theeffect of parental schooling includes both a direct transfer from the given parent and an indirect transferfrom the other parent due to assortative mating effects. The regression approach allows controllingfor a variable of spousal education as an additional regressor, but usually ignores the endogeneity ofthis variable because it is difficult to find different IVs for each parent. Additionally, the regression
model relies on an implicit linear separability assumption of both parents’ years of schooling.2
The objective of this paper is to estimate the causal effect of a combination of parents’ schoolingon their child’s schooling in Japan using a nonparametric bounds approach. To do so, I consider aset of treatment vectors of mother’s and father’s education that takes semi-ordered multiple values. I
rely on monotonicity and semi-monotonicity assumptions to acquire informative bounds, such as themonotone treatment response (MTR) of Manski (1997), monotone treatment selection (MTS), andmonotone instrumental variables (MIV) of Manski and Pepper (1998, 2000), which are weaker than
the assumptions required to obtain point estimates. Due to the nonparametric bounds with multiple
treatments framework, this paper identifies the causal effect of parents’ educational attainment on thechild’s schooling, allowing for the separate effects of maternal and paternal education.
1See Black and Devereux (2011) for a review.
2See Holmlund, Lindahl, and Plug (2011) for a review.

1

The main results show that the tightest bounds on the average effects of parents’ schooling are
positive, but lower than the point estimates that rely on an exogenous selection of parents’ schooling.
These results suggest that the regression estimates, mainly provided in previous Japanese studies,
have an upward bias from the true causal effects of parents’ schooling. This paper shows that the
combination of weak monotone assumptions is too conservative to identify a point estimate, but isstill credible and produces informative bounds on the effects of the parents’ education on the child’s
schooling even when considering spousal schooling.
This paper provides new insights into the intergenerational effects of education in controlling for
spousal education. In previous studies that utilize the regression approach, Chevalier et al. (2013),
for instance, estimate the separate causal effects of parental schooling using a raising of the minimum
school leaving age in the U.K. as an instrument. Their identification depends on changes in parental
schooling induced by the compulsory schooling reform, only analyzing a local treatment effect of theparents at the margin.3 Oreopoulos, Page, and Stevens (2006) propose an alternative way to control for
spousal education. They use the sum of mother’s and father’s completed education as an explanatory
variable, instead of including each parent’s education separately. This allows to directly estimate theeffect of a one-year increase in either parent’s schooling. This specification implicitly assumes that theeffects of father’s and mother’s education are similar, although it is unclear whether this assumption
is satisfied in general. This paper’s specification of multiple treatments avoids the discussion on themost adequate way to control for spousal effects, but directly evaluates the effects of a combinationof both parents’ schooling. Furthermore, the nonparametric bounding approach is general enoughto identify the average treatment effect (ATE), thus allowing for the heterogeneous effects of eachparent’s schooling.
This paper relates to the literature using a nonparametric bounding approach for the returns to
education and skills (e.g., Ginther, 2000; Manski and Pepper, 2000; Gonzalez, 2005; Giustinelli, 2011;
Huang, van den Brink, and Groot, 2012; Okumura and Usui, 2014; Mariotti and Meinecke, 2015).
For example, De Haan (2011) first applies a nonparametric bounding method to examine the effectsof parent’s schooling on the child’s schooling using U.S. data. She considers ordered multivalued
treatments of each parent’s educational level and shows that the mother’s or father’s college degree has
a positive effect on the child’s schooling. Moreover, the estimated bounds are significantly different
from zero, but substantially lower than the ordinary least squares (OLS) estimates. This paper confirms
these findings with semi-ordered multivalued treatments of a pair of both parents’ educational levels
using Japanese data.
This paper also contributes to the intergenerational transmission of inequality literature by analyzing the causal impacts of parental schooling in Japan. Previous empirical studies have found asignificant positive correlation between parents’ and child’s schooling in Japan, but have paid limited
attention to the causal relationship.4 Researchers require micro data that contain twins, adopted chil3Dickson, Gregg, and Robinson (2016) also use an exogenous change in the minimum school leaving age in the U.K. to
identify the effects on the child’s educational attainment, but mainly focus on the direct impacts of the leaving age change.
4For example, see Tachibanaki (1988) and Yamada (2011). See also Kariya (2001) as an example of sociological2dren, or plenty of information about individual characteristics to estimate the causal effects. However,
it is difficult to access such data sets in Japan.5 This limited accessibility to informative micro data
prevents researchers from controlling individual heterogeneity and magnifies the inevitable concern
for point identification under strong assumptions on treatment selection. For this reason, I derive setsof nonparametric bounds under relatively weak and testable assumptions that share broad consensus
in the literature.
The remainder of this paper is organized as follows. Section 2 defines the parameter of interestand describes the identification assumptions. Section 3 explains the data set. Section 4 presents anddiscusses the results. Finally, Section 5 draws the conclusions.

2

MethodThis paper employs a nonparametric technique to obtain the bounds on the causal effect of parents’
education on theri child’s education, following the setup of Manski and Pepper (2000). There is a
probability space (J, Ω, P) of individuals. Each member j of population J has observable covariates
x j ∈ X and a response function y j (·) : T → Y , mapping the mutually exclusive and exhaustive
treatments t ∈ T into real-valued outcomes y j (t) ∈ Y , where treatment T is the level of schooling theparent completed, thus assumed to be an ordered set. Y represents the years of schooling or university
graduation of child j, and x = (w, v), X = W × V . Each value of (w, v) defines an observable subpopulation of children. I assume that the outcome space Y has the greatest lower bound Ymin ≡ min Y
and the least upper bound Ymax ≡ max Y .6
Child j has a realized treatment z j ∈ T and a realized outcome y j ≡ y j (z j ), both observable.
The latent outcomes y j (t), t , z j are not observable. A researcher learns the distribution P(x, z,y) of covariates, realized treatments, and realized outcomes by observing a random sample of the
population. To simplify the notation, subscript j will be dropped except when it required.
The parameter of interest is the ATE between the parent’s education level t ∈ T and t 0 ∈ T, s.t.
t > t 0:AT E(t, t 0 ) ≡ E[y(t) − y(t 0 )] = E[y(t)] − E[y(t 0 )].

(1)

The linearity of expectations provides the second equality. The empirical problem comes down
to learning E[y(·)] from the empirical evidence of the distribution P(x, z, y) of covariates, realized
treatments, and realized outcomes with assumptions. The fact that E[y|z = t] = E[y(t)|z = t] and theresearch on the relationship between educational attainment and family background.
5See Oshio and Seno (2007) for a review of studies on the economics of education in Japan.
6When I analyze the effects on the child’s years of schooling, I assume Ymin = 9 and Ymax = 18 years of schoolingfrom the construction of the data set.

3

law of iterated expectations gives

E[y(t)] = E[y|z = t] · Pr (z = t) + E[y(t)|z , t] · Pr (z , t).

(2)With a data set, researchers can estimate E[y|z = t], Pr (z = t), Pr (z , t), but cannot estimate
E[y(t)|z , t] without adding any assumptions. Manski (1989, 1990) shows that worst-case bounds
on E[y(t)] can be identified if the support of the dependent variable is bounded:

E[y|z = t] · Pr (z = t) + Ymin · Pr (z , t)
≤ E[y(t)] ≤(3)

E[y|z = t] · Pr (z = t) + Ymax · Pr (z , t).
The width of bounds is (Ymax − Ymin ) · Pr (z , t), thus the bounds are informative if Pr (z , t) < 1.
These bounds on E[y(·)] provide the lower (upper) bound on the ATE(t, t 0) by subtracting the upper(lower) bound on E[y(t 0 )] from the lower (upper) bound on E[y(t)]. Hence, the identification region
H {·} for the ATE is the interval:

H {E[y(t)] − E[y(t 0 )]}= {E[y|z = t] · Pr (z = t) + Ymin · Pr (z , t) − E[y|z = t 0] · Pr (y = t 0 ) + Ymax · Pr (z , t 0 ),E[y|z = t] · Pr (y = t) + Ymax · Pr (z , t) − E[y|z = t 0] · Pr (z = t 0 ) + Ymin · Pr (z , t 0 )}.
This interval contains the value zero. Its width is (Ymax −Ymin ) ·[Pr (z , t)+Pr (z , t 0 )]. The worstcase bounds are important at first, because they include all results that depend on different assumptions
about E[y(t)|z , t]. However, the worst-case bounds are too wide to obtain an informative resultin this paper. Therefore, some additional assumptions are necessary to tighten the bounds. In theremainder of this section, I explain how assumptions tighten these bounds. I consider the exogenous
treatment selection (ETS) assumption and the following three assumptions: (1) the MTR, (2) MTS
and (3) MIV. Additionally, I apply the semi-monotonicity version of these three assumptions.

2.1

Exogenous Treatment Selection

The ETS assumes that the realized treatment z is statistically independent of the latent outcomes y(·),
and can be expressed as:
For each t ∈ T, and all (u1, u2 ) ∈ T × T

4E[y(t)|z = u1 ] = E[y(t)|z = u2 ],(4)orE[y(t)] = E[y|z = t].

(5)Under the ETS assumption, the parameter of interest becomes: E[y|z = t] − E[y|z = t 0]. The
ETS assumption or covariates x conditional version of it, E[y(t)|x] = E[y|z = t, x], is implicitly
imposed in most empirical literature. This assumption implies that treatments are randomly assigned
to the population and excludes the existence of heterogeneous treatment effects with essential heterogeneity (Heckman, Urzua, and Vytlacil, 2006). However, this does not seem to be appropriate in
the intergenerational transmission of education. For example, one could consider a situation whereparents received different educational levels because they differ substantially from one another due
to attributes or skills acquired in their earlier life stages, but this heterogeneity is unobservable byresearchers. If the heterogeneity also determines the child’s schooling, the realized treatment is endogenous in the analysis. Moreover, it is difficult to use micro data with sufficient information on
individual characteristics in Japan. In this case, the researchers cannot fully control for heterogeneity,
and thus the endogenous selection of parents’ education violates the ETS assumption.
However, one advantage of assuming the ETS is that it yields point identification. The ETS
estimates are equivalent to the coefficients obtained by regression of the child’s schooling on dummy
variables for each parental educational level, with no other covariates. This paper shows the point
estimates under the ETS assumption for comparison with bounds estimates.

2.2

Monotone Treatment Response

Manski (1997) proposes an MTR assumption that states response functions are weakly increasing.
That is,For each j ∈ J and all (t 1, t 2 ) ∈ T × T s.t. t 1 ≤ t 2 ,t 1 ≤ t 2 ⇒ y j (t 1 ) ≤ y j (t 2 ).

(6)

The MTR assumes that an increase in parents’ schooling does not decrease their child’s schooling.
The validity of this assumption is suggested in the literature. For instance, the human capital theory(Becker, 1975; Becker and Tomes, 1979; Solon, 1999) suggests a positive impact of increasing parents’schooling on the child’s education. The literature on the returns to schooling shows that an additional
year of schooling has positive effects on both pecuniary and non-pecuniary outcomes (Card, 1999;
Oreopoulos and Salvanes, 2011). These advantages are transmitted to children from several parental
5

investment channels (Carneiro, Meghir, and Parey, 2013), which include time spent by the parents
with their children (Guryan, Hurst, and Kearney, 2008; Del Bono et al., 2016). It is worth noting that
the MTR assumption does not exclude zero effect of parental education, but allows that the positive
correlation between parents’ and child’s schooling has no causal relationship.
Manski (1997) shows that the MTR assumption tightens the worst-case bound:7

t < z j ⇒ Ymin ≤ y j (t) ≤ y j ,t = z j ⇒ y j (t) = y j ,(7)t > z j ⇒ y j ≤ y j (t) ≤ Ymax .
Thus,
E[y|z ≤ t] · Pr (z ≤ t) + Ymin · Pr (z > t)≤E[y(t)] ≤(8)

Ymax · Pr (z < t) + E[y|z ≥ t] · Pr (z ≥ t).

2.3

Monotone Treatment Selection

Manski and Pepper (1998, 2000) weaken equation 4 to an inequality and yield the MTS assumption:
For each t ∈ T and all (u1, u2 ) ∈ T × T s.t. u1 ≤ u2

u1 ≤ u2 ⇒ E[y(t)|z = u1 ] ≤ E[y(t)|z = u2 ].

(9)

If the MTS assumption holds, then,u < t ⇒ Ymin ≤ E[y(t)|z = u] ≤ E[y|z = t],u = t ⇒ E[y(t)|z = u] = E[y|z = t],
u > t ⇒ E[y|z = t] ≤ E[y(t)|z = u] ≤ Ymax .
It follows that
7For a full derivation of the MTR bounds, see Manski (1997).

6

(10)

Ymin · Pr (z < t) + E[y|z = t] · Pr (z ≥ t)
≤E[y(t)] ≤(11)

E[y|z = t] · Pr (y ≤ t) + Ymax · Pr (z > t).
Manski and Pepper (1998, 2000) also show that combining the MTR and MTS assumptions canhave substantial identifying power.8 Let the MTR and MTS assumptions 6 and 9 hold. Then, thebounds reduce toE[y|z < t] · Pr (z < t) + E[y|z = t] · Pr (z ≥ t)
≤ E[y(t)] ≤(12)

E[y|z = t] · Pr (z ≤ t) + E[y|z > t] · Pr (z > t).
According to the interpretation of Manski and Pepper (1998, 2000), the MTS assumption indicatesthat parents who select higher levels of schooling have a weakly higher mean response function thanthose who select lower levels. This implies that a genetic transmission of unobserved abilities fromparents to children provides the correlation of the educational levels. Researchers commonly assume
that unmeasured inherent differences are a key source of the endogeneity of education. Therefore,
the MTS assumption is consistent with arguments in the literature. Furthermore, Manski and Pepper(1998, 2000) suggest a simple test for the validity of the joint MTR-MTS assumption. Under thisassumption:
For all (u1, u2 ) ∈ T × T, s.t. u1 ≤ u2,
E[y|z = u1 ] = E[y(u1 )|z = u1 ] ≤ E[y(u1 )|z = u2 ] ≤ E[y(u2 )|z = u2 ] = E[y|z = u2 ].

(13)

The MTS assumption implies the first inequality and the MTR assumption gives the second. Under
this hypothesis, E(y|z = u) must be a weakly increasing function of u. Hence the researcher should
reject the hypothesis if the mean outcomes of the child’s schooling are not weakly increasing in therealized parental education levels.

2.4

Monotone Instrumental Variables

The recent literature on causal intergenerational schooling effects has moved from estimating OLS under the ETS assumption, and relies on alternative identification assumptions, such as an IV assumption(Holmlund, Lindahl, and Plug, 2011). Using the notation in this paper, the familiar mean-independence
8For a full derivation of the MTS and the MTR-MTS bounds, see Manski and Pepper (1998, 2000).

7

form of the IV assumption can be written as follows,
For each t ∈ T and all (u1, u2 ) ∈ V × V ,E[y(t)|v = u1 ] = E[y(t)|v = u2 ].

(14)

If the treatment effects are homogeneous, the IV identifies the parameter of interest. Even if theeffects are heterogeneous, researchers can recover the parameter of interest or a related interpretable
one using an IV with additional assumptions on the selection model.9 Unfortunately, it is difficult tofind an IV that satisfies the mean-independence conditions. Instead of assuming mean-independence,
Manski and Pepper (1998, 2000) introduce a weaker assumption to replace the equality in 14 by an
inequality yielding a weakly monotone relation between the IV and the mean response function.
For each t ∈ T and all (u1, u2 ) ∈ V × V ,u1 ≤ u2 ⇒ E[y(t)|v = u1 ] ≤ E[y(t)|v = u2 ].

(15)

In case the MIV assumption holds, the bounds are:

X

(

)
Pr (v = u) · max u1 ≤u E[y|z = t, v = u1 ]·Pr (z = t|v = u1 ) + Ymin · Pr (z , t|v = u1 )

u∈V

(16)

≤ E[y(t)] ≤
X

(

)

Pr (v = u) · minu2 ≥u E[y|z = t, v = u2 ]·Pr (z = t|v = u2 ) + Ymax · Pr (z , t|v = u2 ) .
u∈VThe MIV and the MTR-MTS assumptions make distinct contributions to the identification. Manski
and Pepper (1998, 2000) suggest that combining these assumptions yields particularly interestingbounds.10
9See Imbens and Angrist (1994), and Heckman and Vytlacil (2007) for further details.
10For a full derivation of the MIV and the MTR-MTS-MIV bounds, see Manski and Pepper (1998, 2000). Following theliterature, the MTS indicates a conditional MTS assumption when combining with (semi-)MIV assumptions. See Lafférs(2013) for more details.

8

X

(

Pr (v = u) · max u1 ≤u E[y|z < t, v = u1 ] · Pr (z < t|v = u1 )u∈V

+ E[y|z = t, v = u1 ] · Pr (z ≥ t|v = u1 ))

≤ E[y(t)] ≤
X

(

(17)

Pr (v = u) · minu2 ≥u E[y|z = t, v = u2 ] · Pr (z ≤ t|v = u2 )


u∈V

)
+ E[y|z > t, v = u2 ] · Pr (z > t|v = u2 ) .

2.5

Semi-Monotone Instrumental Variables

Manski and Pepper (1998) also suggest ways to combine multiple scalars of IV or MIV assumptions.
Here, I use two MIVs simultaneously to tighten the bounds following De Haan (2011). To do so,
I assume that the pair of MIVs, (v a, v b ), is a two-dimensional semi-monotone instrumental variable(SMIV):For each t ∈ T and all [(u1a, u1b ), (u2a, u2b )] ∈ (V a × V b ) × (V a × V b ) s.t. u1a ≤ u2a and u1b ≤ u2b⇒ E[y(t)|(v a, v b ) = (u1a, u1b )] ≤ E[y(t)|(v a, v b ) = (u2a, u2b )].

(18)

In this assumption, a pair of MIVs is assumed to be semi-ordered rather than ordered, because it
includes some value pairs that are not ordered.11 The MTR-MTS-MIV bounds in 17 can be extended
to the MTR-MTS-SMIV bounds if the maxima and minima operations are taken over ordered pairs ofvalues.

2.6

Semi-Monotone Treatment Response

The previous studies of the effect of parents’ schooling on child’s schooling have focused on theidentification of a single treatment of one parent and estimated the effect of mother’s and father’s
schooling separately. By contrast, children are affected by a combination of both parents’ schoolingin practice. This indicates that the treatments of parents’ schooling are multiple for the child. In
this paper, I analyze effects of both parents’ schooling simultaneously within the framework of thenonparametric bounding approach.12
11For example, consider that u1a < u2a and u1b > u2b . This does not predict the ordering of E[y(t)|(v a, v b ) = (uka, ukb )], k =1, 2.
12The matching approach is an alternative method analyzing multiple treatments. For example, see Imbens (2000) and
Lechner (2001). See also Flores and Mitnik (2013) as an example of applying the difference-in-difference method for
multiple treatments. Frölich (2004) is a comprehensive survey of the multiple treatments analysis.

9Manski (1997) proposes the assumption of a semi-monotone treatment response (SMTR), wherea semi-ordered vector of multivalued treatments T exists. This suggests that the discussion basedon the single ordered treatment of a parent’s schooling can be extended to a semi-ordered vector of
multiple treatments, which is a combination of each parent’s schooling. This paper considers thetwo-dimensional treatment vectors of mother’s and father’s schooling, (t M , t F ) = t ∈ T. In this case,the SMTR assumption states:
For each j ∈ J and all (t1, t2 ) ∈ T × T s.t. t1 ≤ t2 ,t1 ≤ t2 ⇒ y j (t1 ) ≤ y j (t2 ),(19)

where I define t1 ≤ t2 if and only if t 1M ≤ t 2M and t 1F ≤ t 2F .
The analysis under the MTR assumption still holds under the assumption of SMTR, except for thecase where treatments are not ordered:13t < z j ⇒ Ymin ≤ y j (t) ≤ y j ,t = z j ⇒ y j (t) = y j ,(20)t > z j ⇒ y j ≤ y j (t) ≤ Ymax,t ∅ z j ⇒ Ymin ≤ y j (t) ≤ Ymax,
where ∅ denotes that t and z are not ordered. Then, I have the following SMTR bounds:

E[y|z ≤ t] · Pr (z ≤ t) + Ymin · Pr (z > t) + Ymin · Pr (z∅t)≤E[y(t)] ≤(21)

Ymax · Pr (z < t) + E[y|z ≥ t] · Pr (z ≥ t) + Ymax · Pr (z∅t).

2.7

Semi-Monotone Treatment Selection

The SMIV assumption of Manski and Pepper (1998) suggests an assumption on selection with a semiordered vector of multivalued treatment. The semi-monotone treatment selection (SMTS) assumption
is the special SMIV assumption that the MIVs, v = (v a, v b ) are the realized treatments z. With the
SMTS assumption, the analysis under the MTS assumption is also available, except for the case where
treatments are not ordered. Therefore, I can obtain the SMTR-SMTS bounds by combining the SMTR
and SMTS assumptions, as well as the analysis on the single treatment. Let the SMTR and SMTS
13For a full derivation of the SMTR bounds, see Manski (1997).

10

assumptions hold.14 Then,u < t ⇒ Ymin ≤ E[y|z = u] ≤ E[y(t)|z = u] ≤ E[y|z = t],u = t ⇒ E[y(t)|z = u] = E[y|z = t],(22)

u > t ⇒ E[y|z = t] ≤ E[y(t)|z = u] ≤ E[y|z = u] ≤ Ymax,u ∅ t ⇒ Ymin ≤ y(t) ≤ Ymax .
It follows thatE[y|z < t] · Pr (z < t) + E[y|z = t] · Pr (z ≥ t) + Ymin · Pr (z∅t)≤ E[y(t)] ≤(23)

E[y|z = t] · Pr (z ≤ t)+E[y|z > t] · Pr (z > t) + Ymax · Pr (z∅t).

These bounds can be viewed as a natural extension of the MTR-MTS bounds. The bounds canthen be combined with the MIV and SMIV assumptions, as well as the single treatment bounds since
the assumptions are not mutually exclusive.

3

Data

The main analysis data are the Japanese General Social Surveys (JGSS).15 The JGSS are repeatedcross-sectional data for men and women aged 20–89 on each survey date. This paper uses the surveys
conducted in 2000, 2001, 2002, 2003, 2005, 2006, 2008, and 2010, and pools the respondents fromall waves. From the pooled original data, I exclude observations by following four steps.
First, I restrict the sample by age, excluding the respondents below 25 years old on the survey
year because they might not have completed their academic schooling. Second, I drop the individuals
who answered that their father or mother was absent at the age of 15, because I am unable to control
unobservable characteristics if parental structures are substantially different between those with single
14For a full derivation of MTR-MTS bounds and a further discussion on semi-monotonicity assumptions, see Manski(1997) and Manski and Pepper (1998).
15The Japanese General Social Surveys (JGSS) are designed and carried out by the JGSS Research Center at Osaka
University of Commerce (Joint Usage / Research Center for Japanese General Social Surveys accredited by Minister of
Education, Culture, Sports, Science and Technology), in collaboration with the Institute of Social Science at the Universityof Tokyo.

11

parents and those with couples of parents. Third, I only use the observations for those born after
1940, because the current educational system was enacted after 1947. Finally, I use the individuals
whose observational characteristics are available. Therefore, the remaining sample contains 13669
individuals, born in 1940–1984 and being 25–69 on the date of the survey.
For the educational attainment of the respondents (children) and their parents, I use information
about the level of the last school attended and assign the standard years of schooling in Japan. For
children, completion of their last schooling is available. Following Tanaka (2008), I reduce the numberof years of education by one year from the standard years for those who dropped out before graduating
the last attended school. To examine the robustness of the definition of the outcome variable, I also
use an indicator of university graduation.
In the analysis, I consider the following four schooling levels for parents: (1) junior high school
denotes that the parent completes compulsory schooling or lower secondary education, (2) high school
denotes that the parent completes high school or upper secondary education, (3) some college denotesthat the parent completed a two-year college or some years of higher education, and (4) university
denotes that the parent has a bachelor’s degree. I also use schooling of a parent as an MIV. WhenI obtain bounds on the effect of the mother’s schooling, I use the indicator of father’s university
graduation as an MIV. For father’s schooling, the indicator of mother’s college graduation takes therole of an MIV.
Under the SMIV assumption, I use two additional dummy variables as MIVs. The first additional
MIV is a dummy variable of father’s “regular” worker status that takes the value one if the father was
an executive of a company or an employee with non-terminable contract (joji-koyo no rodosha) when
the child was 15. A father’s more stable working status implies more stable parental income during
the child’s adolescent years. It is natural to assume that a child with more parental income is lesslikely to face credit constraints for the decision of additional schooling. The second additional MIV
is a dummy variable that takes the value one if the child was born after 1975. Taking the birth cohort
as an MIV is equivalent to assuming that children born in more recent years have weakly higher meanoutcome functions than others. This assumption can be attributed to the decreasing trends of numberof children, non-decreasing trends in the number of schools per child, and stable non-decreasing trendsof post-compulsory school or university enrollment rates in recent years in Japan. Table 1 shows somesummary statistics.

4

Results

Before presenting the results of the bounds on the ATE, I test that the combined assumption of the
MTR and the MTS cannot be rejected from the analysis data. Table 2 reports mean schooling outcomesof children by the educational level of parents. All variables show that the MTR-MTS assumption is
not rejected, since the average values of outcome variables are weakly increasing both in the levels of
12the mother’s and father’s schooling.

4.1

Single Treatment Effects on Child’s Years of SchoolingI begin by showing the results of a single treatment of one parent to confirm the findings in De Haan
(2011). Table 3 provides bounds on the ATE of the mother’s and father’s schooling on child’ years ofschooling. Following Manski and Pepper (1998, 2000), Gonzalez (2005), and Giustinelli (2011) alltables below also report 95% confidence intervals obtained by the percentile bootstrap method with
3000 replications.
The column of the ETS assumption reports point estimates of the effects. The effect of the changeof mother’s schooling from junior high school to high school shows an increase in the child’s schoolingby about 1.50 years. The corresponding effect of the father’s schooling increases child’s schoolingby about 1.30 years. The change in mother’s schooling from high school to some college increases
the child’s schooling by about 1.02 years. Increasing father’s schooling from high school to somecollege, increases the child’s schooling by about 0.74 years. These results suggest stronger effects
of maternal over paternal education on the child’s years of schooling. However, the ETS results ofa bachelor’s degree seem to be mixed. Compared to college degrees, mother’s bachelor degree has
smaller effects than the father’s bachelor degree. Increasing the mother’s schooling from junior highschool to a bachelor’s degree, increases the child’s schooling by about 2.95 years. This is larger thanthe corresponding effect of father. Increasing father’s schooling from junior high school to bachelor’s
degree, increases the child’s schooling by about 2.65 years.
From columns (2) and (7), the worst-case bounds are extremely wide. Imposing the MTR
assumption increases significantly the lower bounds by the definition of the assumption that implies
non-negative effects (columns (3) and (8)). The MTS assumption allows tightening upper and lower
bounds compared to the worst-case ones, but is not powerful enough to acquire information on thetreatment effect (columns (4) and (9)). The combination of the MTR and MTS assumptions leads
to substantially tighter bounds than imposing each assumption alone. Unfortunately, the MTR-MTS
assumption cannot provide informative results itself (columns (5) and (10)).
Table 3 also shows the bounds under the MIV assumption and its combinations. The MIV
assumption tightens upper and lower bounds compared to the worst-case bounds, but it is not powerfulenough to obtain informative bounds as well as the MTS assumption (columns (11) and (15)). These
results do not depend on the definition of the MIV (columns (12) and (16)). Columns (13), (14), (17),and (18) show the bounds under the MTR-MTS-MIV assumption. The combination of these threeassumptions leads to informative bounds. I obtain upper bounds that are lower than the ETS resultsin some cases. The ETS point estimates fall outside the confidence intervals when increasing parent’s
schooling from junior high school to university.

13

4.2

Single Treatment Effects on Child’s University Graduation

To examine the robustness of the results in Table 3, I estimate bounds on an alternative definition ofoutcomes. Table 4 displays the results of the effects on a child’s university graduation. The MTR andMTS bounds are rather wide and not very informative; thus these are not shown in Table 4.
At all levels of parental education, the ETS estimates show that additional years of schooling fora parent have significant impact on the probability of the child’s university graduation (columns (1)and (2)). However, the MTR-MTS-MIV upper bounds suggest that the ETS estimates overstate the
ATE for some levels of parental education (columns (4), (5), (9), and (10)). The upper bounds under
the MTR-MTS-MIV assumption suggest much smaller effects of a parent’s bachelor’s degree on thechild’s university graduation when compared with a parent who completed only compulsory or lower
secondary education. Unfortunately, the ETS estimates of the lower categories of a parent’s schoolingreport small effects. Even when imposing the MTR-MTS-MIV assumption, the upper bounds on theeffects cannot exclude the smaller effects of the point estimates for these levels of parental education.

4.3

Single Treatment Effects on Child’s Education under the SMIV assumption

The results under the MTR-MTS-MIV assumption provide informative upper bounds since they aresubstantially smaller than the point estimates obtained under the ETS assumption. However, lower
bounds do not exclude the zero effect of a parent’s schooling on both years of schooling and theprobability of university graduation of the child. Here, I use the two MIVs simultaneously under the
SMIV assumption instead of using the schooling of the other parent and father’s regular job variable
separately.
Panel A of Table 5 shows the effects on a child’s years of schooling. As such, the SMIV assumption
gives informative bounds (columns (5)). The upper bounds are lower than the MTR-MTS-MIV ones,
and indicate that both mother’s and father’s college degrees increases a child’s schooling by about
1.41 years. The upper bounds result from a change of the parent’s schooling from junior high school
to bachelor’s degree, showing the mother’s university graduation increases a child’s schooling by
about 2.09 years and the father’s university graduation by about 2.12 years. These upper bounds aresubstantially lower than the estimates under the ETS assumption.
Combining the two MIVs also plays an important role to tightening the lower bounds. For bothmother’s and father’s schooling, the results show that the effect of an increase in parent’s schooling tocollege degree or more impacts the child’s schooling significantly different from zero. For example,
a change in mother’s schooling from junior high school to bachelor’s degree results in an increase inchild’s schooling by about 0.39 years. The father’s bachelor’s degree increases the child’s schoolingby about 0.43 years.
Panel B of Table 5 shows the effects on the probability of a child’s university graduation. The results
using this outcome variable are similar to those using the child’s years of schooling as the outcome
variable. The MTR-MTS-SMIV bounds give not only more informative upper bounds but also more14

informative lower ones (column (10)). The results of increasing father’s or mother’s schooling to acollege degree or more has a positive effect on the child’s university graduation, being statisticallysignificantly different from zero, but substantially lower than the point estimates under the assumptionof ETS.
To summarize, the combination of the MTR-MTS-MIV assumptions provides informative bounds
on the ATE of a parent’s schooling on the child’s schooling. The main results show that the tightest
upper bounds on the effects are lower than the point estimates under the ETS assumption. These
results suggest that the regression estimates have an upward bias from the true causal effect of aparent’s schooling. The combination of the MTR-MTS-SMIV assumptions also gives informativelower bounds, which suggest positive effects of parental education on the child’s schooling.

4.4

Multiple Treatment Effects on Child’s Years of Schooling and University
Graduation

Here, I provide the results of analyses on multiple treatments of both parents’ schooling. The partial
identification approach discussed in the method section provides bounds on the ATE of multiple
treatments under different semi-monotonicity assumptions, as well as the single treatment case. Table
6 shows the multiple treatments effects of parents’ schooling on the child’s schooling. The SMTR andthe SMTS bounds are too wide to provide informative results, and not shown in the table.16
Panel A of Table 6 provides the effects on a child’s years of schooling. I begin by examining
the estimated ATE under the assumption of ETS for a useful benchmark. Children whose parents
complete college or more have about 2.27 more years of schooling than those whose parents completehigh school or less (column (1)). Compared to the point estimates under the ETS assumption, theconservative worst-case bounds are [-4.3261 5.8333], being too wide to acquire information on the
ATE of parents’ schooling (column (2)).
By combining different sets of assumptions, the results clearly illustrate more informative bounds
on the ATE. The estimated bounds are [0, 2.2697] under the SMTR-SMTS assumption and [0, 2.1060]
under the SMTR-SMTS-MIV assumption with an MIV of the father’s regular job or [0, 2.2388] withan MIV for a child was born after 1975 (columns (3), (4), and (5)). These upper bounds are smaller
than the estimates under the ETS assumption. While both of these MIV assumptions substantiallyreduce the upper bounds, there still remains uncertainty about the lower bounds on the ATE. Under
the combined SMTR-SMTS and SMIV assumptions, the bounds narrow to [0.2397, 1.2077] and theconfidence intervals are (0.1913, 1.3243), which excludes zero (column (6)). Therefore, a set ofsemi-monotonicity assumptions provides the non-negative impact of parents’ education on the child’s
schooling.
Panel B of Table 6 shows the effects on the probability of a child’s university graduation. The
16In this table, I only use the two levels of parents’ schooling (college degree or more and high school or less), because
the number of observations is too small to estimate using four levels of parents’ schooling.

15

results of this outcome variable are similar to the results of the child’s years of schooling. Under the
ETS assumption, the point estimates indicate that children whose parents complete college or moreare more likely graduate university than those whose parents complete high school or less (column(7)). The difference in the probability is 42%. Compared to the point estimates under the ETS
assumption, the conservative worst-case bounds are [-30.17, 82.71]%, which includes the negative
effects of parents’ schooling (column (8)).
The estimated bounds are [0, 42.00]% under the SMTR-SMTS assumption and [0, 38.39]% under
the SMTR-SMTS-MIV assumption with an MIV of father’s regular job or [0, 41.43]% with an MIV
for a child was born after 1975 (columns ((9), (10), and (11)). These upper bounds are smaller than theestimates under the ETS assumption. Under the SMTR-SMTS-SMIV assumption, the bounds narrow
to [4.54, 19.84]% and the confidence intervals are (3.53, 22.69), which excludes zero (column (12)).
This suggests that parents’ college degrees are at least somewhat beneficial for the child’s university
graduation. However, this estimated bound is strictly smaller than the point estimates under the ETS
assumption, which is sometimes implicitly assumed in the literature.
To summarize, the combination of the semi-monotonicity assumptions provides informative
bounds on the ATE of parents’ schooling on the child’s schooling in the multiple treatments framework.
The main results show that the tightest upper bounds on the effects are substantially lower than the
point estimates that assume ETS. These results suggest that the regression estimates have an upwardbias from the true causal effect of the parents’ education. The combination of the SMTR-SMTS-SMIV
assumptions also gives informative lower bounds, which suggests non-zero positive effects of parents’
education on the child’s schooling.

5

Conclusion

Quantifying the causal effects of parents’ schooling on the child’s schooling is a goal of recent
empirical literature on intergenerational effects of education. The literature applies several strategies
to identify the causal effects, such as identical twins, adopted children, and IV methods. Nevertheless,the empirical evidence on the causal relationship between parents’ and their child’s schooling is
moderately convincing, because these methods identify the effects for a limited population, and thedifferent methods have inconsistent results. The findings from a regression approach have shownlarge, positive, and statistically significant estimates, but implicitly assume that parents’ schooling is
randomly assigned to the child. This does not seem to be appropriate if unobserved heterogeneity
determines both parents’ and the child’s educational levels.
The main contributions of this paper are twofold. First, it estimates the causal effects of parents’schooling on their child’s schooling in Japan using a nonparametric bounds method, and provides the
ATE bounds without relying on an invalid exogenous selection assumption. Second, it presents resultsof the multiple treatments of both parents’ schooling as an application of the multiple treatments
16

framework. To obtain informative bounds, this paper imposes relatively weak and partially testable
monotonicity and semi-monotonicity assumptions on treatment response, selection, and IVs.
The tightest bounds under the combination of these three assumptions show that the obtained
lower bounds on the ATE are positive and significantly different from zero, but the upper bounds
are lower than the point estimates that rely on the assumption of the exogenous selection of parents’schooling. This suggests that the previous studies using OLS overestimate the true causal effectof parents’ education. This implies that a combination of semi-monotone assumptions is useful toacquire informative bounds on the intergenerational effects of education as semi-ordered multivalued
treatments of the parents’ schooling.

17

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20

Table 1: Summary Statistics
mean
sd
min
Chlid’s years of schooling
13.0893 2.1713
9
Chlid’s university graduation
0.2452 0.4302
0Mother’s years of schooling
9.7783 2.5953
6
Mother’s some college
0.0750 0.2634
0Mother’s university graduation 0.0258 0.1584
0
Father’s years of schooling10.2526 3.1694
6
Father’s some college
0.0681 0.2519
0Father’s university graduation
0.1231 0.3286
0Father had a regular job
0.5872 0.4924
0Child was born after 1975
0.0941 0.2920
0Female0.5406 0.4984
0

max
18
1
18
1
1
18
1
1
1
1
1

Notes: This table reports summary statistics of the analysis data. Number of observations is 13,669. Father hada regular job indicates that the father was an executive of a company or was a regular employee (joji-koyo norodosha) when the child was 15 years old.

21

Table 2: Mean of Child’s Schooling Outcomes by Parent’s Education Level
MotherFatherChlid’s years
Child’s
Chlid’s years
Child’s
Education level of parentof schooling university of schooling university
Junior high school
12.17530.118412.1779
0.1179High school
13.6801
0.309513.4729
0.2691
Some college
14.6966
0.522014.2095
0.4135
University
15.12220.664814.8247
0.5639Notes: This table reports mean schooling outcomes of children by educational level of parents for the test of the
MTR-MTS assumption.

22

23

Notes: This table reports average treatment effects of parent’s education on child’s years of schooling. Numbers between parentheses are 95% confidence
intervals calculated by the percentile bootstrap method. Levels of parental education are (1) JH = junior high school, (2) HS = high school, (3) Coll =some college, (4) Univ = bachelor’s degree or more. Number of observations is 13,669.

Table 3: Bounds of ATE of Parent’s Schooling on Child’s Years of Schooling
Assumption
ETS
Worst-Case
MTR
MTS
MTR MTS
LB
UB
LB
UB
LB
UB
LB
UB
LB
UB
Effect of Mother’s Schooling
(1)
(2)
(3)
(4)
(5)
E[y(t=HS)] - E[y(t=JH)]
1.5049-4.2558 5.6509
05.3291 -3.7843 1.9400
01.6182(1.4365 1.5743) (-4.2673 5.6327) (0 5.3626) (-3.8194 2.0043) (0 1.6873)
E[y(t=Coll)) - E(y(t=HS)]
1.0165-6.7095 6.7336
05.1737 -4.5414 3.2915
0
1.7316(0.8969 1.1355) (-6.7224 6.7343) 0 5.2075) (-4.5855 3.4028) (0 1.8479)
E[y(t=Univ)) - E(y(t=Coll)]
0.4256-8.5946 8.4987
0
4.9942 -5.6240 5.5483
02.0438(0.1902 0.6548) (-8.6021 8.4932) (0 5.0297) (-5.7323 5.7433) (0 2.2469)
E[y(t=Univ)) - E(y(t=JH)]
2.9469-6.1168 7.4401
07.4401 -6.1168 2.9469
0
2.9469(2.7452 3.1443) (-6.1874 7.4581) (0 7.4631) (-6.1402 3.1480) (0 3.1562)
Effect of Father’s Schooling
(6)
(7)
(8)
(9)
(10)
E[y(t=HS)] - E[y(t=JH)]
1.2950-4.7572 5.9639
0
5.3148 -3.9018 2.1607
01.5116(1.2222 1.3728) (-4.7708 5.9647) (0 5.3485) (-3.9380 2.2265) (0 1.5806)
E[y(t=Coll)) - E(y(t=HS)]
0.7366-7.0819 7.1973
05.3335 -4.3424 3.2763
01.4125(0.6015 0.8681) (-7.0943 7.1987) (0 5.3671) (-4.3933 3.3848) (0 1.5240)
E[y(t=Univ)) - E(y(t=Coll)]
0.6153-8.0247 8.2542
05.2369 -4.9590 4.8285
0
1.8111(0.4633 0.7619) (-8.0270 8.2639) (0 5.2702) (-5.0624 4.9196) (0 1.9110)
E[y(t=Univ)) - E(y(t=JH)]
2.6468-5.5845 7.1362
0
7.1362 -5.5845 2.6468
02.6468(2.5472 2.7423) (-5.5978 7.1416) (0 7.1612) (-5.6104 2.7476) (0 2.7480)

24

Notes: This table reports average treatment effects of parent’s education on child’s years of schooling. Numbers between parentheses are 95% confidence
intervals calculated by the percentile bootstrap method. Levels of parental education are (1) JH = junior high school, (2) HS = high school, (3) Coll = somecollege, (4) Univ = bachelor’s degree or more. MIVs are the indicator of other parent’s university / college graduation for mother’s / father’s educationand father’s regular job status. Number of observations is 13,669.

Table 3: Bounds of ATE of Parent’s Schooling on Child’s Years of Schooling (cont.)
Assumption
MIV
MIV
MTR MTS MIV
MTR MTS MIVMIV Other parent’s univ Father’s regular job Other parent’s univ Father’s regular job/ college grad/ college grad
LB
UB
LB
UB
LB
UB
LB
UB
Effect of Mother’s Schooling
(11)
(12)
(13)
(14)
E[y(t=HS)] - E[y(t=JH)]-4.2558 5.4734 -4.2558 5.2105
01.416201.4958(-4.2877 5.5076) (-4.2862 5.2594) (01.5011)(0
1.5648)E[y(t=Coll)) - E(y(t=HS)]-6.7095 6.1976 -6.5188 6.6702
01.412501.6587(-6.7327 6.2405) (-6.5492 6.6945) (01.5514)(0
1.7822)E[y(t=Univ)) - E(y(t=Coll)] -8.0586 8.0606 -8.5312 8.4728
0
1.417801.8613(-8.0974 8.0996) (-8.5442 8.4839) (01.9128)(02.1019)E[y(t=Univ)) - E(y(t=JH)]-6.1168 6.8245 -6.1168 7.1645
02.194802.6893(-6.1405 6.8711) (-6.1405 7.2055) (02.7014)(02.9373)
Effect of Father’s Schooling
(15)
(16)
(17)
(18)
E[y(t=HS)] - E[y(t=JH)]-4.6679 5.8243 -4.7572 5.4669
01.364901.4038(-4.6989 5.8570) (-4.7869 5.5166) (01.4448)(01.4715)E[y(t=Coll)) - E(y(t=HS)]-7.0819 6.4940 -6.9310 7.1391
0
1.218001.2909(-7.1026 6.5455) (-6.9591 7.1615) (01.3513)(01.4148)E[y(t=Univ)) - E(y(t=Coll)] -7.4106 7.0613 -7.9665 8.1252
01.577501.6869(-7.4599 7.1286) -7.9825 8.1426) (01.6966)(01.8045)E[y(t=Univ)) - E(y(t=JH)]-5.5845 5.8037 -5.5845 6.6612
02.283602.4542(-5.6098 5.8780) (-5.6097 6.7041) (02.4119)(02.5719)

25

Notes: This table reports average treatment effects of parent’s education on child’s university graduation. Numbers between parentheses are 95%
confidence intervals calculated by the percentile bootstrap method. Levels of parental education are (1) JH = junior high school, (2) HS = high school, (3)Coll = some college, (4) Univ = bachelor’s degree or more. MIVs are the indicator of other parent’s university / college graduation for mother’s / father’s
education and father’s regular job status. Number of observations is 13,669.

Table 4: Bounds of ATE of Parent’s Schooling on Child’s University Graduation
Assumption
ETS
Worst-Case
MTR MTS
MTR MTS MIV
MTR MTS MIV
MIV
No
No
No
Other parent’s univ Father’s regular job/ college grad
LB
UB
LB
UB
LB
UB
LB
UB
LB
UB
Effect of Mother’s Schooling
(1)
(2)
(3)
(4)
(5)
E[y(t=HS)] - E[y(t=JH)]
0.1912-0.4539 0.6468
00.216300.181500.2036(0.1772 0.2054) (-0.4669 0.6471) (0 0.2302) (00.1987)(00.2178)E[y(t=Coll)) - E(y(t=HS)]
0.2124-0.6630 0.8306
00.305600.232600.2881(0.1801 0.2446) (-0.6652 0.8323) (0 0.3373) (00.2687)(00.3200)E[y(t=Univ)) - E(y(t=Coll)]
0.1428-0.9470 0.9522
00.423300.279500.3896(0.0829 0.2024) -0.9487 0.9536) (0 0.4736) (00.4072)(00.4491)E[y(t=Univ)) - E(y(t=JH)]
0.5464-0.5703 0.9360
00.546400.383800.5079(0.4977 0.5967) (-0.5764 0.9365) (0 0.5977) (00.5128)(00.5678)
Effect of Father’s Schooling
(6)
(7)
(8)
(9)
(10)
E[y(t=HS)] - E[y(t=JH)]
0.1512-0.4983 0.6930
00.197300.178500.1889(0.1371 0.1663) (-0.5038 0.6987) (0 0.2103) (00.1948)(00.2028)E[y(t=Coll)) - E(y(t=HS)]
0.1445-0.7194 0.8671
00.233000.191700.2036(0.1098 0.1779) (-0.7242 0.8721) (0 0.2611) ( 00.2268)(00.2346)E[y(t=Univ)) - E(y(t=Coll)]
0.1503-0.8906 0.9181
00.337200.279400.3153(0.1100 0.1904) (-0.8943 0.9218) (0 0.3633) (00.3096)(00.3436)E[y(t=Univ)) - E(y(t=JH)]
0.4460-0.5218 0.8917
00.446000.372800.4218(0.4209 0.4702) (-0.5263 0.8965) (0 0.4720) (00.4056)(00.4498)

26

UB

0.3279(0.3072 0.3486)
0.4460(0.4209 0.4702)(6)0.3484(0.3214 0.3755)
0.5464(0.4977 0.5967)

1.8748(1.7930 1.9562)
2.6468(2.5472 2.7423)(1)
1.9083(1.8040 2.0119)
2.9469(2.7452 3.1443)

LB

ETS
No

0
(0
0
(0

1.71411.8095)2.45422.5719)-0.2412(-0.2478-0.5218(-0.5263

0.75880.7656)
0.89170.8965)0(00(00.26180.2863)0.37280.4056)0(00(00.29810.3200)0.42180.4498)0.0246(0.0153
0.0665(0.0557

0.23380.2596)
0.34990.3849)(10)0.23170.2689)0.37730.4734)1.56011.6600)2.28362.4119)Panel B: Effect on Probability of Child University Graduation
(7)
(8)
(9)
-0.2334 0.7666
0
0.242900.32450.0092(-0.2402 0.7738) (00.2797)(00.3533)(0.0007-0.5703 0.9360
00.383800.50790.0567(-0.5764 0.9365) (00.5128)(00.5678)(0.0413000(01.41451.5182)2.11672.2564)5.33355.3685)7.1362
7.1416)0.1308(0.0867
0.4321(0.4221-3.6665-(3.7019-5.5845(-5.5978MTR MTS MIV
MTR MTS MIV
MTR MTS SMIV
Other parent’s univ Father’s regular job Other parent’s univ/college/ college gradand Father’s regular job
LB
UB
LB
UB
LB
UB
LB
UB
Panel A: Effect on Chlid’s Years of Schooling
(2)
(3)
(4)
(5)
-3.8263 5.1737
01.458801.79590.04751.4133(-3.8609 5.2100) (01.5945)(01.9147)(0.0028
1.5528)-6.1168 7.4401
02.194802.68930.38902.0886(-6.1874 7.4581) (02.7014)(02.9373)(0.3175
2.4566)

Worst-Case
No

Notes: This table reports average treatment effects of parent’s education on child’s years of schooling and child’s university graduation. Numbers between
parentheses are 95% confidence intervals calculated by the percentile bootstrap method. Levels of parental education are JH = junior high school, Univ =bachelor’s degree or more, HS− = high school or less, Coll+ = college or more. MIVs are the indicator of father’s regular job status and the indicator that
denotes the child was born after 1975. Number of observations is 13,669.

E[y(t=Univ)) - E(y(t=JH)]

Father’s SchoolingE[y(t=Coll+)) - E(y(t=HS-)]E[y(t=Univ)) - E(y(t=JH)]

Mother’s SchoolingE[y(t=Coll+)) - E(y(t=HS-)]E[y(t=Univ)) - E(y(t=JH)]

Father’s SchoolingE[y(t=Coll+)) - E(y(t=HS-)]E[y(t=Univ)) - E(y(t=JH)]

Mother’s SchoolingE[y(t=Coll+)) - E(y(t=HS-)]

Assumption
MIV

Table 5: Bounds of ATE of Parent’s Schooling on Child’s Years of Schooling and University Graduation

27

0.0454(0.0353(7)0.4200(0.3893 0.4491)

E[y[(t M , t F )=(Coll+, Coll+)]- E[y[ (t M , t F )=(HS-, HS-)]](12)
0.19840.2269)1.20771.3243)(6)

Notes: This table reports average treatment effects of a set of vector of mother’s and father’s education on the child’s years of schooling and the child’s
university graduation. t M and t F denotes the treatment level of mother’s and father’s education respectively. Numbers between parentheses are 95%
confidence intervals calculated by the percentile bootstrap method. Levels of parental education are HS− = high school or less, Coll+ = college or more.
MIVs are the indicator of father’s regular job status and the indicator that denotes the child was born after 1975. Number of observations is 13,669.

0
(0

UB

SMTR SMTS SMIV
Father’s regular job
and Born After 1975
LB
UB

Panel B: Effect on Probability of Child University Graduation
(8)
(9)
(10)
(11)
-0.3017 0.8271
00.420000.383900.4143(-0.3080 0.8332) (0 0.4510) (00.4177)(00.4443)- E[y[(t M , t F )=(HS-, HS-)]]LB

SMTR SMTS MIV
Born After 1975

0.2397(0.1913

SMTR SMTS MIV
Father’s regular job(5)
2.23882.3499)LB

SMTR SMTS
No(1)
2.2697(2.1610 2.3856)

UB

Worst-CaseNo

E[y[(t M , t F )=(Coll+, Coll+)]

LB

ETS
No
UB
LB
UB
LB
UB
Panel A: Effect on Chlid’s Years of Schooling
(2)
(3)
(4)
-4.3261 5.8333
02.269702.1060(-4.3591 5.8669) (0 2.3864) (02.2344)

Assumption
MIV

Table 6: Bounds of ATE of Parents’ Schooling on Child’s Years of Schooling and University Graduation