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Must Try Harder. Evaluating the Role of Effort in Educational Attainment

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This paper is based on the idea that the effort exerted by children, parents and schools affects the outcome of the education process. We test this idea using the National Child Development Study. Our theoretical model suggests that the effort exerted by the three groups of agents is simultaneously determined as a Nash equilibrium, and is therefore endogenous in the estimation of the education production function. Our results support this, and indicate which factors affect examination results directly and which indirectly via effort; they also suggest that affecting effort directly has an impact on results.
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No. 5048
MUST TRY HARDER. EVALUATING
THE ROLE OF EFFORT IN
EDUCATIONAL ATTAINMENT
Gianni De Fraja, Tania Oliveira and
Luisa Zanchi
PUBLIC POLICY
ISSN 0265-8003
MUST TRY HARDER. EVALUATING
THE ROLE OF EFFORT IN
EDUCATIONAL ATTAINMENT
Gianni De Fraja, University of Leicester and CEPR
Tania Oliveira, University of Leicester
Luisa Zanchi, University of Leeds
Discussion Paper No. 5048
May 2005
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Copyright: Gianni De Fraja, Tania Oliveira and Luisa Zanchi
CEPR Discussion Paper No. 5048
May 2005
ABSTRACT
Must Try Harder. Evaluating the Role of Effort in
Educational Attainment*
This paper is based on the idea that the effort exerted by children, parents and
schools affects the outcome of the education process. We test this idea using
the National Child Development Study. Our theoretical model suggests that
the effort exerted by the three groups of agents is simultaneously determined
as a Nash equilibrium, and is therefore endogenous in the estimation of the
education production function. Our results support this, and indicate which
factors affect examination results directly and which indirectly via effort; they
also suggest that affecting effort directly has an impact on results.
JEL Classification: I220 and H420
Keywords: educational achievement, educational attainment, educational
outcomes, effort at school and examination results
Gianni De Fraja
Department of Economics
University of Leicester
Leicester
LE1 7RH
Tel: (44 116) 252 3909
Fax: (44 116) 252 2908
Email: defraja@le.ac.uk
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=113071
Tania Oliveira
Department of Economics
University of Leicester
University Road
Leicester
LE1 7RH
Email: to20@leicester.ac.uk
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=162519
Luisa Zanchi
Economics Division
Leeds University Business School
EES Building
University of Leeds
Leeds
LS2 9JT
Tel: (44 113) 233 4464
Fax: (44 113) 233 4465
Email: lz@lubs.leeds.ac.uk
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=150050
*We would like to thank Karim Abadir, Sarah Brown, Gabriele Fiorentini,
Andrea Ichino, Andrew Jones, Steve Machin, Kevin Reilly, Karl Taylor, and
the audience at the Education Department in Leicester for helpful suggestions
and comments on an earlier draft.
Submitted 13 April 2005
NON-TECHNICAL SUMMARY
This paper is based on the very simple observation that the educational
attainment of students is affected by the effort put in by those participating in
the education process: the schools attended by the students, the students'
parents, and of course the students themselves. Although psychologists and
educationalists have long acknowledged the importance of schools’, parents’
and students’ effort, the economic literature on educational achievement has
so far paid only limited attention to the role of effort as a separate input to the
education process, both at a theoretical and at an empirical level.
We build a theoretical model where educational attainment is positively
affected by students’, parents’ and schools’ effort and where the effort of these
three groups of agents is jointly determined: students respond to the effort
exerted by their parents and their schools, and, correspondingly, schools
respond to the effort exerted by their students and their parents, and parents
to the effort exerted by their children and their children's schools. All agents
have a common interest in the realisation of the best educational outcome for
the students, but the complex interactions among them may lead to
counterintuitive results. For example, students may respond to an increase in
school effort with a lower level of their own effort.
We then estimate the theoretical model using the National Child Development
Study, a very rich dataset which follows a cohort of individuals born in 1958,
from birth until the age of 42. We use information obtained by comprehensive
questionnaires completed when the individuals were 7, 11, and 16. We also
have detailed results of all examinations taken up to the age of 20. We
construct our measures of effort using many indicators of a student's, her
parents’ and her school’s attitudes. For students, for example, we use the
answers given by 16-year-olds to questions such as whether they think that
school is a ''waste of time'', and the teacher's views about the students'
laziness. Other questions regard the parents' interest in their children's
education, measured, for example, by whether they read to their children or
attend meetings with teachers, and the teacher's perception of this interest.
For schools we use variables such as the extent of parental involvement
initiated by the school, whether 16-year old students are offered career
guidance, and the type of disciplinary methods employed. We also include
many other standard explanatory variables. These can refer to individual,
family, or school characteristics, as well as geographical indicators. They
comprise the students’ ability, measured by administered tests independent of
formal examinations and taken at the ages of 7, 11 and 16, the parents’ social
class and education, and the type of school, whether state or private.
Our empirical estimates seem to confirm our theoretical prediction of joint
determination of the effort levels of the three groups of agents. Moreover, our
measures of effort seem appropriate, especially so for students and parents.
For example, we find a trade-off between the quantity of children and their
parents’ effort: a child's number of siblings affects negatively the effort exerted
by that child's parents towards that child's education. Our econometric model
allows us to determine whether explanatory variables influence educational
attainment directly, or indirectly by affecting effort. For example, our results
suggest that family socio-economic conditions influence attainment more
strongly via effort than directly. In this case, policies that attempt to stimulate
parental effort might be effective ways to improve the educational attainment.
Affecting parental effort is likely to be easier than modifying social
background.
We also find that the children's own effort has the least important effect on
educational attainment: schools and parents matter more. Interestingly, the
school's effort matters more than the parents' for girls, and, vice versa, it
matters less than the parents' for boys. This may provide an explanation for
the recent trend of improvement in educational achievement, a trend which is
stronger for girls than for boys in the UK, and which is occurring at a time
when increasing attention is paid to schools’ results, and to the provision of
financial incentives to schools and teachers. To the extent that these
incentives stimulate schools' effort, our analysis indicates that girls'
educational attainment should improve more than boys’.
1Introduction
This paper is based on a very simple idea: the educational achievement of a
student is aected by the eort put in by those participating in the educa-
tion process: schools, parents, and of course the students themselves. This is
natural, and indeed psychologists and educationalists have long been aware of
theimportanceofeort for educational attainment. Student’s eort is usually
proxied by the amount of homew ork undertaken that is unconstrained b y the
scheduling practices of the schools (Natriello and McDill (1986)). However,
empirical research in this area is still far from reaching clear conclusions. This
is due partly to ambiguities in the interpretation of homework: it could be seen
as an indicator of either students’ eort, operating at the individual level, or
teachers’ eort, operating at the class level (Trautwein and Köller (2003)). As
well as students’ eort, the educational psychology literature has also studied
the relationship between school attainment and parental eort. A variety of
dimensions of parental eort has been considered, ranging from parents’ ed-
ucational aspirations for their children, to parent-child communication about
school matters, to education-related parental supervision at home, and to par-
ents’ participation in school activities. As Fan and Chen (2001) note, much
of this literature is qualitative rather than quantitative and most of the quan-
titative studies rely on simple bivariate correlations rather than on regression
analysis. Results are not clear-cut here either: if at all, parental eort appears
to aect educational attainment only indirectly, to the extent that it supports
children’s eort (Hoover-Dempsey et al. (2001)).
The lack of specic data quantifying eort as a separate variable aect-
ing educational attainmenthas also hindered studies carried out by economists.
For example, Hanushek (1992) proxies parental eort with measures of fam-
ily socio-economic status (permanent income and parents’ education lev els).
Intuition conrmed by our results would however suggest that eort and
socio-economic conditions are in fact distinct variables. Indeed, Bec ker and
Tomes’ (1976) theoretical model of optimal parental time allocation suggests a
negative relationship between household income and parental eort.
1
Bones-
røning (1998; 2004) and Cooley (2004) are among the very few authors in the
economics literature who measure the eort exerted by students and parents
1
Theirideaisthatparentstrytomaximisethe welfare of their children, and they may
decide to allocate more time and eort to their children’s education if t hey perceive limits
to th eir ability to transfer income through inheritance; this is m o re likely to b e the case for
low-income families.
1
and estimate its eects on examination results.
Theoretical analyses of the role of eort in the education process are also
scarce.
2
Our paper attempts to ll this gap, by developing a theoretical model
of the determination as a Nash equilibrium of the eort exerted by students,
their parents and their sc hools, and subsequently by estimating empirically the
determinants of the eort levels, the interaction among of them, and the eect
of eort on educational attainment.
We test the theoretical model with the British National Child Dev elopment
Study (NCDS). This is a well suited dataset, as it contains a large number of
variables which can be used as indicators of eort: there are variables which
denote a student’s attitude, for example whether they think that school is a
“waste of time”, and the teacher’s views about the student’s laziness. Other
questions regard the parents’ interest in their c hildren’s education, whether
they read to their children or attend meetings with teachers, and the teacher’s
perception of this interest. For sc hools, we use variables such as the extent of
parental involvement initiated by the school, whether 16-year old students are
oered career guidance, and the type of disciplinary methods used.
Our empirical estimates of the determinants of eort are encouraging: the
theoretical assumption of join t interaction of the eort levels of the three groups
of agents appears to be borne out by the data. Moreover, our measures of eort
seem appropriate, especially so for children and parents. For example, as a by-
product of our analysis, we nd conrmation of Becker’s (1960) intuition that
there is a trade-o between quantity and quality of children: a child’s number
of siblings inuences the eort exerted by that child’s parents towards that
child’s education.
Our econometric model allows us to determine whether explanatory vari-
ables inuence educational attainment directly or indirectly, that is by aecting
eort. For example, our results suggest that family socio-economic conditions
aect attainment more strongly via eort than directly. In this case, policies
2
This contrasts sharply with the extensive literature which studies the role of e ort in
rms; a seminal contribution is the theory of eciency wages (Shapiro and Stiglitz (1984)),
and an extensive survey is provided by Holmstrom and Tirole (1989). There have also been
several attempts to estimate empirically the role of eort in rms: an early test of the eciency
wage hypothesis is Capp elli and Chauvin (1991), who m easured workers’ eort by disciplinary
dismissals. More recently, eort has b e en me asure d by the propensity to quit (Galizzi and
Lang (1998)), by misconduct (Ichino and M aggi (2000)) and by absenteeism (Ichino and
Riphahan (2004)). Peer pressure, measured by the presence of a co-worker in the same room,
also appe ars t o aect a worker’s eort (Falk and Ichino (2003)).
2
that attempt to aect parental eort might be eective ways to impro ve the
educational attainment, since aecting parental eort is likely to be easier than
modifying social background.
3
We also nd that the c hildren’s own eort has
the least eect on educational attainment: schools and parents are more im-
portant. Interestingly, the school’s eort matters more than the parents’ for
girls, and, vice versa, it matters less than the parents’ for boys (see Figure
4 below). This may provide an explanation for the recent trend of improve-
ment in the education achievement in the UK, a trend which is stronger for
girls than for boys, and which is happening when increasing attention is being
paid to schools’ results, and nancial incentives are being provided to sch ools
and teachers. To the extent that these incentives stimulate the schools’ eort,
our results indicate that girls’ education attainment should improve more than
boys.
The paper is organised as follows: the theoretical model is developed in
Section 2; the agents’ strategic behaviour is illustrated in Section 3 with a
graphical analysis of the Nash equilibrium; the empirical model is presented in
Section 4; Section 5 describes the data and the variables used; our results are
summarised in Section 6, and concluding remarks are in the last section.
2 Theoretical Model
We model the in teraction among the pupils at a school, their teachers and
their parents. Pupils attend school, and, at the appropriate age, they leave
with a qualication. This is a variable q taking one of m possible values
q {q
1
,...,q
m
},withq
k1
<q
k
, k =2,...,m. Other things equal, a stu-
dent prefers a better qualication: apart from personal satisfaction, there is
substantial evidence showing a positive association between qualication and
future earnings in the labour market: let u (q) be the utility associated with
qualication q,withu
0
(q) > 0.
When at school, pupils exert eort, which w e denote by e
C
E
C
IR (the
superscript C stands for “child”). The restriction to single dimensionality is
made for algebraic convenience, though it is also supported by the data, see
Section 6 below . e
C
measures how diligent a pupil is, how hard she works and
so on, and has a utility cost measured b y a function ψ
C
¡
e
C
¢
, increasing and
3
One exam ple could be the provision of direct nancial rewards to parents helping their
children with homework, or attending p arenting classes, similarly to the policy of providing -
nancial incentives to d isadvantaged teen-age rs for staying on at school b eyond the compulsory
age (Dearden et a l. (2003)).
3
convex: ψ
0
C
¡
e
C
¢
, ψ
00
C
¡
e
C
¢
> 0. Notice that there is no natural scale to measure
eort, and so the interpretation of the function ψ
C
(and the corresponding ones
for schools and parents), is cost of eort relative to the benetofqualication.
Pupils also dier in ability, denoted by a. A student’s education attainment is
aected by her eort and her ability. Formally, we assume that qualication q
k
is obtained with probability π
k
¡
e
C
,a; ·
¢
(the · represents other inuences on
qualication, discussed in what follow s). We hypothesise, naturally, a positive
relationship between eort and the expected qualication:
P
m
k=1
∂π
k
(
e
C
,a;·
)
e
C
q
k
>
0, and between ability and the expected qualication:
P
m
k=1
∂π
k
(
e
C
,a;·
)
a
q
k
> 0.
A student’s objective function is the maximisation of the dierence between
expected utility and the cost of eort:
m
X
k=1
π
k
¡
e
C
,a; ·
¢
u (q
k
) ψ
C
¡
e
C
¢
.(1)
A studen t’s education attainment depends also on her parents’ eort. Par-
ents may help with the homework, provide educational experiences (such as
museums instead of television), take time to speak to their children’s teachers,
and so on: we denote this eort by e
P
E
P
IR; as before, this is treated as
single dimensional. Consistently with common sense, and with the idea that
the education process is best thought of as a long term process (e.g. Hanushek
(1986) and Carneiro and Heckman (2003)), the variable e
P
should be viewed
as summarising the inuence of parental eort throughout the child’s school
career: the NCDS dataset is well suited to take on board this view, as each
subject is observed at three dates, at age 7, at age 11 and at age 16. Par-
en ts dier also in education, social background and other variable which aect
their children’s education attainment; we capture this by means of a possibly
multidimensional variable, s
P
.
Parents care about their children’s qualication, and so t hey will exert eort
e
P
, even though it carries a utility cost, measured by the function ψ
P
¡
e
P
¢
,
increasing and convex: ψ
0
P
¡
e
P
¢
, ψ
00
P
¡
e
P
¢
> 0. Parents may have more than
one c hild and so they care about the average qualication of all their children:
4
4
Rigorously, we should consider the utility of the qualication, for example u
P
(q).Itisnot
in general obvious which shape the function u
P
(q) should have: some parents may obtain a
higher utility gain if the qualication of a less b right child is increased, than if the qualication
of a more able child is increased equivalently; other parents, who value achieving excellence
more than avoinding failure may take an opposite view; given this p otential ambiguity, it
seems a good approximation to take the average attainment as the objective fu nction .
4
if parents have n children, their payo function is given by:
n
X
j=1
π
k
¡
e
C
j
,a
j
; e
P
j
,s
P
; ·
¢
q
k
ψ
P
³
P
n
j=1
e
P
j
´
,
where e
P
j
is the eort devoted by parents to child j, whose ability is a
j
,and
who exerts eort e
C
j
.
5
A student’s qualication will also be aected by the quality of her school,
the last component of the · in the arguments of the probabilities in (1). The
sc hool inuences its pupils’ attainment through its own eort, measured by a
variable e
S
E
S
IR (again assumed one-dimensional). This captures the
idea that a school can take actions which aect the quality of the education it
imparts. Improving the quality of buildings, classroom equipment and sporting
facilities, using computers appropriately, upgrading teachers’ qualications are
all examples. Other examples are th e teachers’ interest and enthusiasm in their
classroom activities, the time they spend outside teaching hours to prepare
lessons, to assess the students’ work, to meet parents, and so on.
6
Eort has
increasing marginal disutility, and can thus be measured by a function ψ
S
¡
e
S
¢
increasing and convex, ψ
0
S
¡
e
S
¢
, ψ
00
S
¡
e
S
¢
> 0.
To wrap up this discussion, the probability that a student obtains quali-
cation q
k
can therefore be written as
π
k
¡
e
C
,a; e
P
,s
P
; e
S
,s
S
¢
,
where, in analogy to s
P
, s
S
is a vector which captures the school’s exogenously
given characteristics. A school’s objective function is a function which depends
positively on the average
7
qualication of its students and negatively on the
5
The interaction between parental eort and the number o f children was rst proposed by
Becker (1960). We ignore the p otential endogeneity of the number of children. Blake (1989)
is a demog ra phic analys is o f the relationship between family size and achievement.
6
Note that the activities in the rst group are xed b efore the students are enrolled at
school and can therefore be observed by parents prior to applying to the scho ol; while those
in the second g roup are carried out once the students are at school. Since the extent of
school choice was fairly limited in the period covered by our data, this distinction w ill b e
disregarded in what follows. The theoretical analysis of De Fraja and Landeras (2005) argues
that a dierent equilibrium concept should be used according to whether schools and students
choose one after the other or simultaneously: Stacke lberg and Nash equilibrium respectively.
Astheyshow,thisdoesnotaect the qu alitative nature of the interaction.
7
As with parents, the average qualication may not be the most suitable approximation
for t he scho ol’s objective function. Teachers m ay care more abo ut the best or the weakest
students in their class. If this were the case, app rop riate weighting could be included to
account for these biases in the school’s payo fun ction (2).
5
teac hing eort:
m
X
k=1
q
k
H
X
h=1
π
k
¡
e
C
(h) ,a; e
P
(h) ,s
P
; e
S
(h) ,s
S
¢
λ
h
ψ
S
¡
e
S
¢
.(2)
(2) assumes that the eort levels e
C
, e
P
,ande
S
are aected by a n umber of
exogenous variables described by the multi-dimensional vector h:thuse
C
(h)
(respectively e
P
(h), respectively e
S
(h))istheeort level exerted by students
(respectively parents, respectively schools) whose vector of relevant variables
takes value h. h will of course also include ability and other variables which are
also in the vectors s
P
and s
S
, as these can have a direct eect on qualication,
or an indirect eect, via the eort lev el exerted by the participants in the
education process. H is the number of all the possible values which the variables
aecting eort can take, and λ
h
is the proportion of pupils at the school with
this variable equal to h.
Additivity between the disutility of eort and the students’ average qual-
ication is an innocuous normalisation. The relative importance of these two
components of the school’s utility will in general depend on how much teachers
care about the success of their pupils, which in turn can depend on go vernment
policy: there could be incen tives for successful teachers (both monetary and in
terms of improved career prospects; De Fraja and Landeras’s (2005) theoret-
ical model studies the eects of strengthening these incentives). The dataset
we have available, which refers to schools in the late ’60s and early ’70s is not
suited to the study of these eects, since there has been no observable change
in the power of the incentive schemes for schools and teachers in that period.
3 A grap h ic a l ana ly sis of the equ ilibrium
All agents have a common interest in the realisation of a high qualication
for the child, but their interests are not perfectly aligned, and their strategic
behaviour ma y lead to complex interactions among them, with counterin tuitive
outcomes.
In this brief section we illustrate this point in an extremely simple case.
We assume that all students in a given school are identical in terms of ability,
parental status, and number of siblings. This is obviously unrealistic, but
the point here is to illustrate that, ev en with highly special assumptions, the
interaction between the parties may turn out to be extremely complex. We
capture this interaction with the game theoretic concept of Nash equilibrium:
6
eac h party is choosing their eort in order to maximise their utility, taking as
given the choice of eort of the other parties. An equilibrium is given by the
set of values e
C
, e
P
,ande
S
, satisfying the rst order conditions
m
X
k=1
u (q
k
)
∂π
k
¡
e
C
,a; e
P
,s
P
; e
S
,s
S
¢
e
C
ψ
0
C
¡
e
C
¢
=0,(3)
m
X
k=1
q
k
∂π
k
¡
e
C
,a; e
P
,s
P
; e
S
,s
S
¢
e
P
ψ
0
P
¡
e
P
¢
=0,(4)
m
X
k=1
q
k
∂π
k
¡
e
C
,a; e
P
,s
P
; e
S
,s
S
¢
e
S
ψ
0
S
¡
e
S
¢
=0.(5)
where the appropriate second order conditions are satised. (3)-(5) are the
best reply function
8
of each of the three agents: their intersections identify the
Nash equilibria. The graphical analysis is best conducted in two dimensions.
Letthereforetheparentaleort be xed, at e
P
. Total dierentiation of (3) and
(5) gives the slope of the best reply function in the relevant Cartesian diagram
(E
C
× E
S
for xed e
P
):
Ã
m
X
k=1
u (q
k
)
2
π
k
(·)
e
C
e
S
!
de
S
U
00
C
(·) de
C
=0,
Ã
m
X
k=1
q
k
2
π
k
(·)
e
C
e
S
!
de
C
U
00
S
(·) de
S
=0,
where U
00
C
(·)=
P
m
k=1
u (q
k
)
2
π
k
(·)
(e
C
)
2
ψ
00
C
¡
e
C
¢
< 0 is the second derivative of
the child’s payo, and analogously for U
00
S
(·).Fromtheabove:
de
S
de
C
¯
¯
¯
¯
child
BRF
=
U
00
C
(·)
P
m
k=1
u (q
k
)
2
π
k
(·)
e
C
e
S
,(6)
de
S
de
C
¯
¯
¯
¯
school
BRF
=
P
m
k=1
q
k
2
π
k
(·)
e
C
e
S
U
00
S
(·)
.(7)
8
Mathematically, for the representative student (that the we can take a representative
student is show n in De Fraja and Landeras (2005)), this is a function from the product of the
other two eort spaces into the child’s: E
P
× E
S
−→ E
C
. Thisadimension2-manifoldinthe
3-dimensional C artesian space E
C
× E
P
× E
S
. Analogously for the parents and the school.
The intersection of three dimension 2-manifolds is (generically) either em pty, or a dimension
0-manifold, that is a set of isolated points. Existence of at least one Nash equilibrium is
ensured by the fact that each player has a compact and convex strategy space, and that
their payo functions are continuous and quasi-concave in their own strategy (Fudenberg and
Tirole 1991, p 34).
7
e
school’s best
reply function
student’s best
reply function
panel (a) panel (b)
C
e
S
e
school’s best
reply function
student’s best
reply function
C
e
S
E
0
E
1
E
0
E
1
Figure 1: Best reply functions of the representative student and of the school.
Both can have either sign:
9
to see what this implies, consider Figure 1. It
illustrates the best reply functions for the studen t and the school. In panel (a),
the case is depicted where both (6) and (7) are positive at their intersection.
The solid lines are the best reply functions associated with the parameter vector
h taking value h
0
. The dashed lines depict the best reply functions associated
to a dierent set of exogenous variables, say h
1
, associated with a higher value
of the students’ eort, for every given level of the school’s eort, and a higher
value of the school’s eort, for every given level of the students’ eort lev els. For
example, the dashed lines may represent the best reply functions of students
and the school for a student with higher ability and a larger school (the data
suggests that these comparative statics changes are associated to higher eort
levels). Graphically, this is a shift upward (for the school) and eastward (for
the student) of the best reply function. In panel (a) both equilibrium eort
levels are higher: compare E
0
with E
1
.
Consider panel (b), however. It diers from panel (a) only in that the
best reply functions meet at a point where the student best reply function is
negatively sloped. In the case depicted in panel (b), the equilibrium eort levels
that result as a consequence of a dierent value in the exogenous parameters
vector h, associated with higher eort levels results in a lower equilibrium eort
exerted by the student. This is so even though the studen t ’s best reply function
9
Note instead that, in this special case of one student per scho ol, the parents and the
scho ol’s best reply function have the sam e sign.
8
shifts eastward: h
1
is associated to higher values in the studen t’s eort for any
given level of the school’s eort. The reason for the lower equilibrium value
of the student’s eort is the strategic interaction of schools and students. The
vector h
1
would be associated to a higher value of the student’s eort if the
school’s eort were the same. However, the student’s and the school’s eorts
are “strategic substitutes” (Bulow et al 1986), and the student responds to the
higher school eort (associated to the vector h
1
) with a lower level of their
own eort. This, in panel (b) in the diagram, more than compensates the
direct increase in the student’s eort caused by the dierent value of h.This
simple example illustrates the poten tial ambiguity of changes in the exogenous
variables h on the equilibrium eort levels; in more general settings the situation
will be ev en more complex.
4EmpiricalModel
Given this theoretical ambiguity, the overall eect of children’s, parents’ and
sc hool’s eorts on educational attainment, and whether these eort levels are
strategic complements or substitutes, is therefore largely an empirical matter,
towhichweturninthissection.
The educational outcome variable considered here, Q
i
,ischildi’s academic
results over a number of secondary school examinations, normally taken be-
tween the ages of 16 and 18. The explanatory variables are measures of the
eort exerted by the child, her parents and her school, and a suitable set of
controls for heterogeneity in socio-economic, demographic and other relevant
factors. Formally, the academic achievement is specied as:
Q
i
= x
Q0
i
β
1
+ β
2
e
C
i
+ β
3
e
P
i
+ β
4
e
S
i
+ u
i
,i=1,...,n,(8)
where x
Q
i
is a set of control variables for demographic and socio-economic
bac kground factors aecting the educational outcome, e
C
i
, e
P
i
,ande
S
i
are the
measures of the eort exerted by child i,bychildi’s parents and by child i’s
sc hool, and u
i
the error term.
Our theoretical analysis in Sections 2 and 3 suggests that the interaction
between the three types of agents is best captured as a Nash equilibrium. This
implies that the eort levels, which are educational inputs, simultaneously de-
termine each other; together with possible omitted variables, this in turn im-
plies that the error term in the estimation of a standard educational production
function (Hanushek (1986) is correlated with the observed input variables and
9
the estimates of the eect of observed inputs on educational outcome are incon-
sistent. The very ric h set of bac kground variables in our dataset should lessen
the problem of omitted variables.
To address the endogeneity of the eort variables, note that the interde-
pendent system:
e
C
i
= x
C0
i
γ
C
1
+ γ
C
2
e
P
i
+ γ
C
3
e
S
i
+ v
C
i
, i =1,...,n,(9)
e
P
i
= x
P 0
i
γ
P
1
+ γ
P
2
e
C
i
+ γ
P
3
e
S
i
+ v
P
i
, i =1,...,n, (10)
e
S
i
= x
S0
i
γ
S
1
+ γ
S
2
e
C
i
+ γ
S
3
e
P
i
+ v
S
i
, i =1,...,n, (11)
is a linear approximation to the Nash equilibrium. In (9)-(11), x
C
i
, x
P
i
and x
S
i
are the background factors aecting child i’s eort, child i’s parents’ eort, and
the eort of child i’s school, respectively, and v
C
i
, v
P
i
and v
S
i
are error terms,
possibly correlated.
The NCDS dataset contains many variables that capture aspects of indi-
vidual eort levels, e
C
i
, e
P
i
and e
S
i
. Described in detail in Section 5, these take
the form of categorical variables, which have dierent scales and are in gen-
eral non-comparable. We therefore use factor analysis
10
to construct a single
aggregate continuous indicator of the three eort levels.
We next need to ascertain whether the eort variables are endogenous as
suggested in Sections 2 and 3. We do so u sing the Durbin-Wu-Hausman (DWH)
augmented regression test suggested by Davidson and MacKinnon (1993). The
test is performed by obtaining the residuals from a model of each endogenous
right-hand side variable as a function of all exogenous variables, and including
these residuals in a regression of the original model. In our case, we rst
estimate the system
e
C
i
=
e
x
C0
i
δ
C
1
+ δ
C
2
e
P
i
+ δ
C
3
e
S
i
+ r
C
i
, (12)
e
P
i
=
e
x
P 0
i
δ
P
1
+ δ
P
2
e
C
i
+ δ
P
3
e
S
i
+ r
P
i
, (13)
e
S
i
=
e
x
S0
i
δ
S
1
+ δ
S
2
e
C
i
+ δ
S
3
e
P
i
+ r
S
i
, (14)
10
We use the p rincipal factor method. Alternative a pproaches include principal comp o -
nents, principal-components factor analysis and m aximum-likeliho od factor analysis (Harman
(1976), Everitt and Dunn (2001)). Since our original variables are de nedonanordinalrather
than an interval scale, they are not suited to being analysed by the maximum-likelihood fac-
tor metho d, due to the assumption of normality implied by this p roced ure. We have instead
exp erimented using principal components as an alternative to the principal factor method.
The dierence in the results prov ided by the two methods is only of order 10
3
at most. Our
results indicate that retaining only the rst factor is the appropriate strategy for the children’s
and the p a rents’ eort; a second factor should perhaps be retained for the school’s eort, but,
for symmetry and ease of interpretation, we retain only the rst factor for the school as well.
10
where r
C
i
, r
P
i
and r
S
i
are error terms and the vectors
e
x
C
i
,
e
x
P
i
and
e
x
S
i
,arethe
union of the set of variables which form the vectors x
C
i
, x
P
i
and x
S
i
in equations
(9)-(11), with the variables which form the vector x
Q
i
in equation (8) (for
example,
e
x
C
i
are background factors aecting either educational attainment,
or the child’s eort, or both; and similarly for
e
x
P
i
and
e
x
S
i
). We then estimate
the following augmen ted regression:
Q
i
= x
Q0
i
η
1
+ η
2
e
C
i
+ η
3
e
P
i
+ η
4
e
S
i
+ η
5
br
C
i
+ η
6
br
P
i
+ η
7
br
S
i
+ eu
i
, (15)
where br
C
i
, br
P
i
,andbr
S
i
are the residuals obtained from the estimates of (12 )-(14).
According to Davidson and MacKinnon (1993), if the parameters η
5
, η
6
and η
7
are signicantly dierent from zero, then OLS estimates of equation (8) are not
consistent due to the endogeneity of e
C
i
, e
P
i
and e
S
i
. Wetestthenullhypothesis
η
5
= η
6
= η
7
=0applying a likelihood-ratio test and, as we show below in
Section 6, we nd endogeneity of the eort variables.
The estimation method we use is 3SLS, because of the interdependent na-
ture of the eort variables, and the possible dependence of the error terms
across equations. Ideally, the four equations (8)-(11) should be estimated si-
multaneously. However, the dependent variable in equation (16) is discrete,
and cannot therefore be estimated with standard 3SLS methods. We therefore
estimate the educational attainmen t equation (16) using the predicted values
be
C
i
, be
P
i
and be
S
i
obtained from a three-stage least squares estimation of equations
(9)-(11) instead of the three original eort variables:
Q
i
= x
A0
i
β
1
+ β
2
be
C
i
+ β
3
be
P
i
+ β
4
be
S
i
+ u
i
,i=1,...,n. (16)
Equation (16) is estimated as an ordered probit as the examination results
variable Q
i
is a discrete ordered variable, taking eleven possible values. Iden-
tication is achieved by the inclusion in the sets x
U
i
, U = Q, C, P, S,ofsome
statistically signicant variables unique to each of the four equations (8)-(11).
5 Data and variables
The National Child Development Study (NCDS)
11
follows the cohort of indi-
viduals born between the 3rd and the 9th of Marc h 1958, from birth until the
age of 42. We use inform ation obtained by detailed questionnaires when the
11
This dataset is widely used (see www.cls.ioe.ac.uk/Cohort/Ncds/Publications/nwpi.htm).
For a discussion of its features, including ways of dealing with non-response and attrition
problems, see Micklewright (1989) and Connolly et al.(1992).
11
individuals w ere 7, 11, and 16. We also use data from the Public Examinations
Survey, also a part of the NCDS, which gives detailed results of examinations
taken until the age of 20. The dataset contains examination results for 7017
girls and 7314 boys; after eliminating observations with insucient information
we were left with a sample of 5611 girls and 5860 boys.
5.1 Dependen t variables
5.1.1 Eort
Table 1 contains the scoring coecients for the child’s, the parents’ and the
sc hool’s eort indicators obtained from the factor analyses performed separately
for boys and for girls. The scoring factors are the weights actually entering the
construction of the eort indicators. To reduce the loss of information due to
non-response, we impute the factor scores for the cases with missing data in
some of the originally observ ed variables on which the indices are based. The
imputation method is such that the new variable created includes predictions
for the missing values based on the best available subset of otherwise present
data. We ha ve imputed 7%, 13.1% and 6.9% of the c hild’s, the parents’ and
the sc hool’s eort information, respectively.
The child’s eort indicators used to construct the child’s eort measure e
C
i
are the c hild’s answers (at age 16) to questions about her attitude towards
school, wishes and expectations about school leaving age, and the frequency of
reading (a higher value denotes higher eort).
12
This information is comple-
mented by the teacher’s assessment of the child’s eort when the individuals
are 16 (the last row in the rst part of Table 1). Co lum ns 3 and 5 of Table
1 provide the scoring coecients for each of the variables reecting the eort
indicators, namely the weights actually entering the construction of the eort
indices. For the children the variable with the highest scoring coecient is
whether the child likes school or not.
13
The parents’ eort measure e
P
i
is produced using both parents’ interest in
the child, their initiativ e to discuss the childs progress in school, the fathers
role in the management of the child, the parents’ wishes and anxiety over the
child’s school achievement, and how often both parents read to their children.
12
The exact description of how we h ave constructed these and all the other variables is
in an App endix available on request or at www.le.ac.uk/economics/gdf4/curres.htm. This
appe ndix also reports the factor loadings.
13
The proportion of explained variance is ap proximately 1. This provides very stron g
evidence that selecting only one factor is the most appropriate decision.
12
Child's effort
Variable Range Mean
Scoring
Coefficient
Mean
Scoring
Coefficient
School is not a waste of time 0-4 3.3066 0.1502 3.1523 0.1517
I
g
et on with classwork 0-4 2.4068 0.0923 2.2482 0.0997
Homework is not borin
g
0-4 1.6646 0.1354 1.4648 0.1387
It is not difficult to keep m
y
mind on work 0-4 2.2991 0.1259 2.2444 0.1119
I take work seriousl
0-4 3.1059 0.1766 3.0291 0.1650
I like school 0-4 2.5038 0.1994 2.3741 0.2109
There is a point in plannin
g
for future 0-4 3.0085 0.0666 3.0913 0.0533
I am alwa
y
s read
y
to help m
y
teache
r
0-4 2.6583 0.0524 2.3195 0.0478
I often read in m
y
spare time 0-3 2.0848 0.0448 1.8286 0.0543
A
g
e I am likel
y
to leave school 0-3 0.9243 0.1186 0.8868 0.1266
I wish I could leave school at 15 0-2 1.4158 0.1798 1.2814 0.1805
Teacher thinks child is laz
y
or hardworkin
g
0-4 2.4132 0.1288 2.0627 0.1266
Proportion of explained variance
Parents' effort
Variable Range Mean
Scoring
Coefficient
Mean
Scoring
Coefficient
Mother's interest in child's education at a
g
e 7 0-4 2.9828 0.1227 2.9042 0.1203
Father's interest in child's education at a
g
e 7 0-4 1.9504 0.0806 2.0035 0.1011
Parents' initiative to discuss child with teacher at a
g
e 11 0-3 1.0442 0.1326 1.1031 0.1323
Fathers' interest in child's education at a
g
e 11 0-4 2.2679 0.1510 2.3271 0.1574
Mothers' interest in child's education at a
g
e 11 0-4 2.8462 0.1389 2.7436 0.1239
Father's interest in child's education at a
g
e 16 0-4 2.4919 0.2253 2.5030 0.2362
Mother's interest in child's education at a
g
e 16 0-4 2.7763 0.2049 2.6552 0.1903
Parental hopes about child's school leavin
g
a
g
e at a
g
e 11 0-2 1.6942 0.1007 1.7112 0.1162
Parents want further education for child at a
g
e 11 0-2 1.7664 0.0760 1.8210 0.0733
Father's role in mana
g
ement of child at a
g
e 11 0-3 2.3685 0.0420 2.4470 0.0419
Mother reads to child at a
g
e 7 0-3 2.3150 0.0664 2.2961 0.0812
Father reads to child at a
g
e 7 0-3 2.0037 0.0891 1.9788 0.0963
Father's role in mana
g
ement of child at a
g
e 7 0-3 2.3701 0.0488 2.4172 0.0535
Parent's initiative to discuss child with teacher at a
g
e 7 0-1 0.5596 0.0834 0.5705 0.0833
Substantial help from parents to school at a
g
e 7 0-1 0.5191 0.0303 0.5229 0.0251
Parents and teacher discuss child at a
g
e 16 0-3 1.0724 0.1110 1.1680 0.1170
Parent's anxiet
y
over child's school achievement at a
g
e 16 0-4 3.2070 0.0210 2.9779 0.0060
Parents wish child
g
oes to hi
g
her education at a
g
e 16 0-1 0.3219 0.0974 0.3286 0.0870
Proportion of explained variance
School's effort
Variable Range Mean
Scoring
Coefficient
Mean
Scoring
Coefficient
Teachers' initiative to discuss child at a
g
e 11 0-1 0.4274 -0.0112 0.4317 0.0598
Child's a
g
e
g
roup streamed b
y
abilit
y
at a
g
e 11 0-1 0.3140 0.0073 0.3347 0.0019
Paid hours of career
g
uidance 0-1 0.8832 0.0613 0.8774 0.0712
Parent-teacher meetin
g
s to discuss child at a
g
e 16 0-3 2.0272 0.0495 2.0139 0.1174
Parents are shown teachin
g
methods at a
g
e 16 0-3 0.5648 0.0781 0.5891 0.1318
School has parent-teacher association at a
g
e 16 0-1 0.6259 0.0748 0.6445 0.1867
Disciplinar
y
methods-suspension at a
g
e 16 0-2 0.9427 0.1364 0.9978 0.1067
Disciplinar
y
methods-corporal punishment at a
g
e 16 0-2 1.0074 0.1053 1.3376 -0.0407
Disciplinar
y
methods-ph
y
sical/manual activities at a
g
e 16 0-2 0.3679 0.1293 0.4789 0.0999
Disciplinar
y
methods-extra school work at a
g
e 16 0-2 1.6247 0.1995 1.6975 0.0920
Disciplinar
y
methods-detention at a
g
e 16 0-2 1.4329 0.1445 1.4665 0.2313
Disciplinar
y
methods-loss special status at a
g
e 16 0-2 0.5990 0.1127 0.6553 0.0072
Disciplinar
y
methods-exclusion from activities at a
g
e 16 0-2 0.8040 0.1565 0.8360 0.0684
Disciplinar
y
methods-report to parents at a
g
e 16 0-2 1.9212 0.1538 1.9079 0.1414
Disciplinar
y
methods-special reports at a
g
e 16 0-2 1.6275 0.2342 1.6822 0.2354
Streamin
g
in En
g
lish at a
g
e 16 0-1 0.7270 0.1604 0.7408 0.0859
Streamin
g
in maths at a
g
e 16 0-1 0.8652 0.1369 0.8527 0.1596
School has parent-teacher association at a
g
e 7 0-1 0.1667 -0.0066 0.1670 0.0531
Educational meetin
g
s arran
g
ed for PTA at a
g
e 7 0-1 0.5963 -0.0103 0.5943 0.0800
Social functions arran
g
ed for parents at a
g
e 7 0-1 0.5048 -0.0100 0.5068 0.0494
Teachers' initiative to discuss child at a
g
e 7 0-1 0.2319 0.0122 0.2691 0.0355
Proportion of explained variance
Girls Boys
0.4865
1.0210
Boys
0.6020
Boys
1.0332
0.4772
Girls
Girls
0.6265
Table 1: Factor analysis for eort measures
13
Figure 2: Density of eort for child, parents, and schools
From the second part of Table 1, we nd that the parents’ interest in the child’s
education at dierent points in time is the most salient con tributor, while on
the other side of the spectrum, reading to the child and the father’s role in the
management of the child seem to contribute least to the index.
Our measure of the sch ool’s eort, e
S
i
, is constructed (see the third part of
Table 1) from information on the extent of activities which school and teachers
are not statutorily required to perform, for example, whether teachers take the
initiative to discuss a student’s progress with his or her parents, the presence
of a parent-teacher association in the school, whether students receive career
guidance in the school, and the practice of grouping children of similar ability
(streaming) in the school. We also include information on disciplinary methods
used, on the grounds that activities such as detention or additional homework
require also additional work on the teachers’ part. Note also that the child’s
and the parents’ eort indices are constructed with similar scoring coecients
for both boys and girls. This is not so for the school’s eort analysis. For girls,
the variables with the bigger weight are the disciplinary ones, while for boys,
parents’ and teachers’ activities are also important. This may well reect a
dierence in the schools’ behaviour towards bo y s and girls in this period. The
construction of the school’s eort index is less satisfactory than for parents and
children, as sho wn by the smaller proportion of explained variance. Figure 2
illustrates the density of the eort variables we have constructed.
14
5.1.2 Examinations
Q10
Q9
Q8
Q7
Q6
Q5
Q4
Q3
Q2
Q1
Q0
0 0.05 0.1 0.15 0.2 0.25 0.3
Girls Boys
Q10 Three or more A-levels at 9 or 10 points, or three or more Scottish Highers at 9 or more points
Q9 Two A-levels at 9 or 10 points or three A-levels at 8 points or less, or three Scottish Highers at 8 points or less
Q8 Two A-levels or Scottish Highers at 8 points or less
Q7 One A-level or Scottish Higher
Q6 Seven or more GCE O-levels or CSE at grade 1, or seven or more Scottish O-levels
Q5 Five or six GCE O-levels or CSE at grade 1, or five or six Scottish O-levels at A-C grade
Q4 One to four GCE O-levels or CSE at grade 1, or one to four Scottish O-levels at A-C grade
Q3 Five or more CSE at grade 2-5, or five or six or more Scottish O-levels
Q2 One or more CSE at grade 2-3, or three or four Scottish O-levels
Q1 One or more O-levels or CSE at grade 4-5, or one or two Scottish O-levels
Q0 No formal qualification
Figure 3: Frequency of examination results
As well as an extremely detailed list of all the examinations taken b y eac h
student (obtained in 1978 by writing to schools), the dataset also includes a
summary measure of the examination performance. This was created paying
special attention to particular problems such as time and place constraints,
grade equivalence, retakes and double entries (Steedman (1983a ;1983b); see
Galindo-Rueda and Vignoles, 2003, p 10 for a more detailed discussion of the
British education system in the early 70s). We have taken this measure modi-
fying it only slightly, to allow inclusion in the sample of the Scottish students.
14
The educational outcome Q
i
in equation (16) is a categorical variable ranging
14
We put together, in “Q9”, observations of “Two A-levels at 9 or 10 points” and “Three
A-levels at 8 points or less”; there are only 27 observations of the former. Sim ilarly, we have
put together, in “Q1”, “One or more O-levels at grade 4-5” and “One or more CSE at grade
4-5”; there are 70 o bservations of the forme r.
15
from 0, indicating no formal qualication, to 10, reecting 3 or more A-levels
at9to10points. Figure3showsthedistribution of examination results for
boys and girls in the samples used. The proportion of boys that have at least
one A-level result is slightly higher, 17.37 against 16.66 for girls. The mode of
both distributions is “up to four O-levels or CSE with grade 1”.
5.2 Explanatory variables
The summary statistics for the bac kground explanatory variables are reported
in Table 2: individual characteristics rst, then family characteristics, followed
by school, peer group and geographical variables.
The main individual characteristic is ability. This is measured at ages 7, 11,
and 16 by administered tests that are independent of educational qualications.
At 7 there is information on arithmetic and reading scores, at 11 and 16 the
individuals were tested on their reading and mathematical ability, and at 11
they also completed a general ability test. Following the literature on cognitive
ability and students’ attainment, we combine the tests undertaken at the dif-
feren t points in time and on dierent subjects using the principal components
method (see, for example, Galindo-Rueda and Vignoles (2003)). We include
birth weight following some of the literature on lifetime attainments (Conley
et al. (2003); Fry er and Levitt (2002)).
The vector of family background variables includes the number of older
and younger brothers and sisters, and indicators of the mother’s position in the
labour mark et. P arental income is measured when the individuals were 16, and
the household socio-economic status is measured by the father’s (or the father
gure’s) social class at age 11.
15
We have also included the percentage of total
income not earned by the father gure, and whether the household’s accom-
modation is owned by the household. Other variables are parental education
attainment and the frequency of reading of both parents, as distinct from the
variable measuring the frequency of parents reading to their children, which
enters the measure of parental eort.
The school characteristics we use are its size, measured by the log of the
number of pupils, and its type: state or private at ages 7, 11, and 16, and
single-sex, comprehensive, secondary modern or grammar at age 16. We also
include class size, and to capture possible non-linearities in class and school
15
We m anipulated all income information using the pro ce dure develop ed for this dataset
(Micklewright (1986)).
16
Variable Mean Std. Dev. Mean Std. Dev.
Exam result 3.716 2.771 3.542 2.918 - - - -
Child's effort -0.063 0.913 -0.075 0.918 - * * *
Parents' effort -0.125 0.891 -0.127 0.887 * - * *
School's effort -0.015 0.796 -0.026 0.793 * * - *
Ability -0.132 2.243 -0.147 2.259 * * *
0.006 0.006 * * *
Weight at birth 104.763 37.049 108.448 39.713 *
0.089 0.097 *
Older brothers 0.489 0.483 *
0.209 0.222 *
Younger brothers 0.513 0.504 *
0.212 0.223 *
Older sisters 0.447 0.449 *
0.211 0.222 *
Younger sisters 0.478 0.476 *
0.212 0.224 *
Mother in work age 16 0.512 0.513 *
0.215 0.222 *
Mother in work age 7 0.251 0.235 *
0.137 0.147 *
Mother married age 0 0.903 0.907 * *
0.063 0.063 * *
Intermediate
§§
0.159 0.144 * * *
Skilled non-manual
§§
0.079 0.080 * * *
Skilled manual
§§
0.346 0.345 * * *
Semiskilled non-manual
§§
0.018 0.017 * * *
Semiskilled manual
§§
0.127 0.125 * * *
Unskilled
§§
0.046 0.051 * * *
0.178 0.185 * * *
House owner 0.403 0.394 * *
0.202 0.213 * *
Total household income 32.031 27.038 31.399 26.494 * *
0.286 0.293 * *
% of income not from father 0.290 0.336 0.289 0.334 * *
Father has higher education 0.075 0.077 * *
Father has secondary education 0.257 0.245 * *
0.230 0.237 * *
Mother has higher education 0.055 0.046 * *
Mother has secondary education 0.363 0.359 * *
0.213 0.226 * *
Father reads books regularly 0.427 0.423 * *
Father reads books occasionally 0.169 0.166 * *
0.134 0.140 * *
Mother reads books regularly 0.301 0.291 * *
Mother reads books occasionally 0.188 0.185 * *
0.135 0.141 * *
English class size age 16 24.710 7.947 24.043 8.050 *
(English class size age 16)
2
673.728 321.876 642.881 316.650 *
0.050 0.051 *
Maths class size age 16 23.832 8.373 23.765 8.207 *
(Maths class size age 16)
2
638.054 332.037 632.104 326.054 *
0.056 0.052 *
No. children in child's present class age 7 31.254 13.309 30.700 13.688 *
(No. children in child's present class age 7)
2
1153.894 610.691 1129.817 624.551 *
0.116 0.125 *
No. children in child's present class age 11 29.129 14.278 28.748 14.443 *
(No. children in child's present class age 11)
2
1052.319 625.554 1035.040 651.123 *
0.157 0.159 *
Girls Boys
Child's
effort
equation
Parents'
effort
equation
School's
effort
equation
Exam
result
equation
17
Log of school size age 16 6.554 1.098 6.578 1.023 * *
0.021 0.016 * *
Log of school size age 11 4.773 2.175 4.755 2.179 * *
0.163 0.164 * *
Log of school size age 7 4.650 1.994 4.615 2.010 * *
0.143 0.147 * *
Private school age 11 0.032 0.034 * *
0.135 0.139 * *
Private school age 7 0.029 0.026 * *
0.113 0.122 * *
Grammar school age 16 0.123 0.098 * *
Private school age 16 0.034 0.040 * *
Secondary modern age 16 0.204 0.205 * *
0.000 0.000 * *
Single sex school age 16 0.262 0.235 * *
0.012 0.010 * *
Pupils from school go to university 0.534 0.542 * *
0.163 0.146 * *
% of girls studying for O-levels 24.958 33.225 13.895 26.088 * *
0.065 0.295 * *
% of boys studying for O-levels 14.119 26.478 24.928 33.994 * *
0.313 0.063 * *
10%-19%
§
0.174 0.163 * *
20%-29%
§
0.174 0.170 * *
30%-39%
§
0.109 0.123 * *
40%-49%
§
0.069 0.079 * *
50%-59%
§
0.075 0.071 * *
60%-69%
§
0.062 0.057 * *
70%-79%
§
0.027 0.035 * *
80%-100%
§
0.073 0.069 * *
0.155 0.137 * *
% of unemployed or sick † 3.975 5.818 3.879 6.138 *
% of professionals or managers † 10.493 13.183 9.827 12.943 *
% of non-manual workers † 22.527 17.300 21.287 17.724 *
% of skilled manual workers † 22.763 16.796 21.446 17.284 *
% of semi-skilled manual workers † 15.079 12.866 14.434 13.131 *
% of unskilled manual workers † 5.917 7.676 5.746 7.604 *
% of owner occupied households † 35.854 35.914 33.397 35.262 *
% of council tenants † 30.667 38.803 29.006 38.229 *
Average no. persons per room 0.506 0.290 0.476 0.300 *
% of households lacking inside WC † 7.133 14.047 7.191 14.155 *
% of new Commonwealth immigrants † 1.286 5.091 1.278 5.033 *
0.205 0.247 *
North West age 11 0.097 0.088 *
North age 11 0.057 0.060 *
East and West Riding age 11 0.072 0.081 *
North Midlands age 11 0.067 0.067 *
Eastern age 11 0.077 0.077 *
Southern age 11 0.055 0.054 *
South West age 11 0.062 0.056 *
Midlands age 11 0.079 0.080 *
Wales age 11 0.048 0.054 *
Scotland age 11 0.108 0.104 *
0.134 0.139 *
% of comprehensive schools in LEA 0.641 0.299 0.649 0.297 *
0.061 0.058 *
Notes: Standard deviations are not reported for 0/1 dummy variables. * Included as an explanatory variable in the corresponding equatio
n
‡ Dummy for missing values of the variable(s) listed above. § Percentage of the child's classmates with a non-manual father, at age 16.
§§ Father's socio-economic status, at age 11. † Enumeration district-level variables from 1971 Census Small Area Statistics
Table 2: Descriptive statistics
18
size (implying, for example that an increase in size may be a good thing for
small size, and a bad thing for larger size) we include the square of the size.
An important inuence on the school’s quality are the characteristics of the
students in the school, that is the “peer group eect”.
16
To capture it, we
consider both academic and social indicators. The former are the percentage
of boys and girls in the school attended at age 16 who are studying for O-
levels and the proportion who subsequently enrolled into a higher education
course (both indicate a more “academic” peer group). The social peer group
is captured by the proportion of children in the individual’s school class whose
father has a non-manual occupation.
The last rows of the Table report some geographical characteristics. As
well as regional dummies, w e include the proportion of comprehensive schools
in the area, and some social indicators of the enum eration district (this is a
small geographical area comprising around 200 households) where the child
was living at age 16. These variables are taken from the 1971 census, and
correspond to those used by Dearden et al. (2002).
The last four columns in the Table illustrate the model specication we
have chosen: an asterisk in a column indicates that the variable in the row
was used as a regressor for the equation indicated by that column. Dummies
for missing values are used for each of the other variables to capture possible
non-randomness in non-response: these are the unnamed variables in the table,
after each variable or group of variables; for example the 0.089 inthelinebelow
“Weight at birth” indicates that 8.9% of the data in the sample did not report
the value of this variable. All estimations include these dummy variables, but
we do not report their coecients or standard errors in the results to make the
interpretation and the reading of the tables easier.
6Results
Our theoretical foundation is that the eort of the three agents is simultane-
ously determined at the Nash equilibrium. Econometrically, the eort variables
should be endogenous. To ascertain this, we perform the DWH test described
in Section 4 on the parameters of equation (15). We can reject, at conven-
16
This is a well documented phenomenon; see Moreland and Levine (1992) for a survey
from a psychology/education viewpoint, Summers and Wolfe (1977), H enderson et al. (1978)
for early economic empirical studies, and Epp le et al. (2002) and Zimmer and Toma (2000)
for more recent ones. The theoretical analyses of Arnott and Rowse (1987) and de Bartolome
(1990) were among the rst to take the peer group eect exp licitly into account.
19
tional signicance levels, the null hypothesis that the residuals of the eort
equations do not aect examination results,
17
and we therefore conclude that
educational attainment and the eort levels exerted by children, their parents
and their school are indeed simultaneously determined, as posited by the the-
oretical model.
All the results are reported separately for the samples of girls and boys. We
have tested, and found support for, the hypothesis that girls and boys dier
signicantly. We have done so by estimating a more general specication of the
entire model with a gender dummy interacting with each of the explanatory
variables, and testing the joint statistical signicance of the parameters of these
interaction terms in the educational attainment equation, using a log-likelihood
ratio test.
18
In Table 3 we report the results for our three-stage least squares estimates
of equations (9)-(11).
19
In each of the three eort equations, the eort level
exerted by the other two groups of agents is signicant, with the exception of
parental eort on the school eort for girls. This conrms our assumption of
simultaneous endogenous determination of eort levels as a Nash equilibrium.
Also note that a 0 coecient does not necessary falsify the Nash equilibrium
hypothesis, because the intersection of the relevant best reply functions could
happen close to a stationary point of one o f them (as, for example, in point
E
1
in panel (a) in Figure 1). The table suggests that parental and the child’s
eorts are strategic complements: by exerting more eort, parents induce their
child to exert more eort, and, vice versa, parents respond positively to their
children exerting more eort. In other words, there is a “multiplier” eect,
suggesting, for example, that policies aimed at aecting directly the eort ex-
erted by children and parents may prove very eective. On the other hand, the
role of the school eortislessclear-cut:itaects negatively the eort exerted
by children and by girls’ parents, and positively the eort of boys’ parents.
Conversely, schools respond positively to girls’ eort, and negativ ely to boys’,
17
The test statistics of the likeliho od-ratio tests of the null hypo thesis are χ
2
(3) = 7.86
(p-value 0.0491) for the sample of girls, χ
2
(3) = 14.49 (p-value 0.0023) for the sample of boys,
and χ
2
(3) = 21.71 (p-value 0.0001) for the combined sample of girls and boys.
18
The test stati stic o f this likelihood-ratio test is χ
2
(88) = 172.89 (p-value 0.0000). We
prefer to report sep arate samples, rather than the m ore general model w ith the interaction
terms because its very large number of regressors would make the interpretation of coecients
very dicult.
19
We have tried several alternative specications, and we present here only the most parsi-
monious, having tested at various stages for linear restrictions on non-signicant coecients.
Other intermediate results and the data to obtain them are available on request.
20
but positively to boys parents eort. Anecdotal evidence does conrm the
p ossibility of dierential attitude of schools and parents towards boys and girls
in the ’60s and early ’70s.
The striking feature of the children’s eort equation is the paucity of statis-
tically signicant explanatory variables: only the other eort levels and their
own ability and birth weight seem to aect their eort. Clearly, our results are
tentative, constrained by the limitations of the dataset, but a possible interpre-
tation for this nding is that children from dierent backgrounds or in dierent
peer groups do not dier signicantly in their propensity to exert eort. If
conrmed by more targeted studies, this ma y have policy implications for the
type of incentives to provide to pupils in schools.
The paren ts’ eort equation indicates that the presence of (older or younger)
siblings reduces parental eort. This is an interesting result, which also indi-
cates that the variables we have used to measure eort do indeed capture rele-
vant features of parental eort: theoretical considerations suggest that parents
face a trade-o between the number of their children and the attention each
of them receive (Becker (1960); Hanushek (1992)). Social class also appears
relevant. Parental taste for education, as reected by their education and the
frequency of their reading, does positively inuence their own eort. There
is also some indication that the mother’s position in the labour market may
have some eect on parental eort: the eect of the mother being in work is
rather ambiguous, but the percentage of household income not earned by the
father gure has a clear negative inuence on parents’ eort. This conrms
the intuition that parents’ eort is not fully captured b y their socio-economic
status. Household income, on the other hand, aects parental eort only for
boys.
The school’s eort is higher in larger schools, for both boys and girls. The
eects of school type variables are generally stronger for girls than for boys. It
is interesting to note the di
erent eect of the “single-sex” variable in the two
subsamples: it suggests that girls’ only schools exert less eort, and boys’ only
sc hools more eort than co-educational schools; this is in line with our per-
ception of the British educational system at the time. Note that for younger
children (age 7 and 11), private schools exert eort level either not signicantly
dierent or lower than state schools. At age 16, their eortlevelisnotsigni-
cantly dierent from the base school type, the state comprehensive . The eect
of the peer group is statistically signicant, especially for bo y s: schools work
harder which have a larger proportion of c hildren from higher socio-economic
21
Dependent variable
Variable Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.
Constant 0.205 * 0.092 0.238 ** 0.085 -0.032 0.072 -0.147 * 0.074 -3.455 ** 0.165 -3.695 ** 0.144
Child's effort 0.363 ** 0.102 0.669 ** 0.080 0.186 ** 0.065 -0.114 * 0.051
Parents' effort 0.521 ** 0.042 0.418 ** 0.042 -0.063 0.053 0.279 ** 0.044
School's effort -0.094 ** 0.039 -0.199 ** 0.035 -0.121 ** 0.024 0.133 ** 0.026
Abilit
y
0.083 ** 0.009 0.111 ** 0.008 0.088 ** 0.016 0.018 0.014
Wei
g
ht at birth -0.002 ** 0.001 -0.002 ** 0.001
Older brothers -0.052 ** 0.011 -0.033 ** 0.010
Youn
g
er brothers -0.049 ** 0.011 -0.020 0.010
Older sisters -0.074 ** 0.012 -0.054 ** 0.011
Younger sisters -0.049 ** 0.010 -0.044 ** 0.011
Mother in work a
g
e 16 0.036 0.020 0.051 * 0.020
Mother in work a
g
e 7 -0.066 ** 0.020 -0.036 0.019
Houseowne
r
0.163 ** 0.026 0.148 ** 0.022
Total household income 0.000 0.001 0.002 ** 0.001
% of income not from fathe
r
-0.146 ** 0.033 -0.131 ** 0.033
Father has hi
g
her education 0.049 0.037 0.009 0.037
Father has secondar
y
education 0.015 0.022 0.011 0.022
Mother has hi
g
her education 0.145 ** 0.043 0.138 ** 0.043
Mother has secondar
y
education 0.063 ** 0.021 0.020 0.021
Father reads books re
g
ularl
y
0.197 ** 0.023 0.170 ** 0.023
Father reads books occasionall
y
0.144 ** 0.025 0.088 ** 0.025
Mother reads books re
g
ularl
y
0.098 ** 0.022 0.119 ** 0.022
Mother reads books occasionall
y
0.066 ** 0.023 0.079 ** 0.023
Intermediate
§§
-0.031 0.058 -0.011 0.054 -0.055 * 0.048 -0.025 0.047
Skilled non-manual
§§
-0.012 0.065 -0.056 0.060 -0.140 ** 0.055 -0.053 0.053
Skilled manual
§§
0.022 0.060 -0.078 0.055 -0.316 ** 0.051 -0.216 ** 0.050
Semiskilled non-manual
§§
-0.013 0.098 -0.038 0.095 -0.243 ** 0.082 -0.232 ** 0.083
Semiskilled manual
§§
0.011 0.065 -0.100 0.062 -0.276 ** 0.055 -0.206 ** 0.056
Unskilled
§§
0.016 0.082 -0.139 0.074 -0.422 ** 0.068 -0.271 ** 0.067
Lo
g
of school size a
g
e 16 0.544 ** 0.021 0.499 ** 0.018
Sin
g
le sex school a
g
e 16 -0.250 ** 0.025 0.117 ** 0.025
Grammar school a
g
e 16 -0.151 ** 0.035 0.073 * 0.037
Secondar
y
modern a
g
e 16 0.096 ** 0.026 0.032 0.024
Private school a
g
e 16 0.082 0.068 -0.017 0.065
Private school a
g
e 11 -0.159 * 0.069 0.048 0.064
Private school a
g
e 7 -0.015 0.068 -0.154 * 0.068
Pu
p
ils from school
g
o to universit
y
0.051 * 0.025 0.009 0.026
% of
g
irls stud
y
in
g
for O-levels -0.001 * 0.001 -0.001 0.001
% of bo
y
s stud
y
in
g
for O-levels 0.000 0.001 0.000 0.000
10%-19%
§
-0.013 0.040 -0.021 0.037 0.108 ** 0.040 0.157 ** 0.038
20%-29%
§
-0.032 0.041 -0.009 0.038 0.066 0.041 0.223 ** 0.039
30%-39%
§
-0.031 0.044 -0.054 0.040 0.033 0.045 0.167 ** 0.042
40%-49%
§
0.007 0.051 -0.046 0.046 0.175 ** 0.050 0.265 ** 0.047
50%-59%
§
-0.061 0.050 -0.022 0.049 0.124 * 0.050 0.414 ** 0.049
60%-69%
§
-0.066 0.054 -0.009 0.051 0.129 * 0.054 0.277 ** 0.052
70%-79%
§
-0.013 0.068 0.066 0.061 0.089 0.068 0.331 ** 0.062
80%-100%
§
-0.091 0.055 0.082 0.055 -0.140 * 0.058 0.406 ** 0.060
Number of observations
R
2
chi
2
Notes:
§
Percenta
g
e of the child's classmates with a non-manual father, at a
g
e 16.
§§
Father's socio-economic status, at a
g
e 11. * Si
g
nificant at the 5% level. ** Si
g
nificant at the 1% lev
e
0.4199
2020.47**
0.24770.2261
1740.60**
Child's Effort
Girls Boys Girls Boys
Parents' Effort School's Effort
Girls Boys
Other variables included in the regression and not reported are: the absence of a father figure, whether the mother works at birth, the log of the school size at age 7 and 11.
5611 5860 5611 5611 58605860
0.2397 0.1891 0.1852
1526.93**1607.82**3902.05**4370.00**
Table 3: Three-stage least squares estimates of eort equations
22
groups.
Table 4 presents the result of our ordered probit estimates. As it should,
eort strongly improves results. Ability also has, again as one would expect,
a strong independen t positive eect on results. Family background variables,
such as the parents’ education, their taste for reading and their social class
have however a less denite eect than they had on eort, and they appear
to have a weaker inuence than much of the literature suggests (Ermisch and
Francesconi (2001), Dearden et al. (2002)); for example, income is statistically
not signicant. Whether the mother (the father) has higher education aect
positively girls’ (boys’) results, but otherwise parental education is not statisti-
cally signicant. Book reading has, if anything, a negative eect on attainment,
and the social class is overall less signicant than in the eort equation. A nat-
ural interpretation of these results is that family background inuences school
examination’s results indirectly, via parental eort, rather than directly.
With regard to the variables describing the school, we nd that being in a
private school has a direct positive eect for boys but not for girls; on the other
hand, for girls, a state grammar has a direct positive eect, while a secondary
modern has a direct negative eect. Girls in single-sex schools obtain, ceteris
paribus, a better qualication. School size matters only at age 16, and only
for girls, with a negative sign. These variables have the opposite sign in the
sc hool’s eort equation: in Table 3, last columns, single-sex has a negative sign,
and the sc hool size a positive sign. We have included several measures of class
size, at the three dierent ages, and their square, to account for possible non-
linearities: of these only the English class size at age 16, for girls, is statistically
signicant.
20
Theacademicpeergroupeect appears very strong: in terestingly,
it operates within genders: girls are not inuenced by boys and vice versa. This
makes sense, and we take it as a further indication that our model specication
is plausible.
We do not include in Table 4 the census variables, listed in Table 2. Among
them, only the percentage of unemployed or sick in the census enumeration
district are statistically signicant for girls, and only the proportion of owner
occupied houses, the proportion of council tenants, and the average number
of persons per room are statistically signicant for boys. These variables have
20
The estimated co ecients suggest that exam results improve with class size up to 29 and
decreases for class size larger than 30. Though the specic appealing value of the “optimal”
class size may well be a uke, it is interesting to note that the relationship between class size
and achievement in this dataset has often the “wrong” s ign (Levacic and Vignoles 2002)
23
Variable Coef. Std. Err. Coef. Std. Err.
Child's effort 0.439 ** 0.046 0.164 * 0.073
Parents' effort 0.739 ** 0.085 0.977 ** 0.033
School's effort 1.076 ** 0.162 0.333 ** 0.104
Abilit
y
0.331 ** 0.014 0.357 ** 0.014
Houseowne
r
0.001 0.043 -0.109 ** 0.038
Total household income 0.000 0.001 0.000 0.001
% of income not from fathe
r
0.179 ** 0.060 -0.028 0.058
Father has hi
g
her education 0.071 0.070 0.161 * 0.063
Father has secondar
y
education -0.014 0.040 -0.001 0.040
Mother has hi
g
her education 0.213 ** 0.080 -0.063 0.082
Mother has secondar
y
education -0.067 0.038 -0.010 0.037
Father reads books re
g
ularl
y
-0.119 ** 0.045 -0.146 ** 0.042
Father reads books occasionall
y
-0.122 ** 0.048 -0.025 0.048
Mother reads books re
g
ularl
y
-0.023 0.041 -0.109 ** 0.039
Mother reads books occasionall
y
-0.048 0.043 -0.105 * 0.043
Intermediate
§§
-0.151 0.078 -0.230 ** 0.073
Skilled non-manual
§§
-0.190 * 0.088 -0.211 ** 0.080
Skilled manual
§§
-0.111 0.082 -0.065 0.074
Semiskilled non-manual
§§
-0.155 0.138 -0.226 0.143
Semiskilled manual
§§
-0.187 * 0.089 -0.180 * 0.083
Unskilled
§§
-0.232 * 0.117 -0.093 0.102
En
g
lish class size at 16 0.046 * 0.018 0.000 0.018
(
En
g
lish class size at 16
)
2
-0.001 * 0.000 0.000 0.000
Maths class size at 16 0.001 0.016 0.008 0.016
(
Maths class size at 16
)
2
0.000 0.000 0.000 0.000
Lo
g
of school size a
g
e 16 -0.434 ** 0.102 -0.084 0.070
Sin
g
le sex school a
g
e 16 0.353 ** 0.103 -0.093 0.074
Grammar school a
g
e 16 0.211 ** 0.066 0.083 0.063
Secondar
y
modern a
g
e 16 -0.223 ** 0.049 0.018 0.045
Private school a
g
e 16 0.193 0.112 0.617 ** 0.096
Private school a
g
e 11 0.217 0.115 -0.002 0.105
Private school a
g
e 7 -0.125 0.120 -0.055 0.103
Pu
p
ils from school
g
o to universit
y
0.035 0.042 0.108 ** 0.041
% of
g
irls stud
y
in
g
for O-levels 0.005 ** 0.001 -0.002 0.001
% of bo
y
s stud
y
in
g
for O-levels 0.000 0.001 0.005 ** 0.001
% of com
p
rehensive schools in LEA -0.141 * 0.071 -0.071 0.070
µ
1
: boundary between Q0 and Q1
-4.352 0.759 -2.956 0.610
µ
2
: boundary between Q1 and Q2
-3.843 0.758 -2.421 0.610
µ
3
: boundary between Q2 and Q3
-3.315 0.757 -1.966 0.610
µ
4
: boundary between Q3 and Q4
-2.678 0.756 -1.235 0.609
µ
5
: boundary between Q4 and Q5
-1.248 0.755 0.059 0.608
µ
6
: boundary between Q5 and Q6
-0.850 0.754 0.355 0.607
µ
7
: boundary between Q6 and Q7
-0.509 0.754 0.660 0.607
µ
8
: boundary between Q7 and Q8
-0.234 0.753 0.924 0.607
µ
9
: boundary between Q8 and Q9
0.202 0.753 1.338 0.607
µ
10
: boundary between Q9 and Q10
0.782 0.754 1.904 0.607
Number of observations
Pseudo R
2
Wald chi
2
(
90
)
Lo
g
-Likelihood
Notes: §§ Father's socio-economic status, at age 11. * Significant at the 5% level. ** Significant at the 1% level.
Other variables included in the regression and not reported are: whether the mother is married at birth,
the absence of a father figure, the log of school size at age 7 and 11, census variables, and regional dummies.
0.2744
4192.97**
-9226.7-8945.4208
4166.01**
0.2677
Girls Boys
5611 5860
Table 4: Ordered probit estimates of exam results equation
24