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DISCUSSION PAPER SERIES

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Available online at: w

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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 aﬀected by the eﬀort 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

theimportanceofeﬀort for educational attainment. Student’s eﬀort 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’ eﬀort, operating at the individual level, or

teachers’ eﬀort, operating at the class level (Trautwein and Köller (2003)). As

well as students’ eﬀort, the educational psychology literature has also studied

the relationship between school attainment and parental eﬀort. A variety of

dimensions of parental eﬀort 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 eﬀort appears

to aﬀect educational attainment only indirectly, to the extent that it supports

children’s eﬀort (Hoover-Dempsey et al. (2001)).

The lack of speciﬁc data quantifying eﬀort as a separate variable aﬀect-

ing educational attainmenthas also hindered studies carried out by economists.

For example, Hanushek (1992) proxies parental eﬀort with measures of fam-

ily socio-economic status (permanent income and parents’ education lev els).

Intuition — conﬁrmed by our results — would however suggest that eﬀort 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 eﬀort.

1

Bones-

røning (1998; 2004) and Cooley (2004) are among the very few authors in the

economics literature who measure the eﬀort exerted by students and parents

1

Theirideaisthatparentstrytomaximisethe welfare of their children, and they may

decide to allocate more time and eﬀort 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 eﬀects on examination results.

Theoretical analyses of the role of eﬀort in the education process are also

scarce.

2

Our paper attempts to ﬁll this gap, by developing a theoretical model

of the determination as a Nash equilibrium of the eﬀort exerted by students,

their parents and their sc hools, and subsequently by estimating empirically the

determinants of the eﬀort levels, the interaction among of them, and the eﬀect

of eﬀort 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 eﬀort: 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

oﬀered career guidance, and the type of disciplinary methods used.

Our empirical estimates of the determinants of eﬀort are encouraging: the

theoretical assumption of join t interaction of the eﬀort levels of the three groups

of agents appears to be borne out by the data. Moreover, our measures of eﬀort

seem appropriate, especially so for children and parents. For example, as a by-

product of our analysis, we ﬁnd conﬁrmation of Becker’s (1960) intuition that

there is a trade-oﬀ between quantity and quality of children: a child’s number

of siblings inﬂuences the eﬀort exerted by that child’s parents towards that

child’s education.

Our econometric model allows us to determine whether explanatory vari-

ables inﬂuence educational attainment directly or indirectly, that is by aﬀecting

eﬀort. For example, our results suggest that family socio-economic conditions

aﬀect attainment more strongly via eﬀort than directly. In this case, policies

2

This contrasts sharply with the extensive literature which studies the role of e ﬀort in

ﬁrms; a seminal contribution is the theory of eﬃciency 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 eﬀort in ﬁrms: an early test of the eﬃciency

wage hypothesis is Capp elli and Chauvin (1991), who m easured workers’ eﬀort by disciplinary

dismissals. More recently, eﬀort 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 aﬀect a worker’s eﬀort (Falk and Ichino (2003)).

2

that attempt to aﬀect parental eﬀort might be eﬀective ways to impro ve the

educational attainment, since aﬀecting parental eﬀort is likely to be easier than

modifying social background.

3

We also ﬁnd that the c hildren’s own eﬀort has

the least eﬀect on educational attainment: schools and parents are more im-

portant. Interestingly, the school’s eﬀort 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 ﬁnancial incentives are being provided to sch ools

and teachers. To the extent that these incentives stimulate the schools’ eﬀort,

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 qualiﬁcation. This is a variable q taking one of m possible values

q ∈ {q

1

,...,q

m

},withq

k−1

<q

k

, k =2,...,m. Other things equal, a stu-

dent prefers a better qualiﬁcation: apart from personal satisfaction, there is

substantial evidence showing a positive association between qualiﬁcation and

future earnings in the labour market: let u (q) be the utility associated with

qualiﬁcation q,withu

0

(q) > 0.

When at school, pupils exert eﬀort, 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 ﬁnancial rewards to parents helping their

children with homework, or attending p arenting classes, similarly to the policy of providing ﬁ-

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

eﬀort, and so the interpretation of the function ψ

C

(and the corresponding ones

for schools and parents), is cost of eﬀort relative to the beneﬁtofqualiﬁcation.

Pupils also diﬀer in ability, denoted by a. A student’s education attainment is

aﬀected by her eﬀort and her ability. Formally, we assume that qualiﬁcation q

k

is obtained with probability π

k

¡

e

C

,a; ·

¢

(the “ · ” represents other inﬂuences on

qualiﬁcation, discussed in what follow s). We hypothesise, naturally, a positive

relationship between eﬀort and the expected qualiﬁcation:

P

m

k=1

∂π

k

(

e

C

,a;·

)

∂e

C

q

k

>

0, and between ability and the expected qualiﬁcation:

P

m

k=1

∂π

k

(

e

C

,a;·

)

∂a

q

k

> 0.

A student’s objective function is the maximisation of the diﬀerence between

expected utility and the cost of eﬀort:

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’ eﬀort. 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 eﬀort 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 inﬂuence of parental eﬀort 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 diﬀer also in education, social background and other variable which aﬀect

their children’s education attainment; we capture this by means of a possibly

multidimensional variable, s

P

.

Parents care about their children’s qualiﬁcation, and so t hey will exert eﬀort

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 qualiﬁcation of all their children:

4

4

Rigorously, we should consider the utility of the qualiﬁcation, 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 qualiﬁcation of a less b right child is increased, than if the qualiﬁcation

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 payoﬀ 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 eﬀort devoted by parents to child j, whose ability is a

j

,and

who exerts eﬀort e

C

j

.

5

A student’s qualiﬁcation will also be aﬀected by the quality of her school,

the last component of the “ · ” in the arguments of the probabilities in (1). The

sc hool inﬂuences its pupils’ attainment through its own eﬀort, measured by a

variable e

S

∈ E

S

⊆ IR (again assumed one-dimensional). This captures the

idea that a school can take actions which aﬀect the quality of the education it

imparts. Improving the quality of buildings, classroom equipment and sporting

facilities, using computers appropriately, upgrading teachers’ qualiﬁcations 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

Eﬀort 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 qualiﬁ-

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

qualiﬁcation of its students and negatively on the

5

The interaction between parental eﬀort and the number o f children was ﬁrst 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 ﬁrst group are ﬁxed 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 diﬀerent 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,thisdoesnotaﬀect the qu alitative nature of the interaction.

7

As with parents, the average qualiﬁcation 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 payoﬀ fun ction (2).

5

teac hing eﬀort:

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 eﬀort levels e

C

, e

P

,ande

S

are aﬀected 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))istheeﬀort 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 eﬀect on qualiﬁcation,

or an indirect eﬀect, via the eﬀort lev el exerted by the participants in the

education process. H is the number of all the possible values which the variables

aﬀecting eﬀort can take, and λ

h

is the proportion of pupils at the school with

this variable equal to h.

Additivity between the disutility of eﬀort and the students’ average qual-

iﬁcation 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 eﬀects 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 eﬀects, 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 qualiﬁcation

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 eﬀort in order to maximise their utility, taking as

given the choice of eﬀort of the other parties. An equilibrium is given by the

set of values e

C

, e

P

,ande

S

, satisfying the ﬁrst 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 satisﬁed. (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.

Letthereforetheparentaleﬀort be ﬁxed, at e

P

. Total diﬀerentiation of (3) and

(5) gives the slope of the best reply function in the relevant Cartesian diagram

(E

C

× E

S

for ﬁxed 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 payoﬀ, 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 eﬀort 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 payoﬀ 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 diﬀerent set of exogenous variables, say h

1

, associated with a higher value

of the students’ eﬀort, for every given level of the school’s eﬀort, and a higher

value of the school’s eﬀort, for every given level of the students’ eﬀort 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 eﬀort

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 eﬀort

levels are higher: compare E

0

with E

1

.

Consider panel (b), however. It diﬀers 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 eﬀort levels

that result as a consequence of a diﬀerent value in the exogenous parameters

vector h, associated with higher eﬀort levels results in a lower equilibrium eﬀort

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 eﬀort for any

given level of the school’s eﬀort. The reason for the lower equilibrium value

of the student’s eﬀort is the strategic interaction of schools and students. The

vector h

1

would be associated to a higher value of the student’s eﬀort if the

school’s eﬀort were the same. However, the student’s and the school’s eﬀorts

are “strategic substitutes” (Bulow et al 1986), and the student responds to the

higher school eﬀort (associated to the vector h

1

) with a lower level of their

own eﬀort. This, in panel (b) in the diagram, more than compensates the

direct increase in the student’s eﬀort caused by the diﬀerent value of h.This

simple example illustrates the poten tial ambiguity of changes in the exogenous

variables h on the equilibrium eﬀort levels; in more general settings the situation

will be ev en more complex.

4EmpiricalModel

Given this theoretical ambiguity, the overall eﬀect of children’s, parents’ and

sc hool’s eﬀorts on educational attainment, and whether these eﬀort 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

eﬀort 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 speciﬁed 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 aﬀecting the educational outcome, e

C

i

, e

P

i

,ande

S

i

are the

measures of the eﬀort 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 eﬀort 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 eﬀect 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 eﬀort 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 aﬀecting child i’s eﬀort, child i’s parents’ eﬀort, and

the eﬀort 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 eﬀort 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 diﬀerent scales and are in gen-

eral non-comparable. We therefore use factor analysis

10

to construct a single

aggregate continuous indicator of the three eﬀort levels.

We next need to ascertain whether the eﬀort 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 ﬁrst

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 ﬁnedonanordinalrather

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 diﬀerence in the results prov ided by the two methods is only of order 10

−3

at most. Our

results indicate that retaining only the ﬁrst factor is the appropriate strategy for the children’s

and the p a rents’ eﬀort; a second factor should perhaps be retained for the school’s eﬀort, but,

for symmetry and ease of interpretation, we retain only the ﬁrst 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 aﬀecting either educational attainment,

or the child’s eﬀort, 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 signiﬁcantly diﬀerent 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 ﬁnd endogeneity of the eﬀort variables.

The estimation method we use is 3SLS, because of the interdependent na-

ture of the eﬀort 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 eﬀort 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-

tiﬁcation is achieved by the inclusion in the sets x

U

i

, U = Q, C, P, S,ofsome

statistically signiﬁcant 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 insuﬃcient information

we were left with a sample of 5611 girls and 5860 boys.

5.1 Dependen t variables

5.1.1 Eﬀort

Table 1 contains the scoring coeﬃcients for the child’s, the parents’ and the

sc hool’s eﬀort 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 eﬀort 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 eﬀort information, respectively.

The child’s eﬀort indicators used to construct the child’s eﬀort 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 eﬀort).

12

This information is comple-

mented by the teacher’s assessment of the child’s eﬀort when the individuals

are 16 (the last row in the ﬁrst part of Table 1). Co lum ns 3 and 5 of Table

1 provide the scoring coeﬃcients for each of the variables reﬂecting the eﬀort

indicators, namely the weights actually entering the construction of the eﬀort

indices. For the children the variable with the highest scoring coeﬃcient is

whether the child likes school or not.

13

The parents’ eﬀort measure e

P

i

is produced using both parents’ interest in

the child, their initiativ e to discuss the child’s progress in school, the father’s

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

y

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 eﬀort measures

13

Figure 2: Density of eﬀort for child, parents, and schools

From the second part of Table 1, we ﬁnd that the parents’ interest in the child’s

education at diﬀerent 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 eﬀort, 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’ eﬀort indices are constructed with similar scoring coeﬃcients

for both boys and girls. This is not so for the school’s eﬀort 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 reﬂect a

diﬀerence in the schools’ behaviour towards bo y s and girls in this period. The

construction of the school’s eﬀort 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 eﬀort 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 qualiﬁcation, to 10, reﬂecting 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 ﬁrst, 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 qualiﬁcations.

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 diﬀerent 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

ﬁgure’s) social class at age 11.

15

We have also included the percentage of total

income not earned by the father ﬁgure, 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 eﬀort.

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 inﬂuence on the school’s quality are the characteristics of the

students in the school, that is the “peer group eﬀect”.

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 speciﬁcation 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 coeﬃcients or standard errors in the results to make the

interpretation and the reading of the tables easier.

6Results

Our theoretical foundation is that the eﬀort of the three agents is simultane-

ously determined at the Nash equilibrium. Econometrically, the eﬀort 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 ﬁrst to take the peer group eﬀect exp licitly into account.

19

tional signiﬁcance levels, the null hypothesis that the residuals of the eﬀort

equations do not aﬀect examination results,

17

and we therefore conclude that

educational attainment and the eﬀort 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 diﬀer

signiﬁcantly. We have done so by estimating a more general speciﬁcation of the

entire model with a gender dummy interacting with each of the explanatory

variables, and testing the joint statistical signiﬁcance 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 eﬀort equations, the eﬀort level

exerted by the other two groups of agents is signiﬁcant, with the exception of

parental eﬀort on the school eﬀort for girls. This conﬁrms our assumption of

simultaneous endogenous determination of eﬀort levels as a Nash equilibrium.

Also note that a 0 coeﬃcient 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

eﬀorts are strategic complements: by exerting more eﬀort, parents induce their

child to exert more eﬀort, and, vice versa, parents respond positively to their

children exerting more eﬀort. In other words, there is a “multiplier” eﬀect,

suggesting, for example, that policies aimed at aﬀecting directly the eﬀort ex-

erted by children and parents may prove very eﬀective. On the other hand, the

role of the school eﬀortislessclear-cut:itaﬀects negatively the eﬀort exerted

by children and by girls’ parents, and positively the eﬀort of boys’ parents.

Conversely, schools respond positively to girls’ eﬀort, 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 coeﬃcients

very diﬃcult.

19

We have tried several alternative speciﬁcations, and we present here only the most parsi-

monious, having tested at various stages for linear restrictions on non-signiﬁcant coeﬃcients.

Other intermediate results and the data to obtain them are available on request.

20

but positively to boys’ parents’ eﬀort. Anecdotal evidence does conﬁrm the

p ossibility of diﬀerential attitude of schools and parents towards boys and girls

in the ’60s and early ’70s.

The striking feature of the children’s eﬀort equation is the paucity of statis-

tically signiﬁcant explanatory variables: only the other eﬀort levels and their

own ability and birth weight seem to aﬀect their eﬀort. Clearly, our results are

tentative, constrained by the limitations of the dataset, but a possible interpre-

tation for this ﬁnding is that children from diﬀerent backgrounds or in diﬀerent

peer groups do not diﬀer signiﬁcantly in their propensity to exert eﬀort. If

conﬁrmed by more targeted studies, this ma y have policy implications for the

type of incentives to provide to pupils in schools.

The paren ts’ eﬀort equation indicates that the presence of (older or younger)

siblings reduces parental eﬀort. This is an interesting result, which also indi-

cates that the variables we have used to measure eﬀort do indeed capture rele-

vant features of parental eﬀort: theoretical considerations suggest that parents

face a trade-oﬀ 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 reﬂected by their education and the

frequency of their reading, does positively inﬂuence their own eﬀort. There

is also some indication that the mother’s position in the labour market may

have some eﬀect on parental eﬀort: the eﬀect of the mother being in work is

rather ambiguous, but the percentage of household income not earned by the

father ﬁgure has a clear negative inﬂuence on parents’ eﬀort. This conﬁrms

the intuition that parents’ eﬀort is not fully captured b y their socio-economic

status. Household income, on the other hand, aﬀects parental eﬀort only for

boys.

The school’s eﬀort is higher in larger schools, for both boys and girls. The

eﬀects of school type variables are generally stronger for girls than for boys. It

is interesting to note the diﬀ

erent eﬀect of the “single-sex” variable in the two

subsamples: it suggests that girls’ only schools exert less eﬀort, and boys’ only

sc hools more eﬀort 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 eﬀort level either not signiﬁcantly

diﬀerent or lower than state schools. At age 16, their eﬀortlevelisnotsigniﬁ-

cantly diﬀerent from the base school type, the state comprehensive . The eﬀect

of the peer group is statistically signiﬁcant, 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 eﬀort equations

22

groups.

Table 4 presents the result of our ordered probit estimates. As it should,

eﬀort strongly improves results. Ability also has, again as one would expect,

a strong independen t positive eﬀect on results. Family background variables,

such as the parents’ education, their taste for reading and their social class

have however a less deﬁnite eﬀect than they had on eﬀort, and they appear

to have a weaker inﬂuence than much of the literature suggests (Ermisch and

Francesconi (2001), Dearden et al. (2002)); for example, income is statistically

not signiﬁcant. Whether the mother (the father) has higher education aﬀect

positively girls’ (boys’) results, but otherwise parental education is not statisti-

cally signiﬁcant. Book reading has, if anything, a negative eﬀect on attainment,

and the social class is overall less signiﬁcant than in the eﬀort equation. A nat-

ural interpretation of these results is that family background inﬂuences school

examination’s results indirectly, via parental eﬀort, rather than directly.

With regard to the variables describing the school, we ﬁnd that being in a

private school has a direct positive eﬀect for boys but not for girls; on the other

hand, for girls, a state grammar has a direct positive eﬀect, while a secondary

modern has a direct negative eﬀect. Girls in single-sex schools obtain, ceteris

paribus, a better qualiﬁcation. 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 eﬀort 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 diﬀerent ages, and their square, to account for possible non-

linearities: of these only the English class size at age 16, for girls, is statistically

signiﬁcant.

20

Theacademicpeergroupeﬀect appears very strong: in terestingly,

it operates within genders: girls are not inﬂuenced by boys and vice versa. This

makes sense, and we take it as a further indication that our model speciﬁcation

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 signiﬁcant 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 signiﬁcant for boys. These variables have

20

The estimated co eﬃcients suggest that exam results improve with class size up to 29 and

decreases for class size larger than 30. Though the speciﬁc appealing value of the “optimal”

class size may well be a ﬂuke, 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