The Impact of School Resources on Student Attainment: A Multilevel Simultaneous
Equation Modelling Approach
Graduate School of Education
University of Bristol
Anna Vignoles and Andrew Jenkins
Bedford Group for Lifecourse and Statistical Studies
Institute of Education, University of London
Improving educational achievement in UK schools is a priority, and of particular concern is
the low achievement of specific groups, such as those from lower socio-economic
backgrounds. An obvious question is whether we should be improving the outcomes of these
students by spending more on their education. The literature on the effect of educational
spending on pupil achievement has a number of methodological difficulties, in particular the
endogeneity of school resource levels, and the intra-school correlations in student responses.
In this paper, we adopt a multilevel simultaneous equation modelling approach to assess the
impact of school resources on student attainment at age 14. This paper is the first to apply a
simultaneous equation model to estimate the impact of school resources on pupil
achievement, using the newly available National Pupil Database (NPDB).
Keywords: education production function, multilevel simultaneous equation model
*Address for correspondence: Fiona Steele, Graduate School of Education, University of
Bristol, 35 Berkeley Square, Bristol BS8 1JA, Email: firstname.lastname@example.org
For policy-makers and parents alike, improving educational achievement in UK schools is a
policy priority. There is certainly an economic imperative to raise educational achievement,
given that an additional year of education in the OECD area is estimated to increase
economic output by between 3 and 6 percent (OECD, 2004). Currently, the UK spends
around 5 per cent of its annual Gross Domestic Product on education, including primary,
secondary and postsecondary (compared to an OECD mean of 5.6 per cent), and expenditure
has been increasing since the mid 1990s. Nonetheless, spending in UK secondary schools
(US$5933) is below the OECD mean of US$6510 (OECD, 2004). However, lower
expenditure does not necessarily mean lower achievement, at least in aggregate. The UK,
along with countries such as Australia, Finland, Ireland and Korea, spends a lower than
average amount on secondary schooling but its students perform relatively well in
international tests of student achievement, such as the Programme for International Student
Assessment (Machin and Vignoles, 2005). An obvious policy question is therefore whether
an increase in per pupil expenditure on education, or a reduction in the average pupil-teacher
ratio in schools, is a viable means of improving pupil attainment across the board. There are a
number of reasons why this may not in fact be a feasible policy option. One possibility that is
much discussed in the literature, and which has hugely important policy implications, is that
state schools are inefficient in their use of resources, so that higher spending schools do not
systematically have better pupil outcomes (Hanushek, 1997). This paper not only aims to
provide empirical evidence to guide policy-makers on this issue, but also seeks to overcome
some important methodological difficulties that plague many of the previous studies in this
area of research.
Another policy issue of particular concern in the UK is the low achievement of specific
groups of students, such as those from lower socio-economic backgrounds and certain
gender/ ethnic groups. Again, an obvious question is whether we should be improving the
outcomes of these students by spending more on their education. This research question is
explored in our previous work on this issue (Levačić et al., 2005), which used an instrumental
variable approach to examine the relationship between school resourcing levels and the
attainment of different subgroups of English pupils. Here, we adopt a somewhat different
methodology (a multilevel simultaneous equation model) to try to accurately ascertain the
direction and magnitude of any links between school resources1 and the mean educational
attainment of pupils in England.
There is a large and controversial literature analysing the relationship between school
resourcing levels and pupil achievement, dating back to the pioneering work by Coleman et
al. (1966). Much of the US evidence suggests a weak and somewhat inconsistent relationship
between school resources and pupil achievement. (Burtless, 1996; Hanushek, 1997).
However, this view has been disputed by some, including Lane et al. (1996) and Krueger
Largely, the controversy in this literature centres on the extent to which studies that show no
significant relationship between school resources and pupil achievement are able to overcome
a number of methodological difficulties. One major methodological difficulty in the literature
is the problem of the endogeneity of school resources due to the non-random way in which
funds are allocated across schools. In the UK, schools with higher concentrations of lower
attaining students receive more funding per student. If this feature of resource allocation is
ignored, a true positive effect of increasing resources will be understated. In addition, there
may be unobserved characteristics of schools, and also of local education authorities (LEAs),
which influence both resource allocation and student attainment. For example, one factor in
the funding allocation formula used by LEAs is the proportion of socially disadvantaged
students in a school, which is also associated with student outcomes. In the absence of
adequate controls for social background, a true positive resource effect will be diluted or may
even appear negative.
There are a number of potential methods that might be used to overcome this endogeneity
problem, including random assignment. For example, the Tennessee STAR class size
experiment randomly allocated children in primary school to small and large class sizes.
Results from STAR suggest that smaller classes do increase student attainment and that gains
persist to the school leaving age and college (Krueger and Whitmore, 2001). Another method
that is used to overcome the endogeneity problem is a natural experiment. The international
literature using natural experiments, such as rules on class size, or court-imposed policies to
raise spending on schools, has produced mixed results. Angrist and Lavy (1999) and Jepson
1 Per student expenditure and the school pupil-teacher ratio.
and Rivkin (2002) found positive effects of smaller class size on student attainment for Israel
and California respectively. However Hoxby (2000) found no effect of class size and
Dobbelsteen et al. (2002), instrumenting on teacher allocation rules, reported a significant
positive effect of larger class size on attainment for the Netherlands.
Yet another approach to tackling endogeneity is to include a large number of control
variables to reduce the possibility of covariance between resources and any unobserved
variables that affect attainment. For example, Wilson (2000) using extensive data on family
and neighbourhoods for the US found school spending to be positively related to high school
graduation and years of schooling. Another method tried by Hakkinen et al. (2003) is using
panel data over a number of years to difference out school and district effects. They find no
effects on exam scores in Finnish upper secondary schools of changes in per student spending
It is fair to say, however, that the vast majority of school resource effect studies have not
been able to address the endogeneity problem. This is certainly the case in the UK (Levačić
and Vignoles, 2002). UK studies that have made some attempt to address endogeneity have
generally found small but statistically significant positive effects from school resource
variables on educational outcomes. (Dearden et al., 2001; Dolton and Vignoles, 2000;
Dustmann et al., 2003; Iacovou, 2002).
Endogeneity issues are not the only methodological difficulty in this literature. Another
important methodological issue to be considered is the intra-school correlations in student
responses. The need to control for clustering in the analysis of hierarchically structured data
is well known (see, e.g., Goldstein, 2003). One consequence of ignoring clustering is the
underestimation of standard errors due to the decrease in the effective sample size, and in
general the underestimation is most severe for explanatory variables defined at the cluster
level. In the present case, it is especially important to adjust for clustering because the
variables of major interest, measures of school resources, are school-level characteristics.
In this paper, we adopt a multilevel simultaneous equation modelling approach to assess the
impact of school resources on student attainment at age 14. A multilevel model is used to
allow for clustering of student outcomes by school and LEA, and clustering of school
resources by LEA. A simultaneous equation model is used to adjust for the endogeneity of
school resource allocation. In this approach, student attainment and a measure of school
resources are treated as a bivariate response. A multilevel model is defined for each response
with LEA and school level random effects included in each; these random effects may be
correlated across the attainment and resource equations, which allows explicitly for
correlation between the unobserved LEA and school characteristics that influence each
response. Our approach differs from the instrumental variable (IV) method traditionally used
to account for endogeneity in the assumptions made about the level at which selection effects
operate. The standard approach involves estimating equations for the outcome of interest and
the endogenous regressor, either simultaneously or more commonly in two stages, but the
equations are linked via correlated residuals defined at the lowest level of observation, in this
case the student. This method may be inappropriate on two counts: first, it incorrectly treats
school resources as a student-level variable and, second, it does not recognise that
endogeneity arises due to correlation between unobservables at the school or LEA level
rather than at the student level.
This paper is the first to apply a simultaneous equation model to estimate the impact of
school resources on pupil achievement, using the newly available National Pupil Database.
The NPDB contains information on the characteristics and achievement of every pupil in an
English school, as well as characteristics of the schools themselves. The NPDB is
supplemented by information on schools’ levels of resourcing, derived from data submitted to
the Department for Education and Skills by local education authorities. NPDB provides
information on individual students’ attainment at age 14 (Key Stage 3) in 2003 and their
attainment at age 11 (Key Stage 2) in 2000, enabling us to control for prior attainment in our
model. Previous work in this area has been restricted to using either more aggregated data
(school or LEA level data) or relying on the National Child Development Study data set that,
whilst rich, is somewhat dated in terms of providing empirical evidence to inform education
policy today (its sample consists of a cohort born in 1958).
2. Background on the Secondary Education System in England
In England, educational spending on both primary and secondary schooling is administered
by 150 local education authorities (LEAs), which are under local government control.
However, in the years for which our study data were collected, the majority of the money for
education came from central government via a block grant2 to these LEAs for all local
services. LEAs could spend this grant more or less according to their own priorities, and
decide to spend more or less than the amount notionally allocated per pupil in the block grant.
The amount of money received by a particular LEA from central government nominally for
education, which until recently was known as the Education Standard Spending Assessment
(SSA)3, depends on a number of factors that influence the expected educational costs in an
LEA. For example, the education SSA takes account of student numbers, socio-economic
factors (e.g. the number of immigrants in the area, the proportion of the local population in
lower socio-economic groups and the numbers of families on state benefit), density of
population and cost of living in the area.
The fact that socio-economic factors partly determine the SSA implies that in the UK greater
school resources are allocated to areas of greater educational need. This is reinforced by the
fact that the actual block grant given to LEAs takes account of the potential in the LEA to
raise local tax for educational spending. Thus prosperous areas tend to receive less from
central government since they can potentially raise more revenue from local taxation. The
fact that LEAs have some discretion over how to spend the grant they receive4 again
reinforces the point that endogeneity is likely to be a problem in any analysis of the influence
of educational expenditure on pupil achievement.
3.1 The standard multilevel modelling approach
the attainment at age 14 in maths, English or science of student i (i=1, . . .,
; ) in school ( =1, . . .,
) in LEA (k =1, . . .,
K ). The
standard approach to modelling attainment, allowing for clustering at the school and LEA
levels, would be to fit a three-level random effects model. The simplest such model allows
the regression intercept to vary randomly across schools and LEAs:
2 Revenue Support Grant.
3 Now the Education Formula Spending Share.
4 Thus actual expenditure per pupil varies systematically by LEA, depending partly on the political party in
control of the local authority and their educational priorities.
is a vector of explanatory variables defined at the student, school or LEA level, α
is a vector of associated coefficients,
is a measure of school resources with coefficient
β , and , and are residuals for LEAs, schools and students respectively.
Typically, the residuals are assumed to be normally distributed:
), 0 (
) , 0 (
) , 0 (
A further assumption of the standard multilevel model is that the residuals at each level are
uncorrelated with the predictor variables
and . For the reasons given in Sections 1
and 2 above, however, this assumption is questionable because the mechanisms by which
resources are allocated to schools are likely to be related to the unobserved determinants of
student attainment; these unobserved factors may be acting at the school or LEA level or
both, leading to nonzero correlations between
and either or both of and .
3.2 A simultaneous equations model for attainment and resource allocation
One way to allow for the potential endogeneity of resources
with respect to attainment
is to model the resource allocation process jointly with attainment. A two-level random
intercept model for school resources is
where is a vector of explanatory variables defined at the school or LEA level,
γ is a
vector of coefficients, and and are school and LEA level residuals.
Equations (1) and (2) define a simultaneous equations model. The equations are linked via
the school and LEA residuals and must therefore be estimated jointly. At each level, we
assume that the residuals follow bivariate normal distributions, i.e.
and . We denote the
covariances at the school and LEA level by
and respectively. Likelihood ratio
tests may be used to test whether either or , or both, equal zero. A covariance that
is significantly different from zero implies that
is endogenous, and the nature of the
selection effect is given by the direction of the covariance estimate.
In order to identify the simultaneous equations model (1) and (2), the vector
contain at least one variable, called an instrument, which is not contained in
. To qualify
as an instrument, a variable must predict the allocation of resources across schools, but
should not have a direct effect on attainment.
Finding adequate instruments in this area of research is quite problematic (Burtless, 1996).
Given that school funding varies by LEA, and that LEAs are subject to political control, the
political party in control of the local authority is one potential instrument. We argue that
political control of the local authority will affect educational spending in that LEA but will
not directly impact on pupil achievement. The first instrument is therefore a variable
indicating the political control of the local authority, i.e. whether Labour, Conservative,
Liberal or other (including no overall political control by one party). As can be seen in Figure
1, the mean raw expenditure per student is highest in Liberal and Labour controlled local
education authorities, and lowest in Conservative controlled authorities.
It is possible that residents who place greater emphasis on education (and hence whose
children tend to do better in school) will vote for parties that advocate higher educational
spending. However, residents vote for a party that has policies on a number of different
issues, not just educational spending. It is not clear that residents will vote purely, or even
primarily, on the basis of parties’ educational spending plans, especially as in the UK local
elections are generally dominated by national politics. It is therefore unlikely that educational
spending is a major issue in most local elections.
Our second instrument is lagged school size, which is an instrument that has been used by
others in the field (Iacovou, 2002). School size (in terms of pupil numbers) is a key factor
predicting the per capita level of funding in a school. The correlation between lagged school
size and expenditure per student is –0.30 and significant at the 5% level. The correlation
between lagged school size and the student teacher ratio is +0.11 and significant at the 5%
Of course for school size to be an adequate instrument it must not impact directly on pupil
achievement. There is little evidence that school size has an effect on pupil achievement, at
least not in studies that use rich pupil level data such as the NPBD. An argument can be
made that more effective schools tend to be bigger because they attract more pupils, thereby
causing a positive relationship between school size and pupil achievement. However, in our
data we are able to control for this to some extent by including an indicator of how popular
and ‘full’ the school is5. As a further robustness check, we also re-estimated our models using
lagged school capacity, rather than lagged school size. This was on the grounds that school
capacity is simply a function of the physical construction of the school, unrelated to current
student enrolment. There is little change in the results when this alternative instrumental
variable is used.
5 That is the school’s percentage capacity utilization , which is the actual number of students in years 7-11
compared to the maximum physical capacity in terms of student numbers, which is determined by the
Department for Education and Skills.
The simultaneous equations model can be framed as a multilevel bivariate response model.
For each individual, we can define a bivariate response
r =1, 2) where and
. In addition, we define two response indicators as follows:
Equations (1) and (2) can then be written in the form of a single equation for the stacked
In the standard bivariate model, both responses are at the individual level and therefore the
bivariate response vector will be of length
(Goldstein, 2003; Chapter 6). In the present
case, however, the responses are defined at different levels of the hierarchy:
is a student-
level response, while
is at the school level. While we could replicate values of for
students in the same school, it is more computationally efficient to restructure the data so that
there is a single observation of
for each school, leading to a response vector of length
. The explanatory variables in (3) are the two-way interactions between and each
element of , and between and the elements of . The random effects in the
attainment and resource equations are fitted by allowing the coefficient of
to vary across
students, schools and LEAs, and the coefficient of
to vary across schools and LEAs.
We estimated model (3) using MLwiN v2.0 (Rasbash et al., 2004).
The data for this paper come largely from the NPDB. PLASC contains school characteristics
(size, type, pupil-teacher ratio etc.), pupil characteristics (age, gender, ethnicity, eligibility for
free school meals etc.) and pupil achievement data at each key stage of the curriculum (ages
7, 11, 14 and 16). We merged into these data additional information on school expenditure
and political control of the local authority, as well as Census information on the socio-
economic characteristics of each child’s neighbourhood.
Our model estimates the impact of school resources on pupil achievement in English,
mathematics and science at age 14, i.e. Key Stage 3 in 2002/3. This consists of a sample of
430,000 pupils. We control for each pupil’s prior achievement at Key Stage 2 (age 11), i.e. in
1999/2000. The dependent variables are continuous test scores, which vary from 0 to almost
9 for maths, and from 0 up to almost 8 for science and English.
The resource variables we use are all at school level, namely expenditure per student6, the
average student teacher ratio in the school and the ratio of students to non-teaching staff. The
resource variables were averaged over the three years that the sample was in secondary
school. We estimated separate models for the expenditure and the staffing resource variables,
since the majority of school spending is on teachers. Teacher salary costs are on average 61
per cent of secondary schools’ expenditure (OFSTED, 2003). If expenditure per pupil and the
pupil-teacher ratio are included in the same model, then the effect of the pupil-teacher ratio is
biased downwards because a lower pupil-teacher ratio for a given level of spending
automatically implies that there are less resources available for other inputs (Todd and
Full descriptive statistics are given in Table 1.
6 Deflated by an indicator of the cost of living in the area, namely the Area Cost Adjustment.
We begin by examining the extent to which student attainment scores are clustered within
schools and LEAs, and school resources are clustered within LEAs. Table 2 shows estimates
of the residual variance at each level, from which estimates of the intra-school and intra-LEA
correlations have been calculated. The estimates for attainment are from estimating separate
three-level models for attainment at age 14 in maths, science and English, adjusting for
attainment at age 11 in the same subjects. Thus the variance components represent the
variance at each level in progress from entry into secondary school up to age 14. The
estimates for school resources are from fitting separate two-level models to the expenditure
and staffing measures. At this stage of the analysis, no student or school characteristics have
been included in any of the models.
The intra-school correlations for attainment show that there are moderate school effects on
performance in all three subjects, with the strongest effect on English scores: 22% of the total
variance in English progress is due to differences between schools. After taking into account
school effects on progress, LEA effects are very weak. Turning to the school resource
measures, we find that 19% of the total variance in expenditure per student can be explained
by differences between LEAs. This moderately high intra-LEA correlation implies that while
LEAs vary in their mean expenditure per student (averaging across all schools in an LEA),
there is similarity in the expenditure of schools in the same LEA. There is rather less
homogeneity within LEAs in pupil-teacher ratios. This is a reflection of the fact that, whilst
overall per student spending in each school is determined at LEA level, schools themselves
have much more discretion over how this money is spent, and in particular they have some
control over the pupil-teacher ratio in each class and year in the school.
We next consider the evidence for the endogeneity of school resources with respect to student
attainment. Table 3 shows the results from likelihood ratio tests comparing, for each subject
and resource measure, a standard multilevel model and a simultaneous equation model. All
models include a number of controls for student background and school characteristics, as
described in Section 4. In the standard model, the covariances between the school and LEA
residuals across the attainment and resource equations are constrained to equal zero, while in
the simultaneous equation model these covariances are freely estimated. Thus we are testing
the null hypothesis that
, which is a test of the exogeneity of the relationship
between attainment and resources. Rejection of the null implies that school resources are
endogenous to attainment, in which case estimates of the impact of resources on attainment
from the standard multilevel model will be biased. We find strong evidence that both per
student expenditure and the pupil-teacher ratio are endogenous to attainment in science.
There is also evidence that staffing and, at the 10% level, expenditure are endogenous to
maths attainment. We conclude, however, that both resource variables are exogenous to
Having established that both of our school resource indicators are endogenous to attainment
in maths and science, we can examine estimates of the residual correlations to assess the
direction of selection effects and whether they operate at the school or LEA level or both.
The correlation at the LEA level is interpreted as the (residual) association between the LEA
mean level of resources (expenditure or staffing) and LEA mean attainment. A strong
correlation at this level would suggest a selection effect that is driven by the way in which
central government allocates resources to local authorities. The residual correlation at the
school level measures the within-LEA association between school resources and school mean
attainment. A strong correlation at the school level implies a selection effect that is due to the
nature of resource allocation among schools within an LEA, i.e. non-random allocation
within LEAs. A dominant LEA-level correlation would suggest that selection is largely the
result of central government policy and political choice at local level, as Conservative LEAs
tend to be lower spending authorities.
Table 4 shows estimates of the correlation between the school and LEA residuals across the
resource and attainment equations in the simultaneous equation model. We discuss only the
interpretation of the correlations between resources and attainment in maths and science,
since exogeneity tests (Table 3) suggest that resources may be assumed exogenous to English
scores. The school and LEA-level correlations between the residuals for expenditure per
student and attainment in maths and science are negative; these correlations are strongest for
science and, for both subjects, the LEA-level correlation is the largest. A negative correlation
at the LEA level implies that unobserved LEA factors influencing school expenditure are
negatively correlated with the unobserved LEA-level determinants of student attainment.
Equivalently we may conclude that, even after controlling for a rich set of explanatory
variables, there is a negative association between the mean level of expenditure in an LEA
and the LEA mean attainment. A negative selection effect is consistent with the policy of
compensatory funding where schools with greater learning needs receive more funding per
student (see Section 2). The evidence suggests that the selection effect is stronger at the LEA
level, which is as one would expect, given that the expenditure for education that is
notionally allocated to each LEA (the education Standard Spending Assessment discussed in
Section 2) is determined by central government on the basis of a formula that explicitly
includes many factors likely to be highly correlated with pupil attainment. For example,
central government takes the following factors into account when determining the level of
each LEA's education SSA: the proportion of immigrants in the area, the proportion of the
resident population on benefits and indicators of deprivation. The selection effect is greatest
for science, particularly at LEA level. It appears that the socio-economic factors that
determine each LEA's allocation for expenditure on education are also more highly correlated
with science achievement. Further investigation is required as to why this might be the case
but our results clearly indicate that resourcing effects vary across subjects.
The residual correlations between maths and science attainment and the pupil-teacher ratio
follow a similar pattern to those for attainment and expenditure, although the correlations are
now positive because a high pupil-teacher ratio is an indicator of lower resources. However,
the correlations at both levels are stronger than for expenditure, particularly at the school
level. The fact that the selection effect is greater for the pupil-teacher ratio, as compared to
expenditure, indicates that there is more autonomy for schools to determine how they spend
their resources. The large positive selection effect is consistent with the widely held view that
education professionals tend to allocate poorer performing students into smaller class sizes.
This phenomenon may also occur at LEA and school level, whereby schools with lower
performing pupils either are allocated or opt for lower pupil teacher ratios. This would come
about by LEAs systematically attempting to reduce the pupil-teacher ratio in their most
disadvantaged schools and by schools with disadvantaged pupils opting to have a lower
pupil-teacher ratio for a given level of expenditure, as compared to their more prosperous
In Table 5, we demonstrate the impact of adjusting for endogeneity on estimates of the effects
of school resources on student attainment. For each subject and resource indicator,
standardised coefficients are presented for two models: the standard multilevel model
denoted in (1), which assumes that resources are exogenous, and the simultaneous equation
model denoted jointly by (1) and (2), which allows for endogenous resource effects. Based
on the results from either model, we would predict a statistically significant, though small,
improvement in students’ maths and science progress for an increase in the expenditure per
student or a decrease in the pupil-teacher ratio, et ceteris paribus. When we allow for
endogeneity, however, the magnitude of these effects increases substantially. The increase in
effect size is expected due the nature of selection implied by the direction of the residual
correlations between resources and attainment (Table 4).
To assess the effects of school resources on English attainment, we may interpret the
estimates from the standard multilevel model due to the lack of significance of the residual
correlations in the simultaneous equation model (Table 3). We find a counter-intuitive
negative effect of expenditure per student on English progress, and no significant effect of the
pupil-teacher ratio. It has been suggested that the school environment has a lesser effect on
progress in English than in other subjects, partly because the home environment is relatively
more important in determining language development. This might explain why the pupil-
teacher ratio does not have a significant impact on pupil progress in English, particularly at
the relative low levels of pupil-teacher ratio found in the English education system (relative
to world standards). However, it does not explain why expenditure might be negatively
related to English progress.
This paper has adopted a multilevel simultaneous equation modelling approach to determine
the impact of school resources on pupil attainment at age 14. The primary objective of the
paper was to determine whether additional expenditure on education would lead to improved
pupil attainment, clearly an important issue for policy makers attempting to raise standards in
education and improve the performance of low achieving groups. The paper, building on
previous work using an instrumental variable approach (Levačić et al., 2005) addresses a
number of methodological difficulties in this literature, in particular the endogeneity of
school resource levels, and the intra-school correlations in student responses.
In policy terms our results suggest the following. Firstly, additional resources do have a