Efficiency in the higher education sector: technical exploration
Source: data.gov.uk (2013-10-16)

The efficiency scores, as shown in the historgrams, that are derived from the econometric analysis.

Department for Business, Innovation and Skills (BIS)
 
 
BIS RESEARCH PAPER NO. 113 
Efficiency in the Higher Education 
sector: A technical exploration 
SEPTEMBER  2013 

 
 
 
 
EFFICIENCY IN THE HIGHER EDUCATION SECTOR: 
A TECHNICAL EXPLORATION 
 
Jill Johnes 
Geraint Johnes 
Department of Economics 
Lancaster University 
LA1 4YX 
E: xxx@xxx.xx 
T: xx-xxxx-xxxx 
F: xx-xxxx-xxxx 
 
 
 
 
 
 
 
Acknowledgement:  We are grateful to Vasileios Pappas for help in compiling the panel 
data. 
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Efficiency in the Higher Education sector: A technical exploration 
 Contents 
Contents............................................................................................................................................ 3 
Executive summary........................................................................................................................... 5 
1. 
Introduction ............................................................................................................................... 7 
2. 
Review of the literature on costs in higher education in the UK................................................ 9 
3. 
Specification of the cost function: linear versus quadratic ...................................................... 13 
3.1 Linear specification ............................................................................................................... 13 
3.2 Non-linear specification......................................................................................................... 14 
4. 
Cost function estimation and the measurement of efficiency.................................................. 17 
5. 
Empirical analysis ................................................................................................................... 22 
5.1 Model specification................................................................................................................ 22 
5.2 Data....................................................................................................................................... 26 
5.3 Linear model over 3 time periods and for the whole time period .......................................... 27 
5.4 Quadratic model over 3 time periods and for the whole time period..................................... 32 
5.5 Linear latent class model over 3 time periods and for the whole time period ....................... 33 
5.6 Quadratic latent class model over 3 time periods and for the whole time period.................. 38 
5.7 Comparison of results with pre-defined groupings of universities......................................... 40 
5.8 Discussion............................................................................................................................. 44 
6. 
Conclusions............................................................................................................................. 45 
Appendices ..................................................................................................................................... 46 
Appendix 1: Summary of results of previous studies of costs in the UK higher education sector46 
Appendix 2: Alternative non-linear cost function specifications .................................................. 56 
Appendix 3: Data definitions ....................................................................................................... 57 
Appendix 4: Additional results..................................................................................................... 59 
3 

Efficiency in the Higher Education sector: A technical exploration 
 
References.................................................................................................................................. 73 
 
 
4 

Efficiency in the Higher Education sector: A technical exploration 
Executive summary 
 
Costs incurred by a higher education institution in producing its output vary with the levels 
of output that it produces. They also vary with a number of other factors such as quality, 
student demographics, and the nature of the real estate. Even after allowing for all relevant 
factors, costs are likely to vary because institutions differ in their levels of efficiency. It is 
important to study differences in efficiency because this offers lessons about good practice 
that can lead to improvements in the performance of the higher education system as a 
whole. 
Numerous studies analysing the costs of higher education institutions have been 
conducted, including many in the United Kingdom. Advances in statistical methodology 
have allowed these simultaneously to evaluate cost structures and institutional efficiency. 
The most recent studies, using latent class or random parameter stochastic frontier 
models, do this in a context that makes allowance for differences between types of 
university (reflecting, for example, variation in student quality) that are not easily captured 
by the data.  
In all exercises of this type, an important judgement call must be made concerning which 
of the factors affecting costs should be taken into consideration when determining 
institutional efficiency. Clearly allowance should be made for differences in the level of 
output. Arguably allowance should also be made for differences in costs that are due to, 
say, the historical nature of an institution’s real estate, or to the role the institution plays in 
the widening participation agenda. In general, the more refined is the model of costs, the 
more efficient institutions appear to become, because the model contains more information 
that can be used to explain cost differences. There is no objective way of determining how 
much detail should be included in the analysis. 
In this report, various stochastic frontier models are used to evaluate efficiency in English 
higher education institutions over the period 2003/04 to 2010/11, and over three sub-
periods within that time frame. The stochastic frontier approach involves fitting a curve 
through data on costs and a variety of explanatory variables. This is not, however, a line of 
best fit; rather it is an envelope that defines an efficiency frontier – a curve that shows the 
lowest possible costs at which a given set of outputs can be produced. The position of this 
curve can then be used as a benchmark against which the efficiency with which each 
institution produces its output can be determined.  
These are estimates of efficiency, and, like any other statistical estimates, are measured 
with error. It is possible for one estimate to be different from another, but not significantly 
so in the statistical sense. In addition, there is no consensus about what value indicates an 
efficient or inefficient score. Consequently, the results should be interpreted with caution. 
The latent class stochastic frontier approach refines this method by separating the 
institutions into a number of classes based on institutional characteristics that are revealed 
by the data. The statistical method then estimates a cost frontier for each class, thereby 
5 

Efficiency in the Higher Education sector: A technical exploration 
 
allowing each institution’s efficiency to be assessed relative to other institutions of the 
same type.  
An alternative to the latent class approach is to prescribe groups of institutions that, on a 
priori grounds, appear to share similarities with each other. The groups used in the recent 
HEFCE report on the impact of the HE reforms are used to define these clusters of 
institutions.  
The empirical work conducted in this project involves the evaluation of a large number of 
models of institutions’ costs, each of these models producing measures of institutions’ 
efficiency. The models range from a simple specification in which the explanatory variables 
include only linear terms in the outputs produced by each institution, through models that 
include further variables to capture factors such as widening participation and real estate 
characteristics, to models that include a rich variety of interaction terms designed to 
capture economies of scale and scope. The models are estimated first on the assumption 
that the same cost structure applies to all institutions, and secondly on the assumption that 
different groups of institutions exist for which cost structures may differ. 
A key finding of the report is that, once differences between institutions are accounted for1, 
the variation in efficiency scores across institutions is greatly reduced, with a concentration 
of scores above 0.9 (where a score of one represents efficiency). Indeed, the relatively 
small number of institutions with low scores is exclusively made up of small and specialist 
institutions. The results do not, therefore, support the notion that substantial sector-wide 
gains could be made by using efficiency scores as a criterion for resource allocation.  
It may be argued that the more sophisticated models are the ones most appropriate for 
evaluating the efficiency of institutions, since they make most allowance for the different 
circumstances that might influence costs. Nevertheless there are drawbacks associated 
with using these models as a means of understanding costs in higher education. The 
greater sophistication of these models comes at a price. By increasing the number of 
explanatory variables used in the analysis, it becomes more likely that co-movement of 
some of the variables reduces the precision with which the impact of any one of them on 
costs is estimated. Moreover, the estimation of cost models that are specific to distinct 
classes of institutions involves, in effect, a reduction in the sample size used to estimate 
the model for each class. For these reasons, the simpler models reported here have 
considerable merit as means of understanding cost structures.  
The analyses reported in this paper provide a useful starting point in understanding 
differences in efficiency across higher education institutions. The data on which they are 
based, however, are highly aggregated, and fail to capture the detail of how and why 
efficiency scores vary. A more in-depth analysis of institutions that achieve scores at either 
end of the distribution, such as via a case study approach, would be instructive as a 
means of ascertaining the organisational factors underpinning efficient performance. This 
is likely to necessitate the collection of qualitative data of a kind that does not fit easily 
within the statistical approach adopted in this report.  
                                            
1 Even when only using a relatively unrefined latent class modelling procedure 
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Efficiency in the Higher Education sector: A technical exploration 
1. Introduction 
 
Under the new funding mechanism for higher education in England, many students will not 
pay off the whole of their debt within 30 years. The tuition fee charged by providers is not 
necessarily the same, therefore, as the amount paid by customers. The Resource 
Accounting and Budgeting (RAB) cost of the student loans scheme is estimated to amount 
to around 35 per cent of the value of the loan book2, more than under the previous system, 
although there has been a simultaneous reduction in the amount of teaching grant, which, 
in effect, had a RAB cost of 100%. In addition, the fact that students do not make up-front 
payments reduces their price sensitivity. These factors have the potential to produce a 
market failure such that the usual competitive pressures fail to incentivise providers to 
become more efficient. Moreover, the government continues to subsidise both teaching 
and research, and has an interest in the efficient operation of all aspects of higher 
education. An analysis of the cost structure and efficiency of higher education institutions 
(HEIs) is therefore of on-going interest and importance. 
Extensive work has been undertaken on evaluating efficiency in the higher education 
sectors of various countries. Work in the United Kingdom (UK) is of particular relevance 
here (see, for example, Johnes and Taylor 1990; Johnes, J 1996; Johnes et al. 2005; 
Johnes 2008; Thanassoulis et al. 2011). Much of the literature on efficiency measurement 
has emphasised the statistical evaluation of costs (Cohn et al. 1989), since efficiency 
concerns how a given output can be produced at as low a cost as possible. Statistical and 
econometric techniques have been developed which allow efficiency to be evaluated for 
each institution. These statistical methods do not drill down into the detail of how 
institutions do what they do3; rather they offer the analyst both an understanding of how 
costs are determined in higher education institutions as a whole, and a measure of the 
extent to which different institutions manage to produce their outputs efficiently. It allows 
an assessment to be made of the extent to which institutions differ in terms of their 
efficiency, and it also allows an analysis of changes in efficiency over time. At a higher 
level of abstraction, the method provides much the same input into benchmarking 
exercises as do more qualitative exercises, but it offers the advantage of a focus on the 
front-end activities of teaching and research. A number of studies exist which have 
adopted this general approach for UK higher education4 (Glass et al. 1995a; 1995b; 
Johnes, G 1996; Johnes 1997; 1998; Izadi et al. 2002; Johnes et al. 2005; Stevens 2005; 
Johnes et al. 2008b; Johnes and Johnes 2009; Thanassoulis et al. 2011). 
                                            
2 Announced by David Willetts at the Higher Education Policy Institute Spring Conference (15th May 2013) - 
see https://www.gov.uk/government/speeches/david-willetts-minister-for-universities-and-science-hepi-
conference-speech. 
3 Unlike, for example, the Transparent Approach to Costing (TRAC). 
4 Note that there are also notable studies of cost structures of higher education systems of other countries 
such as Japan, Italy, Spain, Portugal, the USA and Germany, respectively (Hashimoto and Cohn 1997; 
Agasisti and Salerno 2007; Johnes and Salas Velasco 2007; Johnes et al. 2008a; Agasisti and Johnes 2009; 
2010; Johnes and Schwarzenberger 2011). 
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Efficiency in the Higher Education sector: A technical exploration 
 
The purpose of this report is to undertake an empirical study of costs and efficiency in 
English higher education using data from 2003/04 to 2010/11. The report is in 6 sections of 
which this is the first. A review of empirical studies of costs in UK higher education is 
presented in section 2. Section 3 examines linear and non-linear specifications of the cost 
function and the implications of particular choices. The methods of estimating cost 
functions are considered in section 4 which also looks at how estimates of efficiency can 
be derived from the cost function. The empirical analysis using data, in turn, from 2003/04 
to 2004/05, 2005/06 to 2007/08, and 2008/09 to 2010/11, is presented in section 5. 
Conclusions are drawn in section 6.  
8 

Efficiency in the Higher Education sector: A technical exploration 
2. Review of the literature on costs 
in higher education in the UK 
 
There is now a considerable literature concerning the cost structure of systems of higher 
education, some of which also examines the efficiency levels of HEIs. The data underlying 
the empirical studies of costs in UK higher education span a period from the 1960s up to 
the early 2000s, and it is this literature which will be predominantly reviewed in this 
section. Additional details of the studies reviewed can be found in Appendix 1.  
The first cost functions to be estimated and published for higher education in the UK relate 
to data from 1968 (Verry and Layard 1975) and include as outputs measures of both 
teaching and research. Six separate linear cost equations are estimated, one for each 
subject area: arts; social sciences; mathematics; physical sciences; biological sciences, 
and engineering. For undergraduate teaching, marginal costs are higher in the ‘laboratory’ 
science subjects than in the ‘classroom’ subjects such as arts, social sciences and 
mathematics. The marginal cost of postgraduate teaching is higher than for undergraduate 
teaching, and (like undergraduate teaching) is more expensive in the sciences than in the 
arts. 
Scale economies (which arise when an expansion of scale of production leads to costs 
rising less than proportionately with output) are observed in all subject areas (apart from 
physical sciences) but are significant only in the social sciences. In fact, the linear nature 
of the cost function makes it inevitable that there should be scale economies so long as 
there exist some fixed costs of provision – and in this respect the finding for physical 
sciences is somewhat curious. 
Though clearly dated, this study acts as a benchmark against which we can compare the 
results of later studies. By including measures of both teaching and research output, the 
Verry and Layard (1975) study is the first to recognise the impact that a multiplicity of 
different types of output has on costs and on the technology of production in universities. 
The linear cost function, however, does not allow for the possibility of synergies that arise 
from the joint production of the multiple outputs, and the separate estimation of cost 
functions by subject does not allow for synergies resulting from joint production across 
subject areas. This means that the methodology adopted by Verry and Layard is incapable 
of explaining why undergraduate education, postgraduate education and research are all 
carried out in the same institution; nor can it explain why different subjects are delivered 
within the same institution. The form of the model that is being estimated does not admit 
the possibility that there might be benefits (in the form of cost savings) arising from such 
joint production. The outputs, moreover, are limited to teaching and research only, and 
there is no consideration that there may be inefficiency in the sector. The omission from 
the sample of two universities (Oxford and Cambridge) because they are outliers raises 
the issue of comparability of the production units in the higher education sector – this is no 
doubt a growing concern as the UK higher education sector changes in its composition.  
9 

Efficiency in the Higher Education sector: A technical exploration 
 
Some of these issues are addressed, though not resolved, in a companion study (Verry 
and Davies 1976); but the seminal work by Cohn et al. (1989) was the real trigger for more 
sophisticated studies of costs of higher education in the UK which gradually got to grips 
with the shortcomings of these early models. Four studies, which use data for the pre-1992 
definition of the university sector (Glass et al. 1995a; 1995b; Johnes, G 1996; Johnes 
1998), address the issue of synergies in higher education production by using complex 
non-linear functional forms for the cost equations and by defining teaching outputs by 
broad subject area. Where undergraduate teaching is split by broad subject, the studies 
find that average costs of teaching within the arts are lower than within the sciences (at 
both undergraduate and postgraduate levels). For a given subject area, however, there 
appears to be little difference in the average costs of undergraduate compared to 
postgraduate teaching (Johnes, G 1996; Johnes 1998). 
With regard to ray returns to scale – that is, returns to scale that arise from a simultaneous 
increase in all types of output being produced – there is evidence that scale economies 
are significant and unexhausted for the typical university (Glass et al. 1995a; 1995b; 
Johnes, G 1996; Johnes 1998). But the results regarding product-specific returns to scale 
– the returns to scale associated with an increase in one output only – are mixed (details 
can be found in Appendix 1) (Glass et al. 1995a; 1995b; Johnes, G 1996; Johnes 1998).  
Evidence regarding global economies of scope – the economies arising from producing all 
outputs together rather than separately – is also mixed. When teaching outputs are split by 
subject group in Johnes, G (1996) and Johnes (1998) we observe global economies of 
scope, but this contrasts with the finding of no significant scope economies when teaching 
output is aggregated across all subjects (Glass et al. 1995a; 1995b). This may be because 
studies where the outputs are more highly aggregated are, in effect, aggregating out the 
possibility of observing scope economies that exist at a finer level of analysis. 
Two of these studies are the first to allow for inefficiency in the estimation of the cost 
function by using frontier estimation methods: stochastic frontier analysis (SFA) (Johnes, 
G 1996; Johnes 1998) - and data envelopment analysis (DEA) - (Johnes 1998)5. Efficiency 
of each HEI is measured on a scale of zero to 1 with the latter representing complete 
efficiency. The studies find that mean efficiency for the higher education sector as a whole 
is over 0.90 (using the DEA method). The estimates of efficiency derived from the two 
frontier estimation methods are positively correlated, although, at 0.133, the magnitude of 
the correlation coefficient is rather low, suggesting that the two frontier estimation methods 
provide different rankings of HEIs based on estimated efficiency. Any estimate of efficiency 
for an individual institution therefore needs to be treated with extreme caution. 
The higher education sector in the UK saw major changes in its composition in 1992, when 
polytechnics were given university status, and later in 2003 when it was announced that 
Colleges of Higher Education would be allowed to apply for university status. Six studies 
have estimated cost functions from data referring to the extended higher education sector 
(Johnes 1997; Izadi et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes and 
Johnes 2009; Thanassoulis et al. 2011).  
                                            
5 See section 4 for details of the frontier methods of estimation: SFA and DEA. 
10 

Efficiency in the Higher Education sector: A technical exploration 
For undergraduate teaching, average costs vary by subject and are highest in the sciences 
(or in medicine followed by other sciences where a more detailed subject split is made) 
and lowest in the non-sciences (Johnes 1997; Izadi et al. 2002; Johnes et al. 2005; Johnes 
et al. 2008b; Johnes and Johnes 2009; Thanassoulis et al. 2011). The cost of 
postgraduate teaching is generally higher than undergraduate teaching in the sciences 
and non-sciences, but lower than undergraduate teaching in medicine (Johnes 1997; Izadi 
et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009; 
Thanassoulis et al. 2011), but no reasons are provided for why this might be the case. 
The findings regarding economies of scale and scope, from the studies based on the 
extended higher education sector, differ from the findings of the earlier studies. Ray 
returns to scale are close to constant or decreasing for the typical university (Johnes 1997; 
Izadi et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009) 
implying, in the latter case, that expanding output leads to an increase in costs. Findings 
on product-specific economies of scale are mixed and depend on choice of data (single-
year or panel data), definition of outputs, the functional form of the cost function and the 
estimation method. Global diseconomies of scope are a consistent finding in these later 
studies (Johnes 1997; Izadi et al. 2002; Johnes et al. 2005; Johnes et al. 2008b; Johnes 
and Johnes 2009). 
Results regarding efficiency in the extended higher education sector also vary by choice of 
data and estimation method. Using only a single year of data, average efficiency for the 
sector as a whole is estimated to be around 0.88 (Izadi et al. 2002). This result, somewhat 
surprisingly given the variety of HEIs included in the 2002 study, is not too dissimilar to 
results based on only 50 pre-1992 HEIs (Johnes 1998). There is a considerable range in 
efficiency, however, from under 0.40 to 0.99, and this is likely a consequence of the 
diversity of the HEIs in the sample. Institutions at the lower end of the distribution of 
efficiencies tend to have characteristics that suggest that their relatively high costs (given 
output) are due to idiosyncrasies that are not adequately captured by the data on outputs, 
and the efficiency scores attached to these institutions therefore needed to be treated with 
caution.  
Studies which use a panel of data over a number of years find that mean efficiency is 
generally lower than in the single-period models. The magnitude also varies by estimation 
method:  mean efficiency across the whole sector is 0.69 on the basis of SFA (Johnes et 
al. 2005; Johnes et al. 2008b), 0.863 when DEA is used (Thanassoulis et al. 2011), and 
0.753 in the case of a random parameter SFA model (Johnes and Johnes 2009). A simple 
SFA, unlike DEA and a random parameter SFA, does not make any allowance for each 
HEI to have a different set of objectives or mission. Thus any efficiency results derived 
using SFA should be interpreted with this in mind.  
There is strong evidence that efficiency varies by HEI type. Colleges of higher education 
appear to be least efficient and post-1992 and some pre-1992 HEIs are typically the most 
efficient (Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009; Thanassoulis 
et al. 2011). The available software has not, however, allowed evaluation of the extent to 
which these differences are statistically significant. 
The expansion of the higher education sector calls into question the comparability of HEIs 
included in the sample used to estimate the cost function. We need to be sure that HEIs 
included in the sample are comparable in terms of their environment such as ‘quality’ of 
11 

Efficiency in the Higher Education sector: A technical exploration 
 
students, input prices, and real estate costs. While one study finds that the proportion of 
students achieving first and upper second class degrees has a positive influence on both 
costs and on efficiency (Stevens 2005), student quality is generally not a significant 
determinant of costs (Verry and Davies 1976; Johnes et al. 2005; Johnes et al. 2008b; 
Johnes and Johnes 2009). It is possible that the random parameter specification used in 
some of these later studies already accounts for persistent quality differences across 
institutions and hence leads to the finding of insignificance of the quality variable.   
Dummy variables such as a London dummy and an Oxbridge dummy, included in models 
to reflect, respectively, differences in input prices and costs of upkeep of ancient buildings, 
are not significant determinants of costs (Johnes et al. 2005; Johnes et al. 2008b; Johnes 
and Johnes 2009). Once again, these results may be a consequence of using a random 
parameter framework.6 
Clearly much work has been undertaken on estimating the cost functions and efficiencies 
of UK HEIs. Many of the earlier studies restrict output to just teaching and research, are 
estimated on the basis of a restricted sample of HEIs, and do not allow for inefficiency in 
higher education production; indeed only four of all the studies reviewed include a 
measure of the third mission outputs of universities, use data which reflect the current 
composition of the English higher education sector  and use a frontier estimation method 
(Johnes et al. 2005; Johnes et al. 2008b; Johnes and Johnes 2009; Thanassoulis et al. 
2011). Thus conflicts in findings from the different studies regarding, for example, average 
costs and economies of scale and scope, are not surprising.  
The diversity observed in the UK higher education sector raises difficulties in estimation 
which have not, to date, been adequately addressed. Previous studies have examined 
costs and efficiency amongst pre-defined mission groups and have found differences 
between them (Johnes et al. 2005; Johnes et al. 2008b; Thanassoulis et al. 2011). But 
these studies are based on preconceived notions of how costs ought to vary. In a later 
study (Johnes & Johnes 2009), a random parameters approach is adopted which 
acknowledges that each university varies in its mission and faces distinct circumstances 
affecting its costs, but which allows the data themselves (rather than the researchers’ 
preconceptions) to determine the nature of each institution – and hence what each 
institution’s cost function should look like. The random parameters frontier estimation 
model is an exciting development. By allowing parameters to vary across institutions, cost 
functions for HEIs that are clearly different from one another can be estimated in a single 
framework and without recourse to separate estimation for pre-determined groups of HEIs. 
The disadvantage is that the model can be difficult to fit; indeed such were the demands 
on the data in this particular study that some of the richness of the previous models was 
lost by amalgamating the medicine and science undergraduate teaching outputs, and by 
dropping the third mission output measure from the equation. The random parameters 
approach might also be viewed as being too permissive in that, in effect, it allows each 
institution to define its own mission.  
                                            
6 Work reported later in the present paper, which does not use a random parameter model, finds a significant 
Oxbridge effect. 
12 

Efficiency in the Higher Education sector: A technical exploration 
3. Specification of the cost 
function: linear versus quadratic 
 
Costs typically increase as output increases. In many contexts, and certainly in the case of 
most HEIs, the output of each producer comes in a multiplicity of forms. Hence a HEI 
might produce graduates in a number of disciplines at a number of levels (such as 
bachelors, masters, doctorates), and might also produce research and engage in 
knowledge transfer across various fields. Each of these distinct outputs has an impact on 
costs. Moreover, these costs are likely to differ across institutions that have different types 
of student intake, and are likely to vary according to the extent to which they produce 
outputs of different quality.  
3.1 Linear specification 
The simplest way to consider the relationship between costs and output is to suppose that 
each unit of each type of output adds a certain (fixed) amount to total costs. This approach 
is appealing in that it suggests a functional form for a cost equation that is particularly 
simple to estimate using statistical methods. That functional form is linear, and given by an 
equation such as 
C =  + T + R 
 
 
 
 
 
 
 
 
 
(1) 
where C denotes total costs and T and R respectively denote the quantity of distinct types 
of output being produced, say teaching and research. The ,  and  terms are known as 
parameters of the model, and they are estimated statistically.  
The simplest way to do this is to conduct a least squares regression, using data for a 
cross-section or panel of institutions; in effect, this involves evaluating a line (or, strictly 
speaking, a plane) of best fit through a scatter plot of data in three dimensions – one 
dimension for costs, and one for each of the output variables. A more appropriate 
estimation technique is the stochastic frontier approach (SFA), which, rather than providing 
the line of best fit, evaluates the envelope of cost below which a certain combination of 
outputs cannot be produced, however efficient the producer. In this context, a latent class 
estimator can be used to estimate a separate cost equation, if desired, for each group of 
institutions (assuming two or more groups). More discussion of estimation methods and 
the implications of choice of method for efficiency estimation are provided in section 4. 
Once the estimation has been conducted, it is straightforward to interpret  as fixed costs 
(the costs that would be incurred even if production of teaching and research were zero),  
as the (marginal) cost associated with each unit of teaching, and  as the (marginal) cost 
associated with each unit of research. In the case of the latent class model, the 
parameters ,  and  will be different for each of the classes – indicating that one group 
of institutions has different fixed costs to the other, and that the (marginal) costs 
associated with teaching and research also differ across the groups.  
13 

Efficiency in the Higher Education sector: A technical exploration 
 
This specification of the cost equation has the considerable merit of simplicity. The model 
has few parameters, and this reduces the likelihood with which statistical problems will be 
met that hamper the estimation. This is a particularly important consideration if the models 
being estimated are rich in terms of the number of explanatory variables being considered 
as potential determinants of costs. Moreover, the estimated parameters from a linear 
specification lend themselves to straightforward interpretation. 
The linear specification of the cost function allows a limited consideration of returns to 
scale. If  is equal to zero, then a given percentage increase in both outputs, T and R, 
leads to the same percentage increase in C, thus implying constant returns to scale. If, on 
the other hand,  is greater than zero, a given percentage increase in both outputs leads 
to a smaller percentage increase in costs. This is because, with the increase in T and R, 
fixed costs can now be spread over a higher number of units of output. In this case we 
observe increasing returns to scale. It is possible also to observe diseconomies of scale 
(where  is less than zero), though this would be somewhat counterintuitive in the case of 
higher education since it would imply that system-wide costs are minimised by organising 
provision through a large number of very small providers.  
While the linear specification of costs can provide some information about returns to scale, 
however, the specification itself is clearly highly restrictive in this regard. If returns to scale 
are increasing for some level of output, then they must be increasing for every level of 
output. Likewise, if they are decreasing (or constant) for some level, they must be 
decreasing (or constant) at every level. This does not correspond to the conventional 
thinking that returns to scale are initially increasing but subsequently constant or 
decreasing as output rises. Neither does it correspond to the stylised facts: while there are 
relatively few very small institutions in existence, the higher education system is not 
dominated by a single very large institution, sweeping up all the available economies of 
scale. It seems more reasonable to consider a specification of the cost equation that is 
capable of accommodating both increasing and decreasing returns to scale at different 
levels of output.  
3.2 Non-linear specification 
The preceding points suggest that a nonlinear specification of the cost function might have 
merit, and this in turn raises the question of what type of nonlinear representation of the 
cost equation might be appropriate. One specification that has been particularly popular in 
the literature is the quadratic cost function7. Using the same two outputs as above, this 
has the form 
                                           
C =  + T + R + T2 + R2 + TR 
 
 
 
 
 
 
(2) 
The squared terms in T and R, and the interaction term (where T and R are multiplied) 
give this equation its quadratic (nonlinear) character. The equation has several appealing 
 
7 Appendix 2 presents some alternative non-linear cost function specifications. The quadratic specification is 
commonly used in the literature, largely because it can be regarded as a linear equation in variables that are 
nonlinear combinations of outputs. This makes estimating the equation considerably more straightforward 
than estimating other non-linear functions. 
14 

Efficiency in the Higher Education sector: A technical exploration 
properties. First, returns to scale can be different at different levels of output. Depending 
on the parameter values, it is possible for there to be economies of scale at low levels of 
output, and diseconomies of scale at very high levels of output, thus providing a rationale 
for the existence of institutions that are neither very small nor very large. Secondly, the 
interaction term allows for the possibility that the joint production of the distinct types of 
output under consideration can yield economies. These are known as economies of 
scope, or synergies. In an important sense, the (potential) existence of economies of 
scope explains the existence of organisations as we know them. In the case of HEIs, such 
economies might explain why research is conducted within the same organisation as 
teaching, why postgraduates receive their training in the same organisations as 
undergraduates, or why tuition and research in the arts are delivered in the same 
organisations as tuition and research in the sciences.  
The quadratic function thus provides a much richer framework within which to analyse 
costs than does the linear function. Both are parametric, and therefore impose restrictions 
on the shape that the cost curve can take. But, since the quadratic nests the linear as a 
special case8, the quadratic function is clearly more general. It is important to note, 
however, that increasing the number of output types has very different implications for the 
two types of specification. In the linear case, increasing the number of outputs by one 
leads to an increase of one in the number of terms on the right hand side of the equation.  
In the quadratic case, increasing the number of outputs by one results in an increase of 
2+x in the number of parameters, where x is the number of output types. This means that, 
even when considering only a modest number of outputs, the specification of the quadratic 
cost function involves many terms on the right hand side of the equation. This can result in 
statistical problems owing to over-parameterisation and multicollinearity9. For this reason, 
the quadratic model is most suitable when a fairly parsimonious model is under 
consideration.  
A further characteristic of the quadratic model is that each output appears more than once 
in the set of explanatory variables, because it appears in squared and interaction terms, 
not just as a linear term. This makes the interpretation of parameters more difficult than in 
the linear model. It is, however, possible to extract information from the model about 
measures that are of policy interest, such as the (marginal and average) cost associated 
with the provision of each type of output. 
Once the quadratic cost function has been estimated – using the same types of statistical 
methods as are used to evaluate the parameters of a linear function – it is straightforward 
to calculate the marginal costs associated with each type of output. This measure 
indicates how much an extra unit of output adds to total costs10. In contrast to the linear 
case, however, where the cost function is quadratic the marginal cost will not be a 
                                            
8 If in the special case that the coefficients , ,  in the quadratic model (equation (2)) are all zero, then 
equation (2) simply becomes the linear model of equation (1) with =, =, and =. Thus equation (2) is 
said to ‘nest’ equation (1). 
9 Multicollinearity occurs when two or more of the variables on the right hand side of the equation are highly 
correlated. It can lead to imprecise estimates of the coefficients of the model, with small changes in model 
specification often leading to large changes in the estimated impact of each variable on left hand side 
variable – costs in this case. 
10 Formally, the marginal cost is found by differentiating costs with respect to the output type of interest. 
15 

Efficiency in the Higher Education sector: A technical exploration 
 
constant. It will depend on the level of output. Owing to the existence of the squared and 
interaction terms, the marginal cost associated with each type of output will vary with the 
amount of that output type and with the amount of other output types produced. Hence the 
quadratic form of the cost function allows the returns to both scale and scope to vary, 
depending on the output profile of an institution. 
The average incremental cost (AIC) associated with the production of a particular output 
type can be calculated as the difference between total costs at the outturn level of output 
and the estimate of what total costs would be if none of the output type of interest were 
produced (all other outputs remaining equal), expressed as a proportion of the outturn 
level of that output type11.  
  
 
                                            
11 See Baumol et al. (1982). 
16 

Efficiency in the Higher Education sector: A technical exploration 
4. Cost function estimation and the 
measurement of efficiency 
 
The previous section made allusions to various statistical methods for estimating cost 
functions (such as least squares regression, stochastic frontier analysis, and latent class 
models). The choice of estimation method implies assumptions regarding the efficiency of 
the group of organisations whose cost function is being estimated, and hence has 
implications for the measurement of efficiency. This section examines the issues of cost 
function estimation and the measurement of efficiency in more detail. 
Organisations of various types have a variety of motivations that lead them to seek to be 
efficient. In sectors characterised by intense competition, efficiency is a prerequisite of 
survival. Elsewhere, efficiency is needed in order to ensure that the objectives of the 
organisation can be maximised. Where organisations are funded, at least in part, by the 
public purse, government has a responsibility to the taxpayer to ensure that resources are 
used efficiently. Yet the evaluation of efficiency is not straightforward. 
Efficiency refers to the process whereby inputs are converted to outputs. The ratio of the 
value of outputs to the value of inputs provides one means whereby efficiency can be 
measured, and requires knowledge of costs, outputs and the estimation of the cost 
function. Early cost function studies used simple least squares regression to estimate a 
line of best fit through the data.  This approach calculates, over a number of organisations, 
the average value of total cost associated with producing a given level of output, and does 
not tell us how cheaply it is possible to produce that output. We know, because we can 
see it happening, that it is possible to produce the output more cheaply than the average. 
In analysing costs and efficiency, therefore, it is important that we should be able to 
identify an envelope below which it is technically impossible for costs to go. Rather than a 
line of best fit (which may be estimated by regression), we need to identify the position of a 
cost frontier, a cost curve that would typically lie below the best fit line. 
In practice, the method used to estimate the parameters of the cost frontier involves a 
modification of the basic least squares regression method. In the least squares method, 
the residuals (the gap between observed values for each data point and the line of best fit) 
are required to follow a normal distribution with a zero mean. This means that the 
observed values are as likely to lie below the estimated cost curve as above it, and that 
the sum of deviations below the line is as great as the sum of deviations above the line. 
This clearly violates the requirement that the line should represent a frontier. For a 
stochastic frontier model12, the requirement that the residuals follow a normal distribution 
is replaced by a specification in which the residuals are made up of two components: one 
is normal, with zero mean, and is designed to capture measurement error; the second 
                                            
12 The stochastic frontier model was introduced by Aigner et al. (1977). 
17 

Efficiency in the Higher Education sector: A technical exploration 
 
component is non-normal (usually a half-normal distribution is assumed13), and is 
designed to capture differences in efficiency across the observed units14. The effect of 
introducing this latter component of the residual is to shift the line so that it becomes a 
frontier rather than a line of best fit. The parameters of the model may be estimated using 
maximum likelihood techniques15. An advantage of this method is that it allows the 
magnitude of the non-normal residual associated with each observation to be calculated; 
this may then be interpreted as a measure of efficiency. More commonly, the ratio of 
predicted costs to the sum of predicted costs and the non-normal residual is used to define 
an efficiency score16. 
These measures of efficiency are interesting and are likely to be instructive at least 
inasmuch as they provoke questions. It should be remembered, however, that they are 
obtained from a statistical exercise in which a line (or plane) is fitted through data that do 
not allow a perfect fit. They are estimates of efficiency, and, like any other statistical 
estimates, are measured with error. The efficiency estimate for each data point (in our 
case, each HEI) may be different from the estimate for other data points, but not (in the 
statistical sense) significantly so. Moreover, there is no consensus about the precise value 
an efficiency score should have in order for the unit to be deemed efficient or inefficient. As 
a result, caution is needed in their interpretation.  
The stochastic frontier estimation approach (like earlier estimation methods) has the 
underlying assumption that all production units under examination are directly comparable. 
It may be the case, however, that the cost associated with a given level of production may 
be higher in one organisation than in another for reasons that may reflect differences in the 
cost and production structures of different organisations rather than differences in 
efficiency. For example, an organisation that enjoys the use of new, purpose-designed 
buildings may enjoy lower costs than one that uses antiquated accommodation. This may 
mean that the former is more efficient than the latter; but equally it may mean that the 
latter produces, as a by-product, intangible outputs (such as architectural heritage) which it 
is obliged to preserve. Likewise it is possible that different institutions differ in terms of the 
quality of students they can attract, and in the quality of graduates that they produce.  It is 
important therefore to recognise the danger that the measurement of efficiency may be 
conflated with the issue of legitimate differences between organisations in cost and 
                                            
13 Alternative distributions for this error component include the truncated normal, the exponential and the 
gamma distributions. The half-normal and exponential distributions have a mode at zero while the gamma 
and truncated normal models have a much wider range of distribution shapes. The parameters of the 
gamma and truncated normal, however, are much more difficult to estimate than for the half-normal and 
exponential distributions. While the values of efficiency scores can be sensitive to the precise distribution 
that they are assumed to follow, when observations are ranked on the basis of efficiency the rankings are 
usually not sensitive to choice of distribution (Coelli et al. 2005). 
14 It is possible to test whether the half-normal distribution of efficiencies (across all units) is significantly 
different from zero using a statistic λ which is calculated from the variance of the random error component 
and the variance of the inefficiency component (Coelli et al. 2005). If λ is not significantly different from zero, 
then there is no significant inefficiency component. 
15 Maximum likelihood estimation involves using an iterative procedure to find the parameters of a given 
statistical model that maximize the likelihood of observing the particular set of data. 
16 The efficiency score typically lies between zero and one, with one representing efficiency. It is possible for 
the score to be outside that range under some circumstances. This can happen if predicted costs are less 
than zero. 
18 

Efficiency in the Higher Education sector: A technical exploration 
production structures. What constitutes legitimacy in this context inevitably involves 
a judgement call. It is possible, of course, to compare like with like, but someone 
has to make a judgement about how alike the members of each cluster of 
organisations have to be in order to be deemed comparable. 
These difficulties have been acknowledged by researchers for some time, and efforts have 
been made to develop methods that allow efficiency to be evaluated while ensuring that 
the organisations that are being compared with each other are indeed comparable. An 
important contribution to this research effort is the development of the latent class 
stochastic frontier model17. This is a statistical model that allows the analyst 
simultaneously to estimate the parameters of the cost structures of two or more groups of 
organisations and to evaluate the efficiency of each organisation in each group, while also 
determining which organisations comprise the membership of each group.   
At the same time, the position and shape of this cost frontier needs to be evaluated 
separately for each of a number of groups of organisations. This is in recognition of the 
fact that different groups of organisations face different challenges and have different 
missions. The structure of costs is not expected to be the same across all organisations, 
simply because the characteristics of these organisations vary widely. There may be good 
reasons to suppose that there are (say) two distinct groups – or what we might call latent 
classes – of organisations included within the data set. The analyst can then set up a 
problem that can be solved to provide estimates of the position and shape of two cost 
curves – one for each group. Part of the solution of this model involves establishing which 
organisations belong in which latent class. The problem is solved using maximum 
likelihood estimation methods (see footnote 15). To be clear, the estimation methodology 
simultaneously provides information about what organisations comprise which group and 
provides estimates of the parameters of the cost equation for each group.  
To summarize, it is possible to combine the stochastic frontier and latent class approaches 
so that (i) cost frontiers (or envelopes) are estimated (ii) yielding measures of the efficiency 
of each organisation in the data set and (iii) establishing which organisations belong in 
each of the latent classes or groups. This is illustrated in Figure 1. This shows a scatter 
plot of points, each of which describes the costs and output levels of a single observation. 
Each observation might represent a decision-making unit or organisation – for example a 
HEI. Where panel data are used, each observation might represent a particular 
organisation in a particular time period. A straightforward latent class analysis of these 
data might involve the analyst in specifying that there are two18 different types of 
organisation in the data set. The latent class model therefore fits two lines to the data. 
These are shown by the two dashed lines. In fitting these two lines, the model also 
determines which observations belong to which of the two latent classes – thus the model 
classifies some of the cost-output pairings into class X and some into class Y. These 
letters are shown as the data points on the diagram, but it should be emphasised that the 
observations are placed in these classes by the maximum likelihood algorithm used in the 
                                            
17 While the latent class model was introduced by Lazarsfeld and Henry (1968), the frontier version of the 
latent class model was much later (Orea and Kumbhakar 2004; Greene 2005). 
18 It is, of course, possible to develop latent class models with more than two classes. More discussion on 
this is provided in section 5. 
19 

Efficiency in the Higher Education sector: A technical exploration 
 
latent class estimation itself; the observations are not placed within one class or the other 
by the analyst.  
The two dashed lines represent the best fit that is associated with the observations (given 
that there are two latent classes), but they do not represent the cost envelope faced by 
organisations within each of these two classes. To find these cost envelopes, the latent 
class method must be used alongside a stochastic frontier model. Doing this moves the 
lines down (and this is not necessarily a parallel shift). The resultant cost envelopes are 
represented by the solid lines. Note that, within each latent class, some observations lie 
below the cost frontier (because of the stochastic error component), but most lie above. 
The preponderance of observations above the frontiers represents inefficiency. The 
technique allows the efficiency of each observation to be evaluated by reference to its 
position relative to the frontier for the latent class to which the observation belongs19. 
Caution should therefore be exercised when interpreting the results from a latent class 
model. In particular, while it is valid to compare HEIs within a group (because they are 
all being evaluated relative to the same frontier) it is not appropriate to make 
comparisons across groups, since the estimated frontier may be different for each 
group20. 
                                            
19 This is done using a method developed by Jondrow et al. (1982). 
20 In any case, the point of a latent class model is that HEIs are different and therefore comparisons should 
not be made across HEIs in different classes. 
20 

Efficiency in the Higher Education sector: A technical exploration 
Figure 1: Illustration of the latent class approach 
 
 
 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
X 
Y 
Y 
Y 
Y 
Y 
Y 
Y 
Y 
Y
  Y 
Y 
costs 
output 
Key: 
 
 
 
 
Line of best fit 
 
 
 
 
Stochastic frontier 
 
21 

Efficiency in the Higher Education sector: A technical exploration 
 
5. Empirical analysis 
 
Several different specifications of the model of costs are reported in the tables that follow. 
These include both linear and nonlinear models; the former benefit from simplicity, but the 
latter have the advantage of allowing more sophisticated analysis of economies of scale 
and of scope. The simplest estimates reported below are based on an assumption that all 
institutions belong to a single class – that, while institutions might differ vastly in both scale 
and in the mix of outputs produced, the underlying technology is common, so that costs 
are determined in the same way in all institutions. The more sophisticated models assume 
that there are two or more latent classes, so that cost structures differ across these 
classes. 
5.1 Model specification 
The explanatory variables in all models include a set of outputs and a number of controls. 
The outputs are: full-time equivalent (FTE) student numbers in each of four categories – 
undergraduates in medicine (UGMED), in other science (UGSCI), and in other subjects 
(which, for conciseness, are referred to as ‘arts’, though this set of subjects also includes 
humanities and social sciences - UGARTS) and the total number of FTE postgraduates 
(PG); research income (RESEARCH); and a measure of income from intellectual property 
(IPINCOME). This last variable is intended to proxy the output of third mission work 
undertaken by institutions. The control variables, used in some models, are the number of 
students at the institution that come from neighbourhoods with low levels of participation in 
higher education (LOWPNO) and the area of the institution’s estate that has listed building 
status (LISTED). A binary variable is also included to identify the ancient institutions 
(Oxford and Cambridge – OXBRIDGE), and, since the data used in the analysis are in the 
form of a panel of institutions over several time periods, year dummies are used to capture 
sector-wide changes over time. A complete list of variables with their precise definitions is 
provided in Appendix 3. All variables measured in monetary units are deflated to 2011 
values by the Office for Budgetary Responsibility’s GDP deflator.  
It is worth making a number of observations about the choice of explanatory variables 
used in the models. First, while undergraduates are disaggregated by broad subject area, 
the same is not done for postgraduates. Considerable efforts were made to evaluate 
models in which postgraduates are disaggregated into subject groups, but these proved to 
be unsuccessful, yielding results that were suggestive of statistical problems. Institutions 
that are major providers of postgraduate education in one area of their activity tend also to 
be highly active in training postgraduates in other areas. Hence several variables in the 
model were highly correlated with one another, thus making it impossible accurately to 
determine the effect on costs of each variable. This problem is known as multicollinearity, 
and it leads to imprecise estimates of the coefficients of the model, with small changes in 
model specification often leading to large changes in the estimated impact of each variable 
on costs. Aggregating across subjects at postgraduate level appears to mitigate this 
problem and results in a considerably more robust model specification. 
22 

Efficiency in the Higher Education sector: A technical exploration 
Secondly, research income is used as a measure of research activity. This is standard in 
the literature, but is nonetheless worth commenting upon. The measurement of research 
undertaken by a university raises questions about how the quantity and the quality (or 
impact, perhaps) of research should be weighted. By using research income as a 
measure, these questions can be finessed. Income provides a measure of the valuation 
that is put on research by clients, and hence implicitly provides the appropriate weights on 
quantity and quality. It is recognised that the clients in this case are not necessarily all 
operating in competitive markets, but nonetheless the use of this measure offers (implicit) 
weights that are not arbitrary.  
Alternative measures of research activity are available and have been considered for use 
in this study. Data on numbers of publications (PUBLICATIONS), and on the number of 
times work from each institution has been cited (CITATIONS), are available from the Web 
of Science. The correlation between research income, publications and citations measures 
of research activity is high, as is demonstrated in Table 1. Early experimentation with 
models similar to those reported below, using the publications and citations variables 
rather than research income, suggested that results are robust with respect to the choice 
of variable used to measure research activity. This being the case, and to be consistent 
with the received literature, the results reported below use the income measure of 
research. 
Table 1: Correlation between various possible measures of research output (2008/09 
to 2010/11) 
Variable 
PUBLICATIONS 
CITATIONS 
RESEARCH 
0.973 
0.776 
PUBLICATIONS 
- 
0.788 
 
Thirdly, alternative measures of third mission activity are available from the Higher 
Education Business and Community Interaction Survey, and were considered for use in 
the present analysis. The number of people attending (ATEVENT), or the number of staff 
days involved in (EVENTS), events such as concerts, exhibitions, public lectures etc. at 
institutions are examples of such measures. Examination of the data on events reveals 
that the quality of the data is poor. Some institutions that are known to have arts centres 
report no attendance at events, for example. Moreover the data for single institutions often 
vary considerably, implausibly so, from year to year. This suggests that the interpretation 
of these variables differs both across time and across institutions. We experimented with 
inclusion of the variables (both individually and together) in early estimations, but the 
results confirmed that they were unfit for use in the statistical analysis.  Therefore, this 
study only reports results where IPINCOME is the measure of third mission activity. 
Fourthly, the control variables used in the models reported below require some 
justification. The character of an institution’s real estate is likely to be a major influence on 
maintenance costs. Two measures of the nature of the estate were considered as 
candidate control variables in the current exercise. The first is the institutions’ self-reported 
figure for the estimated cost of upgrading their real estate to newly refurbished condition 
(UPGRADE). While superficially attractive, these data suffer a major drawback in the 
present context. Institutions which, over the period of analysis, engage in major 
refurbishment works see a fall in the value of this variable while simultaneously increasing 
their expenditures. This causes ambiguity in the impact that the upgrade measure is 
23 

Efficiency in the Higher Education sector: A technical exploration 
 
expected to have on costs. For this reason, the second variable – namely the area of the 
institution’s estate that is accounted for by listed buildings (LISTED) – is preferred.  The 
variable UPGRADE was included in some early estimations but the results were 
unsatisfactory (in that the coefficient was not significant and had a sign which did not 
accord with intuition), and hence our reported results only consider the impact of LISTED 
on costs. 
Fifthly, the inclusion of a dummy variable for the ancient universities is worthy of 
discussion. It is not surprising to find that the cost structures of these universities are 
different from those of other HEIs. The nature of their estate, their organisational 
structures, the balance of their activities (with a relatively heavy concentration on 
postgraduate and research activities), and their positions in international rankings of 
universities all distinguish these universities from others in the country. One option would 
be to exclude them from the analysis, but this resulted in implausible values for some 
coefficients when estimation was based on all other observations, and in a latent class 
model which failed to converge.  It appears appropriate therefore to employ an alternative 
approach of including a dummy variable to identify the Oxbridge institutions21.  
Some further variables were considered for inclusion in the model, but do not appear in the 
preferred specifications reported below. Data from Unistats on graduate earnings, based 
on the Destinations of Leavers of Higher Education (DLHE) survey, provide a market 
based measure of the quality of institutions’ output (NMEAN) which is assumed to reflect 
inter-institution variations in quality of teaching output. There are various problems with 
including this variable in the cost equation. The data are based on survey data with 
different response rates for each institution. In addition, the data refer to graduates’ 
success in the labour market 6 months after graduation, and it is debateable whether this 
provides an adequate reflection of graduate quality.  Finally, there is a problem with using 
average graduate earnings as a control variable in an equation which has total costs as 
the dependent variable. This would suggest that institutions’ fixed costs vary with quality 
(as measured by graduate earnings) but that variable costs do not. This is clearly 
implausible. Nevertheless, graduate earnings were used as a control variable in some 
early specifications of the cost equation (not reported here), and consistently proved to be 
insignificant as a determinant of costs.   
A further measure of quality that was considered for use in the analysis was institutions’ 
performance in the National Student Survey – specifically the percentage positive 
response to the question ‘overall, I am satisfied with the quality of the course’ (NSS). In 
common with the graduate earnings variable, this measure of quality suffers the drawback 
that its inclusion in a cost equation would imply that it can affect fixed but not variable 
costs. Its use as an explanatory variable in some specifications of the cost function 
                                            
21 It would be possible to allow for this in another way. A random parameter model would allow each 
institution, including Oxford and Cambridge, to have a distinct cost structure. Such a model might, however, 
be considered to be too permissive in the sense that it would, in effect, allow a distinct cost function to be 
estimated for each institution, thus allowing each institution to claim that differences in costs are due to 
differences in structure rather than differences in efficiency. The latent class model goes some way towards 
the random parameter specification without allowing quite so much flexibility – it does this by identifying two 
or more separate classes of institution, but, on the evidence of the data used here, this mechanism is not 
refined enough to allocate Oxford and Cambridge to their own class. 
24 

Efficiency in the Higher Education sector: A technical exploration 
produced unsatisfactory results, typically leading to an estimated equation with efficiencies 
which have the wrong skew22. 
Other factors which might be considered of interest, a priori, have not been included in the 
analysis because it has not been possible to obtain satisfactory measures. This is the case 
with, for example, the quality of student intake. We note, however, that the latent class 
approach adopted in some of the work which follows is designed precisely to allow for 
differences between HEIs which are not otherwise observed in the data. 
We end this section by highlighting the points of original contribution of the empirical work 
reported below: 
 We estimate cost equations for English higher education over a period of 8 years of 
data, and also for sub-periods of 2 and 3 years within that period. This is a considerably 
longer period than any used in previous literature: typically, analysis has been been 
based on one year of data, or, in the case of panel data studies, on a maximum of 3 
years of data.  
 We include a measure of third mission output. Attempts have been made to measure 
third mission in some previous work, but the variable used here is more appropriate 
than any previously used. In addition, we consider the interaction of third mission and 
research output in some models reported below, and this is new to the literature. 
 Undergraduate teaching is broken down into 3 subject areas (medicine, other sciences 
and non-sciences). Previous studies have typically divided teaching only into two 
subject areas; where three groups have been used in previous studies the approach 
has not been combined with a third mission measure and its interaction with research. 
Separation of undergraduate teaching into this many groups poses challenges in 
estimation which will be discussed in the context of the results presented below. 
 We investigate for the first time in the literature the effect of other possible determinants 
of costs, in particular, the effect of recruiting students from traditionally low participation 
neighbourhoods (LOWPNO), and the effect of having buildings with potentially costly 
upkeep (LISTED).  
 We investigate the possibility that the cost function varies for distinct groups of 
universities, as revealed by the data, using stochastic frontier latent class estimation. 
This is the first time this approach has been used in the context of English higher 
education. 
                                            
22 The method used to evaluate efficiency, stochastic frontier analysis, requires that variation between 
institutions’ costs that are unexplained by the output and control variables should follow a distribution that 
comprises a normal and a ‘one-sided’ element. The latter should all be positive values (in a cost function 
context), reflecting the extent of inefficiency observed in each institution. To obtain such a one-sided 
component of the residual, the total residual has to be skewed in a certain direction. If the skew goes the 
wrong way, one cannot interpret the one-sided element as a measure of inefficiency. In this case, the 
ordinary least squares estimator provides the best estimate of the cost function, and the data indicate that all 
institutions are efficient. 
25 

Efficiency in the Higher Education sector: A technical exploration 
 
 We compare the cost functions and efficiency derived from the latent class approach 
with the cost functions and efficiency estimated using pre-defined classes of ‘similar’ 
institutions. These pre-defined classes are the same as those used by the Higher 
Education Funding Council for England (HEFCE 2013). 
5.2 Data 
Summary statistics for the variables used in the models for each of the time periods are 
displayed in Table 2. It should be noted that values of n (number of observations on which 
the mean is calculated) vary because of missing data for some variables. 
Table 2: Summary statistics for the data  
 
a)  2008/09 to 2010/11 
 
Mean 
Standard 
deviation 
Minimum 
Maximum 
n 
COST 
156,943.20 
178,620.10 
5,424.49 
1,249,909.00 
387 
UGMED 
1,333.43 
1,457.47 
0.00 
6,839.48 
387 
UGSCI 
2,684.83 
2,344.37 
0.00 
9,249.96 
387 
UGARTS 
5,101.03 
3,703.83 
0.00 
16,162.84 
387 
PG 
2,229.55 
1,890.64 
0.00 
9,457.09 
387 
RESEARCH 
38,928.08 
81,725.65 
0.00 
490,105.10 
387 
IPINCOME 
19,958.34 
28,859.26 
0.00 
163,684.00 
384 
LOWPNO 
220.47 
214.79 
0.00 
1,090.00 
375 
LISTED 
16,624.64 
28,023.97 
0.00 
182,536.00 
367 
 
b) 2005/06 to 2007/08  
 
Mean 
Standard 
deviation 
Minimum 
Maximum 
n 
COST 
136,959.20 
156,131.80 
3,987.54 
1,120,624.00 
394 
UGMED 
1,339.37 
1,510.61 
0.00 
6,851.22 
394 
UGSCI 
2,474.75 
2,560.99 
0.00 
22,110.14 
394 
UGARTS 
4,760.30 
4,153.28 
0.00 
39,186.91 
394 
PG 
1,972.08 
1,688.05 
0.00 
8,668.88 
394 
RESEARCH 
33,316.15 
68,630.43 
0.00 
412,845.00 
392 
IPINCOME 
17,310.54 
22,930.70 
0.00 
122,171.90 
387 
LOWPNO 
205.64 
197.31 
0.00 
1,100.00 
369 
LISTED 
16,005.44 
29,582.77 
0.00 
197,000.00 
349 
 
26 

Efficiency in the Higher Education sector: A technical exploration 
c) 2003/04 to 2004/05 
 
Mean 
Standard 
deviation 
Minimum 
Maximum 
n 
COST 
118,689.00 
128,321.60 
4,698.33 
790,109.10 
263 
UGMED 
1,305.93 
1,523.77 
0.00 
6,890.56 
263 
UGSCI 
2,487.89 
2,713.90 
0.00 
22,322.49 
263 
UGARTS 
4,546.28 
4,407.67 
0.00 
40,237.30 
263 
PG 
1,954.64 
1,631.95 
0.00 
7,927.64 
263 
RESEARCH 
29,874.27 
59,394.44 
0.00 
311,924.80 
259 
IPINCOME 
14,752.30 
20,184.71 
0.00 
100,684.80 
258 
LOWPNO 
219.59 
199.40 
0.00 
975.00 
243 
LISTED 
16,762.90 
30,890.83 
0.00 
197,000.00 
206 
 
d) 2003/04 to 2010/11 
 
Mean 
Standard 
deviation 
Minimum 
Maximum 
n 
COST 
140,016.70 
159,277.10 
3,987.54 
1,249,909.00 
1042 
UGMED 
1,330.20 
1,493.97 
0.00 
6,890.56 
1042 
UGSCI 
2,560.84 
2,522.88 
0.00 
22,322.49 
1042 
UGARTS 
4,841.46 
4,061.73 
0.00 
40,237.30 
1042 
PG 
2,067.03 
1,754.97 
0.00 
9,457.09 
1042 
RESEARCH 
34,582.19 
71,811.05 
0.00 
490,105.10 
1037 
IPINCOME 
17,657.22 
24,773.57 
0.00 
163,684.00 
1029 
LOWPNO 
214.91 
204.55 
0.00 
1,100.00 
986 
LISTED 
16,421.15 
29,245.00 
0.00 
197,000.00 
922 
 
5.3 Linear model over 3 time periods and for the whole time period 
Table 3 reports the coefficient estimates obtained in a stochastic frontier regression of 
costs against linear terms in the various outputs and a set of control variables – including 
year dummies, an Oxbridge indicator, area of real estate comprising listed buildings, and 
the number of students originating from traditionally low participation neighbourhoods. The 
specifications of the model (and the models that follow) allow efficiency to vary across time 
for each institution in the data set. Costs are measured in thousands of pounds, so the 
coefficients on the student number variables each represent the sum (in thousands of 
pounds) that the marginal student adds to total costs. Hence, for example, one extra 
science undergraduate costs a typical university an extra £7,775 per year during the latest 
time period (measured at 2011 prices). It is readily observed that, within each of the (two 
or three year) time periods under investigation, the undergraduates that impose the 
highest costs on institutions are those studying medicine, followed by those studying other 
sciences, followed by those studying other subjects. The higher costs of these subjects are 
recognised by the support given by HEFCE for band A and band B disciplines. 
Postgraduate provision is generally more costly than undergraduate provision (much 
27 

Efficiency in the Higher Education sector: A technical exploration 
 
postgraduate provision involves one-to-one supervision), except undergraduate provision 
in medicine.  
The value of the constant in each equation is worthy of further discussion. A negative 
constant term in a cost equation suggests that fixed costs are negative. In some cases, the 
estimated coefficient is not significantly different from zero. Where a negative constant 
term is significant it becomes somewhat difficult to interpret. It should be noted, however, 
that the cost functions estimated in exercises of this type are based on data for institutions 
whose output exceeds zero. The equations may provide an imprecise guide to what costs 
would be in the hypothetical case of in institution that produced nothing. While they provide 
a good fit to the data over the range of data that is actually observed, the fit outside this 
range may be less satisfactory. A negative constant term does not therefore mean that 
institutions would make a profit by producing nothing; it means, rather, that, within the 
range of data observed, some diseconomies of scale may exist. 
28 

Efficiency in the Higher Education sector: A technical exploration 
Table 3: Linear model with a complete set of controls 
AICs 
2008/09 to 
2010/11 
2005/06 to 
2007/08 
2003/04 to 
2004/05 
2003/04 to 
2010/11 
UGMED 
13.48440 
13.86610 
9.74789 
13.92710 
UGSCI 
7.77511 
7.04032 
5.60877 
7.11761 
UGARTS 
4.57408 
6.65664 
3.95087 
7.13533 
PG 
13.95320 
9.40896 
9.81821 
12.21420 
RESEARCH 
0.81612 
0.83696 
1.18236 
xx-xxxx-xxxx 
IPINCOME 
0.81920 
1.15411 
0.35017 
0.81935 
CONTROLS 
 
 
 
 
2003/04 
 
 
-2,485.99 
-17,257.20 
2004/05 
 
 
 
-14,638.20 
2005/06 
 
 
 
-10,858.80 
2006/07 
 
3,640.36 
 
-6,838.74 
2007/08 
 
10,429.90 
 
-182.20 
2008/09 
 
 
 
xx-xxxx-xxxx 
2009/10 
-2,813.51 
 
 
1437.09 
2010/11 
-4,070.18 
 
 
 
OXBRIDGE 
322,696.00 
184,217.00 
111,966.00 
205,326.00 
LISTED 
0.31 
0.25 
0.22 
0.12 
LOWPNO 
-37.29 
-33.06 
-3.41 
-28.63 
CONSTANT 
-8,426.55 
-39,195.80 
-3,237.03 
-42,488.50 
Is λ significantly different from zero at 
the 5% significance level? 23 
YES 
YES 
YES 
YES 
Notes: 
1. 
Controls: LISTED; LOWPNO; OXBRIDGE; YEAR dummies. 
2. 
Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
The universities of Oxford and Cambridge are distinctive owing to their antiquity, 
organisational structures and academic orientation, and this is reflected in higher costs. 
Those institutions whose real estate includes a higher area covered by listed buildings 
typically have higher costs than others. Finally, those institutions that admit relatively high 
numbers of students from low participation neighbourhoods tend to have lower costs. This 
is, at first sight, a somewhat surprising result. The direction of causality, however, is open 
to debate: it may be that students from low participation neighbourhoods are attracted to 
institutions that have relatively low costs, possibly because of the type of subjects provided 
or because they undertake less research. The correlations between LOWPNO and, 
respectively, UGMED, UGSCI, UGARTS and RESEARCH are 0.51, 0.67, 0.78 and -0.08 
and are therefore consistent with this hypothesis. We do not investigate this further as it is 
not the main issue of interest. 
                                            
23 This row in the table indicates whether or not the one-sided inefficiency term is statistically significantly 
different from zero across all observations. Here they are but in later tables (particularly ones relating to 
latent class models) they are not. These provide us with a check on the confidence with which we can 
interpret the efficiency scores. 
29 

Efficiency in the Higher Education sector: A technical exploration 
 
Estimation of the equations that comprise our models of costs also allows computation of 
efficiency scores for each institution. These are obtained from the one-sided residual in the 
stochastic frontier estimator, and may be expressed as the predicted value of costs divided 
by the predicted value plus the one-sided residual. A score of one thus implies efficiency, 
while lower scores imply the existence of some inefficiency. The distribution of efficiency 
scores suggests that some institutions are more efficient than others. To illustrate, the 
distribution associated with the 2008/09 to 2010/11 model reported in Table 3 is shown in 
Figure 2 (efficiency distributions associated with the other models in Table 3 are reported 
in Appendix 4.1). This indicates that the majority of HEIs have efficiency scores above 0.8, 
but that there is a noticeable tail of institutions which, on this measure, appear to be less 
efficient. Some institutions have an efficiency score of less than zero. This is possible 
where the output levels of the institution are very low24, and indicates that the model of 
costs does not satisfactorily explain the relationship between costs and outputs for such 
small institutions. In Figure 2, the single observation at the bottom end of the efficiency 
distribution is the Rose Bruford College of Theatre and Performance which is a small 
specialist institution. As we shall see later, more refined models of costs tend to produce 
distributions of efficiencies that look rather different from those reported here. 
Figure 2: Histogram of efficiency scores – final year of 2008/09 to 2010/11 (linear 
model) 
 
In Table 4 we investigate the effect of excluding the control variables from the equation. 
The results are broadly similar, though some observations are warranted. First, there is a 
noticeable shift in the coefficient values between the two time periods reported in this table 
– and, indeed, for the 2005/06 to 2007/08 period, between the results obtained in this table 
                                            
24 The efficiency score is defined as predicted costs divided by the sum of predicted costs and the one-sided 
residual. An efficiency score can be less than zero if predicted costs are negative. 
30 

Efficiency in the Higher Education sector: A technical exploration 
and those reported earlier. While, in the latest time period, it costs more to produce a 
marginal science undergraduate than an undergraduate in non-science subjects, the 
reverse is true in the earlier time period. The counterintuitive result obtained here for the 
2005/06 to 2007/08 time period serves as a warning that some of the statistical results are 
not robust to minor changes in specification or modelling strategy. 
Table 4: Linear model with a limited set of controls 
AICs 
2008/09 to 
2010/11 
2005/06 to 
2007/08 
2003/04 to 
2010/11 
UGMED 
13.71790 
14.17640 
14.42160 
UGSCI 
7.34657 
3.17296 
3.89932 
UGARTS 
3.01348 
6.04901 
5.68941 
PG 
18.41520 
13.46680 
16.13060 
RESEARCH 
0.93983 
0.93658 
0.92238 
IPINCOME 
0.53013 
1.11874 
0.87065 
CONTROLS 
 
 
 
2003/04 
 
 
-15,018.50 
2004/05 
 
 
-12,543.60 
2005/06 
 
 
-9,984.87 
2006/07 
 
5,220.86 
-4,594.99 
2007/08 
 
10,904.60 
1,229.65 
2008/09 
 
 
5,046.32 
2009/10 
-3,838.74 
 
1,583.07 
2010/11 
-5,372.17 
 
 
OXBRIDGE 
309,483.00 
186,112.00 
205,629.00 
CONSTANT 
-18,258.00 
-44,053.60 
-41,383.20 
Is λ significantly different from zero at 
the 5% significance level? 
YES 
YES 
YES 
Notes: 
1. 
Controls: OXBRIDGE; YEAR dummies. 
2. 
Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
3. 
The 2003/04 to 2004/05 model does not converge. 
No equation is reported in Table 4 for the 2003/04 to 2004/05 time period. This is because 
the algorithm used to obtain the maximum likelihood results for these periods failed to 
converge. This is likely to be because different solutions to the model (different sets of 
coefficients and different parameterisations of the structure of the residuals) yield similar 
likelihoods. This simply means that the data in these cases are not suitable for estimating 
a model of this kind. Efficiency distributions associated with the models displayed in Table 
4 can be found in Appendix 4.2. 
31 

Efficiency in the Higher Education sector: A technical exploration 
 
5.4 Quadratic model over 3 time periods and for the whole time period 
We now turn to consider the quadratic stochastic frontier model. The model includes as 
explanatory variables: 
 linear terms in all variables 
 squared terms in each of the student number variables, research, and income from 
intellectual property 
 a full set of interaction terms between the student number variables, and between each 
of these and research 
 an interaction term between research and income from intellectual property. 
The model also includes a full set of controls. The estimated coefficients on the control 
variables are similar to those obtained in earlier models, and are not discussed further 
here. 
Rather than report the estimated parameters of the full quadratic model, which are difficult 
to interpret, Table 5 reports the average incremental costs (AICs) associated with each of 
the outputs. We use the definition of average incremental costs given in section 3.2 and 
evaluate at mean values of each of the explanatory variables. It is readily observed that 
these follow a similar pattern to that observed in the linear models described earlier – of 
undergraduates, students in medicine are the most costly, followed by those in other 
sciences. With the exception of the low estimate for non-science undergraduates in the 
2003/04 to 2004/05 period, the estimates of average incremental costs look broadly 
plausible. The costs associated with postgraduates are lower than in the estimates 
provided by the linear model, and those associated with research are higher. It is likely that 
collinearity between these two variables reduced the precision of the estimates. The 
negative estimate of average incremental costs associated with postgraduate provision in 
the final column of the table is suggestive of statistical problems, and should be treated 
with scepticism. The relatively high values associated with undergraduate provision and, 
especially, third mission activity in this column indicates that multicollinearity could be 
adversely affecting the precision of these estimates. 
Table 5: Quadratic model with a complete set of controls 
AICs 
2008/09 to 
2010/11 
2005/06 to 
2007/08 
2003/04 to 
2004/05 
2003/04 to 
2010/11 
UGMED 
16.03379 
15.00020 
9.19486 
17.43360 
UGSCI 
7.85770 
9.44366 
4.59139 
7.21945 
UGARTS 
5.45938 
4.58650 
0.32875 
5.12836 
PG 
5.27499 
2.60062 
7.07272 
-0.80377 
RESEARCH 
1.27087 
1.32466 
1.34063 
1.13070 
IPINCOME 
1.00667 
1.74035 
xx-xxxx-xxxx 
1.75663 
Is λ significantly different from zero 
at the 5% significance level? 
YES 
YES 
YES 
YES 
Note: 
1. Controls: LISTED; LOWPNO; OXBRIDGE; YEAR dummies. 
32 

Efficiency in the Higher Education sector: A technical exploration 
Efficiencies are estimated for HEIs using the 2008/09 to 2010/11 model, and these are 
plotted for the final year only in Figure 3. The distribution of efficiencies indicates that, 
compared with the linear model, the efficiency scores are, in general, higher when the 
quadratic specification of the cost equation is used. This is not surprising – a richer model 
allows more of the differences between institutions to be explained, thus leaving less to be 
accounted for by a residual which is (misleadingly, perhaps) labelled inefficiency. The 
observation at the lower extreme of the distribution is Heythrop College, which is a small 
institution specialising in philosophy and theology.  
Figure 3: Histogram of efficiency scores – final year of 2008/09 to 2010/11 (quadratic 
model) 
 
The effect of excluding the control variables from the quadratic equation is examined; AICs 
are reported in Appendix 4.3 and histograms of efficiencies are reported in Appendix 4.4. 
5.5 Linear latent class model over 3 time periods and for the whole time 
period 
The above results all impose a single structure on costs for all institutions. It is clear, 
however, that institutions of higher education in England are not homogenous in character. 
To allow for this, we turn to the evaluation of latent class variants of the stochastic frontier 
model. In these, we suppose that institutions are all one of two types25. We do not 
constrain any institution to be in the same latent class in all years. Table 6 shows the 
results obtained when a latent class stochastic frontier is applied to a linear model of costs 
                                            
25 We do in fact try estimating models with different numbers of classes, in particular 3- and 4-class models. 
In each case, estimated AICs for one class are implausible, presumably because of the small number of 
observations on which the estimates are based. More details are provided at the end of this section. 
33 

Efficiency in the Higher Education sector: A technical exploration 
 
34 
with a full set of controls. There are several striking features that emerge from the results. 
First, in all time periods, the parameter associated with (non-medical) science 
undergraduates is relatively low in the first latent class, but relatively high in the second. 
This may reflect differences in the precise mix of science subjects provided in institutions 
within each class. Secondly, two parameters are striking by virtue of being surprisingly low 
– those on postgraduates in the first latent class in the 2008/09 to 2010/11 period, and on 
undergraduates in medicine in the first latent class in the 2003/04 to 2004/05 period. It 
may be the case that the institutions in these classes do not produce sufficient medical 
students and postgraduates respectively to provide reliable estimates of these parameters. 
We shall examine the membership of each latent class later. Thirdly, the negative 
coefficient on postgraduates in the first latent class in the 2003/xx-xxxx-xxxx period is 
curious and should be treated with scepticism26. 
                                            
26 Note also that the inefficiency component for this model is not significantly different from zero at the 5% 
significance level (as indicated by the value of λ) and this provides further reason to be cautious about this 
model. 

Table 6: Linear latent class model with a complete set of controls (2 classes) 
2008/09 to 2010/11 
2005/06 to 2007/08 
2003/04 to 2004/05 
2003/04 to 2010/11 
AICs 
Class 1 
Class 2 
Class 1 
Class 2 
Class 1 
Class 2 
Class 1 
Class 2 
UGMED 
10.86470 
7.77427 
9.73153 
6.62318 
2.40610 
9.44584 
6.01711 
10.04800 
UGSCI 
1.93079 
8.47168 
1.74760 
8.64085 
2.53791 
7.05533 
2.99762 
5.61400 
UGARTS 
9.35289 
2.75710 
8.16584 
4.65941 
6.50197 
4.42665 
12.14230 
6.07852 
PG 
0.24627 
18.69390 
10.45850 
5.75412 
13.43170 
8.61365 
-17.48280 
9.70639 
RESEARCH 
1.50834 
0.97330 
1.65369 
0.96609 
1.15465 
1.45594 
1.43784 
1.27254 
IPINCOME 
1.07664 
0.62224 
-0.07846 
1.50001 
0.82643 
-0.11413 
2.93942 
0.52597 
CONTROLS 
 
 
 
 
 
 
 
 
2003/04 
 
 
 
 
-1,114.38 
-1,735.85 
-206,226.00
-11,577.00 
2004/05 
 
 
 
 
 
 
-182,882.00
-9,695.82 
2005/06 
 
 
 
 
 
 
-34,486.50 
-8,308.94 
2006/07 
 
 
7,911.85 
-171.29 
 
 
-42,794.40 
-4,722.14 
2007/08 
 
 
18,511.40 
2,281.25 
 
 
-20,563.60 
581.69 
2008/09 
 
 
 
 
 
 
-33,363.10 
3,819.46 
2009/10 
-1,139.03 
-465.19 
 
 
 
 
-13,166.80 
1,547.01 
2010/11 
1,095.46 
-4,842.08 
 
 
 
 
 
 
OXBRIDGE 
431,387.00 
104,301.00 
379,554.00 
104,248.00 
77,371.10 
43,327.90 
302,035.00 
52,040.60 
LISTED 
-0.10 
0.15 
-0.18 
-0.09 
0.74 
-0.03 
-0.18 
0.01 
LOWPNO 
-31.10 
-4.97 
-0.56 
-9.63 
-36.32 
2.81 
-44.11 
-12.87 
CONSTANT 
-519.58 
-557.62 
-19,537.50 
-924.37 
-965.39 
-2,224.99 
36,444.30 
-1,xx-xxxx-xxxx 
Number in each class 
121 
234 
111 
216 
60 
136 
38 
840 
Is λ significantly different from zero 
at the 5% significance level? 
YES 
YES 
YES 
YES 
NO 
YES 
NO 
YES 
Notes:  
1. Controls: LISTED; LOWPNO; OXBRIDGE; YEAR dummies. 
2. Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
3. The number in each class is the number of observations over the estimation period, not the number of HEIs. Given that HEIs can be in different classes 
in different years, dividing by the number of years on which the analysis is based does not give the number of HEIs. 
35 

As is the case for the simpler models estimated above, it is possible to report the 
distributions of efficiency scores obtained by institutions when applying the latent class 
model. In this instance, however, the efficiency of each institution is evaluated by 
comparing the institution only to others in its own class. Since this is more akin to 
comparing like with like, the average efficiency score is higher than was observed in the 
simpler models. The distributions of efficiency scores for the two latent classes, using the 
model for 2008/09 to 2010/11 are shown in Figure 4.  
For the second latent class, most HEIs have efficiency scores which are above 0.6. For 
the first class, just two HEIs have efficiency scores less than 0.6: the London University 
(Institutes and Activities) and Trinity Laban Conservatoire of Music and Dance. The former 
is unusual in that it comprises a small and highly specialised collection of research 
centres, while the latter specialises in music and dance. 
We do not constrain HEIs to be in the same latent class across all years within any period 
of analysis. In Appendix 4.5 we present a table showing the membership of each of the 
two latent classes in each year of the 2008/09 to 2010/11 period. It is readily observed that 
individual institutions switch between the classes quite frequently. Moreover, it is difficult to 
provide a clear rationale for the membership of the latent classes in any given year, 
beyond noting that the classes are defined by the statistical method. Since the latent 
classes are not easily explained by appeal to intuition, we present an alternative means of 
disaggregating the data by institution type later in this report (see section 5.7). 
Results obtained by estimating linear models with 3 and 4 latent classes (respectively) are 
reported in Appendix 4.6. In each case, one of the latent classes is small and yields 
implausible estimates of average incremental costs. We do not, therefore, report the 
distributions of efficiency scores associated with these analyses. 
36 

Efficiency in the Higher Education sector: A technical exploration 
Figure 4: Histogram of efficiency scores – final year of 2008/09 to 2010/11 (linear 
latent class model) 
a) Latent class 1 
 
b) Latent class 2 
 
Note: These histograms relate to the equations in Table 6 (2008/09 to 2010/11 model), and are drawn for 
the final year of the estimation period. 
37 

Efficiency in the Higher Education sector: A technical exploration 
 
5.6 Quadratic latent class model over 3 time periods and for the whole 
time period 
Now consider the latent class stochastic frontier model with a quadratic specification. As 
earlier, rather than report the coefficients of the full model, which are difficult to interpret, 
we tabulate the average incremental costs, evaluated at mean values of the explanatory 
variables within each class. These appear in Table 7. We note some caveats that should 
attach to the results reported here, particularly for the 2003/xx-xxxx-xxxx period. In this 
case, the model has many explanatory variables and the sample used to estimate the 
parameters in this two year period is small. Indeed, owing to missing values for the control 
variables, this is an issue that applies, to a greater or lesser extent, to all estimates 
obtained for this earliest period. The negative estimates obtained for some average 
incremental costs in the 2005/xx-xxxx-xxxx period suggest that the results for this period, 
too, should be treated with caution27. Our confidence in these results has to be 
conditioned by the observation that we are drilling down to subgroups of institutions
data collected over short periods, and using a complex specification with numerous 
explanatory variables; the numbers of observations available are insufficient to allow 
precise estimates o
, with 
f the parameters. 
                                           
The distributions of efficiency scores associated with the model in Table 7 (2008/09 to 
2010/11 model) are reported in Figure 5 for the final year of the estimation period. The 
majority of HEIs (across both classes) have efficiency scores above 0.9, but a number of 
small specialist institutions once again achieve much lower scores. This poses questions 
about the validity of including these in the analysis. 
 
27 It should also be noted that for each of the 2005/06 to 2007/08 and the 2003/04 and 2004/05 periods, the 
inefficiency component is not significantly different from zero (at the 5% significance level) for either of the 
latent classes as indicated by λ. 
38 

Efficiency in the Higher Education sector: A technical exploration 
Table 7: Quadratic latent class model with complete set of controls (2 classes) 
2008/09 to 2010/11 
2005/06 to 2007/08 
2003/04 to 2004/05 
AICs 
Class 1 
Class 2 
Class 1 
Class 2 
Class 1 
Class 2 
UGMED 
8.72000 
19.5951
8 
8.35074 
8.93250 
3.95822 
4.96233 
UGSCI 
5.25962 
7.18468 
7.70809 
11.1091
7 
0.86012 
8.75251 
UGARTS 
5.88274 
2.17572 
-2.35441 
6.14633 
0.76439 
6.57626 
PG 
7.83897 
1.24167 
-10.07146 
xx-xxxx-xxxx 
-4.89484 
0.37550 
RESEARCH 
1.12587 
1.14127 
xx-xxxx-xxxx 
1.32144 
xx-xxxx-xxxx 
1.30870 
IPINCOME 
1.02964 
0.75173 
2.79682 
0.26046 
1.73449 
0.20759 
Number in 
each class 
236 
119 
132 
195 
100 
96 
Is λ 
significantly 
different from 
zero at the 5% 
significance 
level? 
YES 
YES 
NO 
NO 
NO 
NO 
Notes: 
1. Controls: LISTED; LOWPNO; OXBRIDGE; YEARS. 
2. The 2003/04 to 2010/11 model does not converge. 
3. The number in each class is the number of observations over the estimation period, not the number of 
HEIs. Given that HEIs can be in different classes in different years, dividing by the number of years on 
which the analysis is based does not give the number of HEIs. 
 
Figure 5: Histogram of efficiencies – final year of 2008/09 to 2010/11 (quadratic 
latent class model)  
a) Latent class 1 
 
39 

Efficiency in the Higher Education sector: A technical exploration 
 
b) Latent class 2 
 
Note: 
These histograms relate to the equations in Table 6 (2008/09 to 2010/11 model), and are drawn for the final 
year of the estimation period. 
5.7 Comparison of results with pre-defined groupings of universities 
An obvious alternative to latent class modelling involves prescribing groups within which 
institutions are deemed to share certain characteristics. One such categorisation is the 
HEFCE impact report groups – a categorisation that divides institutions into four types: (1) 
specialist (2) high tariff (3) medium tariff and (4) low tariff. Results obtained by estimating 
linear stochastic frontier models (with only a limited set of controls) for each of these 
groups are reported in Table 8.  
We do not report results for a quadratic model because numbers of observations within 
each group are too low. Also, note that, in contrast to the tables reported earlier, and 
owing to the small sample size represented within each group, the specifications of the 
models reported here do not include control variables (area of estate covered by listed 
buildings, and numbers of students from low participation neighbourhoods).  The results 
are instructive. Most of the specialist institutions are specialising either in medicine or in 
the applied arts. The low parameter on undergraduate provision in other sciences that is 
obtained in all years for this group is therefore unsurprising – few students in these 
institutions fall into this subject category. The high cost associated with non-science 
undergraduates in high tariff institutions (type 2) in the later two time periods should be 
40 

Efficiency in the Higher Education sector: A technical exploration 
41 
treated with caution; with only 28 universities in this group, it is the smallest of the HEFCE 
groupings and many coefficients are imprecisely estimated28. 
The efficiency scores that result from the analysis of HEFCE impact report groups are 
reported, for the final year of the 2008/09 to 2010/11 period, in Appendix 4.7. For groups 3 
and 4 (medium and low tariff groups) the efficiency scores are generally clustered around 
a very high level of efficiency, suggesting that there is little scope to differentiate between 
institutions within each of these groups on the basis of efficiency score. For groups 1 and 2 
(specialist and high tariff groups) the distribution of efficiencies is wide, but it should be 
borne in mind that these are derived from poorly estimated equations where many 
coefficients are not statistically significantly different from zero.  
 
                                            
28 It is also the case that the one-sided inefficiency component is not significantly different from zero as 
signalled by the value of λ. 

Table 8: Linear model with a limited set of controls – pre-defined classes 
2008/09 to 2010/11 
2005/06 to 2007/08 
AICs 
Type 1 
Type 2 
Type 3 
Type 4 
Type 1 
Type 2 
Type 3 
Type 4 
UGMED 
12.17810 
8.26523 
8.41394 
8.83894 
12.09190 
8.14358 
7.10110 
8.79205 
UGSCI 
2.08030 
9.82652 
8.08465 
5.02403 
2.06171 
4.17238 
4.40898 
4.69115 
UGARTS 
12.26320 
14.85030 
3.22720 
6.92543 
11.02950 
12.58070 
4.84906 
7.23211 
PG 
6.41110 
11.35770 
14.60920 
11.08710 
8.04269 
17.63310 
11.47020 
6.33013 
RESEARCH 
1.25448 
1.70680 
1.30524 
1.00174 
1.06033 
1.55152 
1.02540 
1.51193 
IPINCOME 
1.85920 
-0.25084 
xx-xxxx-xxxx 
0.51220 
1.56877 
0.26268 
1.31932 
0.70375 
CONTROLS 
 
 
 
 
 
 
 
 
2003/04 
 
 
 
 
 
 
 
 
2004/05 
 
 
 
 
 
 
 
 
2005/06 
 
 
 
 
 
 
 
 
2006/07 
 
 
 
 
520.84 
11,005.40 
3,766.26 
5,581.02 
2007/08 
 
 
 
 
2,602.96 
20,886.30 
9,953.88 
8,545.21 
2008/09 
 
 
 
 
 
 
 
 
2009/10 
-150.01 
-5,397.57 
-3,048.97 
-5,298.47 
 
 
 
 
2010/11 
-2,489.73 
-3,924.66 
-5,275.52 
-6,620.22 
 
 
 
 
CONSTANT 
-35,205.60 
-152,147.00 
4,518.53 
458.43 
-32,350.50 
-141,177.00 
-10,162.20 
-9,191.27 
Number in class 
111 
84 
96 
87 
106 
83 
109 
82 
Is λ significantly 
different from zero at 
the 5% significance 
level? 
YES 
NO 
NO 
YES 
NO 
NO 
YES 
YES 
 
42 

Efficiency in the Higher Education sector: A technical exploration 
43 
Table 8: continued 
2003/04 to 2004/05 
2003/04 to 2010/11 
AICs 
Type 1 
Type 2 
Type 3 
Type 4 
Type 1 
Type 2 
Type 3 
Type 4 
UGMED 
15.17180 
3.41735 
6.54379 
6.91507 
22.90240 
7.88866 
7.42526 
8.56164 
UGSCI 
4.55565 
5.46450 
3.35649 
7.61685 
4.47340 
8.22235 
5.98942 
5.10155 
UGARTS 
10.50840 
4.51969 
5.51970 
5.79197 
11.97400 
12.32610 
4.11670 
6.80409 
PG 
0.21740 
19.26820 
10.62440 
11.25630 
3.09518 
16.18410 
11.87630 
9.64011 
RESEARCH 
1.36032 
1.61664 
1.12153 
0.79771 
1.06451 
1.61885 
1.12084 
1.18349 
IPINCOME 
xx-xxxx-xxxx 
-0.22979 
0.68564 
0.24722 
2.08913 
-0.24532 
xx-xxxx-xxxx 
0.52735 
CONTROLS 
 
 
 
 
 
 
 
 
2003/04 
-533.96 
-6,326.86 
-1,705.88 
-2,463.74 
-4,017.72 
-20,621.80 
-14,264.10 
-10,991.80 
2004/05 
 
 
 
 
-3,836.98 
-15,402.60 
-12,569.10 
-8,805.18 
2005/06 
 
 
 
 
-2,996.30 
-19,980.70 
-10,042.10 
-6,951.10 
2006/07 
 
 
 
 
-2,488.56 
-7,621.67 
-5,622.40 
-906.64 
2007/08 
 
 
 
 
-460.13 
2,289.81 
534.88 
2,514.16 
2008/09 
 
 
 
 
2,228.48 
6,937.04 
4,736.52 
6,696.75 
2009/10 
 
 
 
 
2,062.71 
-156.11 
2,826.78 
1,518.68 
2010/11 
 
 
 
 
 
 
 
 
CONSTANT 
-21,623.30 
-49,047.60 
-8,504.08 
-9,021.09 
-31,205.70 
-126,630.00 
113.46 
-4,549.38 
Number in class 
62 
55 
88 
50 
279 
222 
293 
219 
Is λ significantly different 
from zero at the 5% 
significance level? 
NO 
NO 
YES 
YES 
NO 
YES 
YES 
YES 
Notes: 
1. Controls: YEAR dummies 
2. Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
3. The number in each class is the number of observations over the estimation period, not the number of HEIs. Given that HEIs can be in different classes in 
different years, dividing by the number of years on which the analysis is based does not give the number of HEIs. 
 

5.8 Discussion 
Numerous models have been presented in this report, and it is useful at this stage to 
present some evaluation of their relative merits. The simplest specification is presented in 
Table 4 – this is the linear model with a limited set of controls, where a single set of 
parameters attaches to all institutions in the sample. The estimated parameters are 
broadly plausible, though those on miscellaneous science and non-science 
undergraduates appear to be quite volatile from sub-period to sub-period. Moreover this 
model could not, for statistical reasons, be estimated for the 2003/xx-xxxx-xxxx period. 
Adding control variables to this model produces the results in Table 3, where the pattern of 
costs associated with undergraduate provision is seen to be stable across subject areas, 
with medicine being the most costly and non-science the least costly subjects. The cost 
attached to postgraduate provision (across all subjects) is broadly on a par with those 
associated with undergraduate medicine.  
The parsimonious specification of the model – where all institutions are assumed to have 
the same structure of costs and where no accommodation is made for the possible 
existence of scale and scope economies – works well as a description of how costs are 
determined in higher education. But it leaves out a lot of information that may be relevant 
to the evaluation of efficiency. So the more refined models, where institutions are divided 
into classes (either by the latent class method or by a priori assignment of institutions into 
groups) may be more informative about efficiency. The distributions of efficiencies that 
emerge from a quadratic specification of a latent class model with only two latent classes 
(in Figure 5) are quite compressed, with a concentration of institutions having efficiency 
scores above 0.9. There is nevertheless a (fairly small) number of institutions with low 
scores. These are all unusual by virtue of being small, specialist institutions.  
The latent class specifications of the model – whether linear or quadratic – also have the 
advantage of allowing some correction for unobserved heterogeneity across institutions. 
Albeit only by allowing two classes of institutions (in the results reported here), this makes 
some allowance for differences in (amongst other things) quality of provision that might 
influence costs. A more refined approach, using random parameters so that the 
distinctiveness of each institution is accommodated, would more fully allow for these 
qualitative differences, but might be viewed as being too permissive, and this approach 
has not, therefore, been employed here. 
44 

Efficiency in the Higher Education sector: A technical exploration 
45 
6. Conclusions 
 
This report shows how statistical models of the determination of costs faced by HEIs can 
be constructed. These can help to explain the relationship between costs and both outputs 
and other characteristics of institutions. They also allow the evaluation of an efficiency 
score for each institution.  
It is important to note that efficiency is a slippery concept. A user of the results of a 
statistical analysis may deem some characteristics of institutions, but not others, to be 
legitimate explanations of cost variations. This issue is further complicated by the fact that 
some of the characteristics that influence costs can be measured whereas others cannot – 
though, using panel data, both observable and unobservable characteristics can be 
allowed for in the calculation of an efficiency score.  
A key finding of the report is that, once differences between institutions are accounted for, 
even by a relatively unrefined latent class modelling procedure, the variation in efficiency 
scores across institutions is greatly reduced. Indeed, the relatively small number of 
institutions with low scores is exclusively made up of small and specialist institutions. The 
results do not, therefore, support the notion that substantial sector-wide gains could be 
made by using efficiency scores as a criterion for resource allocation. 
That said, the fact that frontier models can be estimated without difficulty in this context 
confirms that, in general, the residuals have the right skew. In other words, there does 
exist a distribution of efficiencies across institutions. It is not possible, from the analysis 
reported above, to explain these differences in measured efficiency – the nature of the 
analysis is that efficiency is calculated as an unexplained residual. This suggests that 
there may be advantage in conducting an analysis aimed at comparing institutions with 
higher and lower efficiency scores. Such an analysis would need to gather qualitative 
information of a kind that does not fit easily within the statistical approach adopted in the 
current report. Case studies would allow evaluation of the extent to which organisational 
structures, management styles and other more qualitative characteristics affect 
organisational efficiency. We leave this to future research.  
 

Appendices 
Appendix 1: Summary of results of previous studies of costs in the UK higher education sector 
 
Verry & Layard (1975) 
Verry & Davies (1976) 
Cohn et al (1989) 
Johnes (1990) 
Glass et al (1995a) 
Period 
covered 
1968/69 
1968/69 (approx) 
 
1985, 1986, 1987 
1989/90 
Coverage of 
HEIs 
UK universities except 
Oxford and Cambridge 
UK universities except 
Oxford and Cambridge 
UK polytechnics 
1887 US universities 
45 UK universities 
61 UK universities 
Frontier 
estimation 
No 
No 
No 
No 
No 
Functional 
form 
Linear 
Linear 
Quadratic 
Multiplicative 
Quadratic 
Linear 
Hybrid translog 
Definition of 
costs 
Estimation by 6 
departments: 
Departmental current 
costs 
Estimation by 6 
departments: 
Departmental current 
costs 
Also: Recurrent central 
university costs 
Total cost 
Total general 
expenditure on 
academic  
departments divided 
by FTE students 
Total cost 
Definition of 
outputs 
 No. of UG students in the 
given year 
 No. of PG students in the 
given year (split by 
coursework and research 
in some runs) 
 Annual quality weighted 
hours spent on research 
 No. of UG students in 
the given year 
 No. of PG students in 
the given year (split by 
coursework and 
research in some runs) 
 Annual quality weighted 
hours spent on research 
(sometimes replaced by 
number of articles and 
books) 
 UG FTE enrolment 
 PG FTE enrolment 
 Research grants 
 
 No. FTE Ugs 
 No. FTE PGs 
 1989 RAE 
aggregate research 
score 
Third mission 
No 
No 
No 
No 
No 
Input prices 
No 
No 
Average faculty 
salaries 
No 
Price of capital 
Price of labour 
 
46 

Efficiency in the Higher Education sector: A technical exploration 
 
Verry & Layard (1975) 
Verry & Davies (1976) 
Cohn et al (1989) 
Johnes (1990) 
Glass et al (1995a) 
Quality 
No 
Undergraduate quality is 
considered by using a 
value added measure 
based on A level results, 
degree results, & salaries. 
No 
No 
Subject mix found 
to be an important 
determinant of unit 
costs 
No 
Geographical 
location 
No 
London and Scotland 
considered 
No 
No 
No 
Mission group 
 
 Redbricks 
 New 
 Medical schools 
 ExCats 
 Polytechnics 
Separate estimation 
for public and private 
universities 
No 
Research groups: 
 Top-ranking 
 Middle-ranking 
 Bottom-ranking 
Economies of 
scale 
 Arts: 1.069 
 Social science: 1.197 
 Maths: 1.012 
 Physical science: 0.986 
 Biological science: 
1.045 
 Engineering: 1.060 
Departmental costs 
 Arts: 1.069 
 Social science: 1.197 
 Maths: 1.012 
 Physical science: 0.986 
 Biological science: 1.045 
 Engineering: 1.060 
Recurrent central 
university costs: 
Ray: 1.141 
Public institutions: 
 Ray: 1.045 
 UG: 0.944 
 PG: xx-xxxx-xxxx 
 Research: 1.273 
 
Private institutions: 
 Ray: 1.213 
 UG: 0.954 
 PG: 0.674 
 Research: 0.694 
 
 Ray: 1.131 
 Research: xx-xxxx-xxxx 
 PG: 1.263 
 UG: 2.578 
Economies of 
scope 
Assumed zero 
The multiplicative models 
find the interaction terms 
are generally insignificant 
Public institutions: 
 Ray: -0.064 
 
Private institutions: 
 Ray: 0.179 
 
 Global economies of 
scope not significantly 
different from 0. 
 Product specific for R 
and PG are not 
significantly different 
from 0. 
 UG has significant 
diseconomies of scope. 
 
47 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Verry & Layard (1975) 
Verry & Davies (1976) 
Cohn et al (1989) 
Johnes (1990) 
Glass et al (1995a) 
Average or 
marginal 
costs 
MC Arts:  
 UG = £310 
 PG = £710 
MC Social Science: 
 UG = £310 
 PG = £860 
MC Maths: 
 UG = £350 
 PG = £1470 
MC Physical Science: 
 UG = £480 
 PG = £2,100 
MC Biological Science: 
 UG = £550 
 PG = £1,580 
MC Engineering: 
 UG = £680 
 PG = £1,610 
Departmental costs 
MC Arts:  
 UG = £134 
 PG = £468 
MC Social Science: 
 UG = £133 
 PG = £620 
MC Maths: 
 UG = £118 
 PG = £902 
MC Physical Science: 
 UG = £243 
 PG = £1,533 
MC Biological Science: 
 UG = £310 
 PG = £1,012 
MC Engineering: 
 UG = £441 
PG = £1,049 
 
Central costs 
MC Arts: 
 UG = £171  
 PG = £242 
MC Science: 
 UG = £235 
PG = £564 
 
 
MCs: 
 PG = 0.074 
 UG = 0.350 
Efficiencies 
N/A 
N/A 
N/A 
N/A 
N/A 
 
48 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Glass et al (1995b) 
Johnes (1996) 
Johnes (1998) 
Johnes (1997) 
Izadi et al (2002) 
Period 
covered 
1992 
1989/90 
1989/90 
1994/95 
1994/95 
Coverage of 
HEIs 
53 traditional UK 
universities plus 8 
London colleges 
UK universities 
50 UK universities 
99 UK universities 
99 UK HEIs  
Frontier 
estimation 
No 
 OLS 
 SFA 
 SFA 
 DEA 
No 
SFA 
Functional 
form 
Hybrid translog 
Quadratic 
Quadratic 
CES 
CES 
Definition of 
costs 
Total cost 
Total recurrent 
expenditure 
Total recurrent 
expenditure 
Total expenditure 
Total expenditure 
Definition of 
outputs 
 No. FTE UGs 
 No. FTE PGs 
 1992 RAE aggregate 
research score 
 FTE UGs in Arts 
 FTE UGs in Science 
 FTE PGs in Arts 
 FTE PGs in Science 
 Research income in 
Arts 
 Research income in 
Science 
 FTE UGs in Arts 
 FTE UGs in Science 
 FTE PGs in Arts 
 FTE PGs in Science 
 External research 
grants in Arts 
 External research 
grants in Science 
 UG load in Arts 
 UG load in Science 
 PG load 
 Research grants and 
contracts 
Note that each part-time 
student is assumed to 
be 0.5 times a full-time 
student to measure 
student load. 
 UG load in Arts 
 UG load in Science 
 PG load 
 Research grants and 
contracts  
Note that each part-time 
student is assumed to be 
0.5 times a full-time 
student to measure 
student load. 
Third mission 
No 
No 
No 
No 
No 
Input prices 
Price of capital 
Price of labour 
No  
No 
No 
No 
Quality 
No 
No 
No 
No 
No 
Geographical 
location 
No 
No 
No 
No 
No 
Mission group 
Research groups: 
 Top-ranking 
 Middle-ranking 
 Bottom-ranking 
No 
No 
 Pre-1992 HEIs 
 Post-1992 HEIs 
Results run for each of 2 
groups and no 
significant difference in 
cost function found. 
 Arts-biased 
 Science-biased 
 
49 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Glass et al (1995b) 
Johnes (1996) 
Johnes (1998) 
Johnes (1997) 
Izadi et al (2002) 
Economies of 
scale 
 Ray: 1.16 
 PG: 7.057 
 UG: 2.789 
SFA 
 Ray: 1.05 
 PG: 5.06 
 Research: 2.41 
 All others unity 
 Ray: 1.07 
 Arts UG: 1.32 
 Science UG: 0.93 
 PG: 1.81 
 Research: 1.44 
SFA 
 Ray: 1.61 
 PG Science: 5.03 
 Unit for other outputs 
 Ray: 1.01 
 UG Arts: 1.20 
 UG Science: 1.03 
 PG: 3.34 
 Research: 1.39 
Economies of 
scope 
Not calculated 
Global: 0.18 
Global: -0.08 
Global: 0.17 
Global: -0.63 
Average or 
marginal 
costs 
 PG = 0.118 
 UG = 0.328 
Average incremental 
costs 
 UG Arts = £6,239 
 UG Science = £8,261 
 PG Arts = £4,599 
 PG Science = £8,322 
 Research Arts = 
£4.95 
 Research Science = 
£2.19 
Average 
incremental costs 
 UG Arts = £3,920 
 UG Science = 
£6,090 
 PG = £11,120 
 
Average incremental 
costs 
 UG Arts = £6,239 
 UG Science = £8,275 
 PG Arts = £4,598 
 PG Science = £8,327 
Average incremental 
costs 
 UG Arts = £3,241 
 UG Science = £6,714 
 PG = £22,789 
 Research = £3.20 
Efficiencies 
N/A 
Not reported 
N/A 
Mean DEA efficiency = 
0.91 
Rank correlation 
between SFA and DEA 
efficiencies = 0.133 
Mean efficiency = 0.78 
Minimum = 0.374 
Maximum = 0.991 
 
50 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Stevens (2005) 
Johnes et al (2005) 
Johnes et al (2008) 
Johnes & Johnes 
(2009) 
Thanassoulis et al 
(2011) 
Period 
covered 
1995/96 to 1998/99 
2000/01 to 2002/03 
2000/01 to 2002/03 
2000/01 to 2002/03 
2000/01 to 2002/03 
Coverage of 
HEIs 
80 HEIs in England and 
Wales 
121 HEIs in England 
121 HEIs in England 
121 HEIs in England 
121 HEIs in England 
Frontier 
estimation 
SFA 
 RE 
 SFA 
 DEA 
 RE 
 SFA 
 RE 
 Random Parameter 
RE 
 SFA 
 Random Parameter 
SFA 
NB: Only UG Science 
coefficient is random 
 DEA 
 Malmquist 
Functional 
form 
Translog 
Quadratic for 
parametric models 
Quadratic 
Quadratic 
N/A 
Definition of 
costs 
Total expenditure 
Total operating costs 
(excluding catering and 
student 
accommodation) 
Total operating costs 
(excluding catering and 
student 
accommodation) 
Total operating costs 
(excluding catering and 
student 
accommodation) 
Total operating costs 
(excluding catering 
and student 
accommodation) 
Definition of 
outputs 
 UG Science 
 UG Arts 
 PG 
 Research income 
 FTE UGs in Medicine 
 FTE UGs in Science 
 FTE UGs in Non-
science 
 FTE PGs 
 Quality related 
research funding and 
research grants 
 Income from other 
services rendered 
 FTE UGs in Medicine 
 FTE UGs in Science 
 FTE UGs in Non-
science 
 FTE PGs  
 Quality related 
research funding and 
research grants 
 Income from other 
services rendered 
 FTE UGs in Science 
(inc medicine) 
 FTE UGs in Non-
science 
 FTE PGs 
 Quality related 
research funding and 
research grants 
 FTE UGs in Medicine 
 FTE UGs in Science 
 FTE UGs in Non-
science 
 FTE PGs  
 Quality related 
research funding and 
research grants 
 Income from other 
services rendered 
Third mission 
No 
Yes 
Yes 
No 
Yes 
Input prices 
Average staff costs 
No 
No 
No 
No 
 
51 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Stevens (2005) 
Johnes et al (2005) 
Johnes et al (2008) 
Johnes & Johnes 
(2009) 
Thanassoulis et al 
(2011) 
Quality 
 Average A level score 
 Proportion of firsts and 
upper seconds 
A value added variable 
is added to equation 
but is not significant 
A value added variable 
is added to equation 
but is not significant 
No 
No 
Geographical 
location 
No 
London considered but 
barely significant 
London considered but 
not significant 
No 
No 
Mission group 
No 
 Colleges of higher 
education 
 Pre-1992 HEIs 
 Pre-1992 HEIs with 
medical schools 
 Post-1992 HEIs 
 Colleges of higher 
education 
 Pre-1992 HEIs 
 Pre-1992 HEIs with 
medical schools 
 Post-1992 HEIs 
 Top 5 
 Civics 
 ExCATs and 
Greenfields 
 Other pre-1992 HEIs 
 Post-1992 HEIs 
 Colleges of higher 
education 
 Colleges of higher 
education 
 Pre-1992 HEIs 
 Pre-1992 HEIs with 
medical schools 
 Post-1992 HEIs 
Economies of 
scale 
Not calculated 
Random Effects  
 Ray economies: 1.13 
 UG Medicine: 0.98 
 UG Science: 0.89 
 UG Non-science: 0.86 
 PG: 0.99 
 Research :1.05 
 
Random Effects 
 Ray economies: 1.09 
 UG Medicine: 1.01 
 UG Science: 0.99 
 UG Non-science: 0.95 
 PG: 1.00 
 Research: 1.07 
 
SFA 
 Ray economies: 0.96 
 UG Medicine: 0.98 
 UG Science: 1.01 
 UG Non-science: 1.02 
 PG: 0.87 
 Research: 1.07 
Random parameters 
random effects 
 Ray economies: 1.10 
 UG Science: 0.87 
 UG Non-science: 0.96 
 PG: 1.22 
 Research: 1.04 
 
Random parameters 
SFA 
 Ray economies: 0.98 
 UG Science: 0.98 
 UG Non-science:  
0.79 
 PG: 1.30 
 Research: 1.08 
Not calculated 
 
52 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Stevens (2005) 
Johnes et al (2005) 
Johnes et al (2008) 
Johnes & Johnes 
(2009) 
Thanassoulis et al 
(2011) 
Economies of 
scope 
Not calculated 
Random effects 
 Global economies: 
0.58 
 UG Medicine: 0.10 
 UG Science: 0.23 
 UG Non-science: 0.14 
 PG: 0.07 
 Research: 0.08 
 Third mission: 0.11 
 
Random effects 
 Global economies: 
0.38 
 UG Medicine: 0.06 
 UG Science: 0.17 
 UG Non-science: 0.07 
 PG: 0.03 
 Research: 0.04 
 Third mission: 0.08 
 
SFA 
 Global economies:  
-0.18 
 UG Medicine: -0.05 
 UG Science: 0.07 
 UG Non-science:  
-0.04 
 PG: -0.08 
 Research: -0.09 
 Third mission: -0.04 
Random parameters 
random effects 
Global economies: 0.30 
Random parameters 
SFA 
Global economies: -
0.17 
Not calculated 
 
53 

Efficiency in the Higher Education sector: A technical exploration 
 
 
 
Stevens (2005) 
Johnes et al (2005) 
Johnes et al (2008) 
Johnes & Johnes 
(2009) 
Thanassoulis et al 
(2011) 
Average or 
marginal 
costs 
Not calculated 
Average incremental 
costs 
Random Effects 
 UG Medicine = 
£17,769 
 UG Science = £5,079 
 UG Non-science = 
£3,217 
 PG = £9,569 
SFA 
 UG Medicine = 
£15,973 
 UG Science = £5,506 
 UG Non-science = 
£3,665 
 PG = £6,980 
 
Average incremental 
costs 
Random Effects 
 UG Medicine = 
£21,220 
 UG Science = £6,196 
 UG Non-science = 
£3,308 
 PG = £10,664 
SFA 
 UG Medicine = 
£17,603 
 UG Science = £6,368 
 UG Non-science = 
£3,925 
 PG = £7,574 
 
Average incremental 
costs 
Random Parameter 
Random Effects 
 UG Science = £5,516 
 UG Non-science = 
£2,869 
 PG = £16,215 
Random Parameter 
SFA 
 UG Science = £6,452 
 UG Non-science = 
£3,126 
 PG = £10,527 
 
Average incremental 
costs 
DEA 
 UG Medicine = 
£13,121 
 UG Science = £5,627 
 UG Non-science = 
£4,638 
 PG = £3,828 
SFA 
 UG Medicine = 
£15,973 
 UG Science = £5,506 
 UG Non-science = 
£3,665 
 PG = £6,979 
 
 
54 

Efficiency in the Higher Education sector: A technical exploration 
55 
 
 
Stevens (2005) 
Johnes et al (2005) 
Johnes et al (2008) 
Johnes & Johnes 
(2009) 
Thanassoulis et al 
(2011) 
Efficiencies 
Mean Efficiency 
 Model 1: 0.789 
 Model 2: 0.782 
 Model 3: 0.777 
 
Second stage 
equation 
 Staff variables: 
Higher proportions of 
staff aged 50 or 
more leads to lower 
efficiency. 
 Higher proportions of 
staff who professors, 
SLs, research active 
lead to higher 
efficiency. 
 Student variables: 
Higher proportions of 
students with first 
and upper seconds, 
mature students, 
students of low 
socio-economic 
class lead to higher 
efficiency. 
Final year of study 
 Mean efficiency = 
0.75 
 Post-1992 HEIs = 
0.85 
 Pre-1992 HEIs = 
0.82 
 Colleges of higher 
education = 0.56 
 Mean efficiency = 
0.69 
 Post-1992 HEIs = 
0.84 
 Pre-1992 HEIs = 
0.80 
 Colleges of higher 
education = 0.43 
 Mean efficiency = 
0.753 
 Top 5 = 0.942 
 Civics = 0.919 
 ExCATs and 
Greenfields = 0.844 
 Other pre-1992 HEIs 
= 0.712 
 Post-1992 HEIs = 
0.859 
 Colleges of higher 
education = 0.499 
 Mean DEA efficiency 
= 0.863 
 Mean DEA efficiency 
year 3 = 0.854 
 Mean SFA efficiency 
year 3 = 0.837 
 
 
 

Appendix 2: Alternative non-linear cost function specifications 
In addition to the quadratic specification, two other functional forms can be used to provide 
a non-linear representation of costs. 
i) 
The constant elasticity of substitution (CES) function 
Using the two outputs of teaching (T) and research (R) as in the text, the CES function is 
specified as follows: 
 
ii) The hybrid translog function 
 
All the non-linear specifications considered here and in the text satisfy, at least for certain 
parameter vectors, the three desiderata identified by (Baumol et al. 1982), namely that the 
cost function should: 
 be a ‘proper’ cost function that is consistent with cost minimization given input and 
output cost. In particular it should be a non-negative and non-decreasing function; 
 sensibly predict costs where outputs of some products are zero. This is essential if 
average incremental costs and measures of economies of scale and scope are being 
calculated; 
 not either preclude or enforce the existence of economics (or diseconomies) of scale or 
scope. 
It is easily seen that these alternative specifications are highly nonlinear and more 
complicated than the quadratic specification used in this report. Indeed, it is not possible to 
estimate frontier variants of these models using standard software without undertaking 
considerable work to evaluate the likelihood functions and undertake the programming 
needed to maximise these likelihoods. For this reason, the analysis of non-linear models in 
this report is limited to the quadratic specification.  
 
56 

Efficiency in the Higher Education sector: A technical exploration 
Appendix 3: Data definitions 
Variable name 
Definition 
Units 
Dependent variable 
COST 
Total expenditure minus expenditure on residences and 
catering operations 
£000s in 2011 prices 
Undergraduate teaching 
UGMED 
Undergraduate students (first degree and other) in 
medicine 
FTEs 
UGSCI 
Undergraduate students (first degree and other) in 
sciences other than medicine 
FTEs 
UGARTS 
Undergraduate students in all other subjects 
FTEs 
Postgraduate teaching 
PG 
Postgraduate students in all subjects 
FTEs 
Research 
RESEARCH 
HEFCE R plus income from research grants and 
contracts 
£ in 2011 prices 
PUBLICATIONS 
Publications per HEI from Web of Science 
Number 
CITATIONS 
Citations per HEI from Web of Science 
Number 
Third mission 
EVENTS 
Staff time devoted to public events (free and chargeable)
Events include: public lectures; performance arts; 
exhibitions; museum education and other. 
Days 
ATEVENT 
Number of attendees at public events (free and 
chargeable) 
Events include: public lectures; performance arts; 
exhibitions; museum education and other. 
Number of people 
IPINCOME 
Income from third mission activity i.e. the sum of: 
 
Total income from collaborative research involving 
both public funding and funding from business (a5) 
 
Total value of contract research (excluding any 
already returned in a5) (b8) 
 
Total value of consultancy contracts (c8) 
 
Total value of contracts for facilities and equipment 
related services - organisations involved and income 
(d8) 
 
Total revenue for courses for business and the 
community - CPD courses and CE (excluding those 
funded by the NHS or TDA) (e5) 
 
Total income from regeneration and development 
programmes (f7) 
 
Total revenue from: IP income from SMEs plus IP 
income from other (non-SME) commercial 
businesses plus IP income from other non-
commercial organisations plus sale of shares in 
spin-offs (m7) 
£000s in 2011 prices 
57 

Efficiency in the Higher Education sector: A technical exploration 
 
 
Additional factors 
LOWPNO 
Number of young full-time undergraduate entrants from 
low participation neighbourhoods 
 
LISTED 
Area of a HEI’s buildings which are designated as listed 
Metres squared 
UPGRADE 
Cost of upgrade to upgrade all non-residential space in 
condition D to condition B (D20bC13UPDB) plus the 
cost of upgrade to upgrade all space in condition C to 
condition B (D”)bC13UPCB) where condition B is 
defined as: Sound, operationally safe and exhibiting only 
minor deterioration. 
£ in 2011 prices 
NMEAN 
Mean salary of graduates (from DLHE) 
£ in 2011 prices 
NSS 
The percentage positive response to the question from 
the National Student Survey: ‘Overall, I am satisfied with 
the quality of the course’ 
Percentage  
Dummy variables 
2003/04 
1 if year is 2003/04 
Dummy variable 
2004/05 
1 if year is 2004/05 
Dummy variable 
2005/06 
1 if year is 2005/06 
Dummy variable 
2006/07 
1 if year is 2006/07 
Dummy variable 
2007/08 
1 if year is 2007/08 
Dummy variable 
2008/09 
1 if year is 2008/09 
Dummy variable 
2009/10 
1 if year is 2009/10 
Dummy variable 
2010/11 
1 if year is 2010/11 
Dummy variable 
OXBRIDGE 
1 if HEI is Oxford or Cambridge 
Dummy variable 
 
Note: Subject definitions are as follows 
 Medicine: Medicine & dentistry; Subjects allied to medicine;  
 Other science: Biological sciences; Veterinary science; Agriculture and related subjects; Physical 
sciences; Mathematical sciences; Computer science; Engineering and technology; Architecture, building 
and planning;  
 Non-science: Social studies; Law; Business and administrative studies; Mass communications and 
documentation; Languages; Historical and philosophical studies; Creative arts and design; Education.  
 There are a small number of students in combined subjects. These have been divided and allocated 
according to the proportion of students observed in each subject. 
58 

Efficiency in the Higher Education sector: A technical exploration 
Appendix 4: Additional results 
A4.1 Efficiency distributions associated with linear models with a full set of 
controls presented in Table 3 
 
a) Final year of 2005/06 to 2007/08 model 
 
Note: Efficiencies for 14 HEIs lie outside the range from -1 to +1. 
59 

Efficiency in the Higher Education sector: A technical exploration 
 
b) Final year of 2003/04 to 2004/05 model 
 
c) Final year of 2003/04 to 2010/11 model 
 
Note: Efficiencies for 18 HEIs lie outside the range from -1 to +1. 
  
 
60 

Efficiency in the Higher Education sector: A technical exploration 
A4.2 Efficiency distributions associated with linear models with a limited set of 
controls presented in Table 4 
 
a) Final year of 2008/09 to 2010/11 model 
 
Note: Efficiencies for 17 HEIs lie outside the range from -1 to +1. 
b) Final year of 2005/06 to 2007/08 model 
 
Note: Efficiencies for 20 HEIs lie outside the range from -1 to +1. 
61 

Efficiency in the Higher Education sector: A technical exploration 
 
c) Final year of 2003/04 to 2010/11 model 
 
Note: Efficiencies for 18 HEIs lie outside the range from -1 to +1. 
 
 
A4.3 AICs for the quadratic model with a limited set of controls 
AICs 
2008/09 to 2010/11 
2003/04 to 2004/05 
2003/4 to 2010/11 
UGMED 
17.88119
9.50562
12.58065
UGSCI 
6.25241
3.31588
4.80015
UGARTS 
4.44784
3.89313
4.87373
PG 
13.69599
8.54893
0.28524
RESEARCH 
1.22823
1.19398
1.14478
IPINCOME 
1.13426
0.59380
0.92564
Notes: 
1. Controls: OXBRIDGE; YEAR dummies 
2. The 2005/06 to 2007/08 model does not converge 
 
62 

Efficiency in the Higher Education sector: A technical exploration 
A4.4 Efficiency distributions associated with quadratic models with a limited set of 
controls presented in Appendix 4.3 above 
 
a) Final year of 2008/09 to 2010/11 model 
 
b) Final year of 2003/04 to 2004/05 model 
 
 
63 

Efficiency in the Higher Education sector: A technical exploration 
 
c) Final year of 2003/04 to 2010/11 model 
 
Note: The quadratic model for 2005/06 through 2007/08 fails to converge. Once the data for this period are 
included alongside data for the 2003/04 through 2004/05 and 2008/09 through 2010/11 periods, there is little 
skew in the residuals, and so all efficiency scores are estimated to be close to one. 
64 

Efficiency in the Higher Education sector: A technical exploration 
A4.5 Membership of latent classes 
Institution 
2008-09 
2009-10 
2010-11 
Anglia Ruskin University 
2 
1 
2 
Arts University College at Bournemouth 
1 
2 
1 
Aston University 
2 
2 
2 
Bath Spa University 
2 
1 
2 
Birkbeck College 
1 
1 
1 
Birmingham City University 
2 
1 
2 
Bishop Grosseteste University College Lincoln 
2 
2 
2 
Bournemouth University 
2 
2 
2 
Brunel University 
1 
2 
1 
Buckinghamshire New University 
1 
1 
2 
Canterbury Christ Church University 
2 
1 
1 
Central School of Speech and Drama 
2 
2 
2 
City University 
2 
2 
2 
Courtauld Institute of Art 
2 
2 
2 
Coventry University 
1 
2 
2 
Cranfield University 
1 
 
 
De Montfort University 
1 
1 
1 
Edge Hill University 
1 
1 
1 
Goldsmiths College 
2 
1 
2 
Guildhall School of Music and Drama 
2 
2 
2 
Harper Adams University College 
2 
2 
2 
Heythrop College 
2 
2 
2 
Imperial College of Science, Technology and Medicine 
 
 
2 
Institute of Cancer Research 
 
 
2 
Institute of Education 
1 
1 
1 
King's College London 
2 
2 
1 
Kingston University 
2 
2 
1 
Leeds College of Music 
 
2 
2 
Leeds Metropolitan University 
2 
2 
2 
Leeds Trinity University College 
2 
2 
2 
Liverpool Hope University 
2 
2 
2 
Liverpool Institute for Performing Arts 
2 
2 
2 
Liverpool John Moores University 
2 
2 
2 
London Metropolitan University 
2 
2 
2 
London School of Economics and Political Science 
2 
2 
2 
London School of Hygiene and Tropical Medicine 
 
2 
 
London South Bank University 
1 
1 
2 
Loughborough University 
1 
1 
1 
Manchester Metropolitan University 
2 
1 
1 
Middlesex University 
2 
2 
1 
Newman University College 
2 
2 
2 
Norwich University College of the Arts 
2 
2 
2 
Nottingham Trent University 
2 
2 
2 
Oxford Brookes University 
2 
2 
2 
Queen Mary and Westfield College 
1 
1 
1 
65 

Efficiency in the Higher Education sector: A technical exploration 
 
Institution 
2008-09 
2009-10 
2010-11 
Ravensbourne College of Design and Communication 
2 
2 
2 
Roehampton University 
2 
2 
2 
Rose Bruford College 
2 
2 
2 
Royal Academy of Music 
2 
2 
2 
Royal Agricultural College 
2 
2 
2 
Royal College of Art 
 
 
2 
Royal College of Music 
2 
2 
2 
Royal Holloway and Bedford New College 
2 
2 
2 
Royal Northern College of Music 
2 
2 
2 
Royal Veterinary College 
1 
1 
1 
School of Oriental and African Studies 
2 
2 
2 
School of Pharmacy 
2 
2 
 
Sheffield Hallam University 
2 
2 
2 
Southampton Solent University 
1 
1 
1 
St George's Hospital Medical School 
1 
1 
1 
St Mary's University College, Twickenham 
2 
2 
2 
Staffordshire University 
2 
2 
2 
Thames Valley University 
1 
1 
2 
Trinity Laban Conservatoire of Music and Dance 
2 
2 
1 
University College Birmingham 
2 
2 
1 
University College Falmouth 
1 
1 
1 
University College Plymouth St Mark and St John 
1 
1 
2 
University for the Creative Arts 
1 
1 
1 
University of Bath 
1 
1 
1 
University of Bedfordshire 
1 
2 
1 
University of Birmingham 
2 
2 
2 
University of Bolton 
2 
2 
2 
University of Bradford 
2 
 
2 
University of Brighton 
2 
2 
2 
University of Bristol 
1 
1 
1 
University of Cambridge 
1 
1 
1 
University of Central Lancashire 
2 
2 
1 
University of Chester 
2 
 
2 
University of Chichester 
2 
2 
2 
University of Cumbria 
1 
1 
2 
University of Derby 
1 
1 
1 
University of Durham 
2 
2 
1 
University of East Anglia 
2 
2 
2 
University of East London 
2 
2 
2 
University of Essex 
2 
2 
2 
University of Exeter 
2 
2 
2 
University of Gloucestershire 
2 
2 
2 
University of Greenwich 
1 
2 
1 
University of Hertfordshire 
1 
2 
2 
University of Huddersfield 
1 
2 
1 
University of Hull 
2 
2 
2 
University of Keele 
1 
1 
1 
66 

Efficiency in the Higher Education sector: A technical exploration 
Institution 
2008-09 
2009-10 
2010-11 
University of Kent 
2 
1 
1 
University of Lancaster 
1 
1 
1 
University of Leeds 
2 
2 
1 
University of Leicester 
1 
2 
2 
University of Lincoln 
2 
1 
2 
University of Liverpool 
1 
1 
1 
University of London (Institutes and activities) 
 
 
1 
University of Manchester 
1 
1 
1 
University of Newcastle-upon-Tyne 
2 
2 
2 
University of Northampton 
2 
1 
2 
University of Northumbria at Newcastle 
2 
2 
1 
University of Nottingham 
1 
1 
2 
University of Oxford 
2 
2 
2 
University of Plymouth 
2 
2 
2 
University of Portsmouth 
2 
2 
1 
University of Reading 
1 
1 
1 
University of Salford 
2 
2 
2 
University of Sheffield 
2 
2 
 
University of Southampton 
1 
1 
1 
University of Sunderland 
1 
1 
2 
University of Surrey 
2 
 
2 
University of Sussex 
2 
2 
2 
University of Teesside 
2 
2 
2 
University of the Arts, London 
1 
1 
1 
University of the West of England, Bristol 
1 
2 
2 
University of Warwick 
1 
1 
1 
University of Westminster 
2 
2 
2 
University of Winchester 
 
2 
 
University of Wolverhampton 
2 
2 
2 
University of Worcester 
2 
2 
2 
University of York 
2 
2 
2 
Writtle College 
2 
2 
2 
York St John University 
2 
2 
2 
 
Note:  
1. The information in this table allows a sense check to be conducted on the analysis, in that it shows the 
extent to which institutions that are known to be of broadly similar type tend to be clustered together in 
latent classes. 
2. If an institution has missing data for a particular variable for all three years, they were omitted from this 
analysis. This affected the Conservatoire for Dance and Drama, London Business School, University 
College London and the University of Buckingham. 
67 

Efficiency in the Higher Education sector: A technical exploration 
 
A4.6 Results from 3- and 4-class latent class models: linear specification with only 
a limited set of controls 
 
We estimate latent class models with 3 and with 4 classes, using a linear model with no 
controls in the latest run of 3 years. In each case, latent class 1 is small, and in each case 
this class produces implausible coefficients, while the coefficients for the other latent 
classes look broadly sensible.  
a) 3-class linear latent class model 
 
2008/09 to 2010/11 
AICs 
Class 1 
Class 2 
Class 3 
UGMED 
-14.61370 
4.43692 
13.23580 
UGSCI 
-1.12589 
7.98182 
4.40658 
UGARTS 
5.85165 
xx-xxxx-xxxx 
6.67897 
PG 
-16.22300 
15.41280 
9.11164 
RESEARCH 
1.88377 
1.09513 
1.22724 
IPINCOME 
1.29048 
1.19655 
0.46838 
Controls 
 
 
 
2009/10 
8,689.96 
-3,019.64 
-1,255.80 
2010/11 
12,905.60 
-1,380.47 
-4,880.82 
OXBRIDGE 
315,210.00 
61,296.50 
79,624.30 
CONSTANT 
104,531.00 
-162.52 
7,270.85 
Number in class 
18 
254 
112 
Is λ significantly different from 
zero at the 5% significance level?
NO 
NO 
NO 
Notes: 
1. Controls: OXBRIDGE; YEAR dummies 
2. Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
3. The number in each class is the number of observations over the estimation period, not the number of 
HEIs. Given that HEIs can be in different classes in different years, dividing by the number of years on 
which the analysis is based does not give the number of HEIs. 
68 

Efficiency in the Higher Education sector: A technical exploration 
b) 4-class linear latent class model 
 
2008/09 to 2010/11 
AICs 
Class 1 
Class 2 
Class 3 
Class 4 
UGMED 
-11.38990 
xx-xxxx-xxxx 
13.90920 
14.66870 
UGSCI 
3.48366 
9.76754 
2.81820 
7.69351 
UGARTS 
7.74338 
3.87148 
4.74366 
3.75975 
PG 
-23.71430 
12.90010 
18.58490 
xx-xxxx-xxxx 
RESEARCH 
2.23325 
1.10102 
1.11549 
1.08442 
IPINCOME 
0.25371 
1.26308 
xx-xxxx-xxxx 
0.97486 
Controls 
 
 
 
 
2009/10 
11,443.20 
-1,442.06 
-2,210.09 
-982.70 
2010/11 
20,637.40 
-1,692.57 
-1,627.01 
-982.17 
OXBRIDGE 
316,556.00 
62,387.10 
106,659.00 
97,948.70 
CONSTANT 
103,672.00 
378.39 
-142.69 
8,289.91 
Number in each group 
20 
231 
102 
31 
Is λ significantly different from 
zero at the 5% significance level?
NO 
NO 
NO 
NO 
Notes: 
1. Controls: OXBRIDGE; YEAR dummies. 
2. Coefficients in bold are statistically significantly different from zero at the 5% significance level. 
3. The number in each class is the number of observations over the estimation period, not the number of 
HEIs. Given that HEIs can be in different classes in different years, dividing by the number of years on 
which the analysis is based does not give the number of HEIs. 
 
69 

Efficiency in the Higher Education sector: A technical exploration 
 
A4.7 Distributions of efficiency scores obtained from the linear model in which 
institutions are classified by HEFCE impact report groups – final year of 
2008/09 to 2010/11 model 
 
These histograms relate to the equations in Table 8 (2008/09 to 2010/11 models), and are 
drawn for the final year of the estimation period. 
a) Group 1 - specialist 
 
Note: Efficiencies for 16 HEIs lie outside the range from -1 to +1. 
70 

Efficiency in the Higher Education sector: A technical exploration 
b) Group 2 – high tariff 
 
c) Group 3 – medium tariff 
 
71 

Efficiency in the Higher Education sector: A technical exploration 
 
d) Group 4 – low tariff 
 
 
72 

Efficiency in the Higher Education sector: A technical exploration 
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