Chapter 2

THE MEASUREMENT OF EFFICIENCY

This chapter is about the measurement of efciency of production units. It
opens with a section concerning basic denitions of productivity and efciency.
After that, an historical and background section follows, reporting some of the
most important contributions until around 90s. Then, the axiomatic under-
pinning of the Activity Analysis framework used to represent the production
process is described in the economic model section. Afterwards, efcient fron-
tier models are classied according to three main criteria: specication (or not)
of the form of the frontier; presence of noise in the estimation procedure; type
of data analyzed (cross-section or panel data). The presentation of the most
known nonparametric estimators of frontiers (i.e., Data Envelopment Analysis
(DEA) and Free Disposal Hull (FDH)) is subsequently. Finally, a section sum-
marizing recent developments in nonparametric efciency analysis concludes
the chapter.
2.1 Productivity and Efciency
According to a classic denition (see e.g. Vincent 1968) productivity is the
ratio between an output and the factors that made it possible. In the same way,
Lovell (1993) denes the productivity of a production unit as the ratio of its
output to its input.
This ratio is easy to compute if the unit uses a single input to produce a
single output. On the contrary, if the production unit uses several inputs to
produce several outputs, then the inputs and outputs have to be aggregated so
that productivity remains the ratio of two scalars.
We can distinguish between a partial productivity, when it concerns a sole
production factor, and a total factor (or global) productivity, when referred to
all (every) factors.
14 The measurement of efciency
Similar, but not equal, is the concept of efciency. Even though, in the ef-
ciency literature many authors do not make any difference between productivity
and efciency. For instance, Sengupta (1995) and Cooper, Seiford and Tone
(2000) dene both productivity and efciency as the ratio between output and
input.
Instead of dening the efciency as the ratio between outputs and inputs, we
can describe it as a distance between the quantity of input and output, and the
quantity of input and output that denes a frontier, the best possible frontier for
a rm in its cluster (industry).
Efciency and productivity, anyway, are two cooperating concepts. The
measures of efciency are more accurate than those of productivity in the sense
that they involve a comparison with the most efcient frontier, and for that they
can complete those of productivity, based on the ratio of outputs on inputs.
Lovell (1993) denes the efciency of a production unit in terms of a com-
parison between observed and optimal values of its output and input. The
comparison can take the form of the ratio of observed to maximum potential
output obtainable from the given input, or the ratio of minimum potential to
observed input required to produce the given output. In these two comparisons
the optimum is dened in terms of production possibilities, and efciency is
technical.
Koopmans (1951; p. 60) provided a denition of what we refer to as tech-
nical efciency: an input-output vector is technically efcient if, and only if,
increasing any output or decreasing any input is possible only by decreasing
some other output or increasing some other input.
Farrell (1957; p. 255) and much later Charnes and Cooper (1985; p. 72) go
back over the empirical necessity of treating Koopmans denition of technical
efciency as a relative notion, a notion that is relative to best observed practice
in the reference set or comparison group. This provides a way of differentiating
efcient from inefcient production units, but it offers no guidance concerning
either the degree of inefciency of an inefcient vector or the identication of
an efcient vector or combination of efcient vectors against which comparing
an inefcient vector.
Debreu (1951) offered the rst measure of productive efciency with his coef-
cient of resource utilization. Debreus measure is a radial measure of technical
efciency. Radial measures focus on the maximum feasible equiproportionate
reduction in all variable inputs, or the maximum feasible equiproportionate
expansion of all outputs. They are independent of unit of measurement.
Applying radial measures the achievement of the maximum feasible input
contraction or output expansion suggests technical efciency, even though there
may remain slacks in inputs or surpluses in output. In economics the notion of
efciency is related to the concept of Pareto optimality. An input-output bundle
is not Pareto optimal if there remains the opportunity of any net increase in
Productivity and Efciency 15
outputs or decrease in inputs. Pareto-Koopmans measures of efciency (i.e.,
measures which call a vector efcient if and only if it satises the Koopmans
denition reported above, coherent with the Pareto optimality concept) have
been analysed in literature. See e.g., Fare (1975), Fare and Lovell (1978) and
Russell (1985, 1988, 1990) among others.
Farrell (1957) extended the work initiated by Koopmans and Debreu by
notingthat productionefciencyhas a secondcomponent reectingthe abilityof
producers to select the right technically efcient input-output vector in light of
prevailing input and output prices. This led Farrell to dene overall productive
efciency as the product of technical and allocative efciency. Implicit in
the notion of allocative efciency is a specic behavioral assumption about the
goal of the producer; Farrell considered cost-minimization in competitive inputs
markets, although all the behavioral assumptions can be considered. Although
the natural focus of most economists is on markets and their prices and thus on
allocative rather than technical efciency and its measurement, he expressed
a concern about human ability to measure prices accurately enough to make
good use of allocative efciency measurement, and hence of overall economic
efciency measurement. This worry expressed by Farrell (1957; p. 261) has
greatly inuenced the OR/MS work on efciency measurement. Charnes and
Cooper (1985; p. 94) cite Farrell concern as one of several motivations for the
typical OR/MS emphasis on the measurement of technical efciency.
It is possible to distinguish different kind of efciency, such as scale, alloca-
tive and structural efciency.
The scale efciency has been developed in three different ways. Farrell
(1957) used the most restrictive technology having constant returns to scale
(CRS) and exhibiting strong disposability of inputs. This model has been de-
veloped in a linear programming framework by Charnes, Cooper and Rhodes
(1978). Banker, Charnes and Cooper (1984) have shown that the CRS measure
of efciency can be expressed as the product of a technical efciency measure
and a scale efciency measure. A third method of scale uses nonlinear speci-
cation of the production function such as Cobb-Douglas or a translog function,
from which the scale measure can be directly computed (see Sengupta, 1994
for more details).
The allocative efciency in economic theory measures a rms success in
choosing an optimal set of inputs with a given set of input prices; this is dis-
tinguished from the technical efciency concept associated with the production
frontier, which measures the rms success in producing maximumoutput from
a given set of inputs.
The concept of structural efciency is an industry level concept due to Farrell
(1957), which broadly measures in what extent an industry keeps up with the
performance of its own best practice rms; thus it is a measure at the industry
level of the extent to which its rms are of optimumsize i.e. the extent to which
16 The measurement of efciency
the industry production level is optimally allocated between the rms in the
short run. A broad interpretation of Farrells notion of structural efciency can
be stated as follows: industry or cluster A is more efcient structurally than in-
dustry B, if the distribution of its best rms is more concentrated near its efcient
frontier for industry A than for B. In their empirical study, Bjurek, Hjalmarsson
and Forsund (1990) compute structural efciency by simply constructing an
average unit for the whole cluster and then estimating the individual measure of
technical efciency for this average unit. On more general aggregation issues,
see Fare and Zelenyuk (2003) and Fare and Grosskopf (2004, p. 94 ff).
2.2 A short history of thought
The theme of productive efciency has been analysed since Adam Smiths
pin factory and before
1
. However, as we have seen in the previous section,
a rigorous analytical approach to the measurement of efciency in production
originated only with the work of Koopmans (1951) and Debreu (1951), empir-
ically applied by Farrell (1957).
An important contribution to the development of efciency and productivity
analysis has been done by Shephards models of technology and his distance
functions (Shephard 1953, 1970, 1974). In contrast to the traditional produc-
tion function, direct input and output correspondences admit multiple outputs
and multiple inputs. They are thus able to characterize all kinds of technologies
without unwarranted output aggregation prior to analysis. The Shephard direct
input distance function treats multiple outputs as given and contracts inputs vec-
tors as much as possible consistent with technological feasibility of contracted
input vector. Among its several useful properties, one of the most important is
the fact that the reciprocal of the direct input distance function has been pro-
posed by Debreu (1951) as a coefcient of resource utilization, and by Farrell
(1957) as a measure of technical efciency. This property has both a theoretical
and a practical signicance. It allows the direct input distance function to serve
two important roles, simultaneously. It provides a complete characterization
of the structure of multi-input, multi-output efcient production technology,
and a reciprocal measure of the distance from each producer to that efcient
technology.
The main role played by the direct input distance function is to gauge tech-
nical efciency. Nevertheless, it can also be used to construct input quantity
indexes (Tornqvist, 1936; Malmquist, 1953) and productivity indexes (Caves,
Christensen, and Diewert, 1982). Similarly, the direct output distance func-
tion introduced by Shephard (1970) and the two indirect distance functions of
Shephard (1974) can be used to characterize the structure of efcient production
1
This section is based on Fare, Grosskopf and Lovell (1994), pp. 1-23; and Kumbhakar and Lovell (2000),
pp. 5-7.
A short history of thought 17
technology in the multi-product case, to measure efciency to that technology,
and to construct output quantity indexes (Bergson,1961; Moorsteen, 1961) and
productivity indexes (Fare, Grosskopf, and Lovell, 1992).
Linear programming theory is a milestone of efciency analysis. The work
of Dantzig (1963) is closely associated with linear programming since he con-
tributed to the basic computational algorithm (the simplex method) used to
solve this problem. Charnes and Cooper (1961) made considerable contribu-
tions to both theory and application in the development of linear programming,
and popularize its application in DEA in the late 70s (see Charnes, Cooper and
Rhodes, 1978). Forsund and Sarafoglou (2002) offer an interesting historical
reconstruction of the literature developments subsequent to Farrells seminal
paper that lead to the introduction of the DEA methodology.
The use of linear programming and activity analysis can be found in the work
of Leontief (1941, 1953) who developed a special case of activity analysis which
has come to be known as input-output analysis. Whereas Leontiefs work was
directed toward constructing a workable model of general equilibrium, ef-
ciency and productivity analysis is more closely related to the microeconomic
production programming models developed by Shephard (1953, 1970, 1974),
Koopmans (1951, 1957) and Afriat (1972). In these models observed activities,
such as the inputs and outputs of some production units, serve as coefcients
of activity or intensity variables forming a series of linear inequalities, yielding
a piecewise linear frontier technology.
The work of Koopmans and Shephard imposes convexity on the reference
technology, therefore, the DEA estimator relies on the convexity assumption.
The Free Disposal Hull (FDH) estimator, that maintains free disposability while
relaxes convexity, was introduced by Deprins, Simar and Tulkens (1984).
By enveloping data points with linear segments, the programming approach
reveals the structure of frontier technology without imposing a specic func-
tional form on either technology or deviations from it.
Frontier technology provides a simple means of computing the distance to
the frontier - as a maximum feasible radial contraction or expansion of an ob-
served activity. This means of measuring the distance to the frontier yields an
interpretation of performance or efciency as maximal-minimal proportionate
feasible changes in an activity given technology. This explanation is consis-
tent with Debreus (1951) coefcient of resource utilization and with Farrells
(1957) efciency measures. However, neither Debreu nor Farrell formulated
the efciency measurement problem as a linear programming problem, even
though Farrell and Fieldhouse (1962) envisaged the role of linear program-
ming. The full development of linear programming techniques took place later.
Boles (1966), Bressler (1966), Seitz (1966) and Sitorius (1966) developed the
piecewise linear case, and Timmer (1971) extended the piecewise log-linear
case.
18 The measurement of efciency
Linear programming techniques are also used in production analysis for non-
parametric tests
2
on regularity conditions and behavioral objectives. Afriat
(1972) developed a series of consistency tests on production data by assuming
an increasing number of more restrictive regularity hypotheses on production
technology. In so doing he expanded his previous work on utility functions
(Afriat 1967) based on the revealed preference analysis (Samuelson, 1948).
These tests of consistency, as well as similar tests of hypotheses proposed
by Hanoch and Rothschild (1972), are all based on linear programming formu-
lations. Diewert and Parkan (1983) suggested that this battery of tools could be
used as a screening device to construct frontiers and measure efciency of data
relative to the constructed frontiers. Varian (1984, 1985, 1990) and Banker and
Maindiratta (1988) extended the Diewert and Parkan approach. In particular,
Varian seeks to reduce the all-or-nothing nature of the tests - either data pass
a test or they do not - by developing a framework for allowing small failures to
be attributed to measurement in the data rather than to failure of the hypothesis
under investigation.
All these studies use nonparametric linear programming models to explore
the consistency of a dataset, or a subset of a dataset, with a structural (e.g.
constant return to scale) or parametric (e.g. Cobb-Douglas) or behavioral (e.g.
cost minimization) hypothesis. These tools, originally proposed as screening
devices to check for data accuracy, provide also guidance in the selection of
parametric functional forms as well as procedures useful to construct frontiers
and measure efciency. The problemof nonparametric exploration of regularity
conditions and behavioral objectives has been treated also by Chavas and Cox
(1988, 1990), Ray (1991), and Ray and Bhadra (1993).
Some works have indirectly inuenced the development of the efciency
and productivity analysis. Hicks (1935, p.8) states his easy life hypothesis
as follows: people in monopolistic positions [...] are likely to exploit their
advantage much more by not bothering to get very near the position of maximum
prot, than by straining themselves to get very close to it. The best of all
monopoly prots is a quite life. The suggestion of Hicks, i.e. the fact that
the absence of competitive pressure might allow producers the freedom to not
fully optimize conventional objectives, and, by implication, that the presence
of competitive pressure might force producers to do so, has been adopted by
many authors (see e.g. Alchian and Kessel, 1962, and Williamson, 1964).
Another eld of work, related to efciency literature, is the property rights
eld of research, which asserts that public production is inherently less ef-
cient than private production. This argument, due originally to Alchian (1965),
states that concentration and transferability of private ownership shares create
2
Here and below when we use the word test between quotation mark we mean qualitative indicators that are
not real statistical test procedures.
The economic model 19
an incentive for private owners to monitor managerial performance, and that
this incentive is diminished for public owners, who are dispersed and whose
ownership is not transferable. Consequently, public managers have wider free-
dom to pursue their owns at the expense of conventional goals. Thus Niskanen
(1971) argued that public managers are budget maximizers, de Alessi (1974)
argued that public managers exhibit a bias toward capital-intensive budgets,
and Lindsay (1976) argued that public managers exhibit a bias toward visi-
ble inputs. However, ownership forms are more varied than just private or
public. Hansmann (1988), in facts, identies investor-owned rms, customer-
owned rms, worker-owned rms, as well as rms without owners (nonprot
enterprisers). Each of them deals in a different way with problems associated
with hierarchy, coordination, incomplete contracts and monitoring and agency
costs. This leads to the expectation that different ownership forms will generate
differences in performance.
3
As a more micro level is concerned, Simon (1955, 1957) analyzed the per-
formance of producers in the presence of bounded rationality and satisfying
behavior. Later Leibenstein (1966, 1975, 1976, 1978, 1987) argued that pro-
duction is bound to be inefcient as a result of motivation, information, mon-
itoring, and agency problems within the rm. This type of inefciency, the
so called X-inefciency has been criticized by Stigler (1976) and de Alessi
(1983) among others since it reects an incompletely specied model rather
than a failure to optimize.
The problem of model specication - including a complete list of inputs
and outputs, and perhaps conditioning variables as well, a list of constraints,
technological, and other (e.g. regulatory) is a difcult issue to face. Among
others, Banker, Chang and Cooper (1996) analyse the effects of misspecied
variables in DEA. Simar and Wilson (2001) propose a statistical procedure to
test for the relevance of inputs/outputs in DEA models.
This literature suggests that the development of efciency analysis is par-
ticularly useful if and when it could be used to shed empirical light on the
theoretical issues outlined above.
2.3 The economic model
In this paragraph we describe the main axioms on which the economic model
underlined the measurement of efciency is based on.
4
Much empirical evidence suggests that although producers may indeed at-
tempt to optimize, they do not always succeed. Not all producers are always so
successful in solving their optimization problems. Not all producers succeed
3
This expectation is based on a rich theoretical literature. See e.g. the classical survey by Holmstrom and
Tirole (1989).
4
See also Fare and Grosskopf (2004), pp.151-161.
20 The measurement of efciency
in utilizing the minimum inputs required to produce the outputs they choose
to produce, given the technology at their disposal. In light of the evident fail-
ure of at least some producers to optimize, it is desirable to recast the analysis
of production away from the traditional production function approach toward
a frontier based approach. Hence we are concerned with the estimation of
frontiers, which envelop data, rather than with functions, which intersect data.
In this setting, the main purpose of productivity analysis studies is to evaluate
numerically the performance of a certain number of rms (or business units or
Decision Making Units, DMU) from the point of view of technical efciency,
i.e. their ability to operate close to, or on the boundary of their production set.
The problem to be analyzed is thus set in terms of physical input and output
quantities.
We assume to have data in cross-sectional form, and for each rm we have
the value of its inputs and outputs used in the production process. Measuring
efciency for any data set of this kind requires rst to determine what the
boundary of the production set can be; and then to measure the distance between
any observed point and the boundary of the production set.
Given a list of p inputs and q outputs, in economic analysis the operations of
any productive organization can be dened by means of a set of points, , the
production set, dened as follows in the Euclidean space R
p+q
+
:
= {(x, y) | x R
p
+
, y R
q
+
, (x, y) is feasible}, (2.1)
where x is the input vector, y is the output vector and feasibility of the vec-
tor (x, y) means that, within the organization under consideration, it is physi-
cally possible to obtain the output quantities y
1
, ..., y
q
when the input quantities
x
1
, ..., x
p
are being used (all quantities being measured per unit of time). It is
useful to dene the set in terms of its sections, dened as the images of a
relation between the input and the output vectors that are the elements of .
We can dene then the input requirement set (for all y ) as:
C(y) = {x R
p
+
|(x, y) }. (2.2)
An input requirement set C(y) consists of all input vectors that can produce the
output vector y R
q
+
.
The output correspondence set (for all x ) can be dened as:
P(x) = {y R
q
+
|(x, y) }. (2.3)
P(x) consists of all output vectors that can be produced by a given input vector
x R
p
+
.
The production set can also be retrieved from the inputs sets, specically:
= {(x, y) | x C(y), y R
q
+
}. (2.4)
The economic model 21
Furthermore, it holds that:
(x, y) x C(y), y P(x), (2.5)
which tells us that the output and input sets are equivalent representations of
the technology, as is .
The isoquants or efcient boundaries of the sections of can be dened in
radial terms (Farrell, 1957) as follows. In the input space:
C(y) = {x|x C(y), x C(y), , 0 < < 1)} (2.6)
and in the output space:
P(x) = {y|y P(x), y P(x), > 1}. (2.7)
The axiomatic approach to production theory (Activity Analysis framework)
assumes that the technology (production model) satises certain properties or
axioms. These properties can be equivalently stated on , P(x), x R
p
+
,
C(y), y R
q
+
.
Some economic axioms (EA) are usually done in this framework (on these
concepts see also Shephard, 1970).
EA1: No free lunch. (x, y) if x = 0, y 0, y = 0.
5
This axiom states that inactivity is always possible, i.e., zero output can be
produced by any input vector x R
p
+
, but it is impossible to produce output
without any inputs.
EA2: Free disposability. Let x R
p
+
and y R
q
+
, with x x and y
y, if (x, y) then ( x, y) and (x, y) .
This is the free disposability assumption, named also the possibility of destroy-
ing goods without costs, on the production set .
The free disposability (also called strong disposability) of outputs can be
stated as follows: y
1
P(x), y
2
y
1
then y
2
P(x) or equivalently y
1
y
2
then C(y
2
) C(y
1
). The free disposability of inputs can be dened as below:
x
1
C(y), x
2
x
1
then x
2
C(y) or equivalently x
1
x
2
then P(x
1
)
P(x
2
).
The free disposability of both inputs and outputs is as follows:
(x, y) , if x

x and y

y then (x

, y

) .
We have also a weak disposability of inputs and outputs:
5
Here and throughout inequalities involving vectors are dened componentwise, i.e. on an element-by-
element basis.
22 The measurement of efciency
Weak disposability of inputs:
x C(y) 1, x C(y) or P(x) P(x);
Weak disposability of outputs:
y P(x) [0, 1], y P(x) or C(y) C(y).
The weakdisposabilitypropertyallows us tomodel congestionandoveruti-
lization of inputs/outputs.
EA3: Bounded. P(x) is bounded x R
p
+
.
EA4: Closeness. is closed, P(x) is closed, x R
p
+
, C(y) is closed,
y R
q
+
.
EA5: Convexity. is convex. The convexity of can be stated as follows:
If (x
1
, y
1
), (x
2
, y
2
) , then [0, 1] we have :
(x, y) = (x
1
, y
1
) + (1 )(x
2
, y
2
) .
EA6: Convexity of the requirement sets. P(x) is convex x R
p
+
and C(y) is convex y R
q
+
.
If is convex, then the inputs and outputs sets are also convex, i.e. EA5 implies
EA6.
A further characterization of the shape of the frontier relates to returns to
scale (RTS). According to a standard denition in economics, RTS express the
relation between a proportional change in inputs to a productive process and
the resulting proportional change in output. If an n per cent rise in all inputs
produces an n per cent increase in output, there are constant returns to scale
(CRS). If output rises by a larger percentage than inputs, there are increasing
returns to scale (IRS). If output rises by a smaller percentage than inputs, there
are decreasing returns to scale (DRS). Returns to scale can be described as
properties of the correspondence sets C(y) and/or P(x). We follow here the
presentation of Simar and Wilson (2002, 2006b). The frontier exhibits constant
returns to scale (CRS) everywhere if and only if:
(x, y) s.t. x C(y) then x C(y), > 0
or equivalently
6
,
> 0, C(y) = C(y).
6
Analogous expressions hold in terms of P(x): > 0, P(x) = P(x).
The economic model 23
Constant Returns to Scale in the neighborhood of a point (x, y) s.t. x
C(y) are characterized by C(y) = C(y) for some > 0.
Increasing Returns to Scale in the neighborhood of a point (x, y) s.t. x
C(y) implies that (x, y) for < 1.
Decreasing Returns to Scale in the neighborhood of a point (x, y) s.t. x
C(y) implies that (x, y) for > 1.
A frontier that exhibits increasing, constant and decreasing returns to scale
in different regions is a Variable Returns to Scale (VRS) frontier.
The assumptions we have introduced here are intended to provide enough
structure to create meaningful and useful technologies. Generally speaking, we
will not impose all of these axioms on a particular technology, rather we will
select subsets of these assumptions that are suitable for the particular problem
under study.
***
Turning back to the production set itself, the above denitions allow us to
characterize any point (x, y) in as:
input efcient if x C(y)
input inefcient if x C(y)
output efcient if y P(x)
output inefcient if y P(x).
Fromwhat stated above, DMUs are efcient, e.g. in an input-oriented frame-
work, if they are on the boundary of the input requirement set (or, for the output
oriented case, on the boundary of the output correspondence set). In some cases,
however, these efcient rms may not be using the fewest possible inputs to
produce their outputs. This is the case where we have slacks. This is due to the
fact that the Pareto-Koopmans efcient subsets of the boundaries of C(y) and
P(x), i.e. eff C(y) and eff P(x), may not coincide with the Farrell-Debreu
boundaries C(y) and P(x), i.e.
7
:
eff C(y) =

x | x C(y), x

C(y) x

x, x

= x

C(y), (2.8)
eff P(x) =

y | y P(x), y

P(x) y

y, y

= y

P(x). (2.9)
7
We give an illustration in Section 2.5 in Figure 2.2 where we describe DEA estimators of efcient frontier.
24 The measurement of efciency
Once the efcient subsets of have been dened, we may dene the efciency
measure of a rm operating at the level (x
0
, y
0
) by considering the distance
from this point to the frontier. There are several ways to achieve this but a
simple way suggested by Farrell (1957), in the lines of Debreu (1951), is to use
a radial distance from the point to its corresponding frontier. In the following
we will concentrate our attention on radial measures of efciency. Of course, we
may look at the efcient frontier in two directions: either in the input direction
(where the efcient subset is characterized by C(y)) or in the output direction
(where the efcient subset is characterized by P(x)).
The Farrell input measure of efciency for a rm operating at level (x
0
, y
0
)
is dened as:
(x
0
, y
0
) = inf{|x
0
C(y
0
)} = inf{|(x
0
, y
0
) }, (2.10)
and its Farrell output measure of efciency is dened as:
(x
0
, y
0
) = sup{|y
0
P(x
0
)} = sup{|(x
0
, y
0
) }. (2.11)
So, (x
0
, y
0
) 1 is the radial contraction of inputs the rm should achieve
to be considered as being input-efcient in the sense that ((x
0
, y
0
)x
0
, y
0
) is a
frontier point. In the same way (x
0
, y
0
) 1 is the proportionate increase of
output the rm should achieve to be considered as being output efcient in the
sense that (x
0
, (x
0
, y
0
)y
0
) is on the frontier.
It is interesting to note that the efcient frontier of , in the radial sense, can
be characterized as the units (x, y) such that (x, y) = 1, in the input direction
(belonging to C(y)) and by the (x, y) such that (x, y) = 1, in the output
direction (belonging to P(x)). If the frontier is continuous, frontier points are
such that (x, y) = (x, y) = 1. The efcient frontier is unique but we have
two ways to characterize it.
It is sometimes easier to measure these radial distances by their inverse,
known as Shephard distance functions (Shephard, 1970). The Shephard input
distance function provides a normalized measure of Euclidean distance from a
point (x, y) R
p+q
+
to the boundary of in a radial direction orthogonal to y
and is dened as:

in
(x, y) = sup{ > 0|(
1
x, y) } ((x, y))
1
, (2.12)
with
in
(x, y) 1, (x, y) . Similarly, the Shephard output distance
function provides a normalized measure of Euclidean distance from a point
(x, y) R
p+q
+
to the boundary of in a radial direction orthogonal to x:

out
(x, y) = inf{ > 0|(x,
1
y) } ((x, y))
1
. (2.13)
For all (x, y) ,
out
(x, y) 1. If either
in
(x, y) = 1 or
out
(x, y) = 1
then (x, y) belongs to the frontier of and the rm is technically efcient.
A taxonomy of efcient frontier models 25
As pointed out in Simar and Wilson (2001), no behavioral assumptions are
necessary for measuring technical efciency. From a purely technical view-
point, either the input or the output distance function can be used to measure
technical efciency - the only difference is in the direction in which distance to
the technology is measured. The way of looking at the frontier will typically
depend on the context of the application. For instance, if the outputs are ex-
ogenous and not under the control of the Decision Makers (e.g. as in most of
the public services), input efciency will be of main interest, since the inputs
are the only elements under the control of the managers. But even in this case,
both measures are available.
2.4 A taxonomy of efcient frontier models
The analysis of the existent literature is a necessary step for the advancement
of a discipline. This is particularly true for the eld of efciency and produc-
tivity research that in the last decades has known an exponential increasing
in the number of methodological and applied works. For a DEA bibliogra-
phy over 1978-1992, see Seiford (1994, 1996) and for an extension till 2001
see Gattou, Oral and Reisman (2004). In Cooper, Seiford and Tone (2000)
about 1,500 DEA references are reported. Other bibliographic studies include:
Emrouznejad (2001) and Taveres (2002).
As a consequence, a comprehensive review of the overall literature would
require another whole work. Therefore, the aim of this section is to propose
a general taxonomy of efcient frontier models that gives an overview on the
different approaches presented in literature for estimating the efcient frontier
of a production possibility set. Here the review could be biased toward the
nonparametric approach, due to our commitment and involvement with non-
parametric methods most. Anyway, we give several references also on the
parametric approach that could be useful for those interested in it.
In the previous section we described the economic model underlying the
frontier analysis framework based on the Activity Analysis Model. This model
is based on some representations of the production set on which we can
impose different axioms. Nevertheless, the production set , the boundary of
the input requirement set C(y) and of the output correspondence set P(x),
together with the efciency scores in the input and output space, (x, y) and
(x, y), are unknown.
The econometric problemis thus howto estimate , and then C(y), P(x),
(x, y), (x, y), froma randomsample of production units X = {(X
i
, Y
i
) | i =
1, ..., n}.
26 The measurement of efciency
Starting fromthe rst empirical application of Farrell (1957) several different
approaches for efcient frontier estimation and efciency score calculation have
been developed.
8
In Figure 2.1 we propose an outline of what we believe have been the most
inuential works in productivity and efciency analysis, starting from the pi-
oneering work by Farrell (1957). Of course, our outline is far from being
complete and all-inclusive. Figure 2.1 shows some of the articles, books and
special issues of journals (i.e. Journal of Econometrics JE, Journal of Produc-
tivity Analysis JPA, European Journal of Operational Research, EJOR) that
have mainly inuenced the writing of this work, trying to balance them accord-
ing to the adopted approach.
As it is evident from Figure 2.1 we have taken into consideration mainly the
nonparametric approach as we believe that thanks to its last developments, it
can be considered as being very exible and very useful for modeling purpose.
We mayclassifyefcient frontier models accordingtothe followingcriteria:
9
1 The specication of the (functional) form for the frontier function;
2 The presence of noise in the sample data;
3 The type of data analyzed.
Based on the rst criterium (functional form of the frontier) is the classi-
cation in:
Parametric Models. In these models, the attainable set is dened trough
a production frontier function, g(x, ), which is a known mathematical
function depending on some k unknown parameters, i.e. R
k
, where
generally y is univariate, i.e. y R
+
. The main advantages of this
approach are the economic interpretation of parameters and the statistical
properties of estimators; more critical are the choice of the functiong(x, )
and the handling of multiple inputs, multiple outputs cases (for more on
this latter aspect see Section 4.7 below where we introduce multivariate
parametric approximations of nonparametric and robust frontiers).
Nonparametric Models. These models do not assume any particular func-
tional form for the frontier function g(x). The main pros of this approach
are the robustness to model choice and the easy handling of multiple in-
puts, multiple outputs case; their main limitations are the estimation of
unknown functional and the curse of dimensionality
10
, typical of nonpara-
metric methods.
8
For an introduction see e.g., Coelli, Rao and Battese (1998) and Thanassoulis (2001).
9
These criteria follow Simar and Wilson (2006b), where a comprehensive statistical approach is described.
10
The curse of dimensionality, shared by many nonparametric methods, means that to avoid large variances
and wide condence interval estimates a large quantity of data is needed.
A taxonomy of efcient frontier models 27
Figure 2.1. An overview of the literature on efcient frontier estimation.
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28 The measurement of efciency
Based on the second criterium (presence of noise) is the classication in:
Deterministic Models, which assume that all observations (X
i
, Y
i
) belong
to the production set, i.e.
Prob{(X
i
, Y
i
) } = 1
for all i = 1, ..., n. The main weakness of this approach is the sensitivity
to super-efcient outliers. Robust estimators are able to overcome this
drawback.
Stochastic Models, in which there might be noise in the data, i.e. some
observations might lie outside . The main problem of this approach is
the identication of noise from inefciency.
Based on the third criterium (type of data analyzed) is the classication in:
Cross-sectional Models, in which the data sample is done by observations
on n rms or DMUs (Decision Making Units):
X = {(X
i
, Y
i
)|i = 1, ..., n}
Panel Data Models, in which the observations on the n rms are available
over T periods of time:
X = {(X
it
, Y
it
) | i = 1, ..., n; t = 1, ..., T}.
Panel data allow the measurement of productivity change as well as the
estimation of technical progress or regress.
Generally speaking, productivity change occurs when an index of outputs
changes at a different rate than an index of inputs does. Productivity change
can be calculated using index number techniques to construct a Fisher (1922)
or Tornqvist (1936) productivity index. Both these indices require quantity and
price information, as well as assumptions concerningthe structure of technology
and the behavior of producers. Productivity change can also be calculated using
nonparametric techniques to construct a Malmquist (1953) productivity index.
These latter techniques do not require price information or technological and
behavioral assumptions, but they require the estimation of a representation of
production technology. Nonparametric techniques are able not only to calculate
productivity change, but also to identify the sources of measured productivity
change.
A survey of the theoretical and empirical work on Malmquist productivity
indices can be found in Fare, Grosskopf and Russell (1998). On the theo-
retical side the survey includes a number of issues that have arisen since the
Malmquist productivity index was proposed by Caves, Christensen and Diewert
A taxonomy of efcient frontier models 29
(1982). These issues include the denition of the Malmquist productivity in-
dex; although all are based on the distance functions that Malmquist employed
to formulate his original quantity index, variations include the geometric mean
form used by Fare, Grosskopf, Lindgren and Roos (1989) and the quantity in-
dex form by Diewert (1992). The survey of the empirical literature presents
studies on the public sector, banking, agriculture, countries and international
comparisons, electric utilities, transportation, and insurance. See also Lovell
(2003), and Grosskopf (2003) for an historical perspective and an outline of the
state of the art in this area.
Although productivity change is not the main focus of FDH, it can be inferred
from information on efciency change and technical change that is revealed by
FDH. The technique was developed by Tulkens that named it sequential FDH.
For an illustration of the sequential FDH see Lovell (1993, pp. 48-49). On this
topic see also Tulkens and Vanden Eeckaut (1995a, 1995b).
By combining the three criteria mentioned above, several models have been
studied in literature:
Parametric Deterministic Models, see e.g. Aigner and Chu (1968), Afriat
(1972), Richmond (1974), Schmidt (1976) and Greene (1980) for cross-
sectional and panel data;
Parametric Stochastic Models, most of these techniques are based on the
maximumlikelihood principle, following the pioneering works of Aigner,
Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977). For
a recent review see Kumbhakar and Lovell (2000). In the context of
panel data, stochastic models (see Schmidt and Sickles, 1984, and Corn-
well, Schmidt, and Sickles, 1990) have semiparametric generalizations,
in which a part of the model is parametric and the rest is nonparamet-
ric (see Park and Simar, 1994; Park, Sickles and Simar, 1998; and Park,
Sickles and Simar, 2003a, b).
Nonparametric Deterministic Models for cross-sectional and panel data.
Traditional references on these models include: Fare, Grosskopf and
Lovell (1985, 1994), Fried, Lovell and Schmidt (1993), and Charnes,
Cooper, Lewin and Seiford, 1994. Recent and updated references are
Cooper, Seiford and Tone (2000), Ray (2004) and Fare and Grosskopf
(2004).
Nonparametric Stochastic Models for cross-sectional data (see Hall and
Simar, 2002; Simar, 2003b; Kumbhakar, Park, Simar and Tsionas, 2004)
and panel data (see Kneip and Simar, 1996; and Henderson and Simar,
2005).
The mainly used approaches in empirical works are the nonparametric (de-
terministic) frontier approach and the (parametric) stochastic frontier approach.
30 The measurement of efciency
In the following, when we refer to nonparametric frontier approach we indicate
the deterministic version of it; when we talk about stochastic frontier approach
we refer to its parametric version.
The nonparametric frontier approach, based on envelopment techniques
(DEA FDH), has been extensively used for estimating efciency of rms as
it relays only on very few assumptions for . On the contrary, the stochastic
frontier approach (SFA) allows the presence of noise but it demands parametric
restrictions on the shape of the frontier and on the Data Generating Process
(DGP) in order to permit the identication of noise from inefciency and the
estimation of the frontier. Fried, Lovell and Schmidt (2006) offer an updated
presentation of both approaches. A statistical approach which unies paramet-
ric and nonparametric approaches can be found in Simar and Wilson (2006b).
2.5 The nonparametric frontier approach
In this section we introduce the most known nonparametric estimators of
efcient frontiers.
As we have seen in Section 2.3 devoted to the presentation of the economic
model, we can equivalently look at the efcient boundary of from the input
space or from the output space.
The input oriented framework, based on the input requirement set and its
efcient boundary, aims at reducing the input amounts by as much as possible
while keeping at least the present output levels. This is also called input-
saving approach to stress the fact that the outputs level remains unchanged
and input quantities are reduced proportionately till the frontier is reached.
This is a framework generally adopted when the decision maker can control the
inputs but has not the control of the outputs. For instance, this is the case of
public enterprises which are committed to offer some public services and are
interested in the management of the inputs, in the sense of their minimization.
Alternatively, we can take into account the output space and look at the output
correspondence set and its efcient boundary. The output oriented framework
looks at maximize output levels under at most the present input consumption.
This approach is also known as output-augmenting approach, because it holds
the input bundle unchanged and expand the output level till the frontier is
reached. In practice, whether the input or output-oriented measure is more
appropriate would depend on whether input conservation is more important
than output augmentation.
For the relation existent among input and output efciency measures, see
Deprins and Simar (1983).
The main nonparametric estimators available are the Data Envelopment
Analysis (DEA) and the Free Disposal Hull (FDH) which we describe in the
subsections that follow.
The nonparametric frontier approach 31
2.5.1 Data Envelopment Analysis (DEA)
The DEA estimator of the production set, initiated by Farrell (1957) and op-
erationalized as linear programming estimators by Charnes, Cooper and Rhodes
(1978), assumes the free disposability and the convexity of the production set
. It involves measurement of efciency for a given unit (x, y) relative to the
boundary of the convex hull of X = {(X
i
, Y
i
), i = 1, ...., n}:

DEA
=

(x, y) R
p+q
+
| y
n

i=1

i
Y
i
; x
n

i=1

i
X
i
, for (
1
, ...,
n
)
s.t.
n

i=1

i
= 1;
i
0, i = 1, ...., n

(2.14)

DEA
is thus the smallest free disposal convex set covering all the data.
The

DEA
in (2.14) allows for Variable Returns to Scale (VRS) and is often
referred as

DEAV RS
(see Banker, Charnes and Cooper, 1984). It may be
adapted to other returns to scale situations. It allows for:
Constant Returns to Scale (CRS) if the equality constrained

n
i=1

i
= 1
in (2.14) is dropped;
NonIncreasingReturns toScale (NIRS) if the equalityconstrained

n
i=1

i
= 1 in (2.14) is changed in

n
i=1

i
1;
Non Decreasing Returns to Scale (NDRS) if the equality constrained

n
i=1

i
= 1 in (2.14) is modied in

n
i=1

i
1.
The estimation of the input requirement set is given for all y by:

C(y) =
{x R
p
+
|(x, y)

DEA
} and

C(y) denotes the estimator of the input
frontier boundary for y.
For a rm operating at level (x
0
, y
0
) the estimation of the input efciency
score (x
0
, y
0
) is obtained by solving the following linear program (here and
hereafter we consider the VRS case):

DEA
(x
0
, y
0
) = inf

| (x
0
, y
0
)

DEA

(2.15)

DEA
(x
0
, y
0
) = min

| y
0

n

i=1

i
Y
i
; x
0

n

i=1

i
X
i
; > 0;
n

i=1

i
= 1;
i
0; i = 1, ...., n

. (2.16)

(x
0
, y
0
) measures the radial distance between (x
0
, y
0
) and ( x

(x
0
|y
0
), y
0
)
where x

(x
0
|y
0
) is the level of the inputs the unit should reach in order to
32 The measurement of efciency
be on the efcient boundary of

DEA
with the same level of output, y
0
,
and the same proportion of inputs; i.e. moving from x
0
to x

(x
0
|y
0
) along
the ray x
0
. The projection of x
0
on the efcient frontier is thus equal to
x

(x
0
|y
0
) =

(x
0
, y
0
)x
0
.
For the output oriented case, the estimation is done, mutatis mutandis, fol-
lowing the previous steps. The output correspondence set is estimated by:

P(x) = {y R
q
+
|(x, y)

DEA
} and

P(x) denotes the estimator of the
output frontier boundary for x.
The estimator of the output efciency score for a given (x
0
, y
0
) is obtained
by solving the following linear program:

DEA
(x
0
, y
0
) = sup

| (x
0
, y
0
)

DEA

, (2.17)

DEA
(x
0
, y
0
) = max

| y
0

n

i=1

i
Y
i
; x
0

n

i=1

i
X
i
; > 0;
n

i=1

i
= 1;
i
0; i = 1, ...., n

. (2.18)
In Figure 2.2 we display the DEA estimator and illustrate the concept of slacks
through an example. If we look at the left panel assuming that all rms produce
the same level of output, we can see that the DMU E could actually produce 1
unit of y with less input x
1
, i.e., it could reduce x
1
by one unit (from 4 to 3)
moving from E to D. This is referred to as input slack: although the DMU is
technical efcient, there is a surplus of input x
1
.
11
In general, we say that there
is slack in input j of DMU i, i.e., x
j
i
, if:
n

i=1

i
x
i
< x
j
i

(x
i
, y
i
) (2.19)
is true for some solution value of
i
, i = 1, ..., n (see Fare, Grosskopf and
Lovell, 1994, for more details).
The same kind of reasoning can be done for the output oriented case, i.e. the
DMU L could increase the production of y
1
moving from L to M. See Figure
2.2, right panel for a graphical illustration.
Slacks may happen for DEA estimates (as shown in Figure 2.2), as well as
for FDH estimates (presented in the next section). It is interesting to note that
if the true production set has no slacks, than slacks are only a small sample
problem. Nevertheless, it is always useful to report slacks whenever they are
11
Remember the possibility of destroying goods without costs underlying the frontier representation of
the economic model.
The nonparametric frontier approach 33
Figure 2.2. Input and Output slacks.
there. It is left to the analyst to decide if it is better to correct for the slacks or
just point them.
Once the efciency measures have been computed, several interesting analy-
sis could be done, such as the inspection of the distribution of efciency scores
and the analysis of the best performers or efcient facet of the frontier closer
to the analysed DMU, generally called peer-analysis, to study the technical
efcient units and try to learn from them.
2.5.2 Free Disposal Hull (FDH)
The FDH estimator, proposed by Deprins, Simar and Tulkens (1984), is a
more general versionof the DEAestimator as it relies onlyonthe free disposabil-
ity assumption for , and hence does not restrict itself to convex technologies.
This seems an attractive property of FDH since it is frequently difcult to nd
a good theoretical or empirical justication for postulating convex production
sets in efciency analysis. At this purpose, Farrell (1959) indicates indivisibil-
ity of inputs and outputs and economies of scale and specialization as possible
violations of convexity. It is important to note also that if the true production
set is convex then the DEA and FDH are both consistent estimators; however,
as pointed later in this section, FDH shows a lower rate of convergence (due to
the less assumptions it requires) with respect to DEA. On the contrary, if the
true production set is not convex, than DEA is not a consistent estimator of the
production set, while FDH is consistent.
The FDHestimator measures the efciency for a given point (x
0
, y
0
) relative
to the boundary of the Free Disposal Hull of the sample X = {(X
i
, Y
i
), i =
1, ...., n}. The Free Disposal Hull of the set of observations (i.e. the FDH
Slacks DEA input oriented
O
D
X
2
Y
2
X
1
Y
1
C(y)
E
3 4
L M
P(x)
Slacks DEA output oriented
O
34 The measurement of efciency
estimator of ) is dened as:

FDH
=

(x, y) R
p+q
+
| y Y
i
; x X
i
, (X
i
, Y
i
) X

. (2.20)
It is the union of the all positive orthants in the inputs and of the negative orthants
in the outputs whose origin coincides with the observed points (X
i
, Y
i
)
X (Deprins, Simar and Tulkens, 1984). See Figures 2.3 and 2.4 where the
FDH estimator is compared with the DEA estimator of the input and output
requirement sets, respectively.
The efciency estimators, in this framework, are obtained (as for the DEA
case) using a plug-in principle, i.e., by substituting the unknown quantities (in
this case ) by their estimated values (here

FDH
, for the DEA case

DEA
).
The estimated input requirement set and the output correspondence set are
the following:

C(y) = {x R
p
+
|(x, y)

FDH
},

P(x) = {y R
q
+
|(x, y)

FDH
}.
Their respective efcient boundaries are:

C(y) = {x|x

C(y), x

C(y)0 < < 1},

P(x) = {y|y

P(x), y

P(x) > 1}.
Hence, the estimated input efciency score for a given point (x
0
, y
0
) is:

FDH
(x
0
, y
0
) = inf

| x
0


C(y
0
)

= inf

| (x
0
, y
0
)

FDH

, (2.21)
and the estimated output efciency score of (x
0
, y
0
) is given by:

FDH
(x
0
, y
0
) = sup

| y
0


P(x
0
)

= sup

| (x
0
, y
0
)

FDH

. (2.22)
It is clear that for a particular point (x
0
, y
0
), the estimated distance to the
frontiers are evaluated by means of the distance, in the input space (input
oriented) from this point to the estimated frontier of the input requirement
set (

C(y)), and in the output space (output oriented) by the distance from
(x
0
, y
0
) to the estimated frontier of the output correspondence set (

P(x)).
The nonparametric frontier approach 35
It is worthwhile to note that the FDH attainable set in (2.20) can also be
characterized as the following set:

FDH
=

(x, y) R
p+q
+
| y
n

i=1

i
Y
i
; x
n

i=1

i
X
i
,
n

i=1

i
= 1;

i
{0, 1}, i = 1, ..., n

. (2.23)
Therefore the efciencies can be estimated by solving the following integer
linear programs; for the input-oriented case we have:

FDH
(x
0
, y
0
) = min

| y
0

n

i=1

i
Y
i
; x
0

n

i=1

i
X
i
,
n

i=1

i
= 1;

i
{0, 1}, i = 1, ..., n

, (2.24)
and for the output-oriented case:

FDH
(x
0
, y
0
) = max

| y
0

n

i=1

i
Y
i
; x
0

n

i=1

i
X
i
,
n

i=1

i
= 1;

i
{0, 1}, i = 1, ..., n

.(2.25)
The latter expressions allow to make the comparison easier between the FDH
and the DEA estimators (compare for instance (2.23) with (2.14)).
Figure 2.3 illustrates the estimation of the input requirement set C(y) and
of its boundary C(y) through FDH and DEA methods. The dashed line rep-
resents the FDH estimation of C(y), while the solid line shows the DEA
estimation of it. The squares are the observations. The DEA and FDH esti-
mates of efciency score of production unit B, in Figure 2.3, are respectively:

DEA
(x
0
, y
0
) = |OB

|/|OB| 1,

FDH
(x
0
, y
0
) = |OB

|/|OB| 1.
In Figure 2.4 we show the FDH and DEA estimation of the output corre-
spondence set P(x) and its boundary P(x). The dash-dotted line represents
the FDH estimator of P(x), while the solid line the DEA estimator of it.
The black squares, as before, represent the DMUs. For rm B, the estimates
of its efciency score, in output oriented framework, are:

FDH
(x
0
, y
0
) =
|OB

|/|OB| 1,

DEA
(x
0
, y
0
) = |OB

|/|OB| 1.
Practical computation of the FDH
In practice, the FDH estimator is computed by a simple vector compari-
son procedure that amounts to a complete enumeration algorithm proposed in
Tulkens (1993), which is now explained.
36 The measurement of efciency
Figure 2.3. FDH and DEA estimation of C(y) and C(y).
Figure 2.4. FDH and DEA estimation of P(x) and P(x).
Input Oriented Framework
.
.
O
A
B=(Xo,Yo)
C
B
B
X
1
X
2
C(y)
C(y)
O
Y
1
Y
2
P(x) P(x)
Output oriented framework
B = (Xo,Yo)
B
B
The nonparametric frontier approach 37
For a DMU (x
0
, y
0
), in a rst step, the set of observations which domi-
nates it is determined, and then the estimate of its efciency score, relative to
the dominating facet of

is computed. In the simplest case, with a technol-
ogy characterized by one input and one output, the set of observations which
dominate (x
0
, y
0
) is dened as:
D
0
=

i|(X
i
, Y
i
) X, X
i
x
0
, Y
i
y
0

. (2.26)
The input oriented efciency estimate is done through:

FDH
(x
0
, y
0
) = min
iD
0

X
i
x
0

, (2.27)
and the output oriented efciency is computed via:

FDH
(x
0
, y
0
) = max
iD
0

Y
i
y
0

. (2.28)
It has to be noted that as X
i
x
0
then

FDH
1. As for the input-oriented
case, from the fact that Y
i
y
0
follows that

FDH
1.
In a multivariate setting, the expression (2.21) can be computed through:

FDH
(x
0
, y
0
) = min
iD
0

max
j=1,...,p

X
i,j
x
j
0

, (2.29)
where X
i,j
is the j
th
component of X
i
R
p
+
and x
j
0
is the j
th
component of
x
0
R
p
+
.
It is a maximin procedure (for the input oriented framework): the max
part of the algorithm identies the most dominant DMUs relative to which a
given DMU is evaluated. Once the most dominant DMUs are identied, slacks
are calculated from the min part of the algorithm.
The multivariate computation of expression (2.22) is done by:

FDH
(x
0
, y
0
) = max
iD
0

min
j=1,...,q

Y
i,j
y
j
0

(2.30)
where Y
i,j
is the j
th
component of Y
i
R
q
+
and y
j
0
is the j
th
component of
y
0
R
q
+
.
The FDH estimator has been applied in several contexts. For a detailed
presentation of FDH concepts see Vanden Eeckaut (1997).
Recently, some authors have raised explicit doubts about the economic mean-
ing of FDH, but from the exchange between Thrall (1999) and Cherchye, Ku-
osmanen and Post (2000), published on the Journal of Productivity Analysis, it
38 The measurement of efciency
emergedthat FDHcanbe economicallymore meaningful thanconvexmonotone
hull, also under non-trivial alternative economic conditions.
Hence, FDH technical efciency measures remain meaningful for theories
of the rm that do allow for imperfect competition or uncertainty (see e.g.
Kuosmanen and Post, 2001, and Cherchye, Kuosmanen and Post, 2001).
One of the main drawbacks of deterministic frontier models (DEA /FDH
based) is the inuence of super-efcient outliers.
This is a consequence of the fact that the efcient frontier is determined by
sample observations which are extreme points. Simar (1996) points out the need
for identifyingandeliminatingoutliers whenusingdeterministic models. If they
cannot be identied, the use of stochastic frontier models is recommended.
See Figure 2.5 for an illustration of the inuence of outliers in case of FDH
estimation. The same is valid for the DEA case. If point A is an extreme point,
outlying the cloud of other points, the estimated efcient frontier is strongly
inuenced by it. In fact, in Figure 2.5, the solid line is the frontier that envelops
point A, while the dash-dotted line does not envelop point A.
Figure 2.5. Inuence of outliers on the FDH estimation of the production set .
We will come back on this problem in Chapter 4 where we propose robust
nonparametric approaches based on various nonparametric measures less inu-
enced by extreme values and outliers, which have also nice statistical properties.
A
y
x

FDH

FDH frontier of
FDH frontier without A
Free disposability
Recent developments in nonparametric efciency analysis 39
2.6 Recent developments in nonparametric efciency
analysis
In the following, we recall briey some streamof works that have contributed
to the latest advancement of the nonparametric efciency literature.
12
Sensitivity of results to data variation and discrimination (in a DEA
framework)
The focus of studies on sensitivity and stability is the reliability of classica-
tion of DMUs into efcient and inefcient performers. Most analytical methods
for studying the sensitivity of results to variations in data have been developed
in a DEA framework.
After a rst stream of works concentrated on developing solution methods
and algorithms for conducting sensitivity analysis in linear programming, a
second current of studies analysed data variations in only one input or one
output for one unit at a time. A recent stream of works makes it possible to
determine ranges within which all data may be varied for any unit before a
reclassication from efcient to inefcient status (or vice versa) occurs, and
for determining ranges of data variation that can be allowed when all data are
varied simultaneously for all DMUs. For a review and some references see
Cooper, Li, Seiford, Tone, Thrall and Zhu (2001).
As we have seen above, DEA models have a deterministic nature, meaning
that they do not account for statistical noise. Some authors (e.g., Land, Lovell
and Thore, 1993; Olesen and Petersen, 1995) have proposed the application
of the chance-constrained programming to the DEA problem in order to over-
come its deterministic nature. The basic idea is that of make DEA stochastic
by introducing a chance that the constraints on either the envelopment problem
or the multiplier problem may be violated with some probability. However,
the chance-constrained efciency measurement requires a large amount of data
in addition to inputs and outputs. Moreover, it is based on a strong distribu-
tional assumption on the process determining the chance of a constrained to
be violated. The analyst in fact has to provide also information on expected
values of all variables for all DMUs, and variance-covariance matrices for each
variable across all DMUs. An alternative to this approach is given by a fuzzy
programming approach to DEA and FDH efciency measurement.
There is an increasing number of studies that apply the fuzzy set theory
in productivity and efciency contexts. In some production studies, the data
that describe the production process cannot be collected accurately due to the
fact that measurement systems have not been originally designated for the pur-
12
See also Lovell (2001) and Fried, Lovell and Schmidt (2006) for a presentation of some recent fruitful
research areas introduced in parametric and nonparametric approaches to efciency analysis.
40 The measurement of efciency
pose of collecting data and information that are useful for production studies.
Sengupta (1992) was the rst to introduce a fuzzy mathematical programming
approach where the constraints and objective function are not satised crisply.
Seaver and Triantis (1992) proposed a fuzzy clustering approach for identify
unusual or extreme efcient behavior. Girod and Triantis (1999) implemented
a fuzzy linear programming approach, whilst Triantis and Girod (1998), and
Kao and Liu (1999) used fuzzy set theory, to let the traditional DEA and FDH
account for inaccuracies associated with the production plans. A fuzzy pair-
wise dominance approach can be found in Triantis and Vanden Eeckaut (2000)
where, a classication scheme that explicitly accounts for the degree of fuzzi-
ness (plausibility) of dominating units is reported.
According to a classication proposed by Angulo-Meza and Pereira Estellita
Lins (2002), the methods for increasing discrimination within efcient DMUs
in a DEA setting can be classied into two groups:
Methods with a priori information. In these methods, the information pro-
vided by a decision-maker or an expert about the importance of the variables
can be introduced into the DEA models. There are three main methods devoted
to incorporating a priori information or value judgments in DEA:
Weight restrictions. The main objective of the weight restrictions methods
is to establish bounds within which the weights can vary, preserving some
exibility/ uncertainty about the real value of the weights.
13
Preference structure models. These models have been introduced by Zhu
(1996) within a framework of non-radial efciency measures. In this
approach, the target for inefcient DMUs is given by a preference structure
(represented through some weights) expressed by the decision-maker.
Value efciency analysis. This method, introduced by Halme, Joro, Ko-
rhonen, Salo and Wallenius (2000), aims at incorporate the decision-
makers value judgements and preferences into the analysis, using a two
stage procedure. The rst stage identies the decision makers most pre-
ferred solutions through a multiple objective model. The second stage
consists in the determination of the frontier based on the most preferred
solutions chosen.
Methods that do not require a priori information. These family of models
aims at increase discrimination in DEA without the subjectivity, the possibility
of biased or wrong judgements, typical of the methods that introduce a priori
13
See Allen, Athanassopoulos, Dyson and Thanassoulis (1997), and Pedraja-Chaparro, Salinas-Jimenes,
Smith and Smith (1997) for a review of some methods within this approach, including direct weight restric-
tions, cone ratio models, assurance region and virtual inputs and outputs restrictions.
Recent developments in nonparametric efciency analysis 41
information. The main methods that minimize the intervention of the experts
are:
Super efciency. Andersen and Petersen (1993) proposed this method to
rank efcient DMUs.
Cross-evaluation. The main idea of this method is to use DEA in a peer-
evaluation instead of a classical self evaluation evaluated by the clas-
sical DEA models.
Multiple objective approach. A Multiple Criteria Data Envelopment
Analysis has been proposed by Li and Reeves (1999) to solve the prob-
lems of lack of discrimination and inappropriate weighting schemes in
traditional DEA.
Extensions to the basic DEA Models
Directional distance functions have been introduced by Chambers, Chung
and Fare, (1996) and are based on Luenberger (1992) benet functions. These
functions represent a kind of generalization of the traditional distance functions.
Their application leads to measures of technical efciency fromthe potential for
increasing outputs while reducing inputs at the same time. In order to provide
a measure of directional efciency, a direction, along which the observed
DMU is projected onto the efcient frontier of the production set, has to be
chosen. This choice is arbitrary and of course affects the resulting efciency
measures. In addition, those measures are no more scale-invariant. See Fare
and Grosskopf (2004) for more details on these new directions in efciency
analysis.
Examples of the literature that try to link DEA with a theoretical foundation
or that try to overcome and generalize the economic assumptions underlying
DEA include: Bogetoft (2000) which links the theoretically oriented agency,
incentives and contracts literature with the more practical oriented efciency
measurement literature; and Briec, Kerstens and Vanden Eeckaut (2004a, b)
which extend the duality properties to non-convex technologies and propose
congestion-based measures in this framework.
Producers face uncertainty about technology reliability and performance.
The structure of technology and the existence and magnitude of inefciency are
sensitive to the treatment of risk and uncertainty. On productivity measurement
under uncertainty see Chambers and Quiggin (2000) and Chambers (2004).
Statistical inference in efciency analysis
All what we have seen in the previous description of recent developments
does not allowfor a statistical sensitivity analysis, neither for rigorous statistical
testing procedures. This is because the previous literature does not relies on a
42 The measurement of efciency
statistical model; there is not, in fact, a denition of the Data Generating Process
(DGP) and there is no room for statistical inference based on the construction
of condence intervals, estimation of the bias, statistical tests of hypothesis and
so on.
There is instead a new approach, recently developed, which aims exactly at
the analysis of the statistical properties of the nonparametric estimators, trying
to overcome most limitations of traditional nonparametric methods and allow-
ing for statistical inference and rigorous testing procedures. This literature is
the main focus of this book. To the review of the statistical properties of non-
parametric frontier estimators we devote the following Chapter 3. Chapter 4
deals in detail with a family of robust nonparametric measures of efciency,
which are more resistent to the inuence of outliers and errors in data while
having good statistical properties which let inference feasible in this complex
framework. Finally, Chapter 5 illustrates and develop further the topic of con-
ditional and robust measures of efciency and an alternative way to evaluate
the impact of external-environmental variables based on conditional measures
of efciency.
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