Cost Efficiency of French Soccer League Clubs using a Finite Mixture Model
Carlos Pestana Barros
ISEG – School of Economics and Management, Technical University of Lisbon, Rua Miguel Lupi, 20,
1249-078 Lisbon, Portugal; Email: [email protected]
Julio del Corral
Departamento de Economía-Fundación Observatorio Económico de Deporte; Universidad de Oviedo;
Avda. del Cristo s/n; 33006 Oviedo, Spain, Email: [email protected]
Juan Prieto-Rodriguez
Departamento de Economía-Fundación Observatorio Económico de Deporte; Universidad de Oviedo;
Avda. del Cristo s/n; 33006 Oviedo, Spain, Email: [email protected]
This versión: June 2009
Comments welcome
Cost Efficiency of French Soccer League Clubs using a Finite Mixture
Model
Abstract: This paper evaluates the operational activities of French soccer clubs, using a
finite mixture model that allows us to control for unobserved heterogeneity. In doing so,
a stochastic frontier latent class model, which allows for the existence of different
technologies, is adopted to estimate cost frontiers. This procedure not only enables us to
identify different groups of French soccer clubs, but also permits us to analyze their cost
efficiency. The main finding is that these clubs are divided into two groups. Some
managerial implications are derived from the research findings.
Keywords:cost efficiency, latent class model, soccer clubs
JEL codes: L83, C23
2
1. Introduction
In order to improve managerial practices, analysis of a firm’s performance could be of
importance. To this end, it is of value to study firms’ and managers’ efficiency when
inputs are converted into outputs. To do so, the economic literature proposes the
estimation of technological frontiers (e.g., production and cost functions) and the
comparison between such frontiers and the performance of the firm allows the
compilation of its efficiency score.
This subject has attracted much research in the domain of team sports such as
football (i.e., Hofler and Payne, 1996; Hadley et al., 2000; Einolf, 2004), baseball (i.e.,
Porter and Scully, 1982; Koop, 2002; Smart, Winfree and Wolfe, 2008), basketball (i.e.,
Zak, Huang and Siegfried, 1979; Hofler and Payne, 1997) and hockey (Kahane, 2005).
Similarly, the efficiency of soccer clubs and soccer coaches has been analyzed in
respect of most of the major European soccer leagues. For example, the efficiency of the
English Premier League has been analyzed by Dawson, Dobson and Gerrard (2000A),
Carmichael, Thomas and Ward (2001), Barros and Leach (2006A, 2006B) and Haas
(2003A). The Spanish soccer league was analyzed for instance by Espitia-Escuer and
García-Cebrian (2004). Finally, the German league has been analyzed by Kern and
Süssmuth (2005). Two major national leagues are conspicuously absent from this
substantial body of applied research on efficiency in present-day soccer, namely, the
Italian and the French leagues.
The present paper aims to fill a part of this gap in the literature by analyzing the
efficiency of the French professional soccer league first division, or League 1.
Specifically, panel data is used to analyze the cost efficiency of the French League 1
clubs over the period 2002/2003-2005/2006.
3
Two alternative frameworks are used in efficiency analysis. First, there are nonparametric techniques, specifically, the Data Envelopment Analysis (DEA). This
approach was followed, for instance, by Porter and Scully (1982), Fizel and D'Itri
(1996, 1997), Haas (2003A, 2003B), Espitia-Escuer and García-Cebrian (2004) and
Barros and Leach (2006B). Secondly, there are parametric techniques, in which a
functional form must be assumed. These techniques have been applied to sports by
Scully (1994), Hofler and Payne (1997), Dawson, Dobson and Gerrard (2000),
Carmichael, Thomas and Ward (2001), Kahane (2005) and Barros and Leach (2006A).
This paper adopts a stochastic frontier analysis. However, an important feature
in the literature on efficiency in sports, despite the framework used, is that almost all the
above studies has neglected the existence of heterogeneity in the data1 by assuming that
all clubs use the same technology. If this assumption is wrong, it could lead to the
overestimation of the inefficiency of some clubs, since technology differences could be
interpreted as inefficiency (Orea and Kumbhakar, 2004). Hence, the paper advocates
using a stochastic frontier latent class model2 to control for unobserved heterogeneity
(Orea and Kumbhakar, 2004; Greene, 2005A). These models assume that there are a
finite number of classes, that they use different technologies and that each unit can be
assigned to a particular group, using the estimated probabilities of class membership.
Moreover, the number of different groups is tested.
Therefore, this paper contributes to the literature by analyzing the French Soccer
League 1 by means of a cost frontier, which allows for the existence of different
technologies in the sample.
1
The existence of heterogeneity in the data is a hot topic. Some techniques to avoid heterogeneity
problems include quantile regression (Benz, Brandes and Franck, 2009), finite mixture models (Barros,
del Corral and García-del-Barrio) and random coefficients model (Barros and García-del-Barrio, 2008).
2
Latent class models are also called finite mixture models in the literature.
4
The remainder of the paper is organized as follows. In section 2, the contextual
setting is described; in the section 3, a literature survey is presented; in section 4, the
methodology is explained; section 5 contains the data and empirical specification; the
results are presented in section 6 and finally, the concluding remarks are made in
section 7.
2. Contextual Setting
In the European context, League 1 of the French Professional Soccer League
does not rank in the elite category in terms of financial power, spectator numbers and
success in the European club competitions. Despite the French national team’s conquest
of the 1998 FIFA World Cup and the UEFA Euro 2000, then its appearance and defeat
in the 2006 World Cup Final with an ageing team, the country’s domestic league is not
able to compete with England, Spain, Germany or Italy as a destination for the sport’s
star players.
On the other hand, the country’s excellent club resources, policy and system for
producing and nurturing young French talent, together with trainees from Africa, Latin
America and Asia signify that France has become a leading supplier of players to the
international market over the past 15 years. Today, it exports its stars, as well as up and
coming young players, to all the other European countries and beyond. As a result of
globalization and the post-Bosman rule governing the contracting of players, the League
1 clubs, as in every country, have polyglot squads of players. On the field, Olympique
Lyonnais currently dominates the league, winning the championship for a record seven
consecutive seasons from 2002 to 2008. In terms of support and television revenue at
least, Olympique Lyonnnais’ main rivals are Paris Saint Germain and the Olympique de
Marseille. French professional soccer is generally well planned and managed, with
5
competent organizational structures, including responsible organs and instruments for
resolving disputes and disciplinary matters. Table 1 displays some financial and
sporting data on the 20 clubs that comprised the League 1 in the 2005-2006 season.
INSERT TABLE 1
It can be observed that there is a high heterogeneity among theses clubs in terms
of attendance, wages and results measured by the final position attained in the league.
This fact serves as a first insight as to the relevance of our approach. Further details on
the French Soccer League can be found in Andreff (2007) and Gouguet and Primault
(2006).
3. Literature Survey
As mentioned above, there are two main approaches to measure efficiency: the
econometric or parametric approach and the non-parametric approach. With regard to
the first, there are several papers that have applied the econometric approach to
efficiency analysis in soccer. For instance, Dawson, Dobson and Gerrard (2000A)
analyzed the managerial efficiency of English soccer, estimating several production
frontiers. They used as output the winning percentage and as input several measures of
player quality. Dawson, Dobson and Gerrard (2000B), using a similar approach,
provided comprehensive empirical evidence on the robustness of estimates of coaching
efficiency in English soccer to the full range of robustness available estimation
methods, as well as alternative input-output specifications but they did not use a latent
class model, which would have been a more flexible approach. Carmichael, Thomas and
Ward (2001) analyzed the efficiency of the English Football Association Premiership
clubs, using as output the number of points gained during the season. Barros and Leach
(2006B, 2007) estimated cost stochastic frontiers for the English Premiership, using as
outputs both the points achieved in the season and spectator attendance. Ascari and
6
Gagnepain (2007) estimated average wage equations in Spanish soccer, using a
stochastic frontier model to evaluate empirically the consequences of the rent-seeking
behavior of clubs on their costs.
Looking beyond soccer, several papers have applied the econometric approach
to efficiency analysis in sports economics. For example, Zak, Huang and Siegfried
(1979) explored production efficiency in the National Basketball Association, using a
deterministic frontier. Porter and Scully (1982) studied the managerial efficiency of
baseball managers with a deterministic approach. Scully (1994) showed that coaching
tenure was related to managerial efficiency in basketball, baseball and soccer, using
survival analysis. He had previously calculated the managerial efficiency, using
deterministic as well as stochastic frontier models. Hofler and Payne (1997) applied a
stochastic frontier model to the NBA, using the number of wins as output. Barros,
Garcia-del-Barrio and Leach (2009) analyzed the cost efficiency of Spanish soccer
clubs using a random parameter model. Kahane (2005) investigated the relationship
between inefficiency and discriminatory hiring practices in the National Hockey League
(NHL), using a stochastic frontier model. He used as output the proportion of potential
points gained in the regular season. Finally, Barros, del Corral and Garcia-del-Barrio
(2008) identified three segments in the Spanish Soccer League using a latent class
model in a cost frontier framework.
Among the papers adopting a non-parametric approach, we find Espitia-Escuer
and García-Cebrian (2004), in which the efficiency of Spanish First Division soccer
clubs is decomposed into technical efficiency and scale efficiency using DEA. They
used as output the number of points achieved in the league season. Fizel and D’Itri
(1997) applied the DEA technique to measure the managerial efficiency in college
basketball. They used the winning percentage as output and ex-ante quality measures of
7
players as inputs. Finally, Haas (2003B) examines the efficiency of the US Major
Soccer League.
4. Methodology
In order to characterize the production process of firms, the dual approach (i.e., cost
functions or profit functions) is typically used, rather than a primal approach (i.e.
production function). A cost frontier represents the minimum expenditure required to
produce any output, given input prices (Kumbhakar and Lovell, 2000; p. 33). Therefore,
a cost frontier envelops the data in such a way that all clubs must lie on the frontier or
above it. Thus, a cost frontier is specified as:
C = C * (w, y, T )
(1)
where C is cost, w is input prices, y is output and T represents the state of the
technology. Based on the cost frontier definition, the stochastic frontier analysis
(Aigner, Lovell and Schmidt, 1977; Meeusen and van den Broeck, 1977) offers an
analytical framework to estimate cost frontiers by maximum likelihood. Using this
approach, a stochastic cost frontier is specified as:
C = C * ( w, y, t ) ⋅ exp(ε ); ε = v + u ;
u≥0
(2)
where ε is the error term which is composed by two components. The symmetric
component, v, captures statistical noise and is assumed to follow a distribution centered
at zero, while u is a non-negative term that reflects inefficiency and is assumed to
follow a one-sided distribution (i.e. truncated normal, half-normal, exponential). When
u = 0, the club is producing on the cost frontier (i.e., at minimum cost), whereas a
positive u indicates that the club cost is above the minimum cost. A cost efficiency
8
index can be defined as the ratio of the minimum feasible cost (C*) and the observed
cost (C)3:
CE =
C*
C ( y, w, t ) ⋅ exp(v )
=
= exp(− u )
C
C ( y, w, t ) ⋅ exp(u + v )
(3)
Since C* must always be lower than or equal to C, the cost efficiency index is
bounded between 0 and 1, and achieves its upper bound when a club is producing its
output level at minimum cost (i.e., C=C*), given input prices and the technology.
Furthermore, given that the estimation procedure of equation (2) yields merely the
residual ε, rather than the inefficiency term u, the latter must be calculated indirectly,
using the Jondrow et al. (1982) formula, which is the conditional expectation of uit,
conditioned on the realized value of εit.
Equation (2) can be written as a latent class model:
ln C it = C * ( w, y, t ) + vit
j
j
+ u it
j
(4)
where subscript i denotes club, t indicates time and j represents the group or class that
the club belongs to. The vertical bar means that there is a different model for each class
j. Moreover, it is assumed that each club belongs to the same group in all periods4. This
assumption can be considered as restrictive, but necessary for identification purposes
and only implies that managerial behavior is similar along the period.
An important issue in these models is how to determine the number of classes.
The usual procedure is to estimate several models with different numbers of groups and
then use a statistical test in order to choose the preferred model. Hence, it can be used
information criteria, such as the Akaike Information Criterion (AIC) or the Schwarz
Bayesian Information Criterion (SBIC). These statistics are calculated using the
following expressions:
3
4
In the appendix is shown a graph that shows there are calculated cost efficiencies.
Further technical details on the estimation procedure are provided in the appendix.
9
SBIC = −2 ⋅ log LF ( J ) + log(n) ⋅ m
(5)
AIC = −2 ⋅ log LF ( J ) + 2 ⋅ m
(6)
where LF(J) is the value that the likelihood function takes for J groups, m is the number
of parameters used in the model and n is the number of observations. The favored
model will be that for which the value of the statistic is lowest.
5. Data and Empirical Specification
To estimate the cost frontier, the paper uses an unbalanced panel data on the French
Soccer League 1 over the period 2003-2006, available on the league’s website
(www.lfp.fr/actualiteLFP/dncg.asp). It should be noted that the data gathered comes
from all the teams which were present in League 1 (i.e., 20 teams each season) in the
seasons analyzed. However, due to the promotion and relegation system5, an
unbalanced panel data set was obtained to estimate the cost frontier.
We include one output and two input prices in our specification. Although the
selection of the output of sports clubs could be controversial we have chosen the
number of points achieved in a season (e.g., Carmichael, Thomas and Ward, 2001;
Espitia-Escuer and García-Cebrián, 2004; Barros and Leach, 2007; Haas, 2003A).
Furthermore, the paper includes two input prices: a proxy for the price of labor
measured as total wages divided by the number of players, and a proxy for the price of
capital measured by dividing amortization and reconstruction expenditure by net assets
and liabilities. Table 2 shows the descriptive statistics of the variables used in the
empirical analysis.
5
Contrary to US professional sports leagues, which have a closed structure, most European leagues are
organized in such a way that at the end of each season, clubs are transferred between divisions. The top
two or three clubs in each division at the end of the season are promoted to the division above and, at the
same time, the lowest two or three clubs are relegated to the division below. See Noll (2003) for a study
which analyzes the differences between these two systems.
10
INSERT TABLE 2
Figure 1 shows the relation between cost and points achieved during the four
seasons of the dataset. As expected, in all seasons there exists a positive relation
between cost and performance (Hall, Szymanski and Zimbalist, 2002).
INSERT FIGURE 1
The empirical specification is the Cobb-Douglas. We estimate an average cost
frontier. This is done by using as the dependent variable the average cost rather than
total cost. It should be noticed that if the equation to estimate is in logs the only
difference that arises is in the parameter of points in the right hand side. When it is
estimated an average cost function a positive sign in the output coefficient will indicate
decreasing returns to scale and a negative sign will indicate increasing returns to scale..
The paper also includes time dummies for each season. Moreover, a dummy for
promoted teams is also included. Thus, the equation to estimate is:
 cost it
ln
 points it
4
∑λ
t =2
t

 = β 0 + β 1 ⋅ ln ( po int s it ) + β 2 ⋅ ln ( p l it ) + β 3 ⋅ ln ( p k it ) +

⋅ Dt + β 4 ⋅ D promoted + (vit + u it )
(7)
where league points at the end of the season is the output, pl denotes price of labor, pk is
the price of capital, v is a random error which reflects the statistical noise and is
assumed to follow a normal distribution centred at zero, and u reflects inefficiency and
is assumed to follow a half-normal distribution.
6. Results
The latent class model in equation (7) was estimated by maximum likelihood using
Limdep 9.0 (Greene, 2007). The model with two groups was the preferred one
11
according to SBIC and AIC criteria. Table 3 displays the estimations of the two class
model and the standard stochastic cost frontier, which assumes that only one cost
frontier represents all the data in the sample. As expected, total variance is smaller when
two different groups are allowed. In fact, latent class models classify observations by
reducing within group variance in order to maximize the value of the total likelihood
function.
INSERT TABLE 3
The estimated coefficients have the expected signs, as price elasticities are
positive but the capital price elasticity is negative for Group 2. Therefore, the higher the
labor and capital prices, the higher the cost needed to obtain a certain number of points.
On the other hand, the coefficients of points are negative, reflecting that the average
costs are decreasing. Likewise, the coefficient of the points for the Group 1, which is
composed mainly of high-budget clubs, is lower in absolute terms than for Group 2.
An important result that supports the latent class model estimation is that the
differences of the input prices coefficients among groups are statistically significant,
suggesting two different technologies to obtain points in the French Soccer League.
Likewise, it can be observed that the group with the highest labor elasticity is Group 2.
Clubs were assigned into groups using the posterior probability of class
membership6. Table 4 shows the means of some representative variables for the groups
obtained in the latent class model, while Table 5 shows the composition of the groups. It
can be seen that labor price and average cost are clearly higher in Group 1 than in
Group 2. However, the number of points is fairly similar between groups. Moreover, in
both groups the efficiency from the latent class model is higher than the efficiency from
the pooled model. This result suggests that to some extent, the standard stochastic
6
The way to compute the posterior probabilities are indicated in the appendix.
12
frontier, which imposes only one technology, overestimates the inefficiency when
compared with the latent model. Additional, the marginal cost is higher in the Group 1
than in Group 2. This result makes sense, since it is more difficult to increase the
number of points for a high-budget club than for a low-budget club.
Figure 2 shows the kernel densities of the efficiencies from the two latent class
model groups. It can be seen that the clubs in Group 2 are more efficient than those in
Group 1. Lastly, Table 6 shows the estimated cost efficiency in the latent class model
for each team each season.
INSERT TABLE 6
We verify that cost efficiency is constant over time validating Lee and Berri
(2008) who argued that average efficiency should be constant in sports leagues. This
fact is due to the special characteristic of a sports league in which all teams are not able
to increase the output at the same season. Still, individual efficiency can fluctuate over
time. This dynamics happens even for the top club in the league, the Olympique
Lyonnais. Moreover, there are clubs with a limited number of observations since they
were relegated from the league namely S.M. Caen, F.C. Istres, Le Havre A.C., A.S.
Nancy Lorraine, C.S. Sedan-Ardennes and E.S. Troyes Aube Champagne.
The top efficiency is achieved by several clubs in each season, signifying that
several football clubs are simultaneously cost efficient in the same season. The average
value presents a minor club, i.e., the E.S. Troyes Aube Champagne, as the most efficient
club, signifying that cost efficiency is not the same as sport dominance.
7. Conclusions
This article has proposed a simple framework for the comparative evaluation of French
League 1 soccer clubs and the rationalization of their operational activities. The analysis
13
was conducted by means of the implementation of a stochastic frontier latent class
model that allows the incorporation of a broad variety of inputs and outputs while
permitting researchers to account for segments in the sample and the existence of
heterogeneity in the data. The main result is that we find two groups among the French
soccer clubs, both following completely different “technologies”.
This study has drawn attention to the identification of these segments among
French soccer clubs, suggesting that business strategies need to be adapted to the
characteristics of the clubs, i. e., if any club should wish to follow a business strategy
that is more appropriate to clubs in the other group, it will incur a great deal of
inefficiency.
This paper has two main limitations related to the data set. First, the data span is
relatively short. Second, the sample procedure adopted was restricted to a single
country, thus the conclusions cannot be extended beyond the France. Moreover, since
this research is an exploratory study, the intention is not to obtain definitive results for
direct use by football clubs or leagues. Rather, it calls their attention to the value of
identifying heterogeneity among the football clubs, and defining business strategies for
the clusters in order to satisfy the characteristics of each club. In order to draw more
generalised conclusions, a larger data set would be necessary, with the inclusion of
more countries. The limitations of the paper suggest directions for new research.
Firstly, additional research is needed to confirm the results of this paper, as well as to
clarify the above-related issues. Secondly, research concerning football clubs efficiency
in the context of heterogeneity should be expanded to include other European leagues.
14
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18
Table 1. Clubs Averages (2005-2006 Season)
Attendance
Wages
Clubs
(persons) (thousand €)
A.C. Ajaccio
3,414
7,209
A.J. Auxerre
10,668
16,220
F.C. Girondins de Bordeaux
24,247
20,204
Le Mans Union Club 72
11,437
12,433
Racing club de Lens
34,445
18,746
LOSC Lille
13,198
14,757
Olympique Lyonnais
34,465
51,131
Olympique de Marseille
49,200
35,873
F.C. de Metz
16,039
9,277
A.S. Monaco F.C.
11,182
38,864
A.S. Nancy Lorraine
17,163
10,346
F.C. Nantes Atlantic
29,449
24,496
O.G.C. Nice
10,903
9,806
Paris Saint Germain
40,486
31,634
Stade Rennais F.C.
25,000
16,493
A.S. Saint-Etienne
29,111
11,472
F.C. Sochaux Montbeliard
14,257
14,240
Racing Club de Strasbourg
18,983
11,766
Toulouse F.C.
18,875
9,609
E.S. Troyes Aube Champagne
13,795
9,638
19
Points
Position
33
59
69
52
60
62
84
60
29
52
48
45
58
52
59
47
44
29
41
39
18
6
2
11
4
3
1
5
20
10
12
14
8
9
7
13
15
19
16
17
Table 2. Descriptive Statistics of the Data (2003-2006)
Variable
Mean
Standard
deviation
Minimum
Maximum
Cost
Points
Labor price
Capital price
41,287
51
655
0.05
26,156
12
468
0.08
10,682
29
121
0.00
98,945
84
1,869
0.43
Note: The monetary variables are expressed in constant €.
20
Table 3. Estimation Results
Standard SF.
Constant
Points
Labor price
Capital price
Season 2003-2004
Season 2004-2005
Season 2005-2006
Promoted
σ = (σ v2 + σ u2 )
12
Estimated probabilities for
class membership
Log Likelihood Function
Observations
4.442***
(0.77)
-0.632**
(0.31)
0.673***
(0.19)
0.008
(0.06)
-0.061
(0.09)
0.066
(0.12)
0.160
(0.11)
-0.139
(0.16)
2.344
(1.61)
Latent class model
Group 1
Group 2
4.524***
0.896**
(0.05)
(0.35)
-0.429***
-0.816***
(0.06)
(0.04)
0.570***
1.339***
(0.05)
(0.10)
0.002
-0.047**
(0.00)
(0.02)
-0.059*
0.046***
(0.04)
(0.01)
0.042**
0.073**
(0.02)
(0.03)
0.136***
0.156***
(0.02)
(0.01)
-0.270***
0.045***
(0.03)
(0.00)
0.270***
0.171***
(0.03)
(0.03)
0.648
11
80
0.352
45
80
Note: * ,**,*** indicate significance at the 1%, 5% and 10% levels, respectively.
Standard errors are shown in parenthesis.
21
Table 4. Characteristics of the Estimated Groups (Sample Averages)
Cost
Points
Wages
Labor price
Capital price
Average cost
Marginal cost
Pooled efficiency
Latent class model efficiency
Promoted
Relegated
Number of observations
Group 1
Group 2
46,129
(25,710)
52
(13)
19,965
(13,322)
736
(528)
0.06
(0.19)
846
(353)
482
(201)
0.68
(0.09)
0.83
(0.14)
0.16
(0.38)
0.12
(0.34)
54
31,231
(24,589)
49
(10)
13,551
(8,338)
486
(240)
0.03
(0.05)
607
(397)
111
(73)
0.82
(0.15)
0.90
(0.10)
0.12
(0.33)
0.19
(0.40)
26
Note: standard deviations are shown in parenthesis.
22
Table 5. Group Composition
Group 1
F.C. Girondins de Bordeaux
S.M. Caen
F.C. Istres
Le Havre A.C.
Racing Club de Lens
Le Mans Union Club 72
L.O.S.C. Lille Metropole
Olympique Lyonnais
F.C. de Metz
A.S. Monaco F.C.
F.C. Nantes Atlantic
A.S. Nancy Lorraine
Paris Saint-Germain
Stade Rennais F.C.
F.C. Sochaux Montbeliard
Racing Club de Strasbourg
Toulouse F.C.
E.S. Troyes Aube Champagne
Group 2
A.C. Ajaccio
A.J. Auxerre
S.C. Bastia
E.A. de Guingamp
Olympique de Marseille
Montpellier Herault S.C.
O.G.C. Nice
A.S. Saint-Etienne
C.S. Sedan Ardennes
23
Table 6. Cost Efficiency per Teams
2002-2003 2003-2004 2004-2005 2005-2006
A.C. Ajaccio
0.90
0.90
1.00
1.00
A.J. Auxerre
0.94
1.00
1.00
0.79
S.C. Bastia
1.00
1.00
0.86
F.C. Girondins de Bordeaux
0.98
0.78
0.79
0.76
S.M. Caen
0.73
E.A. de Guingamp
0.90
0.88
F.C. Istres
0.90
Le Havre A.C.
0.90
Le Mans Union Club 72
0.80
1.00
Racing Club de Lens
0.74
0.75
0.54
0.68
L.O.S.C. Lille Metropole
0.43
1.00
0.88
0.60
Olympique Lyonnais
0.72
0.83
0.89
1.00
Olympique de Marseille
1.00
0.63
0.85
0.98
F.C. de Metz
0.73
0.83
0.67
A.S. Monaco F.C.
0.83
0.84
0.71
0.99
Montpellier Herault S.C.
0.76
0.93
A.S. Nancy Lorraine
0.99
F.C. Nantes Atlantic
1.00
0.98
0.86
1.00
O.G.C. Nice
0.90
0.85
0.93
0.69
Paris Saint-Germain
0.82
0.96
0.65
0.66
Stade Rennais F.C.
0.84
0.89
1.00
0.97
A.S. Saint-Etienne
0.86
0.90
C.S. Sedan Ardennes
0.90
F.C. Sochaux Montbeliard
0.75
0.69
0.65
0.74
Racing Club de Strasbourg
0.85
0.98
0.98
0.74
Toulouse F.C.
0.78
0.81
0.79
E.S. Troyes Aube Champagne
1.00
1.00
Total
0.86
0.86
0.84
0.85
24
Total
0.95
0.93
0.95
0.83
0.73
0.89
0.90
0.90
0.90
0.68
0.73
0.86
0.87
0.74
0.84
0.84
0.99
0.96
0.84
0.77
0.92
0.88
0.90
0.71
0.89
0.79
1.00
0.85
25
35
45
55
Points
65
Fitted values
75
85
35
45
55
Points
35
65
75
45
55
Points
65
Fitted values
85
25
cost
35
45
55
Points
Fitted values
Figure 1. Cost and Performance Relationship
25
75
85
cost
Season 2005-2006
Cost (thousands €)
10
40
70 100
Cost (thousands €)
10
40
70 100
Fitted values
25
cost
Season 2004-2005
25
Season 2003-2004
Cost (thousands €)
10 40 70 100
Cost (million €)
10
40 70 100
Season 2002-2003
65
75
cost
85
4
3
Density
2
1
0
.4
.6
.8
Efficiency kernels from latent class model
Group 1
Group 2
Figure 2. Efficiency Kernels from Latent Class Model
26
1
Appendix
In this figure is shown how the cost efficiency is calculating assuming that a firm is
producing yA at a cost OC.
C
C(y,w,t)
CE=OC*/OC
A
OC
OC*
yA
y
Figure A1. Cost Efficiency
Next some technical details of the latent class model are explained. Assuming that v is
normally distributed and u follows a half-normal distribution, the likelihood function
(LF) for each club i at time t for group j is (Greene, 2005A):
LFijt = f (C it xit , β j , σ j , λ j ) =
(
Φ − λ j ⋅ ε it j / σ j
Φ (0 )
[
where ε it j = ln C it − β ′j xit , σ j = σ uj2 + σ vj2
]
1
2
)⋅ 1
 ε it j
⋅φ
σ j  σ j




(A1)
, λ j = σ uj σ vj , and φ and Φ denotes the
standard normal density and cumulative distribution function respectively.
The likelihood function for club i in group j is obtained as the product of the likelihood
functions in each period:
T
LFij = ∏ LFijt
(A2)
t =1
The likelihood function for each club is obtained as a weighted average of its likelihood
function for each group j, using as weights the prior probabilities of class j membership.
27
J
LFi = ∑ Pij LFij
(A3)
j =1
The prior probabilities must be in the unit interval: 0 ≤ Pij ≤ 1 . Furthermore, the sum of
these probabilities for each individual must be one:
∑P
ij
= 1 . In order to satisfy these
j
two conditions, we parameterized these probabilities as a multinomial logit. That is:
Pij =
exp(δ j qi )
J
∑ exp(δ
j =1
j
(A4)
qi )
where qi is a vector of variables which are used to split the sample, and δj is the vector
of parameters to be estimated. One group is chosen as the reference in the multinomial
logit. The overall log-likelihood function is obtained as the sum of the individual loglikelihood functions:
N
N
J
T
i =1
i =1
j =1
t =1
log LF = ∑ log LFi = ∑ log ∑ Pij ∏ LFijt
(A5)
The log-likelihood function can be maximized with respect to the parameter set
θ j = (β j , σ j , λ j , δ j ) , using conventional methods (Greene, 2005A). Furthermore, the
estimated parameters can be used to estimate the posterior probabilities of class
membership, using Bayes Theorem:
P( j i ) =
Pij LFij
J
∑ P LF
j =1
ij
(A6)
ij
Moreover, the latent class model classifies the sample into several groups even when
sample-separating information is not available. In this case, the latent class structure
uses the goodness of fit of each estimated frontier as additional information to identify
groups of units (Orea and Kumbhakar, 2004).
28
29