Efficiency Measurement of the English Football Premier League with a Random Frontier Model

Efficiency Measurement of the English Football Premier League
                  with a Random Frontier Model

                                   Carlos Pestana Barros
                       ISEG – School of Economics and Management
                               Technical University of Lisbon
                                    Rua Miguel Lupi, 20
                                 1249-078 Lisbon, Portugal
                       Tf.: 351-1-213922801 / Fax: 351-1-213967971

                                 Pedro Garcia-del-Barrio 1
                ESIrg-Economics, Sport and Intangibles research group and
                         Universitat Internacional de Catalunya
                                    c/ Immaculada 22
                                  08017 Barcelona, Spain
                         Tf.: 34 932541800 / Fax: 34 934187673


      Using the random stochastic frontier model, this paper examines the
      technical efficiency of the English football Premier League from 1998/99 to
      2003/04. The model disentangles homogenous and heterogeneous variables
      in the cost function, which leads us to advise the implementation of
      common policies as well as policies by clusters.

Key words: Football, efficiency, random frontier models, policy implications
JEL-Code: L83, C69

 Correspondent author. The author gratefully acknowledges financial support from the Ministerio de
Ciencia y Tecnologia (SEJ2004-04649, Spain) and (SEJ 2007-67295/ECON).

This paper combines sport and financial data to analyse, with a random frontier model (Greene
2005, 2006), the technical efficiency of the English football Premier League. This model allows
for heterogeneity in the data and is considered the most promising state-of-the-art modelling
available to analyze cost functions (Greene, 2003, 2004, 2005). The advantage of this method
over alternative models is twofold. First, it allows for the error term to combine different
statistical distributions. Second, it uses random parameters; i.e., parameters that describe factors
not linked to observed features on the cost function. The estimation of the random frontier
model disentangles heterogeneous and homogeneous explanatory variables to determine which
of them must be treated in a homogeneous way and which managed by clusters.

        The scope of this research is to account for the fact that English clubs can be identified
as heterogeneous, given that various clusters exist in the league. We estimate a random frontier
model for clubs that played uninterruptedly in the Premier League between seasons 1998/99 and
2003/04. This ensures a balanced panel, which is a pre-requisite to obtain similar average scores
in the period at club level. As the analysis combines sport and financial variables, it permits
verifying if pitch success entails financial success (Cf. Szymanski and Kuypers, 2000, p. 22).

        In the following section, we analyse institutional settings. Section 3 examines the
literature on sport efficiency, while Section 4 explains the theoretical framework. Then, Section
5 presents the data and the main results. Finally, Section 6 discusses the efficiency ranking-list
and concludes with a number of managerial implications.

2. Contextual Setting

The financial underpinning of the Premier League has created four sub-groups of clubs, in terms
of aspirations and likelihood of sport success. First, there is an elite group of three clubs that
dominate the league (Manchester United, Arsenal and more recently Chelsea, which has joined
this privileged status through the vast wealth of Abramovich). Second, there are four or five
aspiring teams that struggle to qualify for the remaining places in European competitions. Then,
we find nine or ten middle-table teams, whose main goal is avoiding relegation. Finally, there is
a group of teams (in which the newly-promoted clubs are usually present) that are engaged in a
fight to retain category. It is not unusual for teams to be relegated after one year, and even to
sink without trace, owing to the financial adjustments that they are obliged to make afterwards.
This emphasizes the importance of middle-rank clubs attracting players commensurate to their
aspirations, in an attempt to prevent drifting into the relegation zone.

        The new financial scenario is related to the increasing success of the UEFA Champions
League. The main clubs enjoy phenomenal revenues generated through broadcasting contracts

and global brand sponsorships. Whilst the league winners get the biggest prizes, qualification to
participate in Europe is considered a financial victory in itself. Of the European leagues, the
Premier is the richest one, attaining revenues of approximately €1.79 billion in the 2002/03
season. Additionally, match-day income in England still represents an important portion of
revenues (around 30%), while this figure is substantially smaller (15-18%) in Italy, Germany
and France.

3. Review of the Literature

There are two main approaches to measure efficiency: the econometric or parametric approach
and the non-parametric or DEA approach. Unlike the econometric stochastic frontier approach,
DEA allows the use of multiple inputs and outputs, but does not impose any functional form on
the data; nor does it make distributional assumptions for the error term. Both methods assume
that the production function of the fully efficient decision unit is known. In reality this is not the
case and the efficient iso-quant has to be estimated from the sample. Under these conditions, the
frontier is relative to the sample considered in the analysis. The stochastic frontier approach has
been applied to various contexts like production (Kumbhakar and Wang, 2005; or Lothgren,
1997). However, as Table 1 shows, the studies published so far applying this methodology to
sports are scarce, which enhances the interest of this paper.

                                                     Table 1
Note that nine papers in total have used DEA, three papers adopted a deterministic econometric
frontier and two papers use the stochastic econometric frontier. In our view, this is                    not
sufficient for such an important issue in the context of sports management, thereby deserving
further research.

4. Theoretical Framework

The approach that we adopt here is the stochastic cost econometric frontier. In its origins, the
random frontier model was proposed by Farrell (1957), and came to prominence with
contributions from Aigner, Lovell and Schmidt (1977), Battese and Corra (1977) and Meeusen
and Van den Broeck (1977). The frontier is estimated econometrically and measures the
difference between the inefficient units and the frontier by the residuals, which are assumed to
have two components: noise and inefficiency. The general frontier cost function proposed is of
the form:
                                       v +u
                      Cit = C ( X ) ⋅ e it it ; ∀ i = 1,2, … N ; ∀ t = 1,2, …T       (1)

    Deloitte & Touche Annual Review of Football Finance 2004. As a reference, the Italian Calcio Serie A
obtained €1,162. More recent information is available but these records are sufficiently illustrative.

Where Cit represents a scalar cost of the decision-unit i under analysis in the t-th period; Xit is a
vector of variables including input prices and output descriptors present in the cost function.
The error term vit is assumed to be i.i.d. and represents the effect of random shocks (noise). It is
independent of uit, which represents technical inefficiencies and is assumed to be positive and to
follow a N(0, σu2 ) distribution. The positive disturbance uit is reflected in a half-normal
independent distribution truncated at zero, signifying that the cost of each club must lie on or
above its cost frontier, implying that deviations from the frontier are caused by factors
controlled by the club.

         The total variance is defined as σ2 = σv2 + σu2. The contribution of the different elements
to the total variation is given by: σv2 = σ2 / (1+ λ2) and σu2 = σ2 λ2 / (1+ λ2); where λ = σu / σv ,
which provides an indication of the relative contribution of u and v to ε = u + v. Because
estimation of equation (1) yields merely the residual ε, rather than u, the latter must be
calculated indirectly (Greene, 2003). For panel data analysis, Battese and Coelli (1988)
employed the expectation of uit conditioned on the realized value of εit = uit + vit, as an estimator
of uit. In other words, E[uit / εit] is the mean productive inefficiency for club i at time t. But the
inefficiency can also be due to clubs heterogeneity, which implies the use of a random effects

                                       cit = ( β 0 + wi ) + ' x it + vit + uit                                       (2)

where the variables are in logs and wi is a time-invariant specific random term that captures
individual heterogeneity. A second issue concerns the stochastic specification of the inefficiency
term u, for which the half normal distribution is assumed. For the likelihood function we follow
the approach proposed by Greene (2005), where the conditional density of cit given wi is:

                                           2         ε it   λε it                                                    (3)
                        f (cit | wi ) =          φ        Φ              , ε it = cit − ( β 0 + wi ) − ' x it
                                          σ          σ       σ

Where φ is the standard normal distribution and Φ is the cumulative distribution function.
Conditioned on wi , the T observations for club i are independent and their joint density is:

                                                               2       ε it   λε it
                        f (ci1 ,..., ciT | wi ) = ∏                φ        Φ                                        (4)
                                                        t =1   σ       σ       σ

The unconditional joint density is obtained by integrating the heterogeneity out of the density,

                                                 2      ε it   λε it                                 T
                                                                                                            2   ε it   λε it   (5)
          Li = f ( ci1 ,..., ciT ) =      ∏σ φ          σ
                                                                     g ( wi )dwi = E w          ii   ∏σ φ       σ
                                       wi t =1                                                       t =1

The log likelihood is then maximized with respect to β0, β, σ, λ and any other parameter
appearing in the distribution of wi. Even if the integral in expression (5) will be intractable, the
right hand side of (5) leads us to propose computing the log likelihood by simulation. Averaging

the expectation over a sufficient number of draws from the distribution of wi will produce a
sufficiently accurate estimate of the integral shown in (5) to allow estimation of the parameters
(see Gourieroux and Monfort, 1996 and Train, 2003). The simulated log likelihood is then given
by the expression:

                                                           1 R      T
                                                                           2   ε it | wir   λε it | wir
             log Ls ( β 0 , , λ , σ , θ ) =
                                              i =1
                                                           R r =1
                                                                    ∏σ φ
                                                                    t =1           σ

where θ includes the parameters of the distribution of wi and wir is the r-th draw for observation
i. Based on our panel data, Table 4 presents the maximum likelihood estimators of model (1) as
found in recent studies carried out by other authors (Greene, 2004 and 2005).

5. Data and Results

To estimate the cost frontier, we use a balanced panel. The sample comprises the twelve clubs
that were uninterruptedly competing at the Premier League from 1998/99 to 2003/04. Frontier
models require identifying inputs (resources) and outputs (transformation of resources). This is
accomplished using the usual criteria: availability of data, findings of previous studies, and
opinions of professionals. The variables have been transformed as described in Table 2, where
monetary variables are expressed in £'000, deflated by GDP deflator and denoted at prices of

                                                                    Table 2

We estimate the stochastic generalized Cobb-Douglas cost function, with three input prices and
three outputs (sales, points and attendance) and the Translog Frontier model. We adopt the log-
log specification to allow for non-linearity of the frontier. In order to capture the specificity of
the two types of capital (funds used and the premises) that clubs require for developing their
activity, we disentangle the analysis into capital-premises and capital-investment. Then, we
impose linear homogeneity in input prices, restricting the parameters in the estimated function:

LgCost it = β 0 + β1Trend + β 2 LgPLit + β 3 LgPK 1it + β 4 LgPK 2 it + β 5 LgSalesit + β 6 LgPoinit + β 7 LgAtt it + ( vit + u it )   (7)

where PL, PK1 and PK2 are respectively the prices of labor, capital-Premises, and capital-
investment. This is the cost frontier model, known in Coelli, Rao and Battese (1998) as the
Error Components Model, as it accounts for causes of efficiency controlled by management.
Next, Table 3 presents the results obtained for the stochastic frontier, using GAUSS and
assuming a half-normal distribution specification for the costs function frontier.

                                                                    Table 3

Regularity conditions require for the cost function to be linearly homogeneous, non-decreasing
and concave in input prices (Cornes, 1992). Attending to the number of observations and

exogenous variables, we use the Cobb-Douglas model with a half-normal distribution, a choice
that is supported by the data. Then, the error components model (Coelli et al., 1998) is adopted.
Having estimated two rival models, homogeneous and heterogeneous Cobb-Douglas frontier
models, we follow the Likelihood test to select the most adequate functional form. The test
compares models with different numbers of parameters by means of the Chi-square distribution,
indicating that the heterogeneous frontier is preferred to the standard model. We also computed
the Chi-square statistic for the general model specification. It also advocates the heterogeneous
frontier, thereby supporting the relevance of adding the variables.

           Finally, in order to differentiate between the frontier model and the cost function, we
consider the sigma square and the lambda variables of the cost frontier model. They are
statistically significant, meaning that the traditional cost function is unable to capture adequately
all the dimensions of the data. Furthermore, the random cost function fits the data well, since
both the R2 and the overall F-statistic (of the initial OLS used to obtain the starting values for
the maximum-likelihood estimation) are higher than the standard cost function. The value of the
parameter lambda is positive and statistically significant in the stochastic inefficiency effects
and the coefficients of the variables have the expected signs. Cost increases alongside the trend,
which indicates that there were no technological improvements during the period to drive costs
down. Moreover, costs significantly increase with the price of labour, the price of capital-
premises and attendance. It also rises with the price of capital-investment and sales; a
relationship that is statistically significant only for the random frontier model. The significant
random parameters vary along the sample. The identification of the mean values of random
parameters implies taking into account heterogeneity when implementing policies for cost

6. Conclusion

Common policies can be defined for English clubs based on the average values of the
homogeneous variables; whereas individual policies by clusters may be prescribed to account
for heterogeneous variables. The model does not specify how many clusters exist in the sample,
an issue which has to be established by non-empirical means, but it identifies their
heterogeneous nature. Given that the scale parameters of the heterogeneous variables are
statistically significant, we recognize their heterogeneous nature, which entails managerial
insights and policy implications.

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Table 1. Survey of the Literature on Frontier models in Sports.
     Papers                 Method                      Units                     Inputs                      Outputs                 Prices
                     Stochastic production       NBA association          Ratios of: field goal %,      Actual number of wins
                     frontier model              clubs, 2001-2002         free throw %, offensive
Hoefler and Payne
                                                                          and defensive rebounds,
                                                                          assists, steals, turnover,
                                                                          blocked shots difference
                     Technical efficiency        Soccer clubs in the      Operational cost              Points, attendance,       Price of labour,
                     effects model               English Premier                                        turnover.                 price of capital-
Barros and Leach
                                                 League                                                 Contextual factors:       players, price
                                                                                                        population, income,       of capital-
                                                                                                        European                  premises
                     Stochastic frontier model   Soccer clubs in the      Operational cost              points, attendance        Price of:
Barros and Leach
                                                 English Premier                                                                  labour, capital,
                                                 League                                                                           and stadiums,
                     DEA-CCR and BCC             Soccer clubs in the      Players, wages, net           Points, attendance and           
Barros and Leach
                     model                       English Premier          assets and stadium            turnover
                                                 League                   facilities
                     DEA-CCR Model and           Soccer clubs in the      Supplies & services           Match, membership,               
                     DEA-BCC model               Portuguese First         expenditure, wage             TV and sponsorship
Barros and Santos                                Division                 expenditure,                  receipts, gains on
(2005)                                                                    amortization                  players sold, financial
                                                                          expenditure, other costs.     receipts, points won,
                                                                                                        tickets sold
                     DEA-CCR and DEA-            12 US soccer clubs       Players wages, coaches        Points awarded,                  
Haas (2003A)         BCC model                   observed in year         wages, stadium                number of spectators
                                                 2000                     utilization rate              and total revenue
                     DEA-CCR and DEA-            20 Premier League        Total wages, coach            Points, spectators and           
Haas(2003B)          BCC model                   clubs observed in        salary, home town             revenue
                                                 year 2000/2001           population
                     DEA-Malmquist index         18 training activities   Number of Trainers,           Number of                        
                                                 of sports federations,   trainers reward, number       participants, number
Barros and Santos
                                                 1999-2001                of administrators,            of courses, number of
                                                                          administrators reward         approvals
                                                                          and physical capital
                     DEA-Allocative model        19 training activities   Number of Trainers,           Number of                 Price of:
                                                 of sports federations,   number of                     participants, number      trainers,
Barros (2003)
                                                 1998-2001                administrators, physical      of courses, number of     administrators,
                                                                          capital                       approvals                 and capital
                     DEA-CCR model in first      147 College              Player talent, opponent       Winning percentages              
Fizel and D’Itri
                     stage and regression        basketball teams,        strength,
                     analysis in second stage    1984-1991
Fizel and D’Itri     DEA-CCR model               Baseball managers        Player talent, opponent       Winning percentages              
(1996)                                                                    strength,
                     A linear program-ming       Major League             Team hitting and team         Team percent wins                
Porter and Scully
                     technique (possibly DEA-    baseball teams,          pitching
                     CCR)                        1961-1980
                     Stochastic Cobb-Douglas     Sample of English        Player age, career            Winning percentages              
                     frontier model              football managers,       league experience,
                                                 1992-1998                career goals, num. of
Dawson, Dobson
                                                                          previous teams, league
and Gerrard (2000)
                                                                          appearances in previous
                                                                          season, goals scored,
                                                                          player divisional status
Hadley,Poitras,      Deterministic frontier      US NFL teams,            24 independent                Team wins                        
Ruggiero and         model                       1969/70-1992/93          variables describing
Knowles (2000)                                                            attack and defence.
                     Hazard functions            English prof. soccer,    Match result, league          Duration (measured
Audas, Dobson and                                1972/73-1996/97,         position, manager age,        by the number of
Goddard (1999)                                   match-level data         manager experience,           league matches
                                                                          player experience             played)
                     Stochastic production       27 NBA teams,            Ratios of: field goal %,      Actual number of wins
                     frontier                    1992-1993                free throw %, turnover,
Hoefler and Payne                                                         offensive rebounds,
(1997)                                                                    defensive rebounds,
                                                                          assists, steals, difference
                                                                          in blocked shots
                     Deterministic and           41 Basketball            Team hitting and team         Win percent                      
Scully (1994)        stochastic Cobb-Douglas     coaches, 1949/50 to      pitching
                     frontier model              1989/90
                     Cobb-Douglas                NBA teams                10 variables of pitch         Ratio of final scores            
Zak, Huang and
                     deterministic frontier                               performance: ratio of
Siegfried (1979)
                     model                                                steals, ratio of assists…

Table 2. Descriptive Statistics of the Data
Variable                              Description                           Minimum    Maximum   Mean
              Logarithm of operational cost in Euros at constant price
LgCost                                                                       6.6685     8.9475   7.4633       0.4104
              Logarithm of price of workers, measured by dividing total
LgPL                                                                        4.61378     6.8152   5.7316       0.3782
              wages between the number of workers
LgPK1-        Logarithm of price of capital-premises, measured by the
                                                                            0.00453     0.3959   0.0689       0.0486
premises      amortizations divided by the value of the total assets
LgPK2-        Logarithm of price capital-investment, measured by the cost
                                                                            3.07E-06    2.1188   0.2438       0.3603
investment    of long term investment divided by the long term debt
              Logarithm of the sales of each club in pound at constant
LgSales                                                                      5.6367     8.3703   7.2507       0.4537
              price 2000=100
              Logarithm of the number of points obtained by each club in
LgPoin                                                                       1.4313     1.9542   1.7216       0.0988
              the league
LgAtt         Logarithm of the number of attendees
                                                                             3.9469     4.9410   4.4003       0.2302

Table 3. Stochastic Cobb-Douglas panel cost frontier (Dependent Variable: Log Cost)
Variables                                     Random Frontier model                 Non Random Frontier Model
Non-random parameters                        Coefficient (t-ratio)                    Coefficient (t-ratio)
Constant (β0)                                       1.0380 (5.480)                          1.1940 (1.442)
Trend (β1)                                          0.0269 (5.709)                          0.0270 (2.680)
LgPL (β2)                                           0.6993 (19.61)                          0.6809 (5.232)
LgPK1 (β3)                                          0.5401 (5.141)                          0.5513 (2.248)
LgPK2(β4)                                                  −                                0.0490 (0.409)
LgSales(β5)                                         0.0540 (2.018)                          0.0521 (0.461)
LgPoin (β6)                                                −                                0.2350 (0.793)
LgAtt (β7)                                                 −                                0.2881 (1.694)
                                             Mean for Random Parameters
LgPK2 (β4)                                          0.6022 (3.957)                                −
LgPoin (β6)                                         0.1975 (2.219)                                −
LgAtt (β7)                                          0.3388 (6.256)                                −
                                  Scale Parameters for Distributions of Random Parameter
LgPK2 (β4)                                          1.4281 (10.41)                                −
LgPoin (β6)                                         0.0202 (4.459)                                −
LgAtt (β7)                                          0.0115 (6.453)                                −
Statistics of the model

σ = [σ V2 + σ U2 ]
                    1/ 2                              0.1362 (29.03)                       0.1225 (1.079)

λ = σU /σV                                            0.2532 (2.706)                       0.8094 (2.132)

Log likelihood                                             75.169                              72.010
Chi Square                                                144.338                             132.214
Degrees of freedom                                           3                                    3
Probability                                                0.000                               0.000
Observations                                                 72                                  72
t-statistics in parentheses (* indicates that the parameter is significant at 1% level).

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