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

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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 cbarros@iseg.utl.pt 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 pgarcia@cir.uic.es Abstract 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 1 Correspondent author. The author gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (SEJ2004-04649, Spain) and (SEJ 2007-67295/ECON).

Introduction 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 1

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 2 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) it 2 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. 2

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 model: 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: T 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, T 2 ε it λε it T 2 ε it λε it (5) Li = f ( ci1 ,..., ciT ) = ∏σ φ σ Φ σ g ( wi )dwi = E w ii ∏σ φ 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 3

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: N 1 R T 2 ε it | wir λε it | wir log Ls ( β 0 , , λ , σ , θ ) = i =1 log R r =1 ∏σ φ t =1 σ Φ σ (6) 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 2000. 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 4

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 control. 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. 5

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Greene, W., 2005, Fixed and random effects in stochastic frontier models, Journal of Productivity Analysis 23, 7-32. Haas, D.J., 2003A, Technical Efficiency in the Major League Soccer. Journal of Sport Economics 4 (3), 203- 215 Haas, D.J., 2003B, Productive Efficiency of English Football Teams – A Data Envelopment Approach. Managerial and Decision Economics 24, 403-410. Hadley, L.; M. Poitras; J. Ruggiero and S. Knowles, 2000, Performance evaluation of National Football League teams, Managerial and Decision Economics 21, 63-70. Hoefler, R.A. and J.E. Payne, 1997, Measuring efficiency in the National Basketball Association, Economics Letters 55, 293-299. Hoefler, R.A. and J.E. Payne, 2006, Efficiency in the National Basketball Association: A Stochastic Frontier Approach with Panel Data, Managerial and Decision Economics 27 (4), 279-85. Kumbhakar, Subal C. and Hung-Jen Wang, 2005, Estimation of Growth Convergence Using a Stochastic Production Frontier Approach, Economics Letters 88 (3), 300-305. Kumbhakar, S.; E. Tsionas; B. Park and L. Simar, L., 2006, Nonparametric stochastic frontiers: A local maximum likelihood approach, Journal of Econometrics (forthcoming). Lothgren, M., 1997, Generalized Stochastic Frontier Production Models, Economics Letters 57 (3), 255-59. Meeusen, W. and J. Van den Broeck, 1977, Efficiency estimation from a Cobb-Douglas production function with composed error, International Economic Review 18, 435-444. Porter, P. and G.W. Scully, 1982, Measuring managerial efficiency: The case of baseball. Southern Economic Journal 48, 642-650. Scully, G.W., 1994, Managerial efficiency and survivability in professional team sports. Managerial and Decision Economics 15, 403-411. Szymanski, S. and T. Kuypers, 2000, Winners and Losers: The Business Strategy of Football. (Viking Books, London, 1999; softcover Penguin Books, 2000). Train, K., 2003, Discrete Choice Methods with Simulation (Cambridge University Press, Cambridge, UK). Zak, T.A.; C.J. Huang and J.J. Siegfried, 1979, Production efficiency: The case of professional basketball, Journal of Business 52, 379-392. 7

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, (2006) 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 (2006c) 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, (2006b) 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 (2006a) 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 (2003) 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, (1997) 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 (1982) 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… 8

Table 2. Descriptive Statistics of the Data Standard Variable Description Minimum Maximum Mean Deviation Logarithm of operational cost in Euros at constant price LgCost 6.6685 8.9475 7.4633 0.4104 2000=100 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 9

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). 10

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