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A Two-Stage Double Bootstrap DEA: The Case of the Top 25 European Football ClubsEfciency Levels George E. Halkos * and Nickolaos G. Tzeremes Department of Economics, University of Thessaly, Volos, Greece This paper analyzes how European football clubscurrent value and debt levels inuence their performance. The Simar and Wilson (J Econometrics, 136: 3164, 2007) procedure is used to bootstrap the data envelopment analysis scores in order to establish the effect of football clubscurrent value and debt levels on their obtained efciency scores. The results reveal that football clubscurrent value levels have a negative inuence on their performances, indicating that football clubshigh value does not ensure higher performance. At the same time, the empirical evidence suggests that football clubsdebt levels do not inuence their efciency levels. Copyright © 2012 John Wiley & Sons, Ltd. 1. INTRODUCTION Several studies have applied efciency analysis on sport teamsperformances. 1 However the economic framework of professional sporting activity is based on the works of Rottenberg (1956), Neale (1964), Jones (1969), and Sloane (1969, 1971, 1976). In addition, the rst empirical evidence in an average production function framework was found in the work of Scully (1974) who investigated the perfor- mance of baseball players. By using the percentage of matches won to model teamsoutput and man- agement, and capital and team spirit as inputs, Scullys empirical work was the rst to apply a production function in order to provide empirical evidence. However, the sporting production process has been modeled by several scholars in a similar way (among others Zech, 1981; Atkinson et al., 1988; Schoeld, 1988). Finally, Carmichael et al. (2000) emphasized that the appropriate establishment of the relationship between the production inputs and outputs used enables us to examine the determi- nants of team success. The application of a frontier production function to measure teamsperformance has been dated back on the works of Zak et al. (1979), Porter and Scully (1982), and Fizel and DItri (1996, 1997). In addition, over the last two decades, several scholars have been applying parametric and nonparametric frontier analysis to establish football teamsperformances and their determinants. Dawson et al. (2000), applying stochastic frontier approach, measured managersefciency for a panel of managers in English soccers Premier League using as output the percentage of matches won and as inputs several player quality variables, for the period of 19921998. Haas (2003a) applied a data envelopment analysis (DEA) measuring team efciency of the US Major *Correspondence to: Department of Economics, University of Thessaly, Korai 43, 38333, Volos, Greece. E-mail: [email protected] Copyright © 2012 John Wiley & Sons, Ltd. MANAGERIAL AND DECISION ECONOMICS Manage. Decis. Econ. 34: 108115 (2013) Published online 21 December 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mde.2597

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MANAGERIAL AND DECISION ECONOMICS

Manage. Decis. Econ. 34: 108–115 (2013)

Published online 21 December 2012 in Wiley Online Library

A Two-Stage Double Bootstrap DEA: TheCase of the Top 25 European Football

Clubs’ Efficiency LevelsGeorge E. Halkos* and Nickolaos G. Tzeremes

Department of Economics, University of Thessaly, Volos, Greece

(wileyonlinelibrary.com) DOI: 10.1002/mde.2597

*CorrespondenKorai 43, 383

Copyright ©

This paper analyzes how European football clubs’ current value and debt levels influencetheir performance. The Simar and Wilson (J Econometrics, 136: 31–64, 2007) procedureis used to bootstrap the data envelopment analysis scores in order to establish the effectof football clubs’ current value and debt levels on their obtained efficiency scores.The results reveal that football clubs’ current value levels have a negative influenceon their performances, indicating that football clubs’ high value does not ensurehigher performance. At the same time, the empirical evidence suggests that footballclubs’ debt levels do not influence their efficiency levels. Copyright © 2012 John Wiley &Sons, Ltd.

1. INTRODUCTION

Several studies have applied efficiency analysis onsport teams’ performances.1 However the economicframework of professional sporting activity is basedon the works of Rottenberg (1956), Neale (1964),Jones (1969), and Sloane (1969, 1971, 1976). Inaddition, the first empirical evidence in an averageproduction function framework was found in thework of Scully (1974) who investigated the perfor-mance of baseball players. By using the percentageof matches won to model teams’ output and man-agement, and capital and team spirit as inputs,Scully’s empirical work was the first to apply aproduction function in order to provide empiricalevidence. However, the sporting production processhas been modeled by several scholars in a similarway (among others Zech, 1981; Atkinson et al.,

ce to: Department of Economics,University of Thessaly,33, Volos, Greece. E-mail: [email protected]

2012 John Wiley & Sons, Ltd.

1988; Schofield, 1988). Finally, Carmichael et al.(2000) emphasized that the appropriate establishmentof the relationship between the production inputsand outputs used enables us to examine the determi-nants of team success.

The application of a frontier production functionto measure teams’ performance has been dated backon the works of Zak et al. (1979), Porter andScully (1982), and Fizel and D’Itri (1996, 1997).In addition, over the last two decades, severalscholars have been applying parametric andnonparametric frontier analysis to establish footballteams’ performances and their determinants.Dawson et al. (2000), applying stochastic frontierapproach, measured managers’ efficiency for apanel of managers in English soccer’s PremierLeague using as output the percentage of matcheswon and as inputs several player quality variables,for the period of 1992–1998.

Haas (2003a) applied a data envelopment analysis(DEA) measuring team efficiency of the US Major

A TWO-STAGE DOUBLE BOOTSTRAP DEA 109

League Soccer. In a DEA setting and for the year2000, Haas used head coaches’ and players’ wagesas inputs, and revenues, points awarded, and num-ber of spectators as outputs. In addition, Haas(2003b), in a similar DEA setting, performed anefficiency analysis for 20 English Premier Leagueclubs for the year 2000–2001. Barros and Leach(2006a, 2006b, 2007) applying a stochastic Cobb–Douglas production frontier and DEA measuredthe performance of football clubs in the EnglishFA Premier League for the periods 1989–1990 to2002–2003. They have applied a combination ofsport and financial data to measure football clubs’efficiency levels. Frick and Simmons (2008) byapplying stochastic frontier approach on the datafor German Premier Soccer League (Bundesliga)showed that managerial compensation impactpositively on team success.

Furthermore, Espitia-Escuer and García-Cebrián(2010) applied several DEA models to measure theefficiency and super efficiency for a sample of footballteams that played in the Champions League (fromdifferent countries) for a period of 5 years. They havefound evidences that only efficient teams (in terms ofuse of their resources) can reach the final. However, ithas been reported that efficient teams have been alsoeliminated in the early stages of the Champions League.

Similar to our study, Barros et al. (2010) byapplying Simar and Wilson’s (2007) DEA bootstrapprocedure analyzed the performance of theBrazilian first league football clubs. More recently,Barros and Garcia-del-Barrio (2011) measured theefficiency of the Spanish football clubs for theseasons 1996–1997 and 2003–2004 by applyingthe two-stage procedure (Simar and Wilson,2007). In their DEA setting, they have usedoperating cost, total assets, and team payroll asinputs, whereas attendance and other receipts asoutputs. In the second stage of their analysis, theyregressed the obtained team efficiency levels onseveral factors using truncated regression and Tobitmodel (for comparison reasons) to explain Spanishclubs’ efficiency variations.

Our study, similarly to the ones already presented,by applying a two-stage DEA bootstrap procedureinvestigates how clubs’ value and debt levelsinfluence their performances. In contrast to the mainresearch stream, instead of using data of a specificnational football league, our study uses a sample ofthe top 25 richest European football clubs andproposes for the first time a composite index formeasuring output.

Copyright © 2012 John Wiley & Sons, Ltd.

2. DATA AND METHODOLOGY

2.1. Description of Variables

In our analysis, we use a sample of the top European foot-ball clubs2 based on their current values. All the data areextracted from Forbes’ list which contains data of soccerteam valuations for 2009 (Forbes, 2009). In our DEAformulation, we use one input and one composite output.The input used is football clubs’ revenues (measured inmillions of $) and one composite output that measuresfootball clubs’ European and domestic trophies. Thecomposite output contains the sum of the number ofEuropean champion’s cups (weighted by 5), Unionof European Football Associations (UEFA) cups/EuroLeague cups (weighted by 4), European cup winner’scups (weighted by 3), Intercontinental cups (weightedby 3), and International Federation of Association Foot-ball (FIFA) Club World cups (weighted by 3).

In addition, the composite output contains also thesum of the number of domestic championships (weightedby 2) and domestic cups (weighted by 1). Both the num-ber of the weighted domestic champions and domesticcups (includes all domestic cups, i.e., super cups, leaguecups, national cups, etc.) are again weighted by FIFAworld ranking score (FIFA, 2010). This extra weighthas been added to reflect the different difficulty levelsof obtaining a domestic cup and/or championship amongthe different European leagues.3 We also assume thatclub revenues are used from the clubs to buy the best(in term of football quality) possible managers andplayers, which can lead to team success (based on world,European, and domestic championships and cups).4

Similarly, a recent study for the English PremierLeague suggests that revenues are related to clubs’success (Carmichael et al., 2010). Then, by applying asecond-stage analysis, we examine in what wayEuropean football clubs’ current value and debt levels(measured in millions of $) affect their obtained effi-ciency levels. Table 1 presents the descriptive statisticsof the variables used in our study. As can be realized,Table 1 reports several variations of the variables usedindicated by the high standard deviation values. Finally,in our DEA setting, we assume an output orientationsuggesting by how much football clubs can increasetheir outputs while keeping the level of inputs fixed.

2.2. Efficiency Measurement

Based on the work by Koopmans (1951) and Debreu(1951), the production set Ψ constraints the produc-tion process and is the set of physically attainablepoints (x,y):

Manage. Decis. Econ. 34: 108–115 (2013)DOI: 10.1002/mde

Table 1. Descriptive Statistics of the Variables Used

External variables Input

Current Value($m) Debt($m) Revenue ($m)Mean 597.080 218.238 274.720St Dev 443.374 338.197 128.008Min 194.000 0.002 128.000Max 1870.000 1284.000 576.000

Output componentsIntercontinental cup FIFA Club World cup Domestic championships

Mean 0.56 0.08 13.80St Dev 1.00 0.28 12.70Min 0.00 0.00 2.00Max 3.00 1.00 51.00

Output componentsEuropean Champions cups UEFA Cups/Euro league cups European Cup Winners cup

Mean 1.600 0.840 0.800St Dev 2.432 1.143 0.913Min 0.000 0.000 0.000Max 9.000 3.000 4.000

Output componentsDomestic cups FIFA country ranking Composite output

Mean 13.48 7.04 27.29St Dev 13.04 8.88 33.82Min 2.00 1.00 1.46Max 57.00 35.00 142.00

St Dev, standard deviation.

G. E. HALKOS AND N. G. TZEREMES110

Ψ ¼ x; yð Þ 2 RNþMþ x can produce yj g�

(1)

where x 2 RNþ is the input vector and y 2 RM

þ is theoutput vector.

Then, the output-oriented efficiency boundary @ Y(x) is defined for a given x 2 RN

þ as

@Y xð Þ ¼ y y 2 Y xð Þ; ly =2 Y xð Þ; 8l > 1j g;f (2)

and the Debreu–Farrell output measure of efficiencyfor a production unit can be defined as

l x; yð Þ ¼ sup l x; lyð Þ 2 Ψj g:f (3)

In equation (3), by construction, l(x,y)≥ 1, andtechnical efficiency is achieved when l(x,y) = 1. Assuggested by several authors (Førsund and Sarafoglou,2002; Førsund et al., 2009), Hoffman’s (1957) discus-sion regarding Farrell’s (1957) paper was the first toindicate that linear programming can be used to findthe frontier and estimate efficiency scores, but onlyfor the single output case. Later, Boles (1967, 1971)developed the formal linear programming problemwith multiple outputs identical to the constant returnsto scale (CRS) model in Charnes et al. (1978) whonamed the technique as DEA.5

Copyright © 2012 John Wiley & Sons, Ltd.

Following Zelenyuk and Zheka (2006, p.149), weapply the assumption of CRS because it enables toobtain greater discriminative power, which in turnwould result in a larger variation of the regressand.In addition, because we examine the 25 Europeanfootball clubs with the highest values, we are notexpecting great differences among their sizes. Thisformulation can be expressed as

Ψ∧CRS¼f x; yð Þ2 RNþM jy≤

Xni¼1

giyi; x≥Xni¼1

gixi for g1 . . . ; gnð Þ

such that gi ≥ 0; i ¼ 1; . . . ; ng (4)

This then can be computed by solving the followinglinear program:

l∧

CRS ¼ sup lf jly≤Xni¼1

gi yi; x≥Xni¼1

gixi for g1 . . . ; gnð Þ

such that gi≥ 0; i ¼ 1; . . . ; ng(5)

2.3. A Bootstrap Approach for Bias Correction ofthe Efficiency Estimator

Simar and Wilson 1998, 2000, 2008 suggested thatDEA estimators were shown to be biased by construc-tion. They introduced an approach based on bootstraptechniques (Efron, 1979) to correct and estimate thebias of the DEA efficiency indicators.6 The bootstrap

Manage. Decis. Econ. 34: 108–115 (2013)DOI: 10.1002/mde

A TWO-STAGE DOUBLE BOOTSTRAP DEA 111

bias estimate for the original DEA estimator l∧CRS x; yð Þ

can be calculated as

BIAS∧

B l∧CRS x; yð Þ

� �¼ B�1

XBb¼1

l�∧

CRS;b x; yð Þ � l∧CRS x; yð Þ

(6)

Furthermore, l�∧

CRS;b x; yð Þ are the bootstrap values,and B is the number of bootstrap replications. Then, abias-corrected (BC) estimator of l(x,y) can be calcu-lated as

l∧∧

CRS x; yð Þ ¼ l∧CRS x; yð Þ � BIAS

∧B l

∧CRS x; yð Þ

� �¼ 2l

∧CRS x; yð Þ � B�1

XBb¼1

l�∧

CRS;b x; yð Þ:(7)

2.4. A Two-Stage Analysis Using a DoubleBootstrap Procedure

Following Simar and Wilson (2007) to account forenvironmental variables (zi) on efficiency scores (li),a double bootstrap procedure must be used in a sec-ond-stage regression analysis in order to produce validestimates. Let us consider the following model:

l∧i ¼ zib

i þ ei (8)

where b is a vector of parameters and ei is the statisticalnoise. According to Simar and Wilson (2007, 2011),when using a conventional method of analysis such asthe ordinary least squares method, two are the main pro-blems that lead to invalid estimates. Firstly, when usingsmall samples, the basic assumption that zi

i is indepen-dent from ei is violated because of the high correlationof inputs/outputs used and the explanatory variables.Secondly, the DEA efficiency scores are expected tobe correlated because the efficiency levels of one foot-ball club are a product of the data of the other clubs ofthe same data set. Therefore, Simar and Wilson (2007,p.42-43) proposed a double bootstrap procedure(Algorithm 2) to avoid the dependency problems andproduce valid estimates of the second-stage regressionanalysis. Synoptically, the algorithm contains thefollowing seven steps:

1. Using the original data, we compute l∧i ¼

l∧xi; ; yið Þ; i ¼ 1; . . . ; n; by applying equation (5).

2. Then, the maximum likelihood estimates b∧and

se∧from the left normal truncated regression of

l∧i on zi (by using only l

∧i > 1) are applied.

3. For each football club i = 1, . . . n, we repeat thenext four steps (a–d) L1 times to obtain

Copyright © 2012 John Wiley & Sons, Ltd.

fl∧ib� gb¼1

L1

; i ¼ 1; . . . ; n : � �

a. For i= 1, . . . n, we draw e�i from N 0; s

e

with left truncation at 1� b’∧zi

� �.

b. Then, we computel�i ¼b’∧zi þ e�i ; i ¼ 1; . . . ; n:

c. We set x�i ¼xi; y�i ¼yil∧i=l

�i for all i¼ 1; . . . ; n:

d. Then, we compute l∧i

�¼ l xi; yijΨ

∧�� �; i ¼

1; . . . ; n; where Ψ∧�

is obtained by replacing

(xi,yi) by x�i ; y�i

� �.

4. We compute the BC estimator l

∧∧i using the boot-

strap estimates in step 3 and the originalestimate l

∧i.

5. Then, we estimate by maximum likelihood the

truncated regression of l

∧∧i on zi to obtain the

b

∧∧; s

∧∧

!.

6. For each football club i = 1, . . . n, we repeat thenext three steps (a–c) L2 times to obtain

b∧ �

; s∧e

�� �b

L2

b¼1

:

a. For i = 1, . . . n, we draw e��i from N 0; s

∧∧

!

with left truncation at 1� b’

∧∧

zi

!.

b. Then,we computel��i ¼b’

∧∧ziþ e��i ; i¼1; . . . ; n:

c. Then, we estimate by maximum likelihood the

truncated regression of l��i on zi to obtain the

b�∧∧

; s

∧∧�

!.

7. Finally, using the bootstrap values from step 6

and the original estimates of b

∧∧; s

∧∧, we

construct confidence intervals for b.

3. EMPIRICAL RESULTS AND CONCLUSIONS

Table 2 presents the results obtained from the effi-ciency analysis assuming the CRS assumption.

Manage. Decis. Econ. 34: 108–115 (2013)DOI: 10.1002/mde

Table 2. Efficiency Scores under the CRS Assumption

a/a Football clubs DEA BC BIAS St Dev LB UB Bound difference

1 Manchester United FC 1.681 1.773 �0.091 0.005 1.950 1.688 0.2622 Real Madrid FC 1.000 1.209 �0.209 0.016 1.466 1.013 0.4543 Arsenal FC 1.752 1.814 �0.062 0.002 1.923 1.760 0.1634 Bayern Munich FC 1.350 1.449 �0.099 0.005 1.621 1.358 0.2635 Liverpool FC 1.141 1.198 �0.057 0.002 1.315 1.145 0.1716 AC Milan FC 1.051 1.112 �0.061 0.002 1.227 1.055 0.1717 Barcelona FC 1.051 1.186 �0.135 0.010 1.415 1.056 0.3598 Chelsea FC 2.380 2.457 �0.077 0.003 2.586 2.388 0.1989 Juventus FC 1.053 1.096 �0.044 0.001 1.185 1.057 0.12910 Schalke 04 FC 1.511 1.560 �0.049 0.001 1.644 1.516 0.12811 Tottenham Hotspur FC 1.342 1.384 �0.042 0.001 1.455 1.347 0.10812 Olympique Lyonnais FC 1.914 2.040 �0.126 0.007 2.212 1.919 0.29313 AS Roma FC 1.902 1.971 �0.069 0.002 2.093 1.909 0.18414 Internazionale Milan FC 1.097 1.142 �0.045 0.001 1.234 1.101 0.13315 Hamburg SV FC 1.269 1.309 �0.040 0.001 1.379 1.273 0.10616 Borussia Dortmund FC 1.130 1.166 �0.036 0.001 1.226 1.134 0.09217 Manchester City FC 1.281 1.374 �0.092 0.003 1.483 1.289 0.19518 Werder Bremen FC 1.279 1.331 �0.052 0.002 1.422 1.282 0.14019 Newcastle United FC 1.429 1.486 �0.058 0.002 1.588 1.432 0.15620 VfB Stuttgart FC 1.375 1.471 �0.096 0.003 1.591 1.381 0.21021 Aston Villa FC 1.092 1.137 �0.046 0.001 1.216 1.095 0.12122 Olympique Marseille FC 1.461 1.523 �0.063 0.002 1.629 1.465 0.16423 Celtic FC 1.079 1.133 �0.054 0.001 1.214 1.085 0.13024 Everton FC 1.115 1.168 �0.052 0.001 1.251 1.120 0.13125 Glasgow Rangers FC 1.065 1.122 �0.075 0.002 1.258 1.081 0.177

Mean 1.352 1.424 �0.073 0.003 1.543 1.358 0.186Std 0.344 0.348 0.038 0.003 0.360 0.344 0.085Min 1.000 1.096 �0.209 0.001 1.185 1.013 0.092Max 2.380 2.457 �0.036 0.016 2.586 2.388 0.454

CRS, constant returns to scale; DEA, data envelopment analysis; BC, bias-corrected; St Dev, standard deviation; LB, lower bound; UB, upper bound.

G. E. HALKOS AND N. G. TZEREMES112

Looking at the descriptive statistics, we realize thatthere is a consistency with previous research on Euro-pean football leagues (Barros and Leach 2006a,2006b, 2007; Barros et al., 2010; Barros and Garcia-del-Barrio 2011) indicated with significant differencesof the original scores (DEA) and the BC efficiencyscores obtained. The standard deviation values are0.34 for the original estimates (DEA) and 0.35 forthe BC estimates. The results indicate that only onefootball club is reported to be efficient (Real MadridFC, i.e., with efficiency score equal to 1 under theoriginal efficiency estimates.

In addition, when looking at the BC results, the fiveEuropean clubs with the highest efficiency scores arereported to be Juventus FC, AC Milan FC, GlasgowRangers FC, Celtic FC, and Aston Villa FC. Whereasthe five European clubs with the lowest efficiencylevels are reported to be Manchester United FC,Arsenal FC, AS Roma FC, Olympique Lyonnais FC,and Chelsea FC. However, it must be noted that thelargest bound differences of the BC efficiency scoresare reported for Real Madrid FC, Barcelona FC,Olympique Lyonnais FC, Bayern Munich FC, andfor Manchester United FC.

Copyright © 2012 John Wiley & Sons, Ltd.

Furthermore, as explained earlier, we apply theapproach of Simar and Wilson (2007) in an estimatedspecification for the regression taking the form of

li ¼ b0 þ b1�Current Valuei þ b2�Debtþ ei (9)

where l represents the DEA model’s efficiency scorespresented in Table 2. In addition, Current Value refersto the football clubs’ current value levels measured inmillions of dollars, whereas Debt refers to footballclubs’ debt levels measured also in millions of dollars.Following Simar and Wilson (2007), we employed abootstrap algorithm of 2000 replications to construct95% confidence intervals. The results of the truncatedbootstrapped second-stage regression are presented inTable 3. It can be seen that the constant term and thefootball clubs’ Current Value levels are statisticallysignificant, whereas the football clubs’ Debt levelsdoes not seem to explain their efficiency variations.In addition, we can observe a negative sign on theCurrent Value coefficient indicating that the higherfootball clubs’ value does not necessary result tohigher efficiency levels.

Finally, in contrast with previous studies, this paperuses a sample of different European football clubs from

Manage. Decis. Econ. 34: 108–115 (2013)DOI: 10.1002/mde

Table 3. Truncated Bootstrapped Second-Stage Regression Results

Bias-adjusted coefficients (2000 bootstrap replications)

Variables Coefficient SE 95% Bootstrap confidence interval

Lower UpperConstant 1.62892* 0.08723 1.45795 1.79989Current Value �0.00052** 0.00020 �0.00091 �0.00012Debt �0.00031 0.00029 �0.00088 0.00026Variance 0.19362* 0.03670 0.12170 0.26555

SE, standard error.Statistically significant at*1%; **5%.

A TWO-STAGE DOUBLE BOOTSTRAP DEA 113

different football leagues and for the first time a compos-ite output measuring football clubs’ long-term successes(rather than just wins, points earned, etc.). Then, byapplying the latest developments in a two-stage boot-strapping DEA modeling, it contributes to the existingliterature in respect to the methodology adopted byproviding an alternative way of how such a procedurecan be applied to estimate teams’ efficiency levels andhow these efficiency levels can then be analyzed byproducing confidence intervals and standard errors in abootstrapped truncated regression setting.

In terms of policy implications, it appears thatwhen comparing the top European football clubs, theirdeterminants of higher efficiency (in terms of the num-ber of domestic and European club trophies) are notbased on their higher revenue and value levels. Thedeterministic nature of DEA methodology proved tobe a vital tool for showing that money alone doesnot ensure football clubs’ success. Other factors suchas managerial efficiency (Fizel and D’Itri, 1996,1997; Dawson et al., 2000), team spirit (Scully,1974), and the reward scheme (Wagner, 2010) maybe more important when comparing the top Europeanfootball clubs with the highest value. This resultconfirms the findings by Espitia-Escuer and García-Cebrián (2010), using a super efficient DEA formula-tion to emphasize the efficiency differences within theefficient football clubs played in the ChampionsLeague competition. In that respect and referring backto our primary DEA formulation, it appears that thecharacteristics of the football club’ owners (or theteam of the decision makers, directors, major share-holders, etc.) are also crucial determinants of theclubs’ success. Most of the times, the owner of a foot-ball club is the primary decision maker who is respon-sible for the allocation of resources (i.e., revenues) andholds the responsibility for the right investments(on players and managers) which in turn can resulton a long-term football clubs’ success.

Copyright © 2012 John Wiley & Sons, Ltd.

Acknowledgements

Thanks are due to Professor Antony Dnes and one anonymous refereefor the helpful and constructive comments on an earlier draft of thepaper. Any remaining errors are solely the authors’ responsibility.

NOTES

1. For a literature review on the subject matter, see Barrosand Garcia-del-Barrio (2008).

2. Nine football clubs are from the English Premier League,six from the German league, four from the Italian league,two from the Spanish league, two from the Frenchleague, and two from the Scottish league. The 25European football club in a descending order on the basisof their current value are: Manchester United FC, RealMadrid FC, Arsenal FC, Bayern Munich FC, LiverpoolFC, AC Milan FC, Barcelona FC, Chelsea FC, JuventusFC, Schalke 04 FC, Tottenham Hotspur FC, OlympiqueLyonnais FC, AS Roma FC, Internazionale Milan FC,Hamburg SV FC, Borussia Dortmund FC, ManchesterCity FC, Werder Bremen FC, Newcastle United FC,VfB Stuttgart FC, Aston Villa FC, Olympique MarseilleFC, Celtic FC, Everton FC, and Glasgow Rangers FC.

3. We assume that it is not of the same difficulty to obtain adomestic championship or cup between the English, theScottish, the Spanish, the German, and the Italian footballleagues. All the weights used in order for the compositeoutput to be constructed are subjective and can be subjectto criticism.

4. Because we have a small sample and a large number ofinputs and outputs could render the DEA model infeasible,we followKao andHung (2008) by constructing a compos-ite output using pre-assign weights and to compose aggre-gate measures making in this way the model more concise.

5. Later, Banker et al. (1984) used convex hull of Ψ∧FDH

(Derpins et al., 1984) to estimate Ψ and thus to allow forvariable returns to scale (VRS) adding the constraintXni¼1

gi ¼ 1 in equations (4) and (5).

6. The essence of bootstrapping efficiency scores has beenhighlighted by several authors. For further applicationsof the bootstrap technique on DEA efficiency scores,see also Simar and Wilson (2002), Zelenyuk and Zheka(2006), Simar and Zelenyuk (2007) and Halkos andTzeremes (2010).

Manage. Decis. Econ. 34: 108–115 (2013)DOI: 10.1002/mde

G. E. HALKOS AND N. G. TZEREMES114

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