Measuring the performance of managers through the season is important for those running football clubs. To do this we can use technical efficiency, defined as the ratio between the observed output and the maximum output attainable with the available resources. As the production function is defined by economists as the maximum output attainable given the available resources, the function can be used to calculate techincal efficiency. Observations located on the production function are “efficient” while those which are beneath the production function are “inefficient”. Obviously, the further from the production function the greater the inefficiency. Given the definition of technical efficiency is bounded between zero and one, the closer to one the more efficient the manager.
There is a consensus among sports economists on using the number of league points scored by a team as the manager’s output, and the quality of a team’s squad as the input. As a measure of the quality of each player in la Liga, I used the market value at the beginning of the 2011-2012 season provided by http://www.transfermarkt.co.uk/. Even though a football squad is usually composed of around 25 players, the number of players that play most regularly (90% of minutes in a season) is about 16: 1 goalkeeper, 6 defenders, 6 midfielders, and 3 forwards. So, I took as the input the total values of the most valuable goalkeeper, the six most valuable defenders, the six most valuable midfielders and the three most valuable forwards for each team
Given that the number of matches during the period varied between managers (some were fired while others were hired during the season), I used the ratio between points obtained and the total possible points (i.e., 3 x the number of matches) as the output. The production function (ie the maximum expected number of points given the value of the squad) is estimated using a stochastic frontier model, in which the Cobb-Douglas functional form has been used. The estimated function is depicted in Figure 1.
Figure 1 The production function
The results show that, as expected, the higher the quality of the squad the higher the points that should be obtained in a specified number of matches. Moreover, the production function is concave. This means that as squad quality increases, the corresponding increase in the number of expected points slows.
Figure 2 shows the scatter plots between the points obtained and quality of the squads for the teams in La Liga. It is important to note that in the estimates the observation unit was the manager, but in order to help the interpretation of these charts team names are used.
Figure 2a La Liga production functions, 2011-12
Figure 2b La Liga production functions, 2011-12 (excluding FC Barcelona and Real Madrid)
Notes: a) Both graphs are the same except for the exclusion of Real Madrid and Barcelona in the second. b) The red line indicates the estimated production function.
As can be seen, the only teams that are on the frontier are Levante and Real Madrid, so these are the only teams that are “fully efficient”. Once the production function has been estimated, it is straightforward to calculate the technical efficiency levels for each manager. These are shown in Table 1.
Table 1 Technical efficiency (TE) levels for managers
|MANAGER||TEAM||MATCHES||POINTS||% POINTS||SQUAD VALUE (€ mil.)||TE|
|J. I Martínez||Levante U.D.||38||55||48.25||25.8||1.00|
|Mourinho||Real Madrid C.F.||38||100||87.72||462||1.00|
|Juanjo||Racing de Santander||12||15||41.67||27.6||0.85|
|Mel||Real Betis Balonpié||38||47||41.23||33.6||0.81|
|Simeone||Atlético de Madrid||22||37||56.06||149||0.81|
|Luis García||Getafe C.F.||19||23||41.23||52.0||0.74|
|Clemente||Sporting de Gijón||16||18||37.50||39.0||0.71|
|Manzano||Atlético de Madrid||16||19||39.58||149||0.57|
|Preciado||Sporting de Gijón||20||18||30.00||39.0||0.57|
|Héctor Cúper||Racing de Santander||13||9||23.08||27.6||0.47|
|Álvaro||Racing de Santander||13||3||7.69||27.6||0.16|
Note: The managers that were fired are in italics.
We can observe from this table that:
- The correlation between the efficiency level and the quality of a team is low (0.23), while the correlation between points achieved and the quality of the a team is rather high (0.76). So in order to measure the performance of some managers, it seems that the technical efficiency index is more accurate than points scored.
- Most managers that were hired during the season, e.g. Jiménez, Caparrós, Juanjo, Abel, show high levels of technical efficiency. In some cases, the higher efficiency of newly-hired coaches saved teams from relegation.
- All managers with a technical efficiency level lower than 0.60 were fired.
Finally, it is important to note that this analysis has not taken into account multi-output technology (ie that teams not only play in the league but also the Spanish cup and European competitions).
Author’s note: I acknowledge the valuable assistance in recording the data from Fernando del Corral and Raúl Laguna.