# Measuring Efficiency in the Summer Olympic Games Disciplines: The Case of the Spanish Athletes

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

#### Specification of the Production Function and Variables

## 3. Empirical Results

#### 3.1. Data

#### 3.2. Results Based in Ordinary Least Squares and SPF

#### 3.3. Efficiency

## 4. Conclusions and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**Summary related to the number of disciplines, the number of observations, and the number of years for the two scenarios considered: Olympic years and all disciplines (in parentheses).

Number of Observations | Number of Disciplines | Number of Years | |
---|---|---|---|

All | 18 (20) | 20 (41) | 4 (12) |

Women | 18 (41) | 20 (18) | 4 (12) |

Men | 23 (55) | 23 (23) | 4 (12) |

**Table 2.**Summary of descriptive statistics of data used considering the different disciplines during the period 2005–2016.

Variable | Mean | St. Dev. | Min. | Max. | |
---|---|---|---|---|---|

Output | All | 6.82609 | 6.31486 | 1.00 | 21.00 |

Women | 3.40 | 3.13553 | 1.00 | 14.00 | |

Men | 4.94444 | 4.47834 | 1.00 | 16.00 | |

Budget | All | 5.1803E6 | 4.9519E6 | 1.0475E6 | 2.2003E7 |

Women | 5.2526E6 | 5.0733E6 | 1.0177E6 | 2.1711E7 | |

Men | 5.8077E6 | 5.2690E6 | 1.0244E6 | 2.1809E7 | |

Scholarship | All | 128.123 | 83.3898 | 18.75 | 279.833 |

Women | 55.6208 | 40.4621 | 15.0833 | 161.167 | |

Men | 85.7731 | 54.473 | 15.8333 | 191.167 |

**Table 3.**Summary of descriptive statistics of data used considering the Olympic years 2008, 2012, and 2016.

Variable | Mean | St. Dev. | Min. | Max. | |
---|---|---|---|---|---|

Output | All | 2.85455 | 2.35245 | 1.00 | 11.00 |

Women | 1.63415 | 1.27977 | 1.00 | 7.00 | |

Men | 2.17073 | 1.71614 | 1.00 | 8.00 | |

Budget | All | 5.8298E6 | 5.4075E6 | 850,761 | 2.5692E7 |

Women | 5.6926E6 | 5.3932E6 | 819,836 | 2.5319E7 | |

Men | 6.4928E6 | 5.8718E6 | 815,986 | 2.5421E7 | |

Scholarship | All | 149.759 | 88.1322 | 23.5 | 367.5 |

Women | 63.7988 | 44.3893 | 11.75 | 214.25 | |

Men | 93.8902 | 57.1326 | 11.75 | 247.00 |

All | Women | Men | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OLS | SFP | OLS | SFP | OLS | SFP | |||||||

Variable | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value |

${\beta}_{0}$ | −6.818 | 0.038 | −6.645 | 0.016 | −3.708 | 0.184 | −2.135 | <0.001 | −3.731 | 0.281 | −3.665 | 0.250 |

${\beta}_{1}$ | 0.348 | 0.149 | 0.394 | 0.059 | 0.182 | 0.375 | 0.078 | <0.001 | 0.221 | 0.367 | 0.258 | 0.247 |

${\beta}_{2}$ | 0.658 | 0.019 | 0.615 | 0.007 | 0.498 | 0.045 | 0.695 | <0.001 | 0.387 | 0.140 | 0.388 | 0.088 |

${\sigma}_{u}$ | 0.142 | 0.002 | 2.114 | <0.001 | 0.208 | 0.009 | ||||||

${\sigma}_{\nu}$ | 0.091 | 0.014 | 0.007 | <0.001 | 0.143 | 0.015 | ||||||

Observations | 23 | 23 | 20 | 20 | 18 | 18 | ||||||

${\ell}_{max}$ | −24.798 | −24.975 | −16.365 | −13.158 | −18.283 | −18.806 |

All | Women | Men | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OLS | SFP | OLS | SFP | OLS | SFP | |||||||

Variable | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value | Coeff | p-Value |

${\beta}_{0}$ | −3.887 | 0.024 | −4.199 | 0.026 | −2.262 | 0.064 | −4.032 | 0.048 | −1.337 | 0.480 | −1.325 | <0.001 |

${\beta}_{1}$ | 0.232 | 0.061 | 0.287 | 0.039 | 0.122 | 0.237 | 0.235 | 0.116 | 0.070 | 0.587 | 0.103 | <0.001 |

${\beta}_{2}$ | 0.233 | 0.125 | 0.267 | 0.074 | 0.275 | 0.025 | 0.363 | 0.018 | 0.185 | 0.218 | 0.215 | <0.001 |

${\sigma}_{u}$ | 0.227 | <0.001 | 0.373 | <0.001 | 0.397 | <0.001 | ||||||

${\sigma}_{\nu}$ | 0.120 | <0.001 | 0.124 | <0.001 | 0.243 | <0.001 | ||||||

Observations | 55 | 55 | 41 | 41 | 41 | 41 | ||||||

${\ell}_{max}$ | −53.101 | −54.514 | −24.566 | −30.682 | −37.855 | −40.295 |

**Table 6.**Technical efficiency for all, women, and men (period 2005–2016) based on data provided in Table 4.

Activity | Technical Efficiency | ||
---|---|---|---|

All | Women | Men | |

Athletics | 0.782416 | 0.884518 | 0.812310 |

Badminton | 0.391189 | 0.323748 | – |

Basketball | 0.283643 | 0.151754 | 0.378020 |

Handball | 0.361354 | 0.247878 | 0.344449 |

Boxing | 0.646577 | – | 0.512122 |

Cycling | 0.983386 | 0.533028 | 0.971078 |

Fencing | 0.560248 | – | 0.567740 |

Gymnastics | 0.802922 | 0.563483 | 0.631612 |

Golf | 0.422304 | – | 0.543999 |

Weightlifting | 0.979878 | 0.990768 | 0.691763 |

Riding | 0.416264 | 0.575962 | – |

Hockey | 0.504114 | 0.208556 | 0.528050 |

Judo | 0.672335 | 0.746668 | 0.286504 |

Olympic fight | 0.461873 | 0.497380 | – |

Swimming | 0.895400 | 0.991153 | 0.637816 |

Canoeing | 0.973994 | 0.698471 | 0.963571 |

Rowing | 0.273845 | 0.401059 | – |

Rugby | 0.274858 | 0.215614 | – |

Taekwondo | 0.921492 | 0.336206 | 0.888649 |

Tennis | 0.987016 | 0.990418 | 0.967453 |

Olympic shot | 0.716979 | 0.722084 | 0.545806 |

Triathlon | 0.824645 | 0.289365 | 0.790250 |

Sailing | 0.645490 | 0.629250 | 0.531064 |

Mean | 0.642705 | 0.549868 | 0.644014 |

Stand. Deviation | 0.250963 | 0.277259 | 0.214627 |

min | 0.273845 | 0.151754 | 0.286504 |

max | 0.987016 | 0.991153 | 0.971078 |

Observations | 23 | 20 | 18 |

**Table 7.**Technical efficiency for all, women, and men (2008, 2012, and 2016) based on data provided in Table 5.

Activity | Olympic | Technical Efficiency | Activity | Olympic | Technical Efficiency | ||||
---|---|---|---|---|---|---|---|---|---|

Year | All | Women | Men | Year | All | Women | Men | ||

Athletics | 2008 | 0.967700 | 0.943627 | 0.903890 | Olympic fight | 2008 | 0.473082 | 0.716555 | – |

Athletics | 2012 | 0.564079 | 0.453555 | 0.556118 | Olympic fight | 2012 | 0.417279 | 0.607966 | – |

Athletics | 2016 | 0.679842 | 0.529508 | 0.599059 | Swimming | 2008 | 0.710183 | 0.491673 | 0.648335 |

Badminton | 2016 | 0.419871 | 0.529054 | – | Swimming | 2012 | 0.960945 | 0.972782 | 0.396360 |

Basketball | 2008 | 0.360631 | 0.239576 | 0.385449 | Swimming | 2016 | 0.924073 | 0.904482 | 0.613924 |

Basketball | 2012 | 0.192984 | – | 0.359248 | Canoeing | 2008 | 0.942985 | 0.542338 | 0.879786 |

Basketball | 2016 | 0.305922 | 0.199187 | 0.342397 | Canoeing | 2012 | 0.966052 | 0.660461 | 0.909266 |

Handball | 2008 | 0.291260 | – | 0.425394 | Canoeing | 2016 | 0.969220 | 0.666904 | 0.910321 |

Handball | 2012 | 0.433781 | 0.301475 | 0.395948 | Rowing | 2016 | 0.399509 | 0.721855 | – |

Handball | 2016 | 0.232063 | 0.305573 | – | Rugby | 2016 | 0.325920 | 0.397904 | – |

Boxing | 2016 | 0.634711 | – | 0.594975 | Taekwondo | 2008 | 0.401910 | – | 0.486333 |

Cycling | 2008 | 0.982030 | 0.576432 | 0.944690 | Taekwondo | 2012 | 0.863927 | 0.475890 | 0.740656 |

Cycling | 2012 | 0.708268 | – | 0.809970 | Taekwondo | 2016 | 0.853728 | 0.468828 | 0.729592 |

Cycling | 2016 | 0.943104 | 0.507327 | 0.877068 | Tennis | 2008 | 0.693408 | 0.604151 | 0.557284 |

Fencing | 2008 | 0.715584 | – | 0.753905 | Tennis | 2012 | 0.796642 | – | 0.883613 |

Gymnastics | 2008 | 0.745366 | 0.365598 | 0.733853 | Tennis | 2016 | 0.975205 | 0.889044 | 0.914704 |

Gymnastics | 2012 | 0.561947 | 0.421137 | 0.442855 | Olympic shot | 2008 | 0.367070 | – | 0.462580 |

Gymnastics | 2016 | 0.605717 | 0.760445 | – | Olympic shot | 2012 | 0.755608 | 0.486301 | 0.678779 |

Golf | 2016 | 0.513099 | – | 0.738237 | Olympic shot | 2016 | 0.533918 | 0.862096 | – |

Weightlifting | 2008 | 0.523615 | 0.741692 | – | Triathlon | 2008 | 0.737580 | – | 0.779243 |

Weightlifting | 2012 | 0.840051 | 0.713533 | 0.532196 | Triathlon | 2012 | 0.646872 | 0.581939 | 0.463382 |

Weightlifting | 2016 | 0.897583 | 0.766744 | 0.568682 | Triathlon | 2016 | 0.365899 | – | 0.449392 |

Riding | 2012 | 0.350499 | 0.511124 | – | Sailing | 2008 | 0.858584 | 0.427862 | 0.807038 |

Riding | 2016 | 0.329976 | 0.447701 | – | Sailing | 2012 | 0.590403 | 0.615145 | 0.361219 |

Hockey | 2008 | 0.508449 | – | 0.417344 | Sailing | 2016 | 0.452861 | 0.635786 | – |

Hockey | 2012 | 0.295025 | – | 0.409758 | |||||

Hockey | 2016 | 0.546375 | 0.371204 | 0.418345 | |||||

Judo | 2008 | 0.831372 | 0.960265 | – | |||||

Judo | 2012 | 0.327100 | – | 0.439772 | |||||

Judo | 2016 | 0.355531 | 0.492962 | – | |||||

Mean | 0.611753 | 0.582139 | 0.617584 | ||||||

Stand. Deviation | 0.237419 | 0.197709 | 0.195349 | ||||||

min | 0.192984 | 0.199187 | 0.342397 | ||||||

max | 0.98203 | 0.972782 | 0.944690 | ||||||

Observations | 55 | 41 | 41 |

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**MDPI and ACS Style**

Gómez-Déniz, E.; Dávila-Cárdenes, N.; Leiva-Arcas, A.; Martínez-Patiño, M.J. Measuring Efficiency in the Summer Olympic Games Disciplines: The Case of the Spanish Athletes. *Mathematics* **2021**, *9*, 2688.
https://doi.org/10.3390/math9212688

**AMA Style**

Gómez-Déniz E, Dávila-Cárdenes N, Leiva-Arcas A, Martínez-Patiño MJ. Measuring Efficiency in the Summer Olympic Games Disciplines: The Case of the Spanish Athletes. *Mathematics*. 2021; 9(21):2688.
https://doi.org/10.3390/math9212688

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, Nancy Dávila-Cárdenes, Alejandro Leiva-Arcas, and María J. Martínez-Patiño. 2021. "Measuring Efficiency in the Summer Olympic Games Disciplines: The Case of the Spanish Athletes" *Mathematics* 9, no. 21: 2688.
https://doi.org/10.3390/math9212688