# Referee Bias and Stoppage Time in Major League Soccer: A Partially Adaptive Approach

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stoppage Time and Television Rights in MLS: An Overview

#### 2.1. Stoppage Time in Soccer

The fourth official indicates the minimum additional time decided by the referee at the end of the final minute of each period of play. The announcement of the additional time does not indicate the exact amount of time left in the match. The time may be increased if the referee considers it appropriate but never reduced (FIFA, 2011: 98).

#### 2.2. MLS and Television

## 3. The Data and Model

_{0}+ β

_{1}Yellow + β

_{2}Red + β

_{3 }Subs + β

_{4}Tied + β

_{5 }Diff1 + β

_{6}Diff2 + β

_{7 }Behind1 + β

_{8 }Behind2 + β

_{9 }Waybehind + β

_{10 }AttPct + β

_{11 }NATV + referee fixed effects + venue fixed effects + ε

_{0}: β

_{7}= β

_{8 }= β

_{9}= 0, versus the alternative that at least one of the coefficients is greater than zero. In previous studies the focus, rightly so, is on testing the null that β

_{7}= 0, however, positive and significant values for any of the coefficients in the group indicates that the home team is being treated more favorably than the visiting team with regards to stoppage time.

_{0}above cannot be rejected, and the null hypothesis, H

_{0}: β

_{4}= β

_{5 }= β

_{6}= β

_{7}can be rejected in favor of the alternative, H

_{A}: β

_{4}> β

_{5 }> β

_{6}> β

_{7}. Thus, absent a home field advantage, the longest extra session should occur following a tie at the end of regulation, the second longest extra session should occur following a one-goal difference at the end of regulation, the third longest extra session should occur following a two-goal difference at the end of regulation, each without regard to which team is ahead or behind.

## 4. Grouped Data and Partially Adaptive Estimation

^{2},

_{1},σ

_{1}

^{2},μ

_{2},σ

_{2}

^{2}) = λϕ(x: μ

_{1},σ

_{1}

^{2}) + (1 - λ)ϕ(x: μ

_{2},σ

_{2}

^{2})

_{i},

_{i}= α + X

_{i}β + ε

_{i}

_{i}= α + X

_{i}β

_{i}based on the normality assumption is given by,

_{ij}= α

_{j}+ X

_{i}β

_{i}; λ,μ

_{i1},σ

_{1}

^{2},μ

_{i2},σ

_{2}

^{2}) = λϕ(y

_{i}; μ

_{i1},σ

_{1}

^{2}) + (1 - λ)ϕ(y

_{i}; μ

_{i2},σ

_{2}

^{2})

_{i}

^{*}= α + X

_{i}β + ε

_{i}

^{2}, while the interval endpoints are,

_{0}= -∞, L

_{1}= 0, and L

_{K}= ∞. Thus, observed y is related to y* as follows:

_{ik}is a set of dummy variables for each observation i, such that I

_{ik}equals 1 if the i

^{th}observation is in the k

^{th}interval, and 0 otherwise.

_{ik}= prob(y

_{i}= k) = F[L

_{k}] - F[L

_{k-1}]

## 5. Results

^{2}We do the same for the venue fixed effects. If, in either case, the null is rejected, we use a forward stepwise procedure to determine which fixed effects to include.

^{3}We also test for the presence of a home field advantage by jointly testing the coefficients of Behind1, Behind2, and Waybehind. If the null cannot be rejected, we omit the group. If the null can be rejected, we retain the entire group in future specifications.

#### 5.1. The 2011 Season

Variable | 2011 | 2012 |
---|---|---|

Stoppage_sec | 260.000 (74.96) | 285.882 (85.39) |

Yellow | 1.837 (1.28) | 1.854 (1.30) |

Red | 0.170 (0.43) | 0.115 (0.35) |

Subs | 4.725 (1.01) | 4.864 (1.08) |

Tied | 0.386 (0.489) | 0.402 (0.49) |

Diff1 | 0.471 (0.50) | 0.421 (0.49) |

Diff2 | 0.118 (0.32) | 0.149 (0.36) |

Bigdiff | 0.026 (0.16) | 0.028 (0.16) |

Behind1 | 0.180 (0.38) | 0.017 (0.38) |

Behind2 | 0.036 (0.19) | 0.046 (0.21) |

Waybehind | 0.007 (0.08) | 0.006 (0.08) |

AttPct | 0.771 (0.24) | 0.825 (0.25) |

NATV | 0.258 (0.44) | 0.334 (0.47) |

^{2}(3) = 25.88 with p-value = 0.00). We find no evidence that attendance as a fraction of stadium capacity has an impact on stoppage time. However, the coefficient of NATV indicates that nationally televised matches are, on average, about 22 seconds longer than untelevised matches (t = 2.32).

Variable | OLS Estimation Results^{1} | Grouped-Data Estimation Results ^{2} | Partially Adaptive Estimation Results |
---|---|---|---|

Intercept 1 | 69.226 (1.90) | 99.208*** (2.76) | 100.382*** (3.65) |

Intercept 2 | ------- | ------- | 85.777*** (3.45) |

Yellow | 8.311 (2.60) | 8.312*** (2.64) | 7.098*** (3.00) |

Red | 20.447 (2.11) | 20.440** (2.14) | 15.236** (2.29) |

Subs | 14.113*** (3.44) | 14.127*** (3.48) | 9.010*** (2.94) |

Tied | 132.338*** (5.07) | 132.354*** (5.14) | 151.508*** (10.10) |

Diff1 | 111.678*** (4.34) | 111.660*** (4.40) | 139.084*** (9.52) |

Diff2 | 103.92*** (3.74) | 103.892*** (3.80) | 138.688*** (8.75) |

AttPct | -20.885 (1.20) | -20.943 (1.22) | -3.308 (0.26) |

NATV | 21.851 (2.29) | 21.881** (2.32) | 12.143* (1.73) |

Sigma1 | 70.360 | 67.233 | 108.255 |

Sigma2 | ------- | ------- | 31.316 |

Mixing weight | ------- | ------- | 0.331 |

-ln L | ------- | 478.494 | 448.665 |

- 1
- Figures in parentheses are absolute values of t-ratios based on robust standard errors.
- 2
- Estimated coefficients in the grouped data regression model and in the partially adaptive regression model are interpreted just like their OLS counterparts; that is, they are marginal effects.
- ***
- Significant at the α = 0.01 level, two-tailed test.
- **
- Significant at the α = 0.05 level, two-tailed test.
- *
- Significant at the α = 0.10 level, two-tailed test.

#### 5.2. The 2012 Season

^{4}The chi-square is 7.22 with 3 degrees of freedom and a p-value of 0.07.

^{5}

Variable | OLS Estimation Results^{1} | Grouped-Data Estimation Results ^{2} | Partially Adaptive Estimation Results |
---|---|---|---|

Intercept1 | 71.321* (1.79) | 108.274*** (2.81) | 171.205*** (4.60) |

Intercept2 | ------- | ------- | 129.599*** (4.03) |

Yellow | 11.913*** (3.48) | 11.614*** (3.52) | 11.919*** (4.25) |

Red | -1.041 (0.08) | -2.468 (0.20) | 6.528 (0.58) |

Subs | 13.314*** (3.29) | 12.537*** (3.19) | 8.665*** (2.82) |

Tied | 136.036*** (4.48) | 134.166*** (4.57) | 120.529*** (4.86) |

Diff1 | 97.818*** (3.17) | 97.104*** (3.26) | 96.473*** (3.82) |

Diff2 | 93.764*** (2.87) | 92.188*** (2.92) | 90.153*** (3.35) |

Behind1 | 29.315* (2.16) | 28.962** (2.20) | 25.855** (2.23) |

Behind2 | 25.749 (1.05) | 26.547 (1.12) | 9.303 (0.48) |

Waybehind | 86.392 (1.39) | 91.844 (1.52) | 82.856* (1.98) |

AttPct | 13.570 (0.78) | 17.717 (1.05) | 1.355 (0.10) |

NATV | -6.084 (0.64) | -16.877* (1.90) | -1.556 (0.20) |

Ref1 | 129.894 (3.69)*** | 127.051*** (3.74) | 92.877*** (4.18) |

Ref6 | 55.325* (1.84) | 55.285* (1.90) | 41.023* (1.74) |

Ref12 | 91.497** (2.61) | 92.949*** (2.76) | 23.224 (1.07) |

Ref22 | -32.403* (1.74) | -31.936* (1.78) | -25.008 (1.60) |

Ref23 | -134.458*** (2.43) | -138.055*** (2.58) | -130.069*** (3.94) |

Ref28 | 64.318** (2.15) | 63.625** (2.20) | 73.532*** (3.43) |

Ref30 | -59.915* (1.99) | -60.103** (2.06) | -33.765*** (3.43) |

Sigma1 | 76.899 | 72.263 | 115.606 |

Sigma2 | ------- | ------- | 43.570 |

Mixing weight | ------- | ------- | 0.273 |

-lnL | ------- | 527.183 | 508.723 |

- 1
- Figures in parentheses are absolute values of t-ratios based on robust standard errors.
- 2
- Estimated coefficients in the grouped data regression model and in the partially adaptive regression model are interpreted just like their OLS counterparts; that is, they are marginal effects.
- ***
- Significant at the α = 0.01 level, two-tailed test.
- **
- Significant at the α = 0.05 level, two-tailed test.
- *
- Significant at the α = 0.10 level, two-tailed test.

^{6}The “nationally televised” effect is not statistically significant in 2012, indicating that differences in the other independent variables in the model accounted for the spread. That is not the case in 2011, as the effect is statistically significant, ceteris paribus. One possible explanation for the finding is that 2011 is the year that the MLS contract with FOX ended, and that some longer matches might have put MLS in a better negotiating position against FOX and/or NBC.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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^{1}All partially adaptive models are estimated using a program written by the authors in the IML language in SAS that is available from the authors upon request.^{2}Alternatively, one could estimate a mixed model but the software does not presently exist to estimate a partially adaptive interval-censored mixed model.^{3}We are grateful to an anonymous referee for this suggestion.^{4}Following a suggestion of a referee, we also tested for home bias by replacing Behind1, Behind2, and Waybehind with the interaction terms AttPct*Behind1, AttPct*Behind2, and AttPct*Waybehind in our models for 2011 and 2012. Our conclusions regarding the absence of home bias in 2011 and the presence of home bias in 2012 were unchanged.^{5}Given that the law governing stoppage time allows referees to account for “wast[ed] time” during regulation play, it is possible that numerous excessive goal celebrations could account for some of the observed stoppage time in a given soccer contest. To account for this, we constructed and added the total number of second-half goals scored to the set of regressors from our most general grouped-data regression model. For 2011, the estimated coefficient on this regressor is 1.05, with a t-ratio of 0.27. For the 2012, the estimated coefficient is −6.08 with a t-ratio of 1.29 (absolute value). Thus, the variable is not significant.^{6}Following the suggestion of a referee, we pooled the 2011 and 2012 data and then divided the sample into nationally televised and not nationally televised matches. We returned to our large model (1), dropped NATV and added a year dummy variable for 2012. For the televised games we found no evidence of a home field advantage. Although the Behind1 coefficient was marginally significant, the “home” group was not (χ^{2}(3) = 3.62 with p-value = 0.3057). For the matches not nationally televised we found some evidence of a home field advantage, surprisingly based on the strength of a large positive and statistically significant Waybehind coefficient. Testing our “home” group here gave a χ^{2}(3) = 7.69 with a p-value of 0.0528. This is consistent with our prior belief that close bias is more important for televised games and home bias is more important for not-televised games.

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## Share and Cite

**MDPI and ACS Style**

Yewell, K.G.; Caudill, S.B.; Mixon, Jr., F.G. Referee Bias and Stoppage Time in Major League Soccer: A Partially Adaptive Approach. *Econometrics* **2014**, *2*, 1-19.
https://doi.org/10.3390/econometrics2010001

**AMA Style**

Yewell KG, Caudill SB, Mixon, Jr. FG. Referee Bias and Stoppage Time in Major League Soccer: A Partially Adaptive Approach. *Econometrics*. 2014; 2(1):1-19.
https://doi.org/10.3390/econometrics2010001

**Chicago/Turabian Style**

Yewell, Katherine G., Steven B. Caudill, and Franklin G. Mixon, Jr. 2014. "Referee Bias and Stoppage Time in Major League Soccer: A Partially Adaptive Approach" *Econometrics* 2, no. 1: 1-19.
https://doi.org/10.3390/econometrics2010001