A Country-Level Efficiency Analysis of the 2016 Summer Olympic Games in Rio: A Complete Picture.

Corral, Julio Del ; Gomez-Gonzalez, Carlos ; Sanchez-Santos, Jose Manuel 等

Introduction

The Olympic Games is viewed as the most prestigious international sporting event due to its history, tradition, global impact, and universal participation. This competition gathers outstanding athletes from more than 200 countries and consists of both team and individual sports. The economics of the Olympic Games has developed a research line that focuses on the efficiencies of the participating nations. In this context, the definition of efficiency is the relationship between the observed output-level and the ideal (or potential) output-level (Greene, 2008), which can be obtained using frontier methods (i.e., DEA and stochastic frontier models). Lozano, Villa, Guerrero, and Cortes (2002) are pioneers in this strand of the literature. (1)

Previous research has used the number of medals or diplomas to determine success and has incorporated the relative value of medals into the analyses (Churilov & Flitman, 2006; Lozano et al., 2002; Zhang, Li, Meng, & Liu, 2009). Nevertheless, the computation of medals, weighted points (e.g., 3-gold; 2-silver; 1-bronze), or diplomas might not be the most appropriate system to define success in the Olympic Games. The qualification of athletes for the Games is a great success by itself for many participating countries, because they can enhance their international image. However, very few studies use the number of participants as the outcome in the Olympic Games. Johnson and Ali (2004) included the number of participants as a dependent variable to identify economic and political factors that determine success. Similarly, Li, Lei, Dai, and Liang (2015) also considered the number of participating athletes as the output in a previous process (the athletes' preparation stage) for competition in the Olympic Games. This paper contributes to this literature by incorporating the number of participating athletes as an output in stochastic frontier models that analyze the efficiency of the participating nations.

The competition in the Summer Olympic Games includes both individual and team sports. Team sports require the participation of more than one athlete, and there are big differences regarding the number of competing athletes (e.g., 18 athletes in field hockey, 12 athletes in volleyball, and two athletes in tennis doubles). Thus, team sports that require a higher number of athletes to compete force the participating countries to prepare more athletes. To consider the difference in the number of athletes required in individual and team sports, we included a weighted number of medals as the output, in which the weight is positively correlated to the number of participants. Thus, this study contributes to the literature by incorporating the importance of medals won in individual and team sports into the analyses of efficiency in the Olympic Games.

Finally, to the best of our knowledge, countries with zero medals are typically excluded from efficiency analyses in the literature because their efficiency is always zero. However, not all countries with zero medals have the same economic resources. Therefore, in this paper, we implement a measure that differentiates the performance of countries with zero medals. For this purpose, we use a probit model to calculate the probabilities of obtaining at least one medal in Rio 2016. Our methodology follows the line of del Corral, Maroto, and Gallardo (2015) and van Ours and van Tujil (2016) that calculate the efficiency of managers as comparing the actual performance with a predicted performance.

Thus, on the one hand, the aim of this paper is to examine the efficiencies of the competing nations in the 2016 Summer Olympics by incorporating the value of participation and medals in team and individual sports. For this purpose, stochastic frontier models are used. The population size of the competing nations and the gross domestic product (GDP) are the inputs, while the number of participants, medals, and weighted medals are the chosen outputs. On the other hand, this paper aims to calculate the efficiency of countries with zero medals using an expected performance measure extracted from betting odds.

Literature Review

The Measurement of performance and rankings in the Olympic Games has attracted the attention of academicians in the fields of economics and operations research. Most of the contributions in this field of study can be classified into two groups: 1) research has analyzed the determinants of medal success in the Olympic Games (explanation and forecasting); and 2) a wide range of papers has evaluated the relative efficiencies of the nations participating in the Olympics.

With regard to the first group, since the contribution of den Butter and van der Tak (1995), a large body of literature, which investigates factors determining performance in the Olympics, has emerged. There is a certain consensus that GDP, population, and the host country's status are the most consistent predictors of Olympic success. However, the list of determinants of success in the Olympic Games is much longer, and includes political regimes (Bernard & Busse, 2004; Rathke & Woitek, 2008), public expenditure on recreation (Forrest, Sanz, & Tena, 2010), expenditure on health (Moosa & Smith, 2004), climatic factors (Hoffmann, Ging, & Ramasamy, 2004; Johnson & Ali, 2004), life expectancy and education (Lui & Suen, 2008), macroeconomic indicators other than GDP (Vagenas & Vlachokyriakou, 2012), poverty and income distribution (Mitchell & Stewart, 2007), regions' cultural traits (Andreff, Andreff, & Poupaux, 2008; Otamendi & Doncel, 2014), geographic situations (Noland & Stahler, 2016; Tcha & Pershin, 2003), sporting traditions (Stamm & Lamprecht, 1999), number of athletes (Moosa & Smith, 2004), prior performance (Celik & Gius, 2014), infrastructures (Condon, Golden, & Wasil, 1999), religion (De Bosscher, Heyndels, De Knop, van Bottenburg, & Shibli, 2008), and previous hosting (Hoffmann, Ging, & Ramasamy, 2002).

In addition, due to the need for understanding the performance of nations in absolute terms, researchers are concerned about the methods to measure achievements in the Olympics effectively and fairly. With the aim of designing an objective system of analysis for the Olympic Games, the nonparametric data envelopment analysis (DEA) model has become increasingly popular. In particular, Lozano et al. (2002) pioneered the application of this method to evaluate the relative efficiencies of participants in the Olympics.

In DEA models, every participating nation is treated as a decision-making unit (DMU), which produces multiple outputs with a certain number of inputs. In this framework, the first problem is that different outputs could be used. Some of the most frequently used in the literature are the total number of medals and the number of medals won on a per capita basis.

Moreover, lexicographic orders, in which winning a gold medal is preferred to winning a silver medal, which in turn is preferred to winning a bronze medal, have been used (i.e., Lins, Gomes, Soares de Mello, & Soares de Mello, 2003; Lozano et al. 2002; Zhang et al., 2009). The aggregation of the outputs "number of medals" as a single indicator can be obtained by means of a weighted sum in which the weights are the measure of the importance of each type of medal. The main difficulty is that the weights involve value judgements about the relative value of medals (a problem of preferences).

Other studies have suggested that participating countries should be grouped into four categories: low income, lower/middle income, upper/middle income, and high income (Li, Liang, Chen, & Morita, 2008). They justified this classification on the grounds that nations may value gold, silver, and bronze medals differently.

Issues that are more controversial have also been addressed using modified versions of DEA models. For example, Lins et al. (2003) showed the need to consider the limited number of medals available to be won. Another problem is that some outputs, such as the number of medals, are integers (Wu, Zhou, & Liang, 2010).

In addition to the aforementioned methodological challenges, the literature has also identified several biases to be corrected. For instance, some evidence suggests that participating nations may have different performance levels in the Summer Olympics and in the Winter Olympics. If Summer and Winter Games are not considered together, assuming similar levels of performance for the participating nations might constitute a bias. Soares de Mello, Angulo-Meza, and Da Silva (2009) provided a ranking using a DEA approach that included the results of both games combined. Lei, Li, Xie, and Liang (2015) also considered the Summer and Winter Olympic Games and their results suggest that the majority of nations have different performance levels in each of them, which are consistent with their geographical features.

More recently, a key issue that has emerged when assessing the performance of nations in the Olympic Games is the distinction between the stages of athlete preparation and athlete competition (Li et al., 2015). The output in the athletes' preparation stage (i.e., the number of participating athletes) is also defined as an intermediate measure that links both stages. In the athlete competition stage, the number of participating athletes is used as the input to produce three final outputs (namely the numbers of gold, silver, and bronze medals). Li et al. (2015) developed several models to measure the efficiencies of nations in these two individual stages. They found that the efficiency in the athlete participation stage is usually higher than it is in the athlete competition stage.

Most of the literature analyzing the efficiency of nations participating in the Olympic Games is based on non-parametric approaches. However, there have also been some attempts to estimate a frontier production function using parametric methods. Rathke and Woitek (2008) used stochastic frontier analysis to identify the determinants of performance differences in the Olympic Games.

To summarize, conventional DEA models and modifications are considered as adequate approaches to measure the efficiency of nations participating in the Olympic Games. Research on the issue offers a set of alternative and complementary measures to achieve results that are more reliable with regard to success and efficiency in the Games. There is no doubt about the useful contributions of the aforementioned studies in two directions. First, the methods can be used as measuring tools for participating countries to identify significant biases towards some of the participants (Churilov & Flitman, 2006). Second, they help to evaluate the performance of nations in comparison to their potential, and enable them to determine realistic goals for future Olympic Games (Wu et al., 2010).

Methodology

Stochastic Frontier Models

A production function is the maximum output attainable given a set of inputs (Greene, 2008). Thus, to estimate a production function, a frontier model must be used. There are two main alternative methods to estimate frontier production functions: DEA (2) and stochastic frontier models. DEA are non-parametric and deterministic. (3) Conversely, stochastic frontier models (4) are parametric and stochastic. (5) Parametric refers to the functional form that needs to be assumed. The most common are the Cobb-Douglas and the translog. Stochastic means that some countries could be above the frontier, which reduces the influence of outliers on the results.

In this paper, we prefer to rely on stochastic frontier methodology because some countries could be labeled as outliers due to missing inputs such as genetics. Moreover, the parametric nature of this methodology allows hypothesis testing, which could be particularly relevant in order to test the input selection.

A stochastic frontier production function can be written as follows:

y=f(x)*exp(e); [epsilon]=v-u

where y represents output, x is a vector of inputs, f(x) represents the technology, and e is a composed error term. The component v captures noise and other stochastic shocks entering into the definition of the frontier (e.g., luck, referees' decisions, and so on), which is assumed to follow a normal distribution centered at zero. The component u is assumed to follow a one-sided distribution, capturing the inefficiency relative to the stochastic frontier. In this paper, it was tested which distribution fits better to the data among half-normal, truncated normal, and exponential by using Wald tests. Based on the assumption that the two components are independent of each other, maximum likelihood estimates can be obtained (Aigner, Lovell, & Schmidt, 1977). In so doing, it is important to set the parameter [lambda]=[[sigma].sub.u]/[[sigma].sub.v], where [[sigma].sub.u] is the standard deviation of the u component, and [[sigma].sub.v] is the standard deviation of the v component. If [[sigma].sub.v] is statistically equal to zero, the model collapses to a determinist model in which all the deviations from the frontier are due to inefficiency while, if [[sigma].sub.u] is statistically equal to zero, there is no inefficiency component. Therefore, if either [lambda]=0 or [lambda] is very large, the empirical application of the model will be problematic and hence the decision is not to use models under these circumstances.

Once the production frontier has been estimated, a technical efficiency index can be computed as the ratio between actual output and potential output. Actual output is f(x) * exp(v-u) while potential output is f(x) * exp(v). Therefore, technical efficiency can be computed as exp(-u), which is specifically calculated as E(exp(-u)|[epsilon]).

Moreover, one of the assumptions of production frontiers is the monotonicity of inputs, which implies that the output is positively related to the levels of each of the inputs. Hence, if any input shows a negative elasticity, such a model is rejected. Lastly, the Cobb-Douglas was tested against the translog using Wald tests for all models.

Performance of Zero-Medal Countries

Frontier methods cannot provide any information on countries with zero level of output (zero medals) other than their efficiency is zero. Thus, in order to get a complete list of the performance of countries participating in the Olympic Games alternative methodologies should be used. In this line, Pieper, Nuesch, and Franck (2014), del Corral et al. (2015), van Ours and van Tujil (2016), and Humphreys, Paul, and Weinbach (2016) proposed to calculate the efficiency of managers as comparing the actual performance with the performance predicted by bookmakers. In this framework, the probability for a country to obtain at least one medal can be estimated using a probit model. After doing so, an "efficiency index" can be created by calculating efficiency=1-p, where p is the predicted probability from the probit model. In this way, countries expected to obtain at least one medal will have lower efficiency values than countries with lower probability of obtaining medals.

Data

Data from the 206 countries participating in the Rio 2016 Summer Olympic Games were obtained from both the International Olympic Committee (IOC) website and the World Bank database. From the IOC website, we collected the number of participants, the number of medals, and the sports in which the medals were awarded. From the World Bank database, (6) we obtained the two main inputs identified in the literature, namely GDP and population. (7)

With regard to the number of participants, it is important to note the following. First, countries face a small maximum number of entries in all individual sports even if they have more potential qualified athletes in the rankings, (8) and therefore, some bias can exist in the analysis. Second, the host country (in this case, Brazil), is invited to participate in all team and individual sports. Hence, the host country is expected to participate with a huge delegation. Another important fact is that several Olympic spots are allotted on a continent basis. This is especially important in Oceania, where Australia and New Zealand are the only countries with a great development in most Olympic sports. Therefore, their likelihood to compete with a large delegation is higher than for other developed countries from other continents. Therefore, their efficiencies regarding the number of participants have to be cautiously interpreted. Moreover, this fact implies that a continent-by-continent analysis could be useful when analyzing the number of participants. Due to space limitations, this analysis is only included at the European level. (9) Finally, it is important to acknowledge that occasionally some countries have refused the participation of some qualified athletes or team squads. (10)

Thus far, the literature has acknowledged that a gold medal is not the same as a bronze medal by incorporating some weights. (11) However, to the best of our knowledge, there is no paper that considers a gold medal to be more valuable in some team sports (e.g., handball or basketball) than in some individual sports. (12) A greater number of influential players is needed to obtain a medal in team sports (instead of only one star). In order to create an alternative output to the number of medals, we have used the following formula: (13)

[mathematical expression not reproducible]

where [n.sub.ij] indicates the number of athletes for participant i from country j, and [d.sub.ij] is a dummy variable that takes the value one if participant ij has obtained at least one medal and zero otherwise. Table 1 shows the team-weighted values of medals for each possible number of participants, as well as some examples.

With regard to the GDP, there are two clear outliers: China and the United States. Dealing with outliers is not straightforward in econometrics, and no clear procedure is preferred. In this study, we believe that the benefit of calculating the efficiency of these two countries is lower than the cost of (possibly) biasing the estimates from the rest of the sample. Hence, we have decided to exclude them from the sample. (14)

Results

Number of Participants

The preferred models for the number of participants are the Cobb-Douglas that includes only GDP using the truncated normal in the world analysis and the Cobb-Douglas that includes both inputs using the exponential distribution for inefficiency in Europe. Figure 1 shows the relationship between the number of participants and GDP for all countries. This figure also includes the stochastic frontier production function, the OTS quadratic regression, and its associated R-squared.

The R-squared for the entire sample is 0.73, which indicates the large and positive relationship between these two variables. Table 3 shows the estimates for the preferred models while Table 4 shows the efficiencies associated with these preferred models.

The Cobb-Douglas estimates show decreasing returns to scale technologies, since the sum of the coefficients are statistically lower to one in both models ([chi square]=374.26 and [chi square]=77.99, respectively). The most efficient countries in the world are Fiji, Cook Islands, Ukraine, and Serbia, while in the European model the most efficient countries are Montenegro, Hungary, and Slovenia. If the analyzed sample consisted of the 15 countries with the highest GDP, the most efficient countries would be Australia, Brazil, and Spain. A possible explanation for the high efficiency of Australia is that the process of qualification in Oceania might be easier than in other continents. An explanation for the high efficiency of Brazil is that it was the host country, which involves direct participation in all sports competitions. However, there is no specific reason that explains Spain's success other than the organization of the sporting system.

Medals

Three different models are estimated with regard to the number of medals. The first one uses the number of medals. The second model uses a weighted measure of the number of medals, in which a gold medal is rated as three, a silver as two, and a bronze as one (labeled as 3-2-1 weighted medals). The last model includes the team-weighted number of medals, which is described in detail in the data section. As in the number of participants' model for the world, the population was rejected as the input when using the models of the three alternative medal outputs. Figure 2 shows the relationship between the number of medals and GDP for countries with at least one medal, since countries with no medal were excluded from the estimates. Finally, Figure 3 shows the relationship between the 3-2-1 weighted number of medals and GDP.

The R-squared values are 0.64 and 0.62 for the number of medals and the 3-2-1 weighted medals, respectively, which is lower than it is for the participation models. Two countries are clearly located above the frontier, namely Russia and the United Kingdom, while India is clearly located below the frontier. Table 5 shows the preferred estimates for the number of medals and the 3-2-1 weighted medals, while Table 6 shows the efficiencies associated with these models.

The results show that the production functions show statistically significant decreasing returns to scale. In this case, the elasticities are close to 0.35. The most efficient countries in the model using the number of medals are Azerbaijan, Jamaica, Russia, the United Kingdom, and Kenya. In the 3-2-1 weighted model, Azerbaijan changed from the first to the fifth position.

Tables 7 and 8 show the preferred model of the team weighted medals and the associated efficiencies, respectively. A very similar list is obtained.

Zero-Medal Countries

The inputs used in this model are the same as those used in the stochastic frontier models (i.e., GDP and population). However, population is not considered in the final model due to the hypothesis testing results.

As expected, the GDP is positively correlated with the likelihood of obtaining a medal. Table 10 shows the efficiencies computed using this method. The results show that the higher the GDP, the lower the efficiency. Thus, most of the countries hold efficiency values in the range of 0.70 and 0.79. However, a few countries with higher GDP hold efficiency values below 0.40. This finding is interesting and has managerial implications for countries participating in the Olympic Games that do not obtain medals but still actively participate in the event.

Discussion

The rankings that result from the implementation of several methods to assess countries' efficiency show significant differences, even when applied to the same Olympic Games. For example, in the Athens 2004 Olympic Games, the United States appeared fourth in the ranking of Zhang et al. (2009) and 64th in Azizi and Wang's (2013) ranking. In Beijing 2008, China appeared 54th in Chiang, Hwang, and Liu's (2011) ranking and third in Calzada-Infante and Lozano's (2016). With regard to Rio 2016, as this is one of the first research papers, the results cannot yet be compared. However, we may use Li et al. (2015) as an immediate reference. The authors analyzed the overall efficiency in London 2012, which included both participation and competition results. The results of this comparison need to be considered very cautiously due to many influential factors. However, we see countries such as Jamaica, Russia, Kenya, and the United Kingdom appearing in high-ranking positions. Moreover, it is also worth noting the case of Montenegro, which was ranked eighth by Li et al. (2015) and fifth in our ranking of efficiency in terms of participation. However, this country is not listed in any of the medal rankings, as Montenegrin athletes did not win any medals in Rio 2016. This example reveals the need for rankings of efficiency that consider not only the results of the competition, but also participation.

This study uses several outputs, which are included in the models of efficiency. The outputs used to determine countries' efficiencies are the number of participants, the number of medals, 3-2-1 weighted medals, and team-weighted medals. When exploring the relationship among the efficiencies, positive and large correlation coefficients (above 0.97) were found for the number of medals and its two weighted systems. However, correlation coefficients decrease (0.63-0.71) if the efficiencies considering the number of participants are compared to the other outputs. This is an interesting finding, and supports the idea of using the number of participants to calculate the efficiency of countries as it provides complementary information to the number of medal.

Finally, more research is needed to further develop these indicators in several directions. First, the number of athletes that a sport requires also affects the total number of participants for countries. For example, as a football (soccer) team will always increase the number of participants by 18 athletes, these 18 entries should be considered differently to 18 entries in athletics (track and field). Therefore, the use of an alternative weighting system that considers the number of participants with respect to the number of participants in each sport can help to develop the analyses of efficiency (15) Second, in this paper we implement an objective measure to weight the number of participants, but it has some limitations. Does it have the same value for a country that earns a medal in the hammer throw than in the 100-meter dash? The answer is probably that a medal in the 100 meters has more value than the medal in the hammer throw, as the worldwide publicity/prestige for the country and the implications for the athlete are very different. Hence, the development of a methodology able to capture this difference is very interesting. Finally, the literature on countries' efficiency in the Olympic Games has neglected to identify the reasons why some countries are more efficient than others, and more importantly the steps for countries to follow in order to become more efficient. In this sense some questions need to be answered: Is it important to have a relevant university sport system? Is it relevant to have foreign coaches? Is it essential to have state-of-the-art sports facilities? The methodology used in Simar and Wilson (2007) or similar frameworks can help to further develop these issues.

Conclusions

The purpose of this study was to get a complete picture about the countries' efficiency in the Rio 2016 Olympic Games. Due to numerous factors that affect the ultimate success and achievement of medals, we consider the participation of athletes to be an important goal for the countries competing in the Olympic Games. Changes in the countries' efficiency rankings were found when the number of participants was included as the output in the models compared to the efficiencies calculated by including the number of medals. Moreover, the high correlation coefficients found between the outputs often used in the literature (i.e., number of medals and its medal-weighted systems) decreased when compared to the number of participants.

This study also contributes to the literature by incorporating the value of medals won in team and individual sports. Team sports require more than one athlete to compete. Thus, countries need to produce a greater number of professional athletes to qualify for international meetings. An effort has been made in this study to include this effect in the analysis of countries' efficiency in the Olympic Games.

The countries that do not obtain medals in the Olympic Games are always excluded from efficiency analyses, as the frontier methods (DEA and stochastic frontier models) do not consider them. In this analysis, we use the predicted probability of obtaining at least one medal to provide these countries with values of efficiency Although many of these countries show similar results, differences do exist. Thus, this analysis has implications for zero-medal countries that do not have the same economic resources.

Furthermore, we believe that some team sports (e.g., basketball) might have a greater impact on nations' prestige than other individual (e.g., archery) and team sports (e.g., canoe double). Hence, further research should place an emphasis on determining the relative importance of sports and disciplines when assessing the final performance of countries. Table A1. Estimates of Participation in All Models World Model 1 Model 2 Model 3 Constant -7.186 (***) 2.978 (***) -7.702 (***) GDP 0.481 (***) 0.494 (***) 0.593 (***) GDP2 0.079 (***) Population -0.142 (***) Population2 GDP- Population [[sigma].sub.u] 0.930 (***) 0.413 0.876 [[sigma].sub.v] 0.377 0.809 (***) 0.387 [gamma] 2.467 (***) 0.511 2.264 Log-likelihood -262.954 -255.397 -258.332 Number of 204 204 204 observations Distribution TN HN TN of u Preferred Yes No No World Europe Model 4 Model 1 Model 2 Constant 4.639 (***) -8.460 (***) 4.911 (***) GDP 0.564 (***) 0.524 (***) 0.531 (***) GDP2 -0.047 -0.063 Population -0.122 (***) Population2 -0.210 (***) GDP- 0.149 (***) Population [[sigma].sub.u] 0.759 0.935 (***) 0.85 [[sigma].sub.v] 0.337 0.402 (***) 0.355 [gamma] 2.252 2.394 (***) 1.584 Log-likelihood -244.686 -51.376 -50.394 Number of 204 50 50 observations Distribution TN HN TN of u Preferred No No No Europe Model 3 Model 4 Constant -6.643 (***) 4.732 (***) GDP 0.166 (*) 0.070 (***) GDP2 -0.117 (***) Population 0.450 (***) 0.484 (***) Population2 -0.213 (***) GDP- 0.164 (***) Population [[sigma].sub.u] 0.488 (***) 0.655 [[sigma].sub.v] 0.308 (***) 1.21E-08 [gamma] 2.467 (***) 5.42E+07 Log-likelihood -38.953 -28.873 Number of 50 50 observations Distribution E E of u Preferred Yes No Note: (*) p<0.10, (**) p<0.05, (***) p<0.01. In parenthesis standard errors. TN-truncated normal, HN-half normal, E-exponential. The efficiencies associated with these models can be requested from the authors by email. Table A2. Estimates of Medals in All Models Number of medals Model 1 Model 2 Model 3 Constant -6.319 (***) 2.364 (***) -6.604 (***) GDP 0.361 (***) 0.257 (**) 0.318 (***) GDP2 0.051 Population 0.084 Population2 GDP. Population [[sigma].sub.u] 1.726 (***) 1.734 (***) 1.717 (***) [[sigma].sub.v] 0.294 (***) 0.274 (***) 0.294 (***) [lambda] 5.869 (***) 6.338 (***) 5.847 (***) Log-likelihood -118.503 -117.992 -118.100 Number of 84 84 84 observations Distribution HN HN HN of u Preferred Yes No No Number of medals 3-2-Iweighted medals Model 4 Model 1 Model 2 Constant 1.225 -5.215 (***) 3.090 (***) GDP -0.002 0.353 (***) 0.201 GDP2 -0.005 0.077 Population 0.259 (**) Population2 -0.504 (***) GDP. 0.259 (**) Population [[sigma].sub.u] 0.071 1.472 (***) 1.922 (***) [[sigma].sub.v] 0.921 (***) 0.242 0.290 (***) [lambda] 0.077 6.083 (***) 6.639 (***) Log-likelihood -112.371 -126.073 -125.996 Number of 84 84 84 observations Distribution HN TN HN of u Preferred No Yes No 3-2-Iweighted medals Model 3 Model 4 Constant -5.361 (***) 3.186 (***) GDP 0.316 (***) -0.018 GDP2 0.007 Population 0.065 0.323 (***) Population2 -0.431 (***) GDP. 0.194 (**) Population [[sigma].sub.u] 1.490 (***) 1.730 (***) [[sigma].sub.v] 0.255 0.354 (***) [lambda] 5.843 (***) 4.881 (***) Log-likelihood -125.875 -121.06 Number of 84 84 observations Distribution TN HN of u Preferred No No Note:(*) p<0.10, (**) p<0.05, (***) p<0.01. In parenthesis standard errors. TN-truncated normal, HN-half normal, E-exponential. The efficiencies associated with these models can be requested from the authors by email. Table A3. Estimates of Team Weighted Medals Model 1 Model 2 Model 3 Constant -6.644 (***) 2.479 (***) -6.785 (***) GDP 0.378 (***) 0.239 (*) 0.358 (***) GDP2 0.066 Population 0.039 Population2 GDP * Population [sigma]_u 1.729 (***) 1.764 (***) 1.723 (***) [sigma]_v 0.321 (***) 0.267 (*) 0.325 (***) [lambda] 5.395 (***) 6.597 (***) 5.298 (***) Log-likelihood -119.71 -118.974 -119.631 Number of obser- 84 84 84 vations Distribution of u HN HN HN Preferred Yes No No Model 4 Constant 1.387 GDP 0.016 GDP2 -0.01 Population 0.226 (*) Population2 -0.513 (***) GDP * Population 0.272 (**) [sigma]_u 0.153 [sigma]_v 0.934 (***) [lambda] 0.163 Log-likelihood -113.864 Number of obser- 84 vations Distribution of u HN Preferred No Note: (***) p<0.01, (**) p<0.05, (*) p<0.1. In parenthesis standard errors. The efficiencies associated with these models can be requested from the authors by email.

References

Aigner, D., Lovell, C. K., & Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6, 21-37.

Alvarez, A., & Arias, C. (2014). A selection of relevant issues in applied stochastic frontier analysis. Economics and Business Letters, 3, 3-11.

Andreff, M., Andreff, W., & Poupaux, S. (2008). Les determinants economiques de la performace olympiques: Prevision des medailles qui seront gagnees aux Jeux de Pekin. Revue d'Economie Politique, 118, 135-169.

Azizi, H., & Wang, Y. M. (2013). Improved DEA models for measuring interval efficiencies of decision-making units. Measurement, 46, 1325-1332.

Bernard, A. B., & Busse, M. R. (2004). Who wins the Olympic Games: Economic resources and medals totals. The Review of Economics and Statistics, 86, 413-417.

Calzada-Infante, L., & Lozano, S. (2016). Analysing Olympic Games through dominance networks. Physica A: Statistical Mechanics and its Applications, 462, 1215-1230.

Celik, O. B., & Gius, M. (2014). Estimating the determinants of Summer Olympic Game Performance. International Journal of Applied Economics, 11, 39-47.

Chiang, C. I., Hwang, M. J., & Liu, Y. H. (2011). Determining a common set of weights in a DEA problem using a separation vector. Mathematical and Computer Modelling, 54, 2464-2470.

Churilov, L., & Flitman, A. (2006). Towards fair ranking of Olympics achievements: The case of Sydney 2000. Computer and Operations Research, 33, 2057-2082.

Condon, E. M., Golden, B. L., & Wasil, E. A. (1999). Predicting the success of nations at the Summer Olympics using neural networks. Computers and Operations Research, 26, 1243-1265.

Cooper, W W, Seiford, L. M., & Zhu, J. (2011). Handbook on data envelopment analysis. New York, NY: Springer Science and Business Media.

de Bosscher, V., Heyndels, B., de Knop, P., van Bottenburg, M., & Shibli, S. (2008). The paradox of measuring success of nations in elite sport. Belgeo. Revue Beige de Geographic, 2, 217-234.

del Corral, J., Maroto, A., & Gallardo, A. (2015). Are former professional athletes and native better coaches? Evidence from Spanish Basketball. Journal of Sports Economics, doi: 527002515595266.

den Butter, E A., &van der Tak, C. M. (1995). Olympic medals as an indicator of social welfare. Social Indicators Research, 35, 27-37.

Forrest, D., Sanz, I., & Tena, J. D. (2010). Forecasting national team medal totals at the Summer Olympic Games. International Journal of Forecasting, 26, 576-588.

Greene, W H. (2008). The econometric approach to efficiency analysis. In H. Fried, K. Lovell, & S. Schmidt (Eds.), The Measurement of productive efficiency and productivity growth (pp. 92-250). Oxford, UK: Oxford University Press.

Hoffmann, R., Ging, L. G, & Ramasamy, B. (2002). Public policy and Olympic success. Applied Economics Letters, 9, 545-548.

Hoffmann, R., Ging, L. G, & Ramasamy, B. (2004). Olympic success and ASEAN countries. Economic analysis and policy implications. Journal of Sports Economics, 5, 262-276. Hossain, M. K., Kamil, A. A., Baten, M. A., & Mustafa, A. (2012). Stochastic frontier approach and data envelopment analysis to total factor productivity and efficiency Measurement of Bangladeshi rice. Plos One, 7(10), e46081.

Humphreys, B. R., Paul, R. J., & Weinbach, A. P. (2016). Performance expectations and the tenure of head coaches: Evidence from NCAA football. Research in Economics, 70, 482-492.

Johnson, D., & Ali, A. (2004). A tale of two seasons: Participation and medal counts at the Summer and Winter Olympic Games. Social Science Quarterly, 85, 974-993.

Kumbhakar, S. C, Wang, H., & Horncastle, A. P. (2015). A practitioner's guide to stochastic frontier analysis using Stata. New York, NY: Cambridge University Press.

Lei, X., Li, Y, Xie, Q., & Liang, L. (2015). Measuring Olympics achievements based on a parallel DEA approach. Annals of Operations Research, 226, 379-396.

Li, Y, Liang, L., Chen, Y, & Morita, H. (2008). Models for measuring and benchmarking Olympics achievements. Omega, 36, 933-940.

Li, Y, Lei, X., Dai, Q., & Liang, L. (2015). Performance evaluation of participating nations at the 2012 London Summer Olympics by a two-stage data envelopment analysis. European Journal of Operational Research, 243, 964-973.

Lins, M. P. E, Gomes, E G., Soares de Mello, J. C. C, & Soares de Mello, A. J. R. (2003). Olympic ranking based on a zero sum gains DEA model. European Journal of Operational Research, 148, 312-322.

Lozano, S., Villa, G., Guerrero, E, & Cortes, P. (2002). Measuring the performance of nations at the Summer Olympics using data envelopment analysis. Journal of the Operational Research Society, 53, 501-511.

Lui, H. K., & Suen, W (2008). Men, money and medals: An econometric analysis of the Olympic Games. Pacific Economic Review, 13, 1-16.

Mitchell, H., & Stewart, E (2007). A competitive index for international sport. Applied Economics, 39, 587-603.

Moosa, I. A., & Smith, L. (2004). Economic development indicators as determinants of medal winning at the Sydney Olympics: An extreme bounds analysis. Australian Economic Papers, 43,288-301.

Noland, M., & Stahler, K. (2016). Asian participation and performance at the Olympic Games. Asian Economic Policy Review, 11, 70-90.

Otamendi, J., & Doncel, L. M. (2014). Medal shares in Winter Olympic Games by sport: Socioeconomic analysis after Vancouver 2010. Social Science Quarterly, 95, 598-614.

Pieper, J., Nuesch, S., & Franck, E. (2014). How performance expectations affect managerial replacement decisions. Schmalenbach Business Review: ZFBF, 66, 5-23.

Rathke, A., & Woitek, U. (2008). Economics and the Summer Olympics: An efficiency analysis. Journal of Sports Economics, 9, 520-537.

Simar, L., & Wilson, P. W (2007). Estimation and inference in two-stage, semi-parametric models of production processes. Journal of Econometrics, 136, 31-64.

Soares de Mello, J. C. C. B., Angulo-Meza, L., & Da Silva, B. P. B. (2009). A ranking for the Olympic Games with unitary input DEA models. IMA Journal of Management Mathematics, 20, 201-211.

Stamm, H., & Lamprecht, M. (1999). Sports organisations in Switzerland. In K. Heinemann (Ed.), Sports clubs in various European countries (pp. 119-142). Schorndorf, Germany: Hofmann.

Tcha, M., & Pershin, V (2003). Reconsidering performance at the Summer Olympics and revealed comparative advantage. Journal of Sports Economics, 4, 216-239.

Vagenas, G., & Vlachokyriakou, E. (2012). Olympic medals and demo-economic factors: Novel predictors, the ex-host effect, the exact role of team size, and the "population-GDP" model revisited. Sport Management Review, 15, 211-217.

van Ours, J. G, & van Tuijl, M. A. (2016). In-season head-coach dismissals and the performance of professional football teams. Economic Inquiry, 54, 591-604.

Wen, M. (2015). Uncertain data envelopment analysis. Berlin-Heidelberg, Germany: Springer-Verlag.

Wu, J., Zhou, Z. X., & Liang, L. (2010). Measuring the performance of nations at Beijing Summer Olympics using integer-valued DEA model. Journal of Sports Economics, 11, 549-566.

Zhang, D., Li, X., Meng, W, & Liu, W. (2009). Measuring the performance of nations at the Olympic Games using DEA models with different preferences. Journal of the Operational Research Society, 60, 983-990.

Endnotes

(1) A more detailed compendium of this body of literature is presented in the Literature Review section.

(2) It is usually referred as DEA. For a detailed overview, please see Cooper, Seiford, and Zhu (2011).

(3) In recent years, several papers have attempted to alleviate the deterministic nature of this methodology. Wen (2015) presented the milestones in the progression of uncertain DEA.

(4) For overviews of the topic, please see Alvarez and Arias (2014), and Kumbhakar, Wang, and Horncastle(2015).

(5) Hossain, Kamil, Baten and Mustafa (2012) explained fairly well the different strengths and weaknesses of both competing approaches: "The main advantage of DEA is that it does not require any information more than input and output quantities. The efficiency is measured relative to the highest observed performance rather than an average. However, a DEA-based estimate is sensitive to Measurement errors or other noise in the data because DEA is deterministic and attributes all deviations from the frontier to inefficiencies. The strength of SFA is that it considers stochastic noise in data and also allows for the statistical testing of hypotheses concerning production structure and degree of inefficiency. Its main weaknesses are that it requires an explicit imposition of a particular parametric functional form representing the underlying technology and also an explicit distributional assumption for the inefficiency terms."

(6) Data for a few countries were not available from the World Bank database. In such cases, the data were obtained using alternative sources.

(7) It is important to note that the inputs for the production frontier for Olympic success should be money invested in sports competitions and people practicing sport. However, no reliable data exist, and GDP and population were used as proxies. In so doing, it was assumed that countries enjoy sports, especially competitive sports, in the same way.

(8) China in table tennis, Spain in male tennis, USA in 100 meters, and Kenya in 3,000 meters steeplechase are some examples of this issue.

(9) Results from the other continents are available upon request.

(10) For instance, South Africa refused to participate in female hockey and some other countries establish additional requirements than those established by the IOC. This is especially relevant in athletics and swimming.

(11) For instance, Lui and Suen (2008), and Mitchell and Stewart (2007) used the 3-2-1 weights for gold, silver, and bronze, while Moosa and Smith (2004) used 0.6-0.3-0.1.

(12) We thank a participant in the 2016 European Conference on Sports Economics for pointing out this issue.

(13) Other possibilities could have also been employed but this one produces sensible and objective differences among the number of participants.

(14) Once again, estimates and results are available upon request.

(15) This idea for future research was introduced by a referee in the peer-review process.

Julio del Corral (1), Carlos Gomez-Gonzalez (1), and Jose Manuel Sanchez-Santos (2)

(1) University of Castilla-La Mancha

(2) University of A Coruna

Julio del Corral, PhD, is an associate professor of economics in the Department of Economics and Finance. His current research focuses on the economics of sport, specifically the topics of demand, competitive balance, productivity, discrimination, and betting.

Carlos Gomez-Gonzalez is a PhD candidate in the Department of Economic Analysis and Finance. He is also a researcher in Group IGOID. His research interests include sport economics and management, with a focus on consumer behavior and discrimination.

Jose Manuel Sanchez-Santos, PhD, is an associate professor of economics in the Department of Economics. His research is developed mainly in the fields of socioeconomics and sport economics and it is particularly focused on social capital, subjective wellbeing, and sport participation.

Authors' Note

The authors would like to thank Daniel Solis for his valuable comments and Maria Moraga, who was not able to work on the paper due to her time-demanding master's in economics program, but her undergraduate thesis analyzing the efficiency in London 2012 was really important for this paper. Table 1. Team Weighting Values Number of athletes Weighted value Example 1 1.00 100 meters 2 1.30 Tennis doubles 3 1.48 Cycling track team sprint 4 1.60 Swimming relays 5 1.70 Rhythmic gymnastics 9 1.95 Synchronized swimming 12 2.08 Volleyball 13 2.11 Waterpolo 15 2.18 Handball 16 2.20 Field hockey 18 2.26 Football Table 2. Descriptive Statistics Obs. Mean Standard Minimum deviation Participants 204 51.24 87.25 1 Gold medals 204 1.15 3.20 0 Silver medals 204 1.24 3.08 0 Bronze medals 204 1.45 3.38 0 Total medals 204 3.83 9.14 0 3-2-1 weighted medals 204 7.36 18.27 0 Team weighted medals 204 4.39 10.77 0 GDP (US dollars 2014) 204 2.21E+11 5.50E+11 3.79E+07 Population (inhabitants) 204 2.77E+07 9.81E+07 9,916 Maximum Participants 470 Gold medals 27 Silver medals 23 Bronze medals 21 Total medals 67 3-2-1 weighted medals 144 Team weighted medals 81.69 GDP (US dollars 2014) 4.12E+12 Population (inhabitants) 1.31E+09 Note: Obs.- number of observations. Table 3. Estimates of Preferred Models for Participation World Europe Constant -7.186 (***) -6.643 (***) GDP 0.481 (***) 0.166 (*) Population 0.450 (***) [[sigma].sub.u] 0.930 (***) 0.488 (***) [[sigma].sub.v] 0.377 0.308 (***) [lambda] 2.467 (***) 2.467 (***) Log-likelihood -262.954 -38.953 Number of observations 204 50 Distribution of u Truncated normal Exponential Note: (*) p<0.10, (**) p>0.05, (***) p>0.01. Table 4. Countries' Efficiency in Participation Country Region Code World Position in top 15 GDP Fiji Oceania FJI 0.82 (1) Cook Islands Oceania COO 0.82 (2) Ukraine Europa UKR 0.80 (3) Serbia Europa SRB 0.76 (4) Belarus Europa BLR 0.75 (5) Montenegro Europa MNE 0.75 (6) New Zealand Oceania NZL 0.73 (7) Jamaica America JAM 0.73 (8) Hungary Europa HUN 0.72 (9) Cuba America CUB 0.71 (10) Croatia Europa HRV 0.67 (11) Australia Oceania AUS 0.66 (12) 1 Mongolia Asia MNG 0.66 (13) Brazil America BRA 0.65 (14) 2 Poland Europa POL 0.65 (15) Georgia Europa GEO 0.60 (16) Lithuania Europa LTU 0.60 (17) Marshall Islands Oceania MHL 0.60 (18) Bahamas, The America BHS 0.60 (19) Kenya Africa KEN 0.59 (20) Tonga Oceania TON 0.58 (21) Argentina America ARC 0.58 (22) Slovenia Europa SVN 0.57 (23) Armenia Europa ARM 0.57 (24) Estonia Europa EST 0.57 (25) Spain Europa ESP 0.57 (26) 3 Netherlands Europa NLD 0.56 (27) Tunisia Africa TUN 0.56 (28) Colombia America COL 0.55 (29) France Europa FRA 0.54 (30) 4 Moldova Europa MDA 0.53 (31) Korea, Dem. People's Rep. Asia PRK 0.53 (32) Russian Federation Europa RUS 0.53 (33) 5 Canada America CAN 0.53 (34) 6 Uzbekistan Asia UZB 0.53 (35) Samoa Oceania WSM 0.53 (36) Palau Oceania PLW 0.53 (37) South Africa Africa ZAF 0.51 (38) Zimbabwe Africa ZWE 0.51 (39) Micronesia, Fed. Sts. Oceania FSM 0.51 (40) Germany Europa DEU 0.51 (41) 7 Czech Republic Europa CZE 0.50 (42) Kazakhstan Asia KAZ 0.50 (43) Italy Europa ITA 0.50 (44) 8 Seychelles Africa SYC 0.49 (45) Azerbaijan Europa AZE 0.49 (46) United Kingdom Europa GBR 0.48 (47) 9 Denmark Europa DNK 0.48 (48) Romania Europa ROU 0.48 (49) Antigua and Barbuda America ATG 0.47 (50) Sweden Europa SWE 0.47 (51) Bulgaria Europa BGR 0.47 (52) Egypt, Arab Rep. Africa EGY 0.46 (53) Kiribati Oceania KIR 0.46 (54) Portugal Europa PRT 0.45 (55) Greece Europa GRC 0.45 (56) Kyrgyz Republic Europa KGZ 0.44 (57) St. Kitts and Nevis America KNA 0.44 (58) Grenada America GRD 0.43 (59) Latvia Europa LVA 0.43 (60) Eritrea Africa ERI 0.41 (61) Bahrain Africa BHR 0.41 (62) Trinidad and Tobago America TTO 0.40 (63) Korea, Rep. Asia KOR 0.40 (64) 10 Honduras America HND 0.40 (65) Senegal Africa SEN 0.39 (66) Japan Asia JPN 0.39 (67) 11 Slovak Republic Europa SVK 0.38 (68) Djibouti Africa DJI 0.36 (69) Belgium Europa BEL 0.36 (70) Morocco Africa MAR 0.35 (71) Algeria Africa DZA 0.35 (72) Ireland Europa IRL 0.35 (73) Lesotho Africa LSO 0.35 (74) Barbados America BRB 0.34 (75) Ethiopia Africa ETH 0.34 (76) Burundi Africa BDI 0.34 (77) Sao Tome and Principe Africa STP 0.33 (78) Virgin Islands (US) America VIR 0.32 (79) Comoros Africa COM 0.32 (80) Venezuela, RB America VEN 0.32 (81) Nauru Oceania NRU 0.32 (82) Cayman Islands America CYM 0.32 (83) Central African Republic Africa CAF 0.32 (84) Guinea-Bissau Africa GNB 0.32 (85) Tuvalu Oceania TUV 0.31 (86) American Samoa Oceania ASM 0.31 (87) St. Vincent and the Grenadines America VCT 0.30 (88) Cameroon Africa CMR 0.30 (89) Switzerland Europa CHE 0.30 (90) Aruba America ABW 0.29 (91) Puerto Rico America PRI 0.29 (92) Vanuatu Oceania VUT 0.29 (93) Turkey Europa TUR 0.29 (94) Uganda Africa UGA 0.29 (95) British Virgin Islands America VGB 0.29 (96) Gambia, The Africa GMB 0.29 (97) Mexico America MEX 0.28 (98) 12 St. Lucia America LCA 0.28 (99) Austria Europa AUT 0.28 (100) Ecuador America ECU 0.28 (101) Finland Europa FIN 0.27 (102) Cabo Verde Africa CPV 0.26 (103) Congo, Rep. Africa COG 0.26 (104) Cyprus Europa CYP 0.26 (105) Nigeria Africa NGA 0.26 (106) Mauritius Africa MUS 0.25 (107) San Marino Europa SMR 0.25 (108) Dominican Republic America DOM 0.24 (109) Norway Europa NOR 0.24 (110) Bermuda America BMU 0.24 (111) Haiti America HTI 0.24 (112) Iran, Islamic Rep. Asia IRN 0.24 (113) Guyana America GUY 0.24 (114) Kosovo Europa KSV 0.23 (115) Qatar Asia QAT 0.22 (116) Israel Europa ISR 0.22 (117) Namibia Africa NAM 0.21 (118) India Asia IND 0.21 (119) 13 Guam America GUM 0.21 (120) Botswana Africa BWA 0.21 (121) Thailand Asia THA 0.21 (122) Chile America CHL 0.21 (123) Bosnia and Herzegovina Europa BIH 0.20 (124) Guatemala America GTM 0.20 (125) Andorra Europa AND 0.20 (126) Angola Africa AGO 0.20 (127) Suriname America SUR 0.20 (128) Solomon Islands Oceania SLB 0.20 (129) Ghana Africa GHA 0.20 (130) Dominica America DMA 0.19 (131) Tajikistan Asia TJK 0.18 (132) Togo Africa TGO 0.18 (133) Rwanda Africa RWA 0.18 (134) Timor-Leste Asia TLS 0.18 (135) Uruguay America URY 0.18 (136) Hong Kong SAR, China Asia HKG 0.17 (137) Malta Africa MLT 0.17 (138) Peru America PER 0.17 (139) Maldives Asia MDV 0.17 (140) Belize America BLZ 0.17 (141) Cote d'lvoire Africa CIV 0.16 (142) Paraguay America PRY 0.16 (143) Bolivia America BOL 0.16 (144) Benin Africa BEN 0.16 (145) Malaysia Asia MYS 0.15 (146) Iceland Europa ISL 0.15 (147) Papua New Guinea Oceania PNG 0.15 (148) Malawi Africa MWI 0.15 (149) Guinea Africa GIN 0.15 (150) Madagascar Africa MDG 0.15 (151) Macedonia, FYR Europa MKD 0.15 (152) China Taipei Asia TPE 0.14 (153) 14 Sierra Leone Africa SLE 0.14 (154) Niger Africa NER 0.14 (155) Iraq Africa IRQ 0.14 (156) Albania Europa ALB 0.14 (157) Vietnam Asia VNM 0.14 (158) Costa Rica America CRI 0.14 (159) Lao PDR Asia LAO 0.13 (160) West Bank and Gaza Africa PSE 0.13 (161) Mali Africa MLI 0.13 (162) El Salvador America SLV 0.13 (163) Gabon Africa GAB 0.13 (164) Mozambique Africa MOZ 0.12 (165) Singapore Asia SGP 0.12 (166) Nepal Asia NPL 0.12 (167) Zambia Africa ZMB 0.12 (168) Turkmenistan Asia TKM 0.12 (169) Burkina Faso Africa BFA 0.12 (170) Panama America PAN 0.11 (171) Cambodia Asia KHM 0.11 (172) Nicaragua America NIC 0.11 (173) Bhutan Asia BTN 0.11 (174) Luxembourg Europa LUX 0.11 (175) Liberia Africa LBR 0.11 (176) Lebanon Africa LBN 0.11 (177) Jordan Africa JOR 0.11 (178) Libya Africa LBY 0.11 (179) Liechtenstein Europa LIE 0.10 (180) Monaco Europa MCO 0.10 (181) Syrian Arab Republic Africa SYR 0.09 (182) Tanzania Africa TZA 0.09 (183) Indonesia Asia IDN 0.09 (184) 15 Sri Lanka Asia LKA 0.09 (185) South Sudan Africa SSD 0.08 (186) Swaziland Africa SWZ 0.08 (187) Myanmar Asia MMR 0.08 (188) Mauritania Africa MRT 0.07 (189) Philippines Asia PHL 0.07 (190) Somalia Africa SOM 0.07 (191) Brunei Darussalam Asia BRN 0.07 (192) United Arab Emirates Africa ARE 0.06 (193) Afghanistan Asia AFG 0.06 (194) Congo, Dem. Rep. Africa COD 0.06 (195) Yemen, Rep. Africa YEM 0.06 (196) Sudan Africa SDN 0.06 (197) Equatorial Guinea Africa GNQ 0.06 (198) Chad Africa TCD 0.05 (199) Saudi Arabia Africa SAU 0.05 (200) Bangladesh Asia BGD 0.05 (201) Oman Africa OMN 0.05 (202) Pakistan Asia PAK 0.04 (203) Guinea Ecuatorial Africa GUE 0.04 (204) Country Europe Fiji Cook Islands Ukraine 0.81 (13) Serbia 0.87 (6) Belarus 0.86 (7) Montenegro 0.91 (1) New Zealand Jamaica Hungary 0.88 (2) Cuba Croatia 0.87 (5) Australia Mongolia Brazil Poland 0.80 (14) Georgia 0.75 (22) Lithuania 0.86 (9) Marshall Islands Bahamas, The Kenya Tonga Argentina Slovenia 0.87 (3) Armenia 0.72 (28) Estonia 0.87 (4) Spain 0.80 (15) Netherlands 0.86 (8) Tunisia Colombia France 0.80 (16) Moldova 0.58 (39) Korea, Dem. People's Rep. Russian Federation 0.58 (40) Canada Uzbekistan Samoa Palau South Africa Zimbabwe Micronesia, Fed. Sts. Germany 0.77 (18) Czech Republic 0.77 (19) Kazakhstan Italy 0.75 (23) Seychelles Azerbaijan 0.62 (34) United Kingdom 0.77 (20) Denmark 0.86 (10) Romania 0.64 (33) Antigua and Barbuda Sweden 0.84 (12) Bulgaria 0.65 (31) Egypt, Arab Rep. Kiribati Portugal 0.73 (24) Greece 0.72 (27) Kyrgyz Republic 0.37 (45) St. Kitts and Nevis Grenada Latvia 0.75 (21) Eritrea Bahrain Trinidad and Tobago Korea, Rep. Honduras Senegal Japan Slovak Republic 0.66 (30) Djibouti Belgium 0.71 (29) Morocco Algeria Ireland 0.78 (17) Lesotho Barbados Ethiopia Burundi Sao Tome and Principe Virgin Islands (US) Comoros Venezuela, RB Nauru Cayman Islands Central African Republic Guinea-Bissau Tuvalu American Samoa St. Vincent and the Grenadines Cameroon Switzerland 0.73 (26) Aruba Puerto Rico Vanuatu Turkey 0.31 (46) Uganda British Virgin Islands Gambia, The Mexico St. Lucia Austria 0.61 (36) Ecuador Finland 0.62 (35) Cabo Verde Congo, Rep. Cyprus 0.56 (41) Nigeria Mauritius San Marino 0.85 (11) Dominican Republic Norway 0.65 (32) Bermuda Haiti Iran, Islamic Rep. Guyana Kosovo 0.28 (47) Qatar Israel 0.45 (43) Namibia India Guam Botswana Thailand Chile Bosnia and Herzegovina 0.24 (48) Guatemala Andorra 0.73 (25) Angola Suriname Solomon Islands Ghana Dominica Tajikistan Togo Rwanda Timor-Leste Uruguay Hong Kong SAR, China Malta Peru Maldives Belize Cote d'lvoire Paraguay Bolivia Benin Malaysia Iceland 0.51 (42) Papua New Guinea Malawi Guinea Madagascar Macedonia, FYR 0.19 (49) China Taipei Sierra Leone Niger Iraq Albania 0.16 (50) Vietnam Costa Rica Lao PDR West Bank and Gaza Mali El Salvador Gabon Mozambique Singapore Nepal Zambia Turkmenistan Burkina Faso Panama Cambodia Nicaragua Bhutan Luxembourg 0.41 (44) Liberia Lebanon Jordan Libya Liechtenstein 0.6 (37) Monaco 0.59 (38) Syrian Arab Republic Tanzania Indonesia Sri Lanka South Sudan Swaziland Myanmar Mauritania Philippines Somalia Brunei Darussalam United Arab Emirates Afghanistan Congo, Dem. Rep. Yemen, Rep. Sudan Equatorial Guinea Chad Saudi Arabia Bangladesh Oman Pakistan Guinea Ecuatorial Note: Ranking of countries in parentheses. The region refers to the continent to which each Olympic National Committee belongs. Table 5. Estimates of Preferred Models for Number and 3-2-1 Weighted Medals Number of medals 3-2-1 weighted medals Constant -6.319 (***) -5.215 (***) GDP 0.361 (***) 0.353 (***) [[sigma].sub.u] 1.726 (***) 1.472 (***) [[sigma].sub.v] 0.294 (***) 0.242 [lambda] 5.869 (***) 6.083 (***) Log-likelihood -118.503 -126.073 Number of observations 84 84 Distribution of u Half normal Truncated normal Note: (*) p<0.10, (**) p<0.05, (***) p<0.01 Table 6. Countries' Efficiencies in Number of Medals and 3-2-1 Weighted Medals Country Eff. number Eff. 3-2-1 Country of medals weighted medals Azerbaijan 0.86 (1) 0.75 (5) Tunisia Jamaica 0.86 (2) 0.88 (1) Romania Russia 0.86 (3) 0.85 (2) Turkey United Kingdom 0.84 (4) 0.84 (3) Bulgaria Kenya 0.78 (5) 0.78 (4) Thailand Uzbekistan 0.77 (6) 0.63 (10) Switzerland Korea, Rep. 0.77 (7) 0.68 (8) Burundi New Zealand 0.77 (8) 0.68 (9) Belgium Georgia 0.75 (9) 0.58 (14) Malaysia Hungary 0.75 (10) 0.73 (6) Fiji Kazakhstan 0.74 (11) 0.55 (16) Cote d'Ivoire France 0.73 (12) 0.62 (11) Bahrain Croatia 0.72 (13) 0.69 (7) Kosovo Cuba 0.69 (14) 0.61 (12) Moldova Germany 0.68 (15) 0.60 (13) Niger Serbia 0.67 (16) 0.55 (15) Tajikistan Ukraine 0.67 (17) 0.51 (18) Norway Australia 0.67 (18) 0.55 (17) Argentina Belarus 0.66 (19) 0.46 (23) Mexico Japan 0.64 (20) 0.48 (22) Egypt Denmark 0.61 (21) 0.42 (24) Venezuela, RB Italy 0.60 (22) 0.49 (20) Estonia Ethiopia 0.58 (23) 0.35 (27) Algeria Netherlands 0.57 (24) 0.50 (19) Trinidad and Tobago Armenia 0.56 (25) 0.49 (21) Vietnam Canada 0.51 (26) 0.31 (30) Ireland Korea, Rep. 0.51 (27) 0.41 (25) Jordan Czech Republic 0.50 (28) 0.28 (32) Indonesia Spain 0.44 (29) 0.36 (26) Israel Brazil 0.43 (30) 0.35 (28) China Taipei South Africa 0.42 (31) 0.33 (29) Dominican Republic Poland 0.40 (32) 0.26 (36) Morocco Sweden 0.39 (33) 0.30 (31) Puerto Rico Lithuania 0.35 (34) 0.17 (48) Qatar Slovenia 0.35 (35) 0.27 (34) Portugal Colombia 0.35 (36) 0.27 (33) Finland Grenada 0.34 (37) 0.26 (38) Philippines Bahamas, The 0.31 (38) 0.23 (39) Singapore Iran, Isl. Rep. 0.30 (39) 0.23 (41) India Greece 0.30 (40) 0.26 (37) United Arab Emirates Mongolia 0.28 (41) 0.16 (52) Austria Slovak Republic 0.27 (42) 0.26 (35) Nigeria Table 7. Estimates of Preferred Models for Team Weighted Medals Constant -6.644 (***) GDP 0.378 (***) [[sigma].sub.u] 1.729 (***) [[sigma].sub.v] 0.321 (***) [lambda] 5.395 (***) Log-likelihood -119.71 Number of observations 84 Note: (*) p<0.10, (**) p<0.05, (***) p<0.01 Table 8. Countries' Efficiency in Team Weighted Number of Medals Country Efficiency Jamaica 0.87 (1) Russia 0.85 (2) United Kingdom 0.84 (3) Azerbaijan 0.84 (4) Serbia 0.80 (5) New Zealand 0.77 (6) Kenya 0.74 (7) Korea, Dem. Peoples Rep. 0.74 (8) Hungary 0.74 (9) Croatia 0.74 (10) Uzbekistan 0.73 (11) Georgia 0.72 (12) Germany 0.71 (13) France 0.71 (14) Australia 0.69 (15) Kazakhstan 0.69 (16) Denmark 0.66 (17) Ukraine 0.65 (18) Cuba 0.65 (19) Belarus 0.64 (20) Italy 0.61 (21) Japan 0.61 (22) Netherlands 0.60 (23) Ethiopia 0.54 (24) Canada 0.54 (25) Armenia 0.53 (26) Bulgaria 0.29 (43) Iran, Islamic Rep. 0.27 (44) Mongolia 0.27 (45) Tunisia 0.25 (46) Belgium 0.24 (47) Malaysia 0.23 (48) Turkey 0.23 (49) Burundi 0.22 (50) Thailand 0.21 (51) Switzerland 0.21 (52) Norway 0.21 (53) Cote d'Ivoire 0.19 (54) Bahrain 0.18 (55) Argentina 0.17 (56) Kosovo 0.17 (57) Moldova 0.17 (58) Niger 0.16 (59) Tajikistan 0.16 (60) Trinidad and Tobago 0.16 (61) Mexico 0.12 (62) Egypt, Arab Rep. 0.12 (63) Venezuela, RB 0.11 (64) Estonia 0.11 (65) Ireland 0.10 (66) Algeria 0.10 (67) Vietnam 0.10 (68) Czech Republic 0.51 (27) Korea, Rep. 0.48 (28) Spain 0.46 (29) Brazil 0.44 (30) South Africa 0.43 (31) Lithuania 0.40 (32) Poland 0.40 (33) Fiji 0.40 (34) Sweden 0.39 (35) Bahamas, The 0.38 (36) Grenada 0.34 (37) Romania 0.32 (38) Slovenia 0.32 (39) Colombia 0.31 (40) Slovak Republic 0.31 (41) Greece 0.29 (42) Indonesia 0.09 (69) Jordan 0.09 (70) China Taipei 0.09 (71) Israel 0.08 (72) Nigeria 0.08 (73) Dominican Republic 0.07 (74) Morocco 0.06 (75) Puerto Rico 0.06 (76) Qatar 0.05 (77) Austria 0.05 (78) Portugal 0.05 (79) Finland 0.05 (80) Philippines 0.04 (81) Singapore 0.04 (82) India 0.04 (83) United Arab Emirates 0.04 (84) Table 9. Probit Estimates. Dependent Variable: 1 if the Country Obtains a Medal in Rio 2016. Variable Coefficient Constant 0.817 (***) GDP 5.51e-12 (***) Log-Likelihood -98.11 Observations 204 Pseudo-R2 0.29 Table 10. "Efficiency Scores" of Countries with Zero Medals Country Efficiency Saudi Arabia 0.00 Hong Kong SAR, China 0.19 Pakistan 0.25 Chile 0.31 Bangladesh 0.40 Peru 0.40 Iraq 0.46 Angola 0.60 Ecuador 0.60 Sudan 0.64 Sri Lanka 0.64 Oman 0.67 Myanmar 0.68 Guatemala 0.68 Luxembourg 0.69 Uruguay 0.70 Panama 0.70 Costa Rica 0.70 Lebanon 0.71 Tanzania 0.72 Syrian Arab Republic 0.72 Ghana 0.73 Turkmenistan 0.73 Yemen, Rep. 0.73 Congo, Dem. Rep. 0.73 Bolivia 0.74 Cameroon 0.74 Libya 0.74 Paraguay 0.75 Latvia 0.75 Uganda 0.75 El Salvador 0.75 Guinea Ecuatorial 0.76 Zambia 0.76 Nepal 0.76 Honduras 0.76 Equatorial Guinea 0.78 South Sudan 0.78 Haiti 0.78 Congo, Rep. 0.78 Benin 0.78 Rwanda 0.78 Guinea 0.78 Kyrgyz Republic 0.78 Malawi 0.78 Monaco 0.78 Somalia 0.78 Bermuda 0.78 Liechtenstein 0.78 Mauritania 0.78 Suriname 0.79 Sierra Leone 0.79 Barbados 0.79 Swaziland 0.79 Togo 0.79 Montenegro 0.79 Eritrea 0.79 Andorra 0.79 Guyana 0.79 Maldives 0.79 Guam 0.79 Aruba 0.79 Lesotho 0.79 Liberia 0.79 Virgin Islands (US) 0.79 Bhutan 0.79 San Marino 0.79 Belize 0.79 Cabo Verde 0.79 Djibouti 0.79 Central African Republic 0.79 Seychelles 0.79 Cyprus 0.76 Afghanistan 0.76 Cambodia 0.76 Papua New Guinea 0.77 Iceland 0.77 Bosnia and Herzegovina 0.77 Brunei Darussalam 0.77 Mozambique 0.77 Botswana 0.77 Gabon 0.77 Zimbabwe 0.77 Senegal 0.77 Mali 0.77 Nicaragua 0.77 West Bank and Gaza 0.77 Lao PDR 0.77 Namibia 0.77 Mauritius 0.77 Albania 0.77 Burkina Faso 0.78 Chad 0.78 Macedonia, FYR 0.78 Madagascar 0.78 Malta 0.78 St. Lucia 0.79 Timor-Leste 0.79 Antigua and Barbuda 0.79 Solomon Islands 0.79 Guinea-Bissau 0.79 Cayman Islands 0.79 St. Kitts and Nevis 0.79 Gambia, The 0.79 British Virgin Islands 0.79 Vanuatu 0.79 Samoa 0.79 St. Vincent and the Grenadines 0.79 American Samoa 0.79 Comoros 0.79 Dominica 0.79 Tonga 0.79 Sao Tome and Principe 0.79 Micronesia, Fed. Sts. 0.79 Palau 0.79 Marshall Islands 0.79 Nauru 0.79 Kiribati 0.79 Cook Islands 0.79 Tuvalu 0.79