On the importance of fixed effects over a short period of time when using sports data: a lesson from home advantage in alpine skiing

ABSTRACT
 Research question The aim of this paper is to compare the effect of competing in a home country between a racer fixed effects model, which assumes a constant ability over the span of a racer’s career, and a racer per season fixed effects model, which has a more plausible assumption that ability may vary between the seasons. Research methods We use data on all 50,046 performances among men and 44,311 performances among women from all the top competitions in alpine skiing that took place in the seasons from 2001–2002 to 2017–2018. We use fixed effects estimations in which we compare the performance of racers when they compete in their home country to performance of the same racers when they compete abroad. Results and findings When using the racer fixed effects model, we find no relationship between competing at home and the likelihood of failing to finish the first run for both genders. However, when using the racer per season fixed effects model, we find that racers have a significantly higher probability of failing to complete the first run when competing abroad. In addition, we find that competing at home has a positive effect on racers’ performance in terms of World Cup points. Implications Given the widespread use of fixed effects models in sports-related studies, as well as in other fields, this paper illustrates the sensitivity of results to the length of the fixed effects. This should help practitioners and scholars to better understand the underlying assumptions that relate to athletes’ abilities over time.


Introduction
When testing different effects among individual athletes, it is necessary to take athletes' abilities into account. In this paper, we emphasize the importance of choosing the correct measure of ability in order to test a well-established home advantage phenomenon.
According to Courneya and Carron (1992), home advantage is 'the consistent finding that home teams in sports competitions win over 50% of the games played under a balanced home and away schedule ' (p. 13). However, it is not a trivial task to study the effect that competing at home has on the performance of an individual athlete in a non-balanced international home-and-away schedule. For example, a naïve approach of correlating a dummy variable of competing in a home country with the performance measure will yield biased estimates. One possible reason for that is that competitions may take place more frequently in countries where the sport is more popular. This means that local athletes are, on average, more skilled than athletes from other countries. Another reason is that an individual's unobserved ability is likely to affect athletes' performance.
One way to control for individual abilities is by using athletes' recent rankings, as has been done in many studies of individual sports, such as tennis (e.g. Koning, 2011), judo (Krumer, 2017), and boxing (Balmer et al., 2005). In those studies, the authors controlled for athletes' abilities by using individual rankings that were based on athletes' individual performance over the period of one year (tennis), two years (judo), or their entire career (boxing). However, very rich sports data allow us to follow the same athletes over time and compare their performance when competing at home or abroad, as has been applied, for example, in studies of speed skating (Koning, 2005), ski jumping (Krumer et al., 2021), and skeleton (Chun & Park, 2021). This type of analysis, which is widely known as the fixed effects model, controls for all time-invariant differences between the individuals. In other words, the fixed effects model makes it possible to compare the same athlete's performance in different situations, assuming that his/her ability does not differ between the cases.
In fact, the fixed effects model is highly popular in sports-related studies that use sports data to test different phenomena other than home advantage (Foellmi et al., 2016;Genakos & Pagliero, 2012;Genakos et al., 2015;Harb-Wu & Krumer, 2019;Sandberg, 2018;Toma, 2017;Zitzewitz, 2006, to mention a few). However, as stated above, the most relevant assumption is that this fixed effects model is helpful if the unobserved ability is constant over time. For example, Toma (2017) employed a fixed effects model on the level of a basketball player to investigate free-throw accuracy using data from the 2002-2013 seasons. Sandberg (2018) investigated possible strategic voting in dressage competitions by using fixed effects on the level of a rider over a period of five years. These (and many other studies) used fixed effects over a relatively long period of time, assuming that athletes' abilities are constant over the whole period of estimation. This seems to be a strong assumption because ability may vary over time due to different preparations between seasons, injuries, or a natural decrease in physical strength. 1 In addition, this assumption is not in line with the literature on peak performance in professional sports. For example, Dendir (2016) found that soccer players achieve their peak performance between the ages of 25 and 27. According to Berthelot et al. (2012), the peak performance in running appears at the age of 26, compared to 21 in swimming and 31 in chess.
Thus, it is reasonable to use shorter fixed effects if the data track the same athletes over several years because this method involves less restrictive assumptions about athletes' abilities. For example, an athlete per season fixed effects model has a more plausible assumption that ability may vary between the seasons. This approach was employed by Harb-Wu and Krumer (2019), who used a per season fixed effects model estimating data from 17 seasons in professional biathlon to study the effect of competing at home on shooting performance. Similarly, Krumer et al. (2021) investigated nationalistic bias in ski jumping by using judge and jumper per season fixed effects in their data, which included seven years of competitions. Moreover, Genakos and Pagliero (2012), as well as Genakos et al. (2015), employed athletes per competition fixed effects, taking advantage of the multi-stage nature of competitions in weightlifting and diving, respectively. 2 The present paper examines home advantage in alpine skiing. Two previous studies have investigated the effect of competing at home in this sport. The first is Bray and Carron (1993), who defined two levels of ability based on racers' previous World Cup points and found a home advantage within each level in terms of the race points. However, that study did not find home advantage in terms of the probability to complete a race. In the other study, Balmer et al. (2001) investigated the home advantage in alpine skiing as a part of a broader study of the effect of hosting Winter Olympic Games. That paper also referred to the importance of controlling for racers' abilities. For that, the authors compared the medal successes of each nation when either hosting or not hosting the Olympic Games. They found that hosting the Olympics had a positive and significant effect on the number of medals and points. Both papers attributed the home advantage in alpine skiing to familiarity with local conditions. We build on the efforts of Bray and Carron (1993) and Balmer et al. (2001) by using a different methodology, in which we compare the performance of the same racers at home and abroad. We investigate each gender separately and our measures of performance include the likelihood of winning a medal (being among the top 3), the World Cup points, and completing the race. In addition, we use a much larger dataset on alpine skiing competitions from all the World Cups, World Championships, and the Olympic Games for the seasons from 2001/2002 to 2017/2018.
As in other sports, it is not plausible to assume that racers' ability in alpine skiing remains constant over the years. The above-mentioned examples emphasize the need to use fixed effects on the level of an athlete over a short period of time. Therefore, beyond studying the effect of competing at home, our main contribution is to raise the importance of the length of the fixed effects model that is widely used in sports management and sports economics literature. For that, we compare the effects of competing at home when using the racer fixed effects model with a more plausible racer per season fixed effects model and even with racer per half a season model. While in some cases the results do not differ between the approaches in terms of the size and statistical significance, some others show a large difference. More specifically, when using racer fixed effects estimation, we find no significant effect of competing at home on the likelihood of failing to complete the first run for both genders. However, when using the racer per season and racer per half a season fixed effects, the size of the home effect is twice as large and significant at conventional levels, suggesting that racers have a significantly higher probability of failing to complete the first run abroad. In addition, we find that, in all the approaches, racers from both genders achieve a significantly larger number of the World Cup points when competing in their home country, which confirms the findings of previous studies regarding the existence of home advantage in professional alpine skiing.

Description of alpine skiing settings
The goal in professional alpine skiing is to complete a given course as quickly as possible. Races take place in four main events: downhill, super giant, giant slalom, and slalom. Downhill and super giant are considered speed events, with downhill having the greatest distances between posts and therefore allowing for the highest speed (Gilgien et al., 2015). Both downhill and super giant are carried out in one run. Giant slalom and slalom are considered technical events (FIS, 2020). The distance between posts in slalom is smaller, and therefore speeds are lower. Technical events are carried out in two runs on two different courses (FIS, 2020). The result is determined by adding the times of both runs. In World Cup races, only racers ranked first to 30th after the first run are allowed to participate in the second run. At the various World Championships, the best 30 or 60 after the first runand, in some years, even all the racers who completed the first runhave been allowed to participate in the second run. At the Winter Olympic Games, all racers who successfully complete their first run are allowed to participate in the second run.
The most important competitions in alpine skiing are the World Cup, the World Championships, and the Winter Olympic Games. The World Cup traditionally starts at the end of October and ends in the beginning of March with the World Cup final. There are 30-40 races per season for each gender, held at different locations worldwide with various combinations of events. The calendars for women and men are independent from each other, meaning that races are not held at the same time and place. The only exceptions are races in Soelden and Levi at the beginning of the season, the World Cup final, the Winter Olympic Games, and the World Championships.
At each World Cup race, athletes ranked first to 30th are rewarded with up to 100 World Cup points, depending on their result. At the end of a season, the athletes who win the most points in all events combined are awarded the large overall World Cup trophy. Winners of the individual events are awarded a small World Cup trophy. At the World Championships and the Winter Olympic Games, which take place every second and fourth year, respectively, the athletes do not receive World Cup points. However, athletes ranked first to third are awarded medals (gold, silver, and bronze). For each race, we have full information on racers' names and nationality, the event's date, location, and type of competition, whether the racer completed all the runs of the race correctly or not, and the final result.

Data and variables
The final dataset consists of 620 men's and 596 women's competitions. The data include performances of 1240 men and 978 female racers. Men's and women's competitions took place in 20 and 16 countries, respectively. Austria and Italy were the most frequent host countries for both genders (see Table 1 for the full list). After removing 1144 observations in which a racer did not start the specific race, our data consist of 50,046 performances among men and 44,311 among women from the two runs.

Variables and descriptive statistics
Our main variable of interest is Home, a dummy variable that receives the value of one if a racer competes in his/her home country. To estimate the possible effect of competing at home on performance in alpine skiing, we have a set of four outcome variables. The first two are dummy variables that receive the value of one if a racer did not finish (complete) the run in the first (DNF1) or in the second run (DNF2). In Table 2 we see that, among men, in 17.2 percent of cases a racer failed to complete the first run when competing in a home country compared to 20.4 percent in case of competing abroad. Among women, in 15 percent of cases a racer failed to complete the first run when competing in her home country compared to 16.1 percent when competing abroad. Note that, as described above, the number of racers in the second run is lower because downhill and super giant are carried out in one run, and because only racers ranked first to 30th after the first run are allowed to participate in the second run in giant slalom and slalom World Cup competitions. We see almost no gap in completing the second run at home or abroad for either gender.
The third outcome variable is a dummy variable that receives the value of one if a racer finished the competition in one of the top three positions, which makes him/her a medalist of a competition (gold, silver, or bronze medal for the first, second, or third places, respectively). We see a higher share of home racers who finish the competition on podium. Our last outcome variable is the number of the World Cup points that a racer achieved in the respective competition. These World Cup points are assigned only in the World Cup competitions, which account for 89.7 percent in our dataset (the other competitions are the Olympic Games and World Championships). For both genders, home racers achieve more World Cup points, on average, than racers from other countries. In addition, since the course is set by one of the coaches of the participating teams (except for the downhill competitions), we coded a dummy variable Compatriot Setter that receives the value of one if racer i competes on a course that was set by his/her compatriot and zero otherwise. Finally, the most observable competition for both genders is slalom, whereas the least frequent one is super giant. The two other competitions are downhill and giant slalom.

Estimation strategy
We are interested in learning the effect of competing in a home country on each one of the four measures of performance that were described in the previous section. A naïve approach of correlating a dummy variable of competing at home with performance will yield biased and inconsistent estimates. This is because racers who compete in their home country are from strong alpine skiing countries, whose racers are, on average, more skilled than racers who arrive from other countries where alpine skiing is less popular. There is also a concern regarding selection into treatment that are driven by the rules of the competitions. This is because, according to FIS (2021), a host country whose quota is fewer than six competitors may add more competitors and participate in each race, to a maximum of six competitors. In addition, an individual's unobserved ability is likely to affect a racer's performance. Furthermore, this individual ability may also vary over the years, due to different preparations between seasons, or injuries. Hence, we need to take the different sources of unobserved heterogeneity into account.
Our panel data follow the same racers over time, which allows us to use a fixed effects model that controls for all time-invariant differences between the individuals. The main advantage of using a fixed effects model is that it takes into account omitted variables under the condition that these variables are constant over time, which allows us to compare the performance of a racer at home to performance of the same racer abroad. The most important variable is ability, which is difficult to model. Thus, by using racer's fixed effects, we assume that the ability is constant. However, it is not plausible to assume that racers' ability is constant over the years. As many of the examples in the introduction show, it is too much to assume that the ability is constant over the seasons; it is more plausible to assume that ability is stable over short periods of time. Moreover, when using, racer per season fixed effects, for example, we assume that not only physical abilitywhich includes strength, speed, or endurancemay change over the years, but also psychological ability to cope with pressure. Moreover, it is also possible that home advantage-proneness may change over time.
In the following, we compare the results of using racer fixed effects (d i ) with shorter lengths of fixed effects such as two seasons, season and half a season (d ip ). Thus, using a fixed effects model, the most basic specification takes the following form: where Y it(p) is one of the four measures of performance, described in the previous section (DNF1, DNF2, Top 3, and World Cup points), that were achieved by racer i in competition/run t of period p (two seasons, season, or half a season, where applicable). 4 The variable Home it(p) is a dummy variable that receives the value of one if racer i competes in his/her home country. The variable Compatriot Setter it(p) is a dummy variable that receives the value of one if racer i competes on a course that was set by his/her compatriot. The model also includes fixed effects for racer (d i ) or racer per period (d ip ). Finally, m t is the specific competition's fixed effects, which allows us to control for all the features of this specific competition that were common for all participants, such as the number of spectators or capacity utilization, the general climate conditions in the area of competition, and the difficulty of the track on the specific day.

Results
Panel A of Table 3 presents the results with racer fixed effects estimation in men's competitions. In Column (1), we see that competing at home has no significant effect on the likelihood of failing to complete the first run. However, when using racer per season fixed effects, as presented in Panel B of Table 3, the results in Column (1) indicate that racers' likelihood of failing to complete the first run at home is 1.4 percentage points lower than when they compete abroad. This effect is significant at the 5 percent level. To put this result into perspective, while the overall rate of failure in the first run is 20 percent, competing at home increases the probability of finishing the run by 7 percent. When comparing the results of failing to complete the second run, which are given in Column (2) of Panels A and B, we find no significant effect for both approaches. We also find that, for both approaches, competing at home has a small positive and statistically significant effect on finishing the competition with a medal (Column 3). Finally, we find that, in the racer per season specification, racers achieve approximately 1.06 more World Cup points at home than abroad (Column 4 of Panel B), which is very similar to the racer fixed effects specification, according to which racers achieve approximately 1.07 more World Cup points at home than abroad (Column 4 of Panel A). This home advantage is likely to be attributed to the familiarity with local conditions, as was also proposed in Bray and Carron (1993) and Balmer et al. (2001). Table 4 presents the results of women's competition. First, as with the men's estimation, when using racer fixed effects over the whole period, we find no significant relationship between the home variable and the likelihood of failing to compete the first run (Column 1 of Panel A). However, when using a more plausible racer per season fixed effects (Panel B), the size of the home effect is more than twice as large as in Panel A, with p = 0.073. As in the men's case, in both approaches we find no significant effect of competing at home on the likelihood of not finishing the second run (Column 2). We also find that the sizes of the home effect on winning the medal are very similar for both approaches. The only difference is the statistical significance, when the coefficient of Notes: The outcome variable in Column 1 is Did Not Finish run 1. The outcome variable in Column 2 is Did Not Finish run 2.
The outcome variable in Column 3 is Finishing the race in Top 3. The outcome variable in Column 4 is the number of World Cup points that were achieved in the specific race. All regressions include unique competitions dummies. In addition, Columns 1 and 2 include a dummy variable of whether or not a respective run was set by a compatriot of the racer. Columns 3 and 4 include dummies of whether a compatriot of the racer set the course in the first or the second run. Standard errors clustered at racer level are in parentheses. **p < 0.05, *p < 0.1.
the home variable in Panel A is significant at the 10 percent level, whereas in Panel B the coefficient of the home variable has p = 0.123. In addition, we find that, in the racer per season specification, female racers achieve approximately 0.83 more World Cup points at home than abroad (Column 4 of Panel B), which is very similar to the racer fixed effects specification, according to which female racers achieve approximately 0.79 more World Cup points at home than abroad (Column 4 of Panel A). Finally, as a robustness check, we also estimate equation (1) by using racer per two seasons (Panel C) and racer per half a season (Panel D) fixed effects. In Table 3, we present the results for men's competitions. We see that the magnitude of the home effect on the likelihood of not completing the first run for the racer per two seasons specification (Column 1 of Panel C) is lower than for the racer per season specification (Panel B) and is only significant at the 10 percent level. However, when using racer per half a season fixed effects, we find that the results in Column (1) of Panel D (racer per half a season fixed effects) and Panel B (racer per season fixed effects) are very similar. All of the other results presented in Columns (2)-(4) do not vary greatly between the different specifications.
In the case of women, presented in Table 4, we again see that the results in Column (1) for racer per half a season fixed effects (Panel D) resemble the results for racer per season fixed effects (Panel B) in terms of both the magnitude and statistical significance. On the contrary, and similarly to the racer fixed effects (Panel A), the racer per two seasons fixed effects (Panel D) produces no significant effect of competing at home on the likelihood of failing to complete the first run. We find that the results in Columns (2)-(3) are very similar. However, the size of the home effect on the World Cup points is slightly The outcome variable in Column 3 is Finishing the race in Top 3. The outcome variable in Column 4 is the number of World Cup points that were achieved in the specific race. All regressions include unique competitions dummies. In addition, Columns 1 and 2 include a dummy variable of whether or not a respective run was set by a compatriot of the racer. Columns 3 and 4 include dummies of whether a compatriot of the racer set the course in the first or the second run. Standard errors clustered at racer level are in parentheses. **p < 0.05, *p < 0.1.
larger for the racer per half a season fixed effects (Column 4 of Panel D) than for the racer per two seasons fixed effects (Column 4 of Panel C), when the former is significant at the five percent level, whereas the latter is only significant at the 10 percent level. To sum up, our analyses illustrate that the definition of the length of the fixed effects may play a significant role in estimating different measures of performance.

Conclusion
In this paper, we estimated the effect of competing in a home country on performance in professional alpine skiing. Using data on 50,046 performances among men and 44,311 performances among women over a period of 17 seasons, our racer fixed effects estimation shows that competing in a home country has no significant effect on the likelihood of finishing the race. However, when using a more plausible racer per season fixed effects model, we find a significantly larger probability of completing the race. We also find home advantage in terms of World Cup points. Given the widespread use of fixed effects model in sports-related studies, the present paper illustrates the sensitivity of results to the length of the fixed effects. It adds to the growing attention in sports management literature given to various methodological issues that may arise in empirical studies emphasizing the vulnerability of results to the use of different empirical models (Gyimesi & Kehl, 2021;Wallrafen et al., 2020).
Such methodological papers should help practitioners in sports management to better understand the underlying assumptions of different empirical models. This is important because the process of policy making or a psychological feedback to athletes should be based on a solid methodology. The possible examples include the identification of referees' biases and psychological effects that should rely on a proper control for athletes' abilities. 5 The main takeaway from our paper is that future studies that use sports data avoid using fixed effects over a long period of time and instead define athlete/team per season fixed effects. This is because the latter relies on more plausible assumptions with regard to athletes' abilities.
Finally, our results are not exclusive for the sports management literature. For example, many papers on international trade and conflict resolution use country fixed effects over long periods of time (e.g. Bormann et al., 2017;Helpman et al., 2004). However, in many countries, governments change every four or five years, taking different political courses. Similarly, studies in finance use firms or advisors fixed effects over the entire period of observation, which may also be very long (Linnainmaa et al., 2021). However, given that firms' management is not constant over time, it is also worthwhile using shorter fixed effects rather than longer ones. The same applies in the literature on education and health, where school (Hanushek & Woessmann, 2011) or hospital (Ho, 2002) fixed effects are often applied over long periods. Notes 1. For example, Flørenes et al. (2009) interviewed 521 alpine athletes who reported 191 acute injuries. In 26.7 percent of the injuries, the athlete was absent 8-28 days from trainings and competitions. In 30.9 percent of the cases, an injury caused an absence of above 28 days. In addition, Stenroos and Handolin (2014) reported an average training pause of 26 weeks after an ACL (anterior cruciate ligament) injury, which according to Flørenes et al. (2009) is the most common injury type, and 17 weeks after a lower leg fracture. 2. Note that the issue of using fixed effects over a long period of time also in teams' sports was discussed in Krumer and Lechner (2017). The authors used data on national soccer teams during the period between 1996 and 2014, stating that their dataset did not even include a single active player in any national team who played in both the 2014 FIFA World Cup (last tournament in their dataset) and in the 1996 UEFA European Championship (the first tournament in their dataset). Therefore, the usage of team fixed effects over such a long period of time is also questionable in team sports. 3. Similarly, six-time Olympic medalist and five-time World Championships medalist Bode Miller from the USA, who was on the podium 79 times in World Cup races, won 16 medals in 39 races in the 2004/2005 season, but could only achieve five podiums out of 30 races in 2013/2014 season. Among women, six-time Olympic medalist and seven-time World Champion, Anja Paerson from Sweden, finished 16 out of her 34 races in the top 10, and also had six podiums in the 2009/2010 season. Two seasons later, however, she finished only three out of her 20 races in the top 10 and did not reach the podium once. 4. Note that one way to use prize/monetary related variables is to transform them to log function. However, in our case, more than 50 percent of the observations have zero World Cup points. One possible solution is to give very small values instead of zero. However, this is likely to create another problem. For example, the log[10] of 0.00001 is −5, the log[10] of 0.01 is −2. Although both original values seem to be close to zero, their logarithms are quite different, which may affect the results. Therefore, we decided not to log-transform the World Cup points. 5. As evidence, Krumer et al. (2021) used jumper per season fixed effects to find nationalistic bias in professional ski jumping. Similarly, Harb-Wu and Krumer (2019) used a similar specification to find a negative effect of competing at home on shooting performance in professional biathlon.