I. INTRODUCTION

The effect of the schedule on the performance of individuals or groups in competitive environments has been widely investigated in the recent behavioral economics literature (Cohen-Zada, Krumer, and Shtudiner 2017; Page and Page 2010; Palacios-Huerta 2014). The increased interest stems from a possible unfair ex-post advantage to one of the contestants caused by psychological or strategic effects driven by the schedule, which, ex ante, appeared to be fair. Therefore, fairness in scheduling actions in competitive environments can be an important economic issue. To be more specific, an unfair schedule may create selection efficiency concerns by reducing a "better" contestant's probability of winning. In addition, it can also harm contestants' future revenues and therefore affect their willingness to exert efforts in the present. Consequently, to maximize both, selection efficiency and effort, it is important for contest designers to minimize any possible advantage to one of the contestants that stems from an unbalanced schedule.

The role of scheduling in tournament settings was discussed in different types of contests. For example, Klumpp and Polborn (2006) described the unfairness of the primary party-based election system used in the United States to nominate a candidate of one of the major parties for the participation in presidential elections. According to the authors, this structure is unfair, since it shifts too much power to the voters in the early primary states. Page and Page (2010) presented a systematic bias in the sequential evaluation of performance, namely that contestants who performed in the later serial position in the popular Idol series had a significantly larger advantage with respect to a positive evaluation by the jury. (1) De Bruin (2005) found similar results for the Eurovision song contest and for the World and European figure skating championships. Similarly, Glejser and Heyndels (2001) found that piano finalists who performed later in the final week of the prestigious Queen Elisabeth music contests obtained a higher rank. However, this relationship was not statistically significant in the violin competition. In another study, Page and Page (2007) showed that there is advantage of playing in the second home leg game in soccer European tournaments. Krumer (2013) explained this result theoretically, assuming that the winner of a first stage has a psychological advantage in the second stage. Finally, to insure the fairness of multistage contests with a sequential order of moves, Palacios-Huerta (2012) proposed to use the so-called Prouhet-Thue-Morse sequence, where the sequence of the first n moves is the exact mirror image of the next n moves.

The aim of this study is to evaluate the effect of midweek matches on the home advantage in the German Bundesliga 1 (hereafter Bundesliga), which is the highest division of German male soccer.2 The relevance of this question stems from the fact that some teams play more midweek matches at their home stadium than others. Therefore, a schedule is considered as fair if ex ante all teams in the Bundesliga have the same probability to convert the home advantage into success, given their individual characteristics, regardless of the weekday.

In general, the home advantage phenomenon is a well-established feature in sports' competitions. Courneya and Carron (1992) defined the home advantage as "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). In their book, Moskowitz and Wertheim (2011) survey 19 different sports leagues covering more than 40 countries between the years 1871 and 2009 and showed that the within league home field advantage "is almost eerily constant through time" (p. 113). The percentage of games won by the home teams in these leagues varied between 53.3% and 69.1%. The home advantage phenomenon can be attributed to crowd noise (Pettersson-Lidbom and Priks 2010), positive psychological states during home games (Terry, Walrond, and Carron 1998), familiarity with facilities (Pollard 2002), increased level of testosterone in the players (Neave and Wolfson 2003), distance between cities (Oberhofer, Philippovich, and Winner 2010), or referee bias (Dohmen and Sauermann 2016; Garicano, Palacios-Huerta, and Prendergast 2005; Sutter and Kocher 2004). Whatever the reason, it is clear that a certain home advantage is a feature of many sporting events.

Despite the fact that home advantage has been widely analyzed in the literature, to the best of our knowledge, there is no study on the effect of playing midweek on the size of the home advantage. It is, however, well documented that midweek matches attract a lower crowd and lower TV ratings in English soccer (Buraimo 2008; Buraimo and Simmons 2015; Forrest and Simmons 2006), U.S. Major League Baseball (Knowles, Sherony, and Haupert 1992), English cricket (Schofield 1983), and the North American National Hockey League (Paul 2003). In a recent study, Wang, Goossens, and Vandebroek (2016) showed that stadium fans in Belgium's soccer Pro Legue show a very high aversion to matches that take place on Wednesdays. Therefore, combining the literatures on the effect of crowds on home advantage and on the relationship between midweek games and the size of the crowd, suggests that the home advantage may differ between weekend and midweek days.

Indeed, based on 2,013 Bundesliga matches in the seasons 2007-2008 to 2016-2017, we find that the home advantage in the midweek matches completely disappears. According to our propensity score matching analysis below, playing in the midweek reduces the difference in points between home and away teams from 0.48 points over the weekend to essentially zero. (3) In our robustness checks, we rule out other possible explanations like fatigue, the size of the crowd, or the effect of a specific weekend day. (4) Interestingly, we also find that both teams commit less fouls in the midweek matches. However, this reduction is significantly higher for home teams than for away teams. Therefore, it appears unlikely that referee bias in favor of away teams occurring in midweek matches explains the midweek effect. Another possible explanation of our results can be linked to the decreased importance of the midweek matches as perceived by the home teams' players. This may reduce the association of testosterone, which is known to enhance performance and aggressiveness, and territoriality, as described by Neave and Wolfson (2003). (5)

To illustrate a possible relationship between an unbalanced schedule and the resulting monetary rewards one can look at the distribution of revenues from TV broadcast contracts in the season 2015-2016. For example, the fourth best team with regard to TV revenues in the Bundesliga 2, SC Paderborn 07, which was relegated from the Bundesliga in the 2014-2015 season, received 9.89 million Euros. In stark contrast, the lowest TV revenues in the Bundesliga were equal to 20.19 million Euros and were earned by SV Darmstadt 98. This means that if SC Paderborn 07 had stayed in the Bundesliga, its revenue from TV alone would have been at least 10.3 million Euros higher (not counting all other revenues from ticketing, advertising, and so on).6 The case of SC Paderborn 07 is interesting also for the reason that in the relegation season, before the last round, this team had one point less than the Hamburger SV, a team that finally survived in the Bundesliga. In that season, each team was scheduled to play three matches in the midweek rounds. SC Paderborn 07 played two midweek matches at home. One of these games was against Hamburger SV in which the latter won. However, Hamburger SV had to play only one midweek match at home. Of course, we do not claim that the main reason for SC Paderborn 07's relegation was the fact that this team played at home in the midweek round against its closest rival in the relegation fight. However, it is clear that the teams had different incentives for winning their last game, which, theoretically, leads to different winning probabilities (Krumer, Megidish, and Sela 2017). Therefore, in a very tight league, where every point is important, an uneven schedule may have an effect on interim rankings and as a result on the final rankings as well (and thus on future revenues).

The remainder of the study is organized as follows: Section II describes the Bundesliga schedule. The data and some descriptive results are presented in Section III. Section IV presents the estimation strategy. The results are contained in Section V. Finally, in Section VI we offer concluding remarks.

II. DESCRIPTION OF THE BUNDESLIGA SCHEDULE

The German Bundesliga consists of 18 clubs. The matches are organized as a round-robin system. Each round consists of nine different matches, such that in each round, each team plays only once. (7) In the first half of the season (typically August to December), every team plays in 17 different matches against all other teams. Half of the matches are at home. In the second half of the season (January to May), each team plays again against all other teams, but teams played at home in the first half of the season are now playing away. Thus, in total, each team plays 34 matches, 17 of them at home, and another 17 away from home. There is a break of about 1 month between the first and the second half of a season without any matches.

The matches usually take place on weekends. A typical weekend round consists of one match on Friday at 20:30 hours, several matches on Saturday, usually at 15:30hours and one game at 18:30 hours. The remaining matches take place on Sundays, usually at 15:30 and 17:30 hours. However, there are some rounds that take place during the midweek days. These matches always take place on Tuesdays and Wednesdays at 20:00 hours. The reason for the existence of these midweek rounds is that the schedule should take into account the winter break (weather conditions), the summer break (players' vacation and recovery time), as well as the international tournaments, such as the FIFA World Cup and UEFA European Championship that take place every 2 years in June and July. Therefore, as there are not enough weekends between August and May, several rounds are played in midweek. In addition, a few regular weekend matches that were postponed due to weather conditions also take place in midweek. (8)

At the end of a season, the final table determines which teams participate in the following season's European club tournaments, such as the Champions League, which is the most prestigious club tournament in Europe, and the Europe League (former UEFA Cup), which also yields big monetary rewards. For example, in the 2015-2016 season, the four highest ranked teams participated in the Champions League (the fourth ranked team had to play an additional international qualification game to participate in the group stage of the Champions League). Teams in the fifth to seventh position played in the Europe League (this may depend also on the outcome of an elimination tournament, called the "DFB-Pokal"). In addition, the two worse ranked clubs relegate to the lower division and the team ranked 16th has to participate in the relegation play-offs against the team that was ranked third in the Bundesliga 2 for the right to play in the Bundesliga in the following year. (9) Note that the different outcomes have financial consequences for the clubs in the next season.

III. DATA AND DESCRIPTIVE RESULTS

A. Data Base

We collected data on all the matches in the Bundesliga from the 2006-2007 season up to the 10th round of the 2016-2017 season (latest data available when revising this study). The data collection starts with the 2006-2007 season, because a large amount of important information was unavailable prior to that season. In total, these data cover 3,150 matches. However, data on 306 matches played in the 2006-2007 season were used to control for the previous season's team's characteristics. In addition, we disregard matches of teams playing the European competitions just before and after these games (UEFA Champions League or UEFA Europa League), because such matches may create different allocation of efforts (for example, saving best players to more important European Cups matches, fatigue, or psychological momentum). We also did not take into account matches in which a home team did not play at its home stadium (e.g., Bayer Leverkusen in season 2008-2009), matches that took place after an international break in which national teams played friendly or qualification games, and matches in which one of the teams was already relegated or already won the title. After dropping these games, 2,013 matches enter the estimation (1,861 weekend games and 152 midweek games).

For every match, there is information on the exact day and hour, attendance, and the final score. For each team in a particular game, we observe the number of shots, shots on target (i.e., all shots that would score in the absence of a goalkeeper), number of fouls, as well as the number of corners and yellow and red cards. In addition, we use data from the Transfermarkt website (www .transfermarkt.com) to proxy the market value of each player of each team in every season. Finally, we have data on the dates of the beginning and the end of work of each coach, as well as data on the capacity of each stadium and on regional economic characteristics. The data were collected from several web sites (see Appendix D for the full list). (10)

B. Descriptive Statistics

To estimate the possible effect of playing midweek, we have a set of possible outcome variables on the level of a single match between home and away teams. The first two are defined as the number of points obtained by home and away teams. In Table 1, we can see that in line with the home advantage phenomenon, a home team attains on average 1.60 points per weekend match, which is significantly higher than the 1.33 points to be expected in the absence of a home advantage. (11) An away team achieves on average 1.14 points per such match. However, in the midweek matches we observe that the home advantage completely disappears.

Next, we define the number of goals scored by home and away teams. Based on these two variables, we also calculate the difference of goals between home and away teams. Table 1 presents, not surprisingly, that on average a home team scores more goals than an away team. Other possible outcome variables are related to the shots and shots on target. Table 1 shows that home teams have higher values for both shots-related variables.

Another dimension, which is interesting to investigate, is related to the aggressiveness of the teams, such as committed fouls, and yellow, and red cards. From the descriptive statistics, we learn that in general home teams commit less fouls, which is probably translated to the lower number of yellow and red cards.

C. Variables

To estimate the effect of midweek matches on performance, we code a dummy variable that is one if a match was played in the midweek and zero otherwise. We also use a rich set of variables that characterizes team value and players' ability, game attendance, and the international and national schedule (and the resulting demands on the international players). In the following, we present some of the most important measures (a more comprehensive list of variables appears in Appendix A).

To approximate teams' abilities, we use teams' monetary values obtained from a popular soccer website, Transfermarkt, which is a reliable data source that provides data on players' market values. As reported by Bryson, Frick, and Simmons (2013), the coverage of Transfermarkt is quite "impressive with information on 190,000 players across 330 football competitions" (p. 611). According to Franck and Nuesch (2012), players' values are estimated by industry experts and take into account salaries, signing fees, bonuses, and transfer fees. The authors found that the correlation between values evaluated by Transfermarkt and Kicker, another highly respected sport magazine in Germany, is as high as 0.89. In addition, Frick (2006) found that the correlation between salary information published by Kicker and actual salaries for two seasons in Bundesliga is 0.8.

The players' values are used to create some additional measures like the distribution of values between and within teams. More specifically, for each team we compute the standard deviation of players' values, the Herfindahl-Hirschman Index (HHI), which is defined as the sum of the squares of the values shares of each player within the team. In addition, we create some other within-team-inequality related variables such as the ratio of different players' values according to their ranking order in the team. For example, one measure is the ratio between the top three players to players ranked 9 to 11 according to their values within a team (see Appendices A and B for more details). (12) It is important to note that the goal of our empirical analysis is to evaluate the effect of midweek matches rather than to determine players' values. We use values-related measures only as covariates, which are supposed to reflect teams' abilities. The teams' values measure strongly correlates with teams' performance, suggesting that we measure teams' abilities quite well. (13) In addition to players' values, we also use several other variables that may reflect the level of ability such as a dummy variable for a team's first season in the Bundesliga after being promoted from Bundesliga 2, whether a team dismissed its coach during a season, and teams' previous season's characteristics, such as shots, corners, and yellow and red cards. (14)

As found previously in the literature, the home advantage is affected by the attendance level. Therefore, we create a measure to reflect the attendance in a match. Our preferred measure, attendance as share of the capacity of the stadium, is the ratio between the number of viewers in a match and the maximal possible capacity of the respective stadium. Table 1 demonstrates that this measure is lower in midweek than in weekend matches. There is also information about the distance between cities and public transport commuting time between cities.

In addition, we obtain information on other schedule-related variables in international competitions such as two pre- and post-World Cup and European Championships months, as well as the months in which the Africa Cups of Nations took place. Furthermore, we take into account different parts of the season, such that the beginning of the season is defined as the first 11 rounds, the middle of the season is defined as rounds 12 to 22, and the end of the season includes rounds 23 to 34. In addition, we split the beginning of the season into several parts.

IV. ECONOMETRICS

A. The Causal Question

We are interested in learning the effect of playing in the middle of the week on the success of the home team in terms of the variables measuring different aspects of the outcome of a Bundesliga soccer game, as described in the previous section. If the allocation of the midweek games over the season were entirely random, then we would compare the means of these variables for midweek matches to the means obtained for weekend matches. The difference would be a consistent estimate of the desired effect. However, scheduling is only partially random, since many other considerations are considered when fixing a league schedule, like weather, players' rest period, European Cups, National teams' tournaments, among others. Furthermore, the distribution of the characteristics shown in Table 1 already point to (small) deviations from randomness. Such deviations need to be taken into account in any estimation strategy, if they are correlated with the outcomes of interest (e.g., Imbens and Wooldridge 2009), which are measures of the success of the home team in our case. (15) Here, the data base available is rich enough in terms of game and team characteristics that we opt for a selection-on-observable strategy to identify the causal effects of interest. As described previously, since the schedule of the Bundesliga has to take into account winter and summer breaks, as well as international tournaments, we expect to capture other schedule-related characteristics, such as the periods of time that are associated with the World Cups, European Championships, and the Africa Cup of Nations. In addition, as suggested by the previous literature on the linkage between midweek matches and the crowd, we use attendance-related variables as well. Finally, we capture other differences related to team, location, and timing by the variables described above.

B. Estimator Used

Since the previous section suggests that controlling for observable characteristics will be sufficient to identify a causal effect, we face two challenges. The first challenge is that we expect that the effect of midweek games may be different for different clubs (e.g., depending on their actual position in the season) and games (early vs. late in the season) in the Bundesliga. Since the exact kind of heterogeneity is unknown, and since a very flexible way of controlling for the various confounding factors appears to be called for, we use a statistical matching approach. To be more specific, we employ the radius-matching-on-the-propensity-score matching estimator with bias adjustment as suggested by Lechner, Miquel, and Wunsch (2011) because it showed its superior finite sample and robustness properties in the large-scale empirical Monte Carlo study conducted by Huber, Lechner, and Wunsch (2013).

The second issue we face is how to exactly specify the propensity score, which is the probability of a weekday match given the relevant characteristics. The problem is that although we have prior knowledge on the kind of variables needed, these considerations are uninformative about exactly which measurements to use (like which functional form, which interactions, or which particular measure of distance, like travel time or kilometers, to mention just a few). In the past, researchers used more or less ad hoc specifications that pass certain checks with respect to the performance of the matching procedures, like so-called balancing tests. However, recent advances in machine learning techniques suggest using more principled variable selection procedures. In particular, we employ the ideas of Belloni, Chernozhukov, and Hansen (2014) of using the Least Absolute Shrinkage and Selection Operator (LASSO). In fact we use the so-called Adaptive LASSO (Buhlmann and van de Geer 2011; Hastie, Tibshirani, and Friedman 2009; Zou 2006) twice to obtain the covariates. LASSO is a statistical procedure suggested by Tibshirani (1996). It reduces the dimension of the model in some optimal sense by adding a specific penalty term to the objective function of the regression-type (linear, logistic, etc.) estimator. The LASSO penalty is the sum of the absolute values of the regression coefficients. This penalty term leads to the fact that many coefficients will be set to zero. Therefore, if the true number of variables belonging to the model is not too large (so-called sparsity assumption), the LASSO estimator has good variable selection properties, and can thus be used to select the variables to be included in the propensity score. A drawback of the LASSO is that its properties deteriorate once the sparsity assumption does not hold. However, in our case this should not be very important since we are not really interested in exactly which variables are included in the propensity score (which is however helpful for its interpretation), but first of all aim at a good prediction of the score, which, intuitively speaking, is sufficient to remove any selection bias due to (observable) omitted variables.

Following Belloni, Chernozhukov, and Hansen (2014), there are two LASSO estimations. The first LASSO estimation concerns the selection equation. However, Belloni, Chernozhukov, and Hansen (2014) point out that using just the selection equation for variable selection may not be sufficient, as it might ignore variables that are mildly related to the treatment (playing midweek), but heavily related to the outcome, and thus should be controlled for. Thus, the second LASSO estimation has the purpose to identify variables that are highly correlated with the outcome (ignoring the midweek variable). The variables used in the matching estimation are the union of variables selected by the LASSO in either of those two steps. Although, there practical applications of these new approaches are so far rare, they provide significant improvements to the ad hoc variable selection procedures used so far.

The inference for the matching estimator is based on the weighted bootstrap (see also the empirical Monte Carlo results on the performance of different inference procedures investigated by Bodory et al. 2016) ignoring the variable selection step (which is justified by the LASSO's oracle properties).

V. RESULTS

Although the purpose of the propensity score estimation is only a technical one, namely to allow the easy purging of the results from selection effects, it is nevertheless interesting to see which variables drive selection. Generally, as already apparent from Table 1, selection effects are limited. They are substantially driven by the lower attendance at midweek games as well as timing effects. The detailed results can be found in Appendix B.

Table 2 shows the key results of this study, namely the effect of playing midweek compared to playing weekends on various outcome variables that may be used to characterize the results of soccer games. The most important one is of course the expected number of points earned (top of table): Playing midweek leads to an effect of about 0.64 points in total, resulting from the home team losing about 0.29 points, while the away team gains about 0.35 points (the asymmetry results from the "3 points rule"). Considering the levels of expected points, it becomes clear that the home team loses all its home advantages in midweek games. In midweek games, the home and the away team (with the same characteristics as the home team) can expect to earn about 1.33 and 1.49 points on average, respectively, while on weekends the (same) home and away teams earn about 1.62 and 1.14 points, respectively. Similar results appear when considering related measures, namely the goals scored and the teams' shots (although not always statistically significant).

Interestingly, when considering one aspect of the playing style by looking at fouls and cards, it turns out that both teams commit fewer fouls in midweek than on weekends. This fact points to a reduced aggressiveness of both teams in midweek games, which might be attributed to the general lower attendance in these games (although the level of attendance is flexibly controlled for).

In addition, we investigate whether the results are robust to including the information about attendance as control variable. If spectators know that midweek games have no or a reduced home advantage, then this fact may reduce their inclination to visit the games in midweek and thus this variable is endogenous and we expect a bias of the result toward zero when controlling for attendance. However, this is clearly not the case as can be seen in Table A3. In fact, the results without conditioning are even somewhat smaller.

We also compare midweek games to games played on Sundays and Fridays. We report the results in the last two columns of Table A3. As we can see, playing on midweek relative to Sunday has a significant and substantial effect of 0.72 points in favor of an away team. As previously, we also find a significantly lower number of fouls committed by a home team in midweek games.

Despite the fact that there are only 205 games played on Friday, we still find that an away team obtains 0.192 points more on midweek relative to Friday with a p value of 8%. As in other specifications, we find again that there are significantly less fouls committed by the home team in midweek games compared to Friday games. The remaining results are in the same direction as in other specifications. However, some of them are not significant at conventional levels due to a much lower number of observations available for estimation.

Finally, the explanation of the midweek effect might be that the teams are more tired because they have a reduced rest period (about half compared to a weekend game without prior midweek game) and thus are less aggressive which might reduce the advantage of the home team. To investigate this issue, we pool midweek games together with weekend games after a midweek game and compare them to weekend games without a prior midweek game. When doing so (Table G1 in Appendix S1, Supporting Information), the effect becomes much smaller and statistically insignificant, which is essentially ruling out that hypothesis.

One possible explanation of our results may be linked to the size of the crowd. Despite the fact that we control for many crowd-related variables, the size of the crowd may have indirect effects on players' performance. As discussed previously, a smaller crowd in the midweek matches is a well-known phenomenon across different sports in different countries, and is found in our data base as well. Therefore, it is likely that home team's players anticipate lower attendance before these matches, such that home team's players may consider midweek matches less important, because fewer viewers will monitor their actions. (16) If so, then our findings on the disappearing home advantage and lower aggressiveness of home teams, as measured by the lower number of its fouls, are in line with the literature on testosterone, which is known to enhance performance and aggressiveness. (17) The point is that the above described anticipation can affect the relationship between testosterone and territoriality that was described by Neave and Wolfson (2003). In fact, these authors provide direct evidence that the level of testosterone among soccer players was significantly higher before home matches than before away matches or training sessions. No difference was found between away matches and training sessions. In addition, they found that the increase in testosterone was higher before matches that were perceived as more important. Similarly, Mazur, Booth, and Dabbs (1992) showed that a pregame increase in testosterone is less likely to occur in the event that is regarded as less important. These findings are in line with Ward (1998) who studied the effect of opening day matches in Major League Baseball on home advantage. In general, these matches are perceived as very important and described as "... more than 'just another game' ..." and as a " ... highly ritualistic and festive occasion ..." Ward (1998, p. 280). Therefore, players and fans in these matches are expected to be more motivated and as a result the home advantage should be greater. Indeed, the author found significantly higher winning probabilities of home teams on these opening day matches than in other matches during a regular season.

VI. CONCLUSION

The main motivation of this study is the potential effect of scheduling on the performance of high-profile agents in a real competitive environment for which good productivity data are available. We find that in the German soccer Bundesliga, the home teams perform significantly worse in midweek matches than in weekend matches. In these games, the home advantage, which is important in many different sports, disappears completely. Since the midweek matches are unevenly allocated among teams, this finding implies that the actual schedule of the Bundesliga favors teams with fewer home (more away) games in midweek, which may be considered as an unfair advantage for those teams. Such unfair scheduling plays a significant role in determining teams' season outcomes. Thus, it may change their income by tens of million Euros per year.

Our results on lower aggressiveness and decreased performance of home teams during their midweek matches, in addition to the smaller crowd during these games, are in line with a biological literature on the relationship between testosterone, the importance of the event, performance, and territoriality. According to this literature, the midweek matches may be perceived by the home teams' players as less important, which may lead to decreased motivation and as a result to a disappearing home advantage.

Despite being driven by different factors, the day of the week effect on performance was also found in completely different settings. In their seminal paper, Gibbons and Hess (1981) showed that there was a negative mean return of financial assets for Mondays and a positive return for Fridays. More recently, Siganos, Vagenas-Nanos, and Verwijmeren (2014) found a significant relationship between Sunday's sentiment and Monday's stock market characteristics, highlighting that the behavioral factors like mood, optimism, and happiness are responsible for their finding. Therefore, since the day of the week effect on performance appears to be a more general behavioral feature than just the effect in soccer, it should be taken into account when evaluating individuals' performance in various fields, such as labor market contests, financial markets, or sports competitions.

In particular, it is worthwhile for the Bundesliga management (DFL) to allocate the midweek matches evenly among teams. For example, if there was a midweek round in the first half of the season, then there should be a midweek round in the second half of the season and it should involve the same teams (even if this means changing the order of games in both halves of the season). This more balanced schedule increases the fairness of the tournament.

Finally, our methodology could be applied to examine other performance-related features such as the effect of different parts of the season, which may include national holidays, or the effect of a dismissed head coach in the Bundesliga. In addition, despite several possible fundamental differences between different leagues, such as the share of midweek matches, existence of a winter break, and relative frequency of matches on different days, we call for further research to investigate the effect of allocation of matches on performance in different sports leagues.

ABBREVIATION

LASSO; Least Absolute Shrinkage and Selection Operator

APPENDIX A: DESCRIPTIVE STATISTICS

The following table contains descriptive statistics for all variables (outcomes, treatment, and controls) relevant for this study.

TABLE A1
Descriptive Statistics (Sample Means)

Variables                          Weekend Games   Midweek Games
                                    (N = 1,861)      (N = 152)

Game outcomes
  Points--home team                    1.60            1.36
  Points--away team                    1.14            1.46
  Goals--home team                     1.56            1.51
  Goals--away team                     1.23            1.39
  Shots--home team                     14.38           13.99
  Shots--away team                     11.93           11.82
  Shots on target--home team           5.13            5.01
  Shots on target--away team           4.21            4.41
  Fouls--home team                     15.91           14.72
  Fouls--away team                     17.02           16.34
  Yellow cards--home team              1.65            1.70
  Yellow cards--away team              1.95            1.86
  Red cards--home team                 0.07            0.09
  Red cards--away team                 0.10            0.10
  Corners--home team                   5.56            5.36
  Corners--away team                   4.42            4.14
Game characteristics
  Attendance                          40,795          40,478
  Attendance as share of               0.91            0.86
    capacity of stadium
  Attendance as share of               0.35            0.25
    capacity of stadium
    if higher than 0.99
  Ln of stadium capacity               10.54           10.51
  Distance between the cities           366             372
    of the teams (km)
  Public transport commuting            194             200
    time between the cities
    (minutes)
  Africa Cup of Nations months         0.12            0.06
  Two months before World Cup or       0.12            0.18
    European Championship
  Two months after World Cup or        0.07            0.24
    European Championship
  Season after World Cup or            0.44            0.55
    European Championship
  Rounds 1-11                          0.27            0.43
    (beginning of a season)
  Rounds 12-22                         0.33            0.20
    (middle of a season)
  Rounds 23-34 (end of a season)       0.40            0.37
  August or September                  0.18            0.36
  Rounds 1-6                           0.17            0.31
Team characteristics
  Value of home team                   86.7            109.7
    (Mil. [euro])
  Value of away team                   90.3            103.9
    (Mil. [euro])
  Standardized value of                -0.19           -0.02
    home team
  Standardized value of                -0.15           0.00
    away team
  First season after                   0.16            0.15
    promotion--home team
  First season after                   0.15            0.11
    promotion--away team
  Ratio of top 3 to ranked 9-11        2.34            2.18
    players' values--home team
  Ratio of top 3 to ranked 9-11        2.34            2.48
    players' values--away team
  HHI of players' values--home         0.06            0.06
    team
  HHI of players' values--away         0.06            0.06
    team
  New coach--home team                 0.18            0.11
  New coach--away team                 0.18            0.14
  Previous season shots--home          14.56           15.04
    team
  Previous season shots--away          11.94           12.02
    team
  Previous season shots on             5.60            6.11
    target--home team
  Previous season shots on             4.53            4.85
    target--away team
  Previous season corners--home        5.64            5.76
    team
  Previous season corners--away        4.38            4.41
    team
  Previous season fouls--home          16.27           15.90
    team
  Previous season fouls--away          17.46           17.17
    team
  Previous season yellow               1.64            1.59
    cards--home team
  Previous season yellow               1.97            1.94
    cards--away team
  Previous season red                  0.08            0.07
    cards--home team
  Previous season red                  0.11            0.11
    cards--away team

Variables                          Sunday Games   Friday Games
                                    (N = 421)      (N = 205)

Game outcomes
  Points--home team                    1.57           1.68
  Points--away team                    1.17           1.05
  Goals--home team                     1.58           1.51
  Goals--away team                     1.30           0.96
  Shots--home team                    14.61          14.37
  Shots--away team                    12.03          11.32
  Shots on target--home team           5.14           5.12
  Shots on target--away team           4.41           3.76
  Fouls--home team                    15.87          15.71
  Fouls--away team                    16.94          16.55
  Yellow cards--home team              1.60           1.63
  Yellow cards--away team              1.94           1.97
  Red cards--home team                 0.07           0.05
  Red cards--away team                 0.10           0.06
  Corners--home team                   5.56           5.70
  Corners--away team                   4.43           4.13
Game characteristics
  Attendance                          41,051         44,036
  Attendance as share of               0.91           0.91
    capacity of stadium
  Attendance as share of               0.33           0.41
    capacity of stadium
    if higher than 0.99
  Ln of stadium capacity              10.55          10.61
  Distance between the cities          357            357
    of the teams (km)
  Public transport commuting           189            186
    time between the cities
    (minutes)
  Africa Cup of Nations months         0.13           0.13
  Two months before World Cup or       0.09           0.08
    European Championship
  Two months after World Cup or        0.09           0.09
    European Championship
  Season after World Cup or            0.43           0.45
    European Championship
  Rounds 1-11                          0.32           0.32
    (beginning of a season)
  Rounds 12-22                         0.33           0.37
    (middle of a season)
  Rounds 23-34 (end of a season)       0.36           0.31
  August or September                  0.22           0.21
  Rounds 1-6                           0.20           0.19
Team characteristics
  Value of home team                   91.0           98.3
    (Mil. [euro])
  Value of away team                   95.0           93.4
    (Mil. [euro])
  Standardized value of               -0.12          -0.09
    home team
  Standardized value of               -0.07          -0.14
    away team
  First season after                   0.14           0.15
    promotion--home team
  First season after                   0.10           0.14
    promotion--away team
  Ratio of top 3 to ranked 9-11        2.33           2.38
    players' values--home team
  Ratio of top 3 to ranked 9-11        2.37           2.30
    players' values--away team
  HHI of players' values--home         0.06           0.06
    team
  HHI of players' values--away         0.06           0.06
    team
  New coach--home team                 0.17           0.13
  New coach--away team                 0.15           0.19
  Previous season shots--home         14.67          14.75
    team
  Previous season shots--away         12.01          11.77
    team
  Previous season shots on             5.69           5.73
    target--home team
  Previous season shots on             4.61           4.43
    target--away team
  Previous season corners--home        5.69           5.68
    team
  Previous season corners--away        4.39           4.32
    team
  Previous season fouls--home         16.22          16.01
    team
  Previous season fouls--away         17.39          17.26
    team
  Previous season yellow               1.64           1.60
    cards--home team
  Previous season yellow               1.97           1.96
    cards--away team
  Previous season red                  0.08           0.07
    cards--home team
  Previous season red                  0.11           0.10
    cards--away team

Variables                            Overall
                                   (N = 2,013)

Game outcomes
  Points--home team                   1.58
  Points--away team                   1.16
  Goals--home team                    1.55
  Goals--away team                    1.24
  Shots--home team                    14.35
  Shots--away team                    11.88
  Shots on target--home team          5.13
  Shots on target--away team          4.22
  Fouls--home team                    15.82
  Fouls--away team                    16.97
  Yellow cards--home team             1.65
  Yellow cards--away team             1.95
  Red cards--home team                0.08
  Red cards--away team                0.10
  Corners--home team                  5.54
  Corners--away team                  4.40
Game characteristics
  Attendance                         40,771
  Attendance as share of              0.90
    capacity of stadium
  Attendance as share of              0.34
    capacity of stadium
    if higher than 0.99
  Ln of stadium capacity              10.54
  Distance between the cities          366
    of the teams (km)
  Public transport commuting           195
    time between the cities
    (minutes)
  Africa Cup of Nations months        0.12
  Two months before World Cup or      0.11
    European Championship
  Two months after World Cup or       0.08
    European Championship             0.45
  Season after World Cup or
    European Championship             0.28
  Rounds 1-11
    (beginning of a season)
  Rounds 12-22                        0.32
    (middle of a season)
  Rounds 23-34 (end of a season)      0.39
  August or September                 0.20
  Rounds 1-6                          0.18
Team characteristics
  Value of home team                  88.4
    (Mil. [euro])
  Value of away team                  91.3
    (Mil. [euro])
  Standardized value of               -0.18
    home team
  Standardized value of               -0.14
    away team
  First season after                  0.16
    promotion--home team
  First season after                  0.15
    promotion--away team
  Ratio of top 3 to ranked 9-11       2.33
    players' values--home team
  Ratio of top 3 to ranked 9-11       2.35
    players' values--away team
  HHI of players' values--home        0.06
    team
  HHI of players' values--away        0.06
    team
  New coach--home team                0.17
  New coach--away team                0.17
  Previous season shots--home         14.60
    team
  Previous season shots--away         11.95
    team
  Previous season shots on            5.64
    target--home team
  Previous season shots on            4.56
    target--away team
  Previous season corners--home       5.65
    team
  Previous season corners--away       4.38
    team
  Previous season fouls--home         16.25
    team
  Previous season fouls--away         17.43
    team
  Previous season yellow              1.64
    cards--home team
  Previous season yellow              1.97
    cards--away team
  Previous season red                 0.08
    cards--home team
  Previous season red                 0.11
    cards--away team

Notes: This table presents the selected characteristics.
Weekend games are the games that took place from Fridays
to Sundays. Midweek games are the games that took place
from Mondays to Thursdays.

APPENDIX B: PROPENSITY SCORE ESTIMATION

Table A2 contains the detailed estimation results of the propensity (after variable selection by the double-LASSO as described in the main text).

TABLE A2
Estimation of Propensity Score (Mean Marginal Effects)

                                                      Midweek
                                      Midweek with    without
Variables                              Attendance    Attendance

Game characteristics
  Attendance (in 10,000s)               0.003 **
  Attendance as share of capacity        0.172
    of stadium
  Attendance as share of capacity        -0.392
    of stadium squared
  Attendance as share of capacity        -0.008
    of stadium if higher than 0.99
  Africa Cup of Nations months          -0.026 *       -0.017
  August or September                  0.225 ***     0.249 ***
  Ln of stadium capacity                -0.114 *       0.020
  Season after World Cup or            0.064 ***     0.058 ***
    European Championship
  Two months before World Cup or       0.130 ***     0.109 ***
    European Championship
  Difference between teams'              0.000         0.000
    values 2 months before
    World Cup or European
    Championship
  Rounds 1-6                           -0.071 ***    -0.073 ***
Team characteristics
  Difference in promotion status         0.013         0.011
  Difference in HHI in the middle      4.068 ***     3.783 ***
    of season
  Difference in new coach in the        -0.036 *       -0.031
    middle of season
  Difference in teams'                   0.000         0.002
    standardized values
  Difference in teams'                   -0.001        0.001
    median values
  Squared difference in teams'         0.001 ***     0.001 ***
    values
  Squared difference in teams'           0.000         0.000
    values in the end of season
  Ln of difference of teams' ratio     -0.097 ***    -0.091 ***
    of top 3 to ranked
    9-11 players' values
  Ln of difference of teams' ratio      0.069 **      0.074 **
    of top 11 to ranked 12-22
    players' values in the
    beginning of season
  Squared difference of teams'           -0.001        -0.001
    ratio of top 11 to ranked
    12-22 players' values
  Difference in standardized             -0.010        -0.011
    values of teams if values
    are positive
  Ratio between teams' values            -0.031        -0.034
    if higher than 5
  Ratio between teams' values if         0.037         0.021
    higher than 3 in the middle
    of season
  Difference between teams' values       -0.009        -0.014
    if ratio between values is
    higher than 2 in the middle
    of season
  Difference between previous           0.015 **       0.010
    season's corners in the end
    of current season
  Ln of previous season's                0.014         -0.012
    corners--home team
  Difference in previous season's        0.001         0.001
    fouls in the end of current
    season
  Previous season's red                  -0.074        -0.091
    cards--home team
  Difference in previous season's        0.003         0.001
    shots in the beginning of
    current season
  Difference in previous season's        0.002         0.004
    yellow cards

                                                        Midweek
                                         Midweek         versus
Variables                             versus Sundays    Mondays

Game characteristics
  Attendance (in 10,000s)                0.012 **       0.017 **
  Attendance as share of capacity         -0.366         -0.570
    of stadium
  Attendance as share of capacity         -0.705         -0.678
    of stadium squared
  Attendance as share of capacity         0.027          -0.078
    of stadium if higher than 0.99
  Africa Cup of Nations months            -0.075         -0.110
  August or September                   0.303 ***      0.334 ***
  Ln of stadium capacity                -0.409 **      -0.703 ***
  Season after World Cup or             0.165 ***      0.216 ***
    European Championship
  Two months before World Cup or        0.293 ***      0.331 ***
    European Championship
  Difference between teams'               0.000          -0.001
    values 2 months before
    World Cup or European
    Championship
  Rounds 1-6                            -0.146 **        -0.127
Team characteristics
  Difference in promotion status          -0.015         0.062
  Difference in HHI in the middle       12.959 ***     12.208 ***
    of season
  Difference in new coach in the        -0.147 **        -0.047
    middle of season
  Difference in teams'                    -0.028         0.002
    standardized values
  Difference in teams'                    0.002          -0.014
    median values
  Squared difference in teams'          0.000 ***        0.000
    values
  Squared difference in teams'            0.000          0.000
    values in the end of season
  Ln of difference of teams' ratio      -0.265 ***     -0.273 ***
    of top 3 to ranked
    9-11 players' values
  Ln of difference of teams' ratio        0.136        0.324 ***
    of top 11 to ranked 12-22
    players' values in the
    beginning of season
  Squared difference of teams'            -0.001       -0.010 **
    ratio of top 11 to ranked
    12-22 players' values
  Difference in standardized              -0.013         -0.049
    values of teams if values
    are positive
  Ratio between teams' values             -0.085         0.062
    if higher than 5
  Ratio between teams' values if          -0.040         0.077
    higher than 3 in the middle
    of season
  Difference between teams' values        -0.004         0.080
    if ratio between values is
    higher than 2 in the middle
    of season
  Difference between previous           0.064 ***        0.022
    season's corners in the end
    of current season
  Ln of previous season's                 -0.008         0.160
    corners--home team
  Difference in previous season's         0.016          -0.023
    fouls in the end of current
    season
  Previous season's red                   -0.274         0.422
    cards--home team
  Difference in previous season's         0.012          -0.001
    shots in the beginning of
    current season
  Difference in previous season's         0.009          0.030
    yellow cards

Notes: Mean marginal effects presented. Inference based
on bootstrapping (99 replications) standard deviation and
using asymptotic normal distribution for inference. The
difference and ratio-related variables are always defined
as difference or ratio between home and away measures.

* Significant at 10%; ** significant at 5%;
*** significant at 1%.

APPENDIX C: RESULTS OF ROBUSTNESS CHECKS

Table A3 shows the results for using different comparisons between midweek games and different definitions of weekend games. More specifically, it shows the comparisons to all weekend games, games played on Sunday, and games played on Friday. Furthermore, in comparison to all weekend games the results are shown when controlling and not controlling for attendance (which may be considered as affected by the midweek effect and thus be endogenous).

TABLE A3
Effects of Different Specifications

                                 Midweek      Midweek
                                   with       without
Variables                       Attendance   Attendance

Points--home team               -0.286 ***     -0.159
Points--away team               0.354 ***     0.207 **
Difference in points            -0.640 ***    -0.330 *
Goals--home team                -0.204 **      -0.099
Goals--away team                  0.047        0.039
Difference in goals              -0.251 *      -0.138
Shots--home team                -0.719 **     -0.588 *
Shots--away team                  -0.295       -0.253
Difference in shots               -0.424       -0.334
Shots on target--home team      -0.502 ***   -0.487 ***
Shots on target--away team        0.102        0.118
Difference in shots on target   -0.604 **    -0.605 ***
Fouls--home team                -1.260 ***   -0.678 **
Fouls--away team                 -0.463 *     -0.485 *
Difference in fouls             -0.797 **      -0.194
Yellow cards--home team          0.158 **     0.168 **
Yellow cards--away team           -0.048      -0.160 *
Difference in yellow cards        0.206       0.328 **
Red cards--home team             -0.021 *      0.010
Red cards--away team              0.039        0.020
Difference in red cards          -0.060 *      -0.010
Attendance as share of                       -0.083 ***
  capacity of stadium
Remaining observations              83           96
  in common support (%)

                                 Midweek      Midweek
                                  versus       versus
Variables                         Sunday       Friday

Points--home team               -0.319 ***     -0.100
Points--away team               0.401 ***     0.192 *
Difference in points            -0.720 ***     -0.292
Goals--home team                -0.224 ***     0.054
Goals--away team                  0.128      0.288 ***
Difference in goals             -0.352 ***    -0.235 *
Shots--home team                -1.049 ***   -0.869 **
Shots--away team                  0.076        0.296
Difference in shots              -1.125 *     -1.166 *
Shots on target--home team      -0.725 ***    -0.325 *
Shots on target--away team        0.175      0.545 ***
Difference in shots on target   -0.900 ***   -0.870 ***
Fouls--home team                -0.908 ***   -1.140 ***
Fouls--away team                  0.078       -0.734 *
Difference in fouls             -0.986 **      -0.407
Yellow cards--home team         0.213 ***     0.169 *
Yellow cards--away team           -0.003       -0.115
Difference in yellow cards       0.216 *      0.284 **
Red cards--home team              0.011       0.033 *
Red cards--away team              0.026        0.030
Difference in red cards           -0.015       0.004
Attendance as share of
  capacity of stadium
Remaining observations              86           85
  in common support (%)

Notes: Mean marginal effects presented. Inference based
on bootstrapping (4,999 replications for the first two
columns and 1,999 replications for the second two columns)
p values. The difference-related variables are defined as
difference between home and away measures. In all the
specifications, we control for attendance-related variables
except for column 2. In the first two columns, we compare
midweek matches to weekend matches by using all the data.
In column 3, we compare midweek matches to matches played
on Sunday. In the last column, we compare midweek matches
to matches played on Friday.

* Significant at 10%; ** significant at 5%;
*** significant at 1%.

APPENDIX D: LIST OF SOURCES

www.uefa.com

www.fifa.com

www.transfermarkt.com

www.football-data.co.uk

www.rsssf.com

www.espnfc.com

https://en.wikipedia.org/wiki/Bundesliga

www.regionalstatistik.de

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SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Appendix S1. Results of additional robustness checks.

(1.) The Idol show is a popular reality television-music competition format where judges and audience select the winner. For additional information, see https://en.wikipedia.org/wiki/Idols_(TV_series) (accessed April I, 2016).

(2.) On average, teams earn 1.6 points per game at home matches, which is almost half a point more compared to playing away.

(3.) Note that the winning team receives three points, while the losing team gets no points. In case of a draw, each team gets one point.

(4.) The elimination of the fatigue explanation is in line with Scoppa (2015), who investigated all FIFA World Cups and UEFA Championships and found no effect of additional rest days on the teams' winning probabilities.

(5.) Territoriality is a protective response to an invasion of one's perceived territory, which is common among animals. For example, Huntingford and Turner (1987) showed that animals attack more readily and with higher toughness when defending a home territory.

(6.) Numbers are taken from http://www .bavarianfootballworks.com/2015/ll/23/9782628/how-much-will-bayern-munich-make-from-bundesliga-tvrevenues (accessed March 8, 2016).

(7.) See Krumer, Megidish, and Sela (2017) for additional details on round-robin structure.

(8.) For example, in two other leagues with unbalanced allocation of matches among different weekdays such as the English Premier League and Championships division (second-highest division in England), in the last nine seasons, 18.2% and 28.9% of the games took place on midweek days, respectively. Moreover, after Saturday and Sunday, the third most frequent day with regard to the number of games in the Premier League is Wednesday. The corresponding day for Championships is Tuesday. Data are available on www.football-data.co.uk (accessed December 5, 2016).

(9.) The relegation play-off format was introduced in the 2008-2009 season. Previously, up to the 2007-2008 season, three teams were directly relegated to the Bundesliga 2.

(10.) More specifically, the data on the attendance are available on www.transfermarkt.com, the data on within game measures, such as shots, yellow cards, number of fouls, and so on, are available on www.football-data.co.uk and www .espnfc.com. The data on regional economic characteristics are available on www.regionalstatistik.de. And the data on the coaches and stadiums were taken from https://en.wikipedia.org/wiki/Bundesliga.

(11.) Assuming an equal probability of a loss, a win, and a draw.

(12.) See Coates, Frick, and Jewell (2016) for discussion on the relationship between players' inequality in salaries and teams' performance.

(13.) The results of the relevant regression analysis are available upon the request.

(14.) See Van Ours and van Tuijl (2016) for discussion on the effects of coach dismissals on team performance.

(15.) Note that the results would be symmetric if we focus on the success of the away team instead.

(16.) For example, Georganas, Tonin, and Vlassopoulos (2015) found some evidence that subjects being observed increase their productivity. In addition, Bernheim and Thomadsen (2005) showed the importance of behavioral implications of anticipatory emotions, whereas Benabou and Tirole (2002) highlighted the role of memory in economic behavior.

(17.) For additional details on positive effect of testosterone on performance in sports see the comprehensive review of Wood and Stanton (2012).

ALEX KRUMER and MICHAEL LECHNER

Krumer: Post-doctoral Fellow, Swiss Institute for Empirical Economic Research (SEW), University of St. Gallen, St. Gallen CH-9000, Switzerland; Phone 0041-712-242-342, Fax 0041-712-242-302, E-mail alexander.krumer@unisg.ch

Lechner: Professor, Swiss Institute for Empirical Economic Research (SEW), University of St. Gallen, St. Gallen CH9000, Switzerland; CEPR, London, UK; CESIfo, Munich, Germany; IAB, Nuremberg, Germany; IZA, Bonn, Germany. Phone 0041-712-242-814, Fax 0041-712-242-302, E-mail michael.lechner@unisg.ch

doi: 10.1111/ecin.12465

TABLE 1
Descriptive Statistics of Selected Variables

                                   Weekend Games    Midweek Games
Variable                            (N = 1,861)       (AT = 152)

Game outcomes
  Points home team                  1.60(1.31)        1.36(1.37)
  Points away team                  1.14(1.26)        1.46(1.38)
  Goals home team                   1.56(1.26)       1.51 (1.30)
  Goals away team                   1.23(1.14)        1.39(1.21)
  Shots home team                  14.38 (4.97)      13.99 (5.20)
  Shots away team                  11.93 (4.59)      11.82 (4.52)
  Shots on target home team         5.13(2.56)       5.01 (2.63)
  Shots on target away team         4.21 (2.36)      4.41 (2.35)
  Fouls home team                  15.91 (4.52)      14.72 (4.34)
  Fouls away team                  17.02 (4.77)      16.34 (5.06)
  Yellow cards home team            1.65(1.19)        1.70(1.14)
  Yellow cards away team            1.95 (1.23)       1.86(1.31)
  Red cards home team               0.07 (0.28)      0.09 (0.29)
  Red cards away team               0.10(0.31)        0.10(0.32)
Game characteristics
  Attendance                      40,795 (15,882)   40,478(17,572)
  Attendance as share of            0.91 (0.12)      0.86 (0.14)
    capacity of stadium
  Africa Cup of Nations                0.12              0.06
    (dummy variable for
    respective month)
  Two months before World Cup          0.12              0.18
    or European Championship
  Two months after World Cup           0.07              0.24
    or European Championship
  Distance between the cities        366(182)          372(181)
    of the teams (km)
Team characteristics
  Value of home team                86.7 (65.4)      109.7(117.5)
    (Mil. [euro])
  Value of away team                90.3 (69.7)      103.9 (87.7)
    (Mil. [euro])
  Standardized value of             -0.19(0.80)      -0.02(1.03)
    home team
  Standardized value of            -0.15 (0.80)      0.00 (0.99)
    away team
  Ratio of top 3 to ranked          2.34 (0.67)       2.18(0.51)
    9-11 players' values--home
    team
  Ratio of top 3 to ranked          2.34 (0.65)      2.48 (0.71)
    9-11 players' values--away
    team
  First season after                   0.16              0.15
    promotion--home team
    (dummy variable)
  First season after                   0.15              0.11
    promotion--away team
    (dummy variable)
  New coach--home team                 0.18              0.11
    (dummy variable)
  New coach--away team                 0.18              0.14
    (dummy variable)

Notes: This table presents average values and standard
deviations (in brackets for nonbinary variables). The
difference and ratio-related variables are defined as
difference or ratio between home and away measures.

TABLE 2
Levels and Effects of Playing Midweek

                                Expected Value    Expected Value
                                 When Playing    When Not Playing
Variables                          Midweek           Midweek

Points home team                    1.330             1.617
Points away team                    1.489             1.135
Difference in points                -0.159            0.482
Goals home team                     1.384             1.588
Goals away team                     1.284             1.237
Difference in goals                 0.100             0.351
Shots home team                     13.728            14.448
Shots away team                     11.490            11.785
Difference in shots                 2.238             2.662
Shots on target home team           4.696             5.198
Shots on target away team           4.322             4.220
Difference in shots on target       0.373             0.977
Fouls home team                     14.534            15.795
Fouls away team                     16.478            16.941
Difference in fouls                 -1.944            -1.147
Yellow cards home team              1.750             1.592
Yellow cards away team              1.880             1.928
Difference in yellow cards          -0.131            -0.336
Red cards home team                 0.047             0.068
Red cards away team                 0.137             0.099
Difference in red cards             -0.091            -0.030

                                   Effect of      Standard Error
Variables                       Playing Midweek   of the Effect

Points home team                  -0.286 ***          0.098
Points away team                   0.354 ***          0.095
Difference in points              -0.640 ***          0.191
Goals home team                   -0.204 **           0.095
Goals away team                    0.047              0.087
Difference in goals               -0.251 *            0.141
Shots home team                   -0.719 **           0.345
Shots away team                   -0.295              0.287
Difference in shots               -0.424              0.493
Shots on target home team         -0.502 ***          0.166
Shots on target away team          0.102              0.123
Difference in shots on target     -0.604 **           0.200
Fouls home team                   -1.260 ***          0.295
Fouls away team                   -0.463 *            0.303
Difference in fouls               -0.797 **           0.370
Yellow cards home team             0.158 **           0.070
Yellow cards away team            -0.048              0.099
Difference in yellow cards         0.206              0.133
Red cards home team               -0.021 *            0.020
Red cards away team                0.039              0.033
Difference in red cards           -0.060 *            0.040

Notes: Average treatment effect. Inference based on
bootstrapping (4,999 replications) p values.
Difference-related variables are defined as difference
between home and away measures.

* Significant at 10%; ** significant at 5%;
*** significant at 1%.