The impact of atmospheric conditions on actual and expected scoring in the NFL.
Paul, Rodney J.
Abstract
This study extends the research on atmospheric conditions and scoring in sporting events by examining components of air density as it relates to National Football League (NFL) games. Statistically significant results were found in relation to the role of humidity in explaining the difference between actual scoring and the betting market total for NFL games. Simple wagering strategies based on humidity, wind speed, and a combination of these factors were shown to reject market efficiency. From game statistics, it appears that humidity may unexpectedly influence the rushing game, leading to greater than expected scoring when humidity is high.
Keywords: market efficiency, betting markets, atmospheric conditions, weather, National Football League
Introduction
Weather and other related atmospheric conditions often tend to influence the flow and outcome of sporting events. Rain or snow, high winds, and brutal temperatures may all play a role in the way a game is played, the flight of a ball, and player fatigue. Recently, atmospheric conditions have been shown to impact the sport of baseball in both pitcher performance compared to expectations (Paul et al., 2014) and in terms of actual and expected runs scored (Paul, Weinbach, & Weinbach, 2014). This paper extends the analysis of the role of atmospheric conditions to both actual and expected scoring, and their interrelationship, for the National Football League (NFL).
The role that air density plays in actual versus expected scoring in Major League Baseball (MLB) stems from original research in physics by Bahill et al. (2009). Air density and its individual components were shown to have a significant effect on runs scored (Paul, Weinbach, & Weinbach, 2014). This factor was not fully incorporated into the totals betting market in baseball, leaving simple strategies based upon air density to yield statistically significant profits. Air density is influenced by temperature, humidity, barometric pressure, and altitude. In addition to air density, similar factors such as wind speed and precipitation were also investigated for their respective roles in the determination of runs scored.
For other outdoor sports, the relationship between air density and points scored is not quite as straightforward. The ability to hit a home run under certain weather conditions does not directly translate to other sports and there tends to be more aspects to scoring in other sports beyond the basic pitcher and batter matchup of baseball. In direct relation to the game of football, atmospheric conditions and related weather variables are likely to play an important role in how the game is played. Scoring can occur in football through passing, running, or kicking the football, which can all be affected by the elements. For instance, passing the football may be quite difficult on windy days and turnovers may be more likely to occur when the weather consists of rain or snow.
Brown and Sauer (1993) noted that weather is a potential factor that would influence the betting market for sports. Track conditions based upon weather was shown to be important as it related to betting patterns, returns, and tests for market efficiency in studies of horse racing wagering markets (Figlewski, 1979). In direct relation to the NFL, Borghesi (2008) illustrated that heat, wind, and rain had negative effects on scoring, which led to strategies that generated statistically significant profits in NFL totals markets.
Outside of studies of betting markets, weather-related effects on financial prices have been studied in a variety of settings. How sunshine and other weather factors contributed to stock returns was undertaken by Hirshleifer and Shumway (2003). Investors were found to respond to these factors, leading to statistically significant changes in market prices. Weather was shown to have a significant impact on orange juice prices (Roll, 1984) and more generally in financial markets in terms of security prices and transaction volume (Roll, 1988).
We directly test if atmospheric conditions impact the number of points scored by both teams in NFL games using a sample of seasons from 2010-2014, investigating which factors, if any, play a key role in the determination of scoring. From there, we use those results to test if these factors are fully incorporated into betting market prices (totals in the over/under market). If the results are similar to baseball, where the full effect was not priced into the totals market, we explore which factors are driving the unexpected results and test for returns to simple strategies involving one or more of these factors. Once completed, to help deepen the understanding of why atmospheric variables may impact scoring more or less than anticipated by the betting market, we use the same explanatory variables to explore their role on individual NFL game statistics, including passing yards, running yards, and turnovers. Through this analysis, we hope to explain the reasons why weather conditions may not only result in more or fewer points scored in an NFL game, but which element(s) of the game is most affected by weather and related conditions and why it is not fully incorporated into betting market prices.
Regression Model of Atmospheric Conditions on Actual and Expected Points Scored
To determine the role, if any, of atmospheric conditions on actual scoring, expected scoring, and betting market total results for the National Football League, we constructed and ran regression models based upon the amount of scoring and the game result compared to the betting market total in the over/under wagering market. Given that a football will travel through the air in the passing game and overall weather conditions likely impact the role in the decision of offenses of whether to pass or run the ball, the role of atmospheric conditions and other related variables are used as explanatory variables to test for their impact and statistical significance as it relates to actual and expected scoring.
The first regression model uses the sum of points scored by both teams in an NFL game as the dependent variable. The independent variables included in the regression model include various measures of atmospheric conditions and air density, team dummies, and a dummy variable for overtime games (where additional scoring beyond that scored during regulation occurs except in the rare cases of a tie). The model was run with and without the betting market total, which represents expected scoring in the game. When the totals betting market is efficient, the total serves as an unbiased predictor of the sum of points scored by both teams in a given game.
The first set of independent variables included in the model come from the components of air density. Bahill et al. (2009) described the calculation of air density for their study of baseball and is noted in equation (1).
Air Density = 1.045 + 0.01045{-0.0035 (Altitude - 2600) - 0.2422 (Temperature - 85) -0.0480 (Relative Humidity - 50) + 3.4223 (Barometric Pressure - 29.92)}. (1)
Each component part may be important for how these variables impact scoring in an NFL game and the variables are included to ascertain their relevance and potential role as it relates to expected and actual scoring. Higher altitudes are likely to favor teams who pass the ball due to the thin air and may play a role in the running game due to exertion under these conditions. If altitude positively influences passing ability and this leads to increased scoring, altitude should have a positive impact on scoring.
Temperature has been studied before in relation to football by Borghesi (2008) and was shown to positively influence scoring by NFL teams. Humidity may also play an important role in this study in terms of physical exhaustion of the players in high humidity environments or through its impact on the ball traveling through the air. Although its overall impact is uncertain, ex-ante, its importance to scoring will be tested through the empirical model. Barometric pressure is also a component of air density, which could play a role in scoring for a football game. Like humidity, we do not have an ex-ante prediction of the sign of this variable, but it is included to be tested for any impact it may have on influencing scoring in an NFL game.
Although not included in air density, wind speed is another weather-related variable that has been shown to play a role in scoring in NFL games (Borghesi, 2008). High wind speeds make it less likely to be able to successfully pass the ball, making scoring less likely. Therefore, wind speed is expected to have a negative effect on scoring in NFL games.
A dummy for overtime games is included to account for higher scoring than would be expected in games due to the teams playing additional time and scoring additional points to determine a winner during a single 15-minute, at most, frame. It is expected that the dummy for overtime games will have a positive effect on scoring. To account for differences from team to team, dummy variables for home and road teams are included in the model to account for which teams are playing in a given game.
The regression model, therefore, takes the following form, incorporating all of the independent variable categories already described.
(Sum of Points Scoredi) = [[alpha].sub.0] + [[beta].sub.1] (Betting Market [Total.sub.i] + [[beta].sub.2] (Temperature) + [[beta].sub.3] (Humidity) + [[beta].sub.4] (Precipitation) + [[beta].sub.5] (Barometric Pressure) + [[beta].sub.6] (Wind Speed) + [[beta].sub.7] (Altitude) + [[beta].sub.8] (OT Game Dummy) + [epsilon] [[beta].sub.i] (Home Team Dummy Variables) + [epsilon] [[beta].sub.j] (Road Team Dummy Variables) + [[SIGMA].sub.i] (2)
All data used on game outcomes, betting market odds, and betting percentages were gathered from Sports Insights (http://www.sportsinsights.com). Data on the weather-related variables were gathered from historical records archived for each city in the National Football League by Weather Underground (http://www.weatherunder-ground.com), which actively captures weather data from a large number of weather stations, providing information on local weather conditions in very close proximity to the stadium. Due to the role of atmospheric conditions in relation to scoring being the focus of the paper, the data set used in this study focuses solely on outdoor stadiums. (1) Retractable-roof stadium games were not included in the sample due to the data set not having information of whether the roof was open or closed during the game. Including all games played in retractable-roof stadiums did not lead to major changes in the results seen in this study, but in focusing on weather-related variables, we restrict our sample to purely outdoor venues.
Summary statistics of the non-binary variables for the regressions and variables that will be used later in the study are shown in Table 1.
The regression results with the sum of the scores of both teams in an NFL game as the dependent variable is shown in Table 2. Four separate regression results are shown. The first two use air density and the last two use the individual components of air density as explanatory variables. Regression results I and III do not include the betting market total as an independent variable, while regression results II and IV include the total as an explanatory variable as it relates to the combined scoring of both teams in an NFL game. Results are presented with heteroskedasticity and autocorrelation-consistent standard errors and covariances due to the presence of these factors following the initial running of the regression model.
From results I and II, it can be seen that air density was not shown to have a significant effect on the combined scoring in the game. The impact was negative, but not at all relevant in relation to statistical significance. When air density was included in the model, both precipitation and wind speed were shown to have negative and significant effects on combined scoring, whether the betting market total was included in the model or not.
When the individual components of air density were included in the model as explanatory variables, the atmospheric variables that were shown to have statistical significance in relation to the total number of points scored in an NFL game were humidity and precipitation. Humidity was shown to have a positive effect on scoring in a football game, while precipitation was shown to have a negative effect. In model specifications II and IV, where the betting market total was included in the model, the total was shown to be statistically significant, as expected. In specification IV, where the individual components of air density and the betting market total were included in the model, humidity and precipitation were found to be significant determinants of scoring. The other variables related to atmospheric conditions (i.e., temperature, barometric pressure, wind speed, and altitude) did not have a significant impact on scoring.
From the regression results of Table 2, it appears that the key variables that affect scoring are humidity, precipitation, and possibly wind speed. Air density as a single explanatory variable was not shown to significantly impact scoring, but it appears its individual components, coupled with other atmospheric condition-related variables, will offer more insight into scoring, expected scoring, and the relation between these two variables for NFL games.
One potential issue in the regression results is that some of the individual atmospheric conditions variables are not independent. Two places where statistically significant correlations were found between the weather-related variables was in precipitation and humidity (positive) and temperature and altitude (negative). Although this makes the isolation of the true variable driving the scoring results (as it relates to weather) more difficult to decipher from the regression results, we will attempt to isolate these factors when observing the betting market returns based on betting simulations in the following section of this paper. In that section, we will investigate simple wagering strategies, based on individual variables (then look at a combination of a couple of variables), that may reject market efficiency and potentially generate positive returns.
To ascertain whether the atmospheric conditions and other weather-related variables are incorporated into market expectations, we investigate whether the independent variables used in the previous section have any explanatory power as it pertains to results in the betting market. Specifically, we test if the weather condition variables used in the previous section have any explanatory power as it relates to whether an NFL game goes over or under the posted closing total in the over/under market. To do this, we ran dummy dependent variable models with the results of the game being an over as the dependent variable. The dependent variable takes a value of 1 when the combined score of both teams exceeds the betting market total (closing price) and a value of 0 when fewer points are scored by both teams than the posted total. Games in which the number of points scored by both teams equals the posted total are considered "pushes" with all monies returned to bettors and are not included in this sample. For comparison purposes, results of both a logit and probit model with the result of over as the dependent variable are shown in Table 3.
When air density is included in the model, it is not found to have a significant effect on if a game goes over the posted betting market total. In both the logit and probit specifications where air density is included in the model, the only significant variable found was wind speed, which was shown to have a negative effect and was significant at the 10% level.
When the individual components of air density were included in the model (both logit and probit specifications), the only independent variable that was shown to have statistical significance as it relates to a game going over in the totals market was humidity. Humidity was found to have a positive and statistically significant effect on the game going over at the 5% level in both model specifications. When humidity is low, it appears that games have a greater chance of going under the posted total, as fewer points are scored than expected. On the other hand, when humidity is high, more points are scored than expected, resulting in a greater number of overs in the totals market.
None of the other independent variables were found to have statistical significance in these regressions. Factors that have been shown to impact the outcome in totals markets in previous studies such as temperature and precipitation were not shown to have a significant impact in this sample. The market may have adjusted over time to temperature and precipitation factors in the wagering market as technology has made this information low-cost and easy to obtain for interested bettors. In most cases, these factors are discussed on NFL pregame shows and are available on many betting websites. Humidity, on the other hand, is seldom discussed and is not generally presented with other weather information on betting sites. Therefore, its impact on the betting market and the overall level of humidity may not be well known to bettors, leaving the possibility of having a significant impact on the betting market for totals.
Although participants in the totals betting market ultimately only care about whether a game actually goes over or under the posted total, it is informative to examine how many more or fewer points are scored compared to expectations (the posted total) due to weather conditions in an NFL game. This analysis may also be helpful to fantasy football players, a large worldwide market, when determining what weather conditions may be optimal for more scoring from their fantasy team players. Therefore, we use the same independent variables as previously used, but instead of using the logit/probit model, we run an ordinary least squares model (presented with HAC standard errors and covariances) with the difference between the combined score of both teams and the posted closing total on the game as the dependent variable. The regression results are presented in Table 4.
The amount to which a game exceeds or does not meet the total in NFL games are influenced by a variety of weather-related factors. Just as in the logit and probit models of whether the game goes over or under the total, air density was not found to have a significant effect, but wind speed was found to have a negative and significant effect, with greater wind speeds leading to lower scoring games relative to the total. In the air density relationship, the level to which the game exceeds or does not meet the total was also shown to be significantly influenced by precipitation, which was found to have a negative and significant effect at the 10% level.
When considering the individual components of air density and other atmospheric conditions on the difference between the combined score in the game and the closing total, humidity was shown to have a positive and significant effect (as in the logit and probit models), while precipitation, barometric pressure, wind speed, and altitude were shown to have a negative effect on the difference between the combined score and the closing total. Although the simple results compared to the total are more important to over/under bettors, these results may be informative to those interested in the role of atmospheric conditions on scoring for fantasy sports.
Betting Market Simulations and Returns
To investigate further the possibilities of atmospheric conditions not fully being incorporated into the betting market total, we use simple betting simulations to determine if any basic strategies won more often than implied by market efficiency during our sample. We separately examine the role of humidity, precipitation, wind speed, and temperature as it relates to the percentage of overs compared to unders in the totals betting market for the NFL. In each case, we look at the extremes as it relates to the individual atmospheric conditions-related variables, testing subsets of the data where the values of the variable were either particularly high, low, or study both extremes. In addition, given the individual results, we investigate situations where humidity and wind speed interact and calculate returns under those conditions.
For each variable, we test the simple strategy of wagering in the totals market when the variable is greater than or less than a certain value. For each case, the number of overs, unders, pushes, win percentage of the strategy, and log likelihood ratio tests for the null hypotheses of a fair bet and no profitability are shown.
The first scenario to be tested relates to humidity. Given that humidity was shown to influence the percentage of games that went over the total in the logit and probit models of the previous section, we test games where humidity was quite high (70% or more, 75% or more, 80% or more) and where it was relatively low (60% or less, 55% or less, and 50% or less). We also calculate the returns to a combined strategy of wagering at both extremes (betting the over when humidity is high and the under when humidity is low). The results are shown in Table 5.
Games played in high humidity tend to have more overs, as compared to unders, when wagering against the betting market total. In subsets of games where humidity was 70% or more, 75% or more, and 80% or more, the over won more than 52.4% of the time, the percentage needed to overcome the commission of the sportsbook, in each case. The null hypothesis of a fair bet could be rejected for the subset of games where the humidity was 75% or more. In low humidity, the under outperformed the over in the totals betting market. In low humidity conditions, defined here as 60% or less, 55% or less, and 50% or less, the under won more than half the time and the win percentage exceeded the necessary threshold to overcome the commission of the sportsbook in each subset. However, the results were not found to be statistically significant in relation to the null hypothesis of a fair bet.
Combining the subsets of high humidity (75% or more) and low humidity (55% or less) games, it was found that betting the over in high humidity and betting the under in low humidity was found to win 55.60% of the time, rejecting the null of a fair bet at the 1% level. As the logit and probit models illustrated in the previous section, the impact of humidity on scoring does not appear to be fully encompassed into the betting market total. Therefore, simple strategies based upon the impact of humidity on scoring are found to win over 50% of the time.
Although there are exceptions, the high humidity results are mainly driven by high scoring games that exceeded the posted total in the cities with the highest average humidity levels in the sample (i.e., Seattle, Green Bay, Cincinnati, Oakland, and Miami). During the sample, games in these cities went over the total in more than 57% of the games (112 overs, 84 unders, 4 pushes). It appears that bookmakers do not fully account for the role of humidity and scoring in these cities. The low-humidity under games, however, were not as attributable to the lowest average humidity NFL cities in our sample (i.e., Denver, New York, Kansas City, San Diego, and Washington), where the under only won slightly more than 51% of the time in these cities (96 overs, 102 unders, and 2 pushes). The low-humidity games where the under was found to win more often than the over was distributed more evenly across the sample than the high-humidity games that were found to go over the posted total.
Precipitation was tested in relation to returns in the totals betting market in a similar fashion to humidity, except we tested the results as the inches of precipitation increased. Given the results from the previous section, we tested if there were more overs or unders based upon how much precipitation occurred on that day and if it ultimately had any impact on actual scoring compared to expected scoring in NFL games.
The level of precipitation does not appear to have significant effects on whether a game goes over the posted total in the over/under betting market. In each of the subsets of precipitation considered--0.01 inches or more, 0.1 inches or more, 0.25 inches or more, and 0.5 inches or more--the null hypothesis of a fair bet could not be rejected. Even though precipitation has a negative and significant effect on the total number of points scored in an NFL game, the level of precipitation is correctly incorporated into the total in the betting market. Given that information on precipitation is embedded in the closing market price of the total, profitable betting strategies based on this information do not appear to exist.
Table 7 uses the same methodology to test returns in the totals betting market when wind speed is either high or low on game days in the NFL.
Wind was another factor that was found to impact the total number of points scored in an NFL game, but not to have a significant effect when it came to totals market betting. The results of the betting simulations generally reinforce this idea, with the notable exception that strategies focused on days with low wind speeds, specifically 6 or less miles per hour, was found to reject the null of a fair bet at the 10% level. Overs won more than 53% of the time when the winds are generally calm during NFL games, leading to the possibility that lack of wind may not fully be incorporated into the betting market total.
Temperature is tested in the same manner as previously described, examining both when the temperature was relatively high and relatively low for our sample of NFL games. Results are shown in Table 8.
In terms of temperature, the only situation where the null hypothesis of a fair bet could be rejected was in the tail of the distribution when games were played in cold temperatures (35 degrees or less). It appears that when the weather is cold, the total is pushed to a level that is too low, resulting in more overs than unders. In all other cases, except for the extreme case mentioned, win percentages are basically around 50% on the over and under, with the null of a fair bet unable to be rejected.
Given the results for both humidity and wind speed, we combine the two strategies and investigate the results when humidity was high (75% or more) coupled with low wind speed days (8 mph or less, 6 mph or less, and 4 mph or less). The results are shown in Table 9.
Combining two of the factors where positive and significant returns are generated compared to a fair bet improves the results even further. In cases of high humidity coupled with low winds, the strategy of wagering on the under wins more than 60% of the time on extremely calm days with high humidity. These results are able to reject both the null of a fair bet and the null of no profitability.
Atmospheric Conditions and Game Statistics
Given the findings of the previous section that simple betting strategies based on humidity and wind speed lead to rejections of the null hypothesis of a fair bet and are high enough in some cases to reject the null of no profitability, it is useful to delve further into the issue and attempt to understand the elements of the game of football where weather and related factors play in leading to the difference between actual and expected scoring based on the betting market totals for NFL games. To that end, we gathered passing yards, rushing yards, and turnover information from each game within our sample. Given this data, we ran a regression model similar to that used when we tested for the effect of atmospheric pressure on points scored, except we used the individual game statistics as the dependent variable and the atmospheric conditions variables (and the others explained in that section) as the explanatory variables in a series of regression models.
Our goal is to attempt to explain where the atmospheric variables impact game play and determine if it is possible to explain why individual factors such as humidity may account for the differences between actual and expected scoring in an NFL game. The regressions were run in the same manner as when we examined the effect of atmospheric pressure on points scored, with dummies to control for the home and visiting teams, and are presented with HAC-consistent standard errors and covariances due to initial concerns about heteroskedasticity and autocorrelation within the regressions. The results are presented in Table 10 for the sum of passing yards, rushing yards, total yards, and turnovers for both teams in each NFL game in our sample.
The yardage and turnover regressions reveal specific results as to how offenses and defenses perform in various atmospheric conditions. Total yards, in addition to yards passing and rushing, are influenced by overtime games in a positive and significant manner, which is to be expected as more plays are added to the game due to additional time. Precipitation and wind speed were also shown to have negative effects on total yardage in an NFL game. Both variables were shown to negatively impact yards gained by the offense.
Turnovers in NFL games were shown to be positively influenced by precipitation and negatively influenced by altitude. Precipitation not only leads to fewer total yards, but also to teams turning the ball over more often due to fumbles and interceptions. Higher altitudes likely lead to fewer turnovers due to the thin air, causing more overthrows, rather than underthrows, on poorly thrown passes, which may make interceptions more difficult for defensive backs to achieve.
In terms of passing yards, temperature was found to have a positive and significant effect, precipitation was shown to have a negative and significant effect, and wind speed was found to have a negative and significant effect. Higher temperatures lead to more passing yardage in a game, but higher wind speeds and more precipitation lead to fewer yards due to the difficulty in completing passes in poor conditions.
Rushing yards were also influenced in a significant manner by temperature, as higher temperatures led to fewer yards rushing. The key atmospheric variable revealed as the difference between actual and expected scoring, humidity, was also shown to play a role in yards rushing. Although relatively small, statistical significance was found in relation to this variable, revealing that higher humidity led to more rushing yards. This result does not appear to be driven by teams in high average humidity markets (i.e., Seattle, Green Bay, Cincinnati, Oakland, and Miami) that may have just had excellent running backs and offensive lines during the timeframe studied as only one team (Seattle) ranked in the top five (fourth) in rushing of the teams not playing their home games in a dome or in a stadium with a retractable roof. The other four teams were at or below team average for rushing in our sample. (2)
Although the actual link to performance is not certain, our belief is that high humidity days may lead to more fatigue on defenses, leading to situations where the rushing game does slightly better than expected. This impact does not appear to be fully incorporated into the betting market total, unlike the impact of precipitation and temperature, leading to more overs on days with high humidity and more unders on days with low humidity.
Discussion and Conclusions
The role of atmospheric conditions on actual and expected scoring was explored for the National Football League. Following research conducted on the role of air density in Major League Baseball (Paul, Weinbach, & Weinbach, 2014) and previous research on weather effects in football, the impact of temperature, humidity, precipitation, barometric pressure, wind speed, and altitude were investigated in terms of overall scoring in NFL games and in relation to the NFL betting market.
There are similarities between the weather-related results for baseball (Paul, Weinbach, & Weinbach, 2014) and football, but also some important differences. In both sports, atmospheric conditions were shown to significantly impact scoring and the importance of weather-related variables were shown to not fully be incorporated into the betting market totals (over/under bets). In baseball, this was captured by the air density statistic. In the NFL, however, air density was not found to be significant, but individual weather-related factors were shown to play a significant role in scoring.
Overall scoring in an NFL game was shown to be significantly influenced by both humidity and precipitation and to a lesser extent by wind speed. Humidity was shown to have a positive effect on overall scoring, while precipitation and wind speed were shown to have a negative effect. Through logit and probit models of scoring compared to expectations, the betting market total was found to properly account for all of the atmospheric and weather-related variables except for humidity. Humidity was shown to have a positive and significant effect on the number of overs at the 5% level. In our sample, high humidity days resulted in more overs and low humidity days resulted in more unders.
The calculation of betting market returns using simple betting simulations confirmed the result of humidity playing an important role in the totals betting market, as high humidity days (75% or more) and low humidity days (55% or less) were both shown to win more than 55% of the time when taking the over and under in these situations, respectively. Although win speed was not shown to be significant in the logit and probit models, betting simulations did reveal that low wind conditions resulted in more overs than unders in our sample. Combining situations that appear favorable for scoring--high humidity and low wind speed--resulted in win percentages of simple strategies of around 60%, rejecting the null hypothesis of both a fair bet and of no profitability.
To explore the rationale behind why atmospheric conditions effect both actual and expected scoring in the NFL, the role of these weather-related variables in explaining game statistics were tested. As it relates to the role of offense and scoring, the sport of football is quite different from baseball. Scoring can be generated through distinct strategies generated through a running game or a passing game. These offensive strategies are influenced in different fashions by atmospheric conditions; therefore, a deeper investigation of offensive statistics is necessary for football compared to the study of baseball (Paul, Weinbach, & Weinbach, 2014) to decipher how weather-related factors impact scoring in the sport.
Regression models were used to explore changes in total passing yards, total running yards, total yards, and turnovers on a game-by-game basis. Temperature was shown to positively influence passing yards, while precipitation and wind speed had a negative effect on passing. Running yards were negatively influenced by temperature, but positively influenced by humidity. With humidity being the variable that is not fully incorporated into the betting market total, it appears that over/unders are set with the near complete understanding of how precipitation, wind speed, and temperature influence passing yards, but underestimates the role of humidity as it relates to running the football. This appears to be the link that influences the rejection of market efficiency, as games on high humidity days tend to be more likely to go over the total and games on low humidity days tend to be more likely to go under the total.
Possible reasons that the market does not fully incorporate weather effects into the betting market total could be due to the relatively nuanced effect that is happening with weather due to the different factors involved (temperature, humidity, wind speed, precipitation, etc.) and that with the availability of better weather data, these inefficiencies may not persist beyond the relatively short time frame (five years) presented in this research. Another rationale is that given that the weather impacts the totals market, the totals market is not as liquid as the sides market, even for a popular betting sport such as the NFL. Therefore, due to less liquidity and lower betting limits in NFL totals markets (Paul & Weinbach, 2002), it may not be worth the transactions costs to attempt to fully account for all of the intricacies of atmospheric conditions when setting totals for NFL games.
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Endnotes
(1) For those curious about totals market results within the sample for games played in domed and retractable roof stadiums, these results are included in Appendix 1. Although overs win slightly more often than unders in both domes and retractable-roof settings, statistically significant results were not found.
(2) A simple regression of city average humidity on rushing yards in the sample revealed a negative coefficient on average humidity (-13.1606), which was not statistically significant as it relates to rushing yards (t-stat of -0.3899). Appendix 1. Totals Market Results from Domed and Retractable Roof Stadia: 2011-2014 Setting Overs Under s Pushes Over Win Percentage Dome 85 74 1 52.46% Retractable Roof 81 75 3 51.92% Dome and 166 149 4 52.70% Retractable Roof Other Games 482 467 11 50.79% Setting Log Log Likelihood Likelihood - Fair Bet - No Profits (50%) (52.4%) Dome 0.7616 0.0743 Retractable Roof 0.2308 NA Dome and 0.9179 0.0128 Retractable Roof Other Games 0.2371 NA The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for a = 0.05), and 6.635 (for [alpha] = 0.01). * is significant at 10%. ** is significant at 5%. *** is significant at 1%.
Rodney J. Paul (1)
(1) Syracuse University
Rodney J. Paul, PhD, is a professor in the Department of Sport Management in the David B. Falk College of Sport and Human Dynamics. His research interests include studies of market efficiency, prediction markets, behavioral biases, and the economics and finance of sport. Table 1. Summary Statistics - Outdoor NFL Games: 2010-2014 variable Mean Median Standard Deviation Temperature 53.19 54.00 15.37 Humidity 67.74 69.00 12.63 Precipitation 0.07 0.00 0.22 Barometric Pressure 30.08 30.07 0.38 Wind Speed 6.69 6.00 3.87 Altitude 542.05 248.00 1040.94 Betting Market Total 43.98 44.00 4.24 Sum of Scores 44.64 44.00 14.03 Rushing Yards 231.60 225.00 62.43 Passing Yards 452.93 443.00 119.45 Total Yards 684.53 674.00 121.10 Turnovers 3.15 3.00 1.81 Table 2. Regression Results - Atmospheric Conditions and Weather Variables on Combined Scoring by Both Teams in NFL Games: 2010-2014 Variable I II III Intercept 42.1012 (***) 19.3050 39.7782 (***) (2.7880) (1.2122) (5.6486) Air Density -1.3102 -2.211 (-0.1087) (-0.1856) Total 0.5772 (***) (4.1840) Temperature -0.0240 (-0.6776) Humidity 0.0821 (**) (1.9536) Precipitation (in.) -5.5852 (***) -4.6694 (**) -7.1127 (***) (-2.6189) (-2.1978) (-3.1145) Barometric Pressure -0.0050 (-1.0885) Wind Speed -0.2297 (*) -0.2398 (*) -0.1952 (-1.8052) (-1.9015) (-1.5042) Altitude -0.3497 (-0.6950) OT Game 1.8777 2.1708 2.0938 (0.9974) (1.1628) (1.1126) Home Team Dummies YES YES YES Road Team Dummies YES YES YES Variable IV Intercept 14.4596 (1.5902) Air Density Total 0.5986 (***) (4.3384) Temperature -0.0210 (-0.6001) Humidity 0.0943 (**) (2.2618) Precipitation (in.) -6.4037 (***) (-2.8244) Barometric Pressure -0.0055 (-1.2139) Wind Speed -0.1984 (-1.5432) Altitude -0.3948 (-0.7923) OT Game -0.3949 (1.3001) Home Team Dummies YES Road Team Dummies YES Statistical significance is shown by (*) -notation: (*) -10%, (**) -5%, (***) -1% Table 3. Logit and Probit Regression Results of Atmospheric Conditions and Weather Variables and Results of Over in the Totals Betting Market: NFL 2010-2014 Variable I - Logit II - Probit III - Logit Intercept 0.9198 0.5269 9.2534 (0.3840) (0.3564) (0.0045) Air Density -1.2899 -0.7597 (-0.6749) (-0.6442) Temperature 0.0002 (0.0420) Humidity 0.0135 (**) (2.0024) Precipitation (in) -0.1606 -0.1008 -0.4269 (-0.4706) (-0.4816) (-1.1588) Barometric Pressure -0.1368 (-0.6192) Wind Speed -0.0349 (*) -0.0214 (*) -0.0305 (-1.7339) (-1.7223) (-1.4518) Altitude -0.6712 (-0.0033) OT Game 0.3367 0.2019 0.3731 (1.1143) (1.0972) (1.2285) Home Team Dummies YES YES YES Road Team Dummies YES YES YES Variable IV - Probit Intercept 3.9134 (0.0033) Air Density Temperature 0.0001 (0.0130) Humidity 0.0086 (**) (2.0624) Precipitation (in) -0.0842 (-0.6151) Barometric Pressure -0.0843 (-0.6151) Wind Speed -0.0186 (-1.4352) Altitude -0.2359 (-0.0020) OT Game 0.2258 (1.2222) Home Team Dummies YES Road Team Dummies YES Statistical significance is shown by (*) -notation: (*) -10%, (**) -5%, (***) -1%. Table 4. Regression Model of Atmospheric Conditions and Weather Variables and the Difference in the Combined Score in the Game and the Closing Total: NFL 2010-2014 Variable I II Intercept 2.6050 -2.5208 (0.1678) (-0.5437) Air Density -2.8797 (-0.2332) Temperature -0.0191 (-0.5284) Humidity 0.1025 (**) (2.3814) Precipitation (in) -3.9984 (*) -5.9281 (***) (-1.8633) -2.6315) Barometric Pressure -0.0058 (***) (-6.3322) Wind Speed -0.2472 (**) -0.2004 (*) (-1.9676) (-1.6421) Altitude -0.4252 (***) (-3.7600) OT Game 2.3856 2.6468 (1.3497) (1.5085) Home Team Dummies YES YES Road Team Dummies YES YES Statistical significance is shown by (*) -notation: (*) -10%, (**) -5%, (***) -1%. Table 5. Totals Betting Market Returns for the NFL Based on Humidity: 2010-2014 Humidity Over Under Push Win % LLR-Fair Bet High Over Win% 80 or more 101 83 2 54.89% 1.7637 75 or more 168 136 4 55.26% 3.3747 (*) 70 or more 238 206 5 53.60% 2.3083 Low Under Win% 60 or less 124 137 1 52.49% 0.6478 55 or less 74 95 1 56.21% 2.6162 50 or less 47 52 1 52.53% 0.2526 Strategy Win% 75 or more & 55 or less 263 210 5 55.60% 5.9512 (**) Humidity LLR-No Profits High 80 or more 0.4662 75 or more 1.0155 70 or more 0.2667 Low 60 or less 0.0013 55 or less 0.9889 50 or less 0.0008 75 or more & 55 or less 1.9747 The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for [alpha] = 0.05), and 6.635 (for [alpha] = 0.01). (*) is significant at 10%. (**) is significant at 5%. (***) is significant at 1%. Table 6. Totals Betting Market Returns for the NFL Based on Precipitation Inches of Over Under Push Under LLR - LLR - No Precipitation Win % Fair Bet Profits 0.5 or more 22 23 0 51.11% 0.0222 NA 0.25 or more 48 40 1 54.55% 0.7283 0.1657 0.1 or more 79 74 1 51.63% 0.1634 NA 0.01 or more 95 85 2 52.78% 0.5558 0.0114 The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for [alpha] = 0.05), and 6.635 (for [alpha] = 0.01). * is significant at 10%. ** is significant at 5%. *** is significant at 1%. Table 7. Totals Betting Market Returns for the NFL Based on Wind Speed Wind Speed Over Under Push Win % LLR - LLR - No (MPH) Fair Bet Profits High Under Win% 10 or more 91 108 4 54.27% 1.4540 0.2858 8 or more 162 181 5 52.77% 1.0530 0.0209 6 or more 256 270 7 51.33% 0.3727 NA Low Over Win% 6 or less 275 237 6 53.71% 2.8229 (*) 0.3640 5 or less 226 197 4 53.43% 1.9897 0.1864 4 or less 170 147 3 53.63% 1.6702 0.1980 The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for [alpha] = 0.05), and 6.635 (for [alpha] = 0.01). (*) is significant at 10%. ** is significant at 5%. *** is significant at 1%. Table 8. Totals Betting Market Returns for the NFL Based on Temperature Temp. Over Under Push Win % LLR - LLR - No (degree F) Fair Bet Profits High Under Win% 75 or more 53 63 2 54.31% 0.8631 0.1735 70 or more 115 110 2 51.11% 0.1111 NA 65 or more 186 181 2 50.68% 0.0681 NA Low Over Win% 45 or less 171 175 6 50.58% 0.0462 NA 40 or less 123 114 3 51.90% 0.3419 NA 35 or less 92 69 2 57.14% 3.2970 (*) 1.4709 The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for [alpha] = 0.05), and 6.635 (for [alpha] = 0.01). (*) is significant at 10%. ** is significant at 5%. *** is significant at 1%. Table 9. Totals Betting Market Returns for the NFL Based on High Humidity and Low Wind Humidity 75%+ Over Under Push Over LLR - LLR - No and Wind Speed Win % Fair Bet Profits (mph): 8 or less 124 98 1 55.86% 3.0520 (*) 1.0785 6 or less 92 66 1 58.23% 4.2980 (**) 2.1793 4 or less 63 41 1 60.57% 4.6892 (**) 2.8291 (*) The log likelihood test statistics have a chi-square distribution with one degree of freedom. Critical values are 2.706 (for [alpha]= 0.10), 3.841 (for [alpha] = 0.05), and 6.635 (for [alpha] = 0.01). (*) is significant at 10%. (**) is significant at 5%. *** is significant at 1%. Table 10. Game Statistics Regressions for the NFL Based on Atmospheric Conditions Variable Sum or Passing Sum or Rushing Sum or Total Yards Yards Yards Intercept 430.4004 (***) 194.8284 (***) 625.2287 (***) (7.5095) (6.0787) (10.7423) Temperature 0.9481 (***) -0.5482 (***) 0.4000 (3.2925) (-3.4041) (1.3677) Humidity -0.0897 0.3381 (*) 0.2484 (-0.2622) (1.7679) (0.7153) Precipitation (in.) -48.4996 (***) 3.3704 -45.1292 (**) (-2.6093) (0.3243) (-2.3910) Barometric Pressure 0.0007 0.0002 0.0009 (0.0186) (0.0114) (0.0246) Wind Speed -2.6734 (**) -0.4585 -3.1419 (***) (-2.5305) (-0.7930) (-2.9286) Altitude -2.5514 0.3014 -2.2500 (-0.6230) (0.1316) (-0.5411) OT Game 53.1712 (***) 33.0609 (***) 86.2321 (***) (3.4715) (3.8599) (5.5441) Home Team Dummies YES YES YES YES Road Team Dummies YES YES YES YES Variable Sum or Turnovers Intercept 5.0497 (***) (5.3678) Temperature 0.0004 (0.0886) Humidity -0.0038 (-0.6751) Precipitation (in.) 0.7471 (**) (2.4489) Barometric Pressure -0.0003 (-0.6139) Wind Speed -0.0075 (-0.4298) Altitude -0.1569 (**) (-2.3354) OT Game -0.3576 (-1.4226) Home Team Dummies YES Road Team Dummies YES Statistical significance is shown by (*) -notation: (*) -10%, (*) -5%, (***) -1%.
