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The impact of atmospheric conditions on the baseball totals market.

Introduction

Bahill et al. (2009) published a paper in the International Journal of Sports Science and Engineering entitled "Effects of Altitude and Atmospheric Conditions on the Flight of a Baseball." This paper illustrated, through both physics and empirical analysis, that altitude and weather affect air density, which impacts how far a batted baseball will travel. Air density consists of four factors: altitude, temperature, humidity, and barometric pressure. Altitude, temperature, and humidity each have an inverse relationship with air density, while barometric pressure is positively related to air density.

The findings of the paper by Bahill et al. (2009) show that altitude is the most important factor determining air density, followed by temperature, barometric pressure, and humidity. The distance a baseball travels is ultimately affected by two countervailing forces, drag force and Magnus force. As air density gets smaller, drag force gets smaller which allows the ball to travel farther, but Magnus force also gets smaller as air density decreases, which decreases how long the ball will be held aloft, ultimately decreasing the distance the ball will travel. In simulations shown in the study by Bahill et al. (2009), drag force dominates Magnus force which results in a baseball traveling farther when air density decreases. This could have has a major impact on the number of home runs in a baseball game. Through the estimates of the authors, a 10% decrease in air density (on a typical July afternoon) will produce a 4% increase in the distance a baseball travels.

The results of the study of Bahill et al. (2009) are likely to have profound importance for the baseball totals betting market. Since atmospheric conditions have an impact on the distance a baseball will travel, which influences the number of home runs and number of runs scored in a baseball game, air density and other weather-related variables could play a role in affecting the game outcome in the totals market. If air density truly impacts scoring in a baseball game, this variable will have an inverse relationship to runs scored in a game. Controlling for the posted total in the over/under market, it is straightforward to test if air density will impact the total number of runs scored in a game and influence the number of overs vs. unders in the baseball totals wagering market. If air density actively plays a role, and is currently unrecognized or inaccurately interpreted by market forces, simple wagering strategies based on atmospheric conditions may generate positive returns in the betting market.

In addition to studying air density, given the availability of other weather condition variables from baseball box scores, we are also able to investigate the role of wind speed, wind direction, and overall weather condition (sunny, cloudy, clear, etc.). We include air density, wind speed and direction, and dummy variables for weather condition in a series of regression models to determine if these factors impact the total number of runs scored in a game, the frequency of overs in the totals market, and the percentage bet on the over to determine the extent to which these factors influence the totals market. From there, we construct simple betting strategies and use betting simulation to determine the returns to these strategies using data from the 2012 baseball season.

Literature Review on Totals Markets and Weather

Baseball totals were first studied by Brown and Abraham (2002), who found some market inefficiencies based upon steaks of overs and unders during the 1997 season, but subsequently found that the market was efficient in other years. Paul and Weinbach (2004) commented on the paper and noted that the market efficiency studies carried out by Brown and Abraham (2002) were incomplete, as their data set did not include the odds adjustment commonly seen in the baseball totals market (where the total itself is typically accompanied by an adjustment either toward the over or the under that makes bettors lay more to win $1 on a particular side of the wager). This elicited a response to the comment by Brown and Abraham (2004) and an evaluation of the debate by Gandar and Zuber (2004), which ultimately concluded that the odds adjustments are necessary for any study of market efficiency involving baseball totals.

The baseball totals market was also studied by Paul et al. (2013), where more detailed betting market data obtained from www.sportsinsights.com was used to investigate the preferences of gamblers in both the sides and totals market. In relation to totals, the under bet was shown to be popular when high-quality pitchers, proxied by Cy Young Award voting, were pitching in a particular game. This result is counter to baseball games where these elite pitchers are not pitching and across other sports where the over is shown to be a much more popular wager than the under.

Studies of the impact of weather on financial markets are not new, both in and out of sports. Roll (1984) studied the impact of weather on orange juice prices and further investigated the role of weather conditions on security prices and transaction volume in the stock market (Roll, 1988). Hirschleifer and Shumway (2003) investigated the role of sunshine and other weather conditions on stock returns and found significant impacts on investor reactions and market prices.

The role of weather as a potential fundamental factor that affects betting markets fits into the research questions posed by Brown and Sauer (1993). Weather and track conditions were suggested as important factors related to betting patterns, returns, and market efficiency in the study of horse racing betting markets as early as the late 1970s (Figlewski, 1979). The impact of weather effects on the totals market has been shown previously in Borghesi (2008). Heat, wind, and rain were shown to have a negative impact on scoring in the National Football League. Bettors were shown to not fully account for these factors when placing totals wagers on the NFL. Statistically significant profits were shown when weather effects were incorporated into an NFL totals market betting strategy.

Regression Model of Atmospheric Conditions on Runs Scored

To determine if atmospheric conditions play a significant role in the number of runs scored in a baseball game, we use a simple regression model with the number of runs scored in a game as the dependent variable. Since the number of runs scored is based upon team performance variables, we use the betting market total (over/under wager) as our control variable for the strength of the scoring prowess of the teams playing and the ability of the starting pitchers in the game. If all of the information about atmospheric conditions is included in the market price, due to the knowledge of these factors by the sports book and/or bettors, then air density, wind speed and direction, and weather condition dummy variables should not have a significant impact on runs scored beyond the adjustment already included when using the market price (total) as an independent variable. All data used on game outcomes, betting market odds, and betting percentages were gathered from www.sportsinsights.com.

Baseball total market bets are generally not placed at flat odds. Most totals include an odds adjustment, with more money needed to be placed on either the over or the under to win one dollar. These odds adjustments are available within our data set and this information is included in the regression model by converting the odds adjustment into the probability of an over for each game. If the odds adjustment is on the over (laying more on the over to win one dollar), the probability of an over is greater than one-half, while if the odds adjustment is on the under (laying more on the under to win one dollar), the probability of an over is less than one-half. This variable is included to capture the additional likelihood of more (fewer) runs than the posted total based on the odds adjustment on the over (under).

The next set of independent variables included in the regression model account for the different atmospheric conditions seen at baseball games. First, directly from the study of Bahill, Baldwin, and Ramberg (2009), we include air density as an independ ent variable. The equation to compute air density is noted on page 119 of their article and is noted below as 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)

As noted in the introduction, air density has a negative relationship with the distance a ball travels and therefore should have a negative impact on the number of runs scored in a game. If the posted total on the game does not fully incorporate the information embedded in air density (altitude, temperature, humidity, barometric pressure), this variable should have a negative and significant effect on the total number of runs scored in a baseball game. Data on the variables used in the calculation of air density were gathered from historical records archived for each city in Major League Baseball by the website www.weatherunderground.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 ballparks.

Beyond air density, we also include wind speed as an independent variable. Wind speed will also influence how far a ball will fly, but it also plays a role in moving the ball while in-air, leading to more difficult fielding conditions. Windy days can lead to more errors by defenders and potentially more hits than otherwise, if winds were calm. The wind speed variable was taken directly from the box score of each game.

In addition to the box score posting the wind speed, it also notes the wind direction. Home runs are likely to be more frequent when the wind is blowing out from home plate toward the outfield fence. The wind direction could be included in the game total, if the market incorporates this information into the closing price. The wind direction variables are constructed as dummies for each of the categories of wind direction noted in the box score. These include wind in (from center field (CF), right field (RF), and left field (LF)), wind out (from CF, RF, and LF), wind right to left, wind left to right, and wind varies (swirling wind). These are all compared to the baseline where the wind condition is listed as "none." To account for game played in dome stadiums, a dome dummy is included, and for stadiums with a retractable roof, a dummy is included for these games when the roof is closed.

The overall weather conditions are also included in the box score of every game, so we included these categories as dummy variables as well. The weather conditions included are all compared to the omitted dummy variable category of partly cloudy. The other weather conditions included as independent variables are sunny, cloudy, clear, overcast, drizzle, and rain. We also ran the regression omitting these categories, but it did not change the statistical significance of the other variables included in the model.

The regression model therefore takes the following form incorporating all of the independent variable categories described above:

(Total Runs Scoredi) = [[alpha].sub.0] + [[beta].sub.1](Betting Market [Total.sub.i]) + [[beta].sub.2](Over Probability Based on Odds Adjustment) + [[beta].sub.3](Air Density) + [[beta].sub.4](Wind Speed) + [summation] [[beta].sub.i](Wind Direction Dummies) + [summation] [[beta].sub.j](Weather Condition Dummies) + [[epsilon].sub.i] (2)

Summary statistics of the non-binary variables are shown in Table 1. Frequencies of the binary variables (wind direction and weather condition) are shown in Appendix 1.

The regression results using ordinary least squares are shown in Table 2. The coefficient is presented along with the t-statistic in parentheses accompanying the individual variable. *-notation is shown to denote statistical significance of the independent variables with *** noting statistical significance at the 1% level, ** noting statistical significance at the 5% level, and * noting statistical significance at the 10% level. Due to the existence of both heteroskedasticity and autocorrelation in the data, we used Newey-West heteroskedasticity- and autocorrelation-constant standard errors and covariances. The results shown in Table 2 reflect the use of these adjusted standard errors and covariances.

The regression results for total runs scored in a game revealed that the total was found to have a positive and significant effect on total runs scored, but it was found to be less than one (0.6940). The probability of an over based on the odds adjustment to the total was found to be statistically insignificant. The two key variables of interest, air density and wind speed, were both found to have statistically significant results in relation to total runs scored in a game, both at the 1% level. The R-squared of the regression is low (as are other similar regressions below) as this reflects the noisy outcomes of sporting events, which are part of the reason that sports are both interesting to watch and to study. Similar levels of R-squared were found in Sauer et al. (1988) and studies that followed which note the large variability in sporting contest outcomes.

Air density was shown to have a negative effect on total runs scored, as the model of Bahill, Baldwin, and Ramberg (2009) explained in their paper on the physics of baseball. The greater the air density during the game, the fewer runs that are typically scored in that contest. Wind speed was shown to have a positive, although much smaller, effect on total runs scored. This is likely due to the difficulties that high winds pose on the defensive players in a baseball game.

The variables related to direction of the wind were not found to have statistically significant results in relation to total runs scored. This either implies that the total on the game fully captures the direction of the wind (i.e., days when the wind is blowing out which likely leads to more home runs) or the wind direction is not an important determinant of total runs scored in a game. The weather conditions listed in the box score for the day of the game were not found to have a statistically significant effect on total runs, except for the category of clear days. Clear days were shown to have a negative effect on total runs scored (statistically significant at the 10% level). Clear days likely give defenders an advantage, particularly in the outfield, where the lack of clouds may help players more clearly see the flying baseball, leading to more fly ball outs in a game.

With air density and wind speed being shown to have statistically significant effects on total runs scored, above and beyond the posted total on the game, the next step is to determine if betting strategies based upon these variables have any possibility of earning positive returns. With this in mind, we next set up a simple logit model with the game outcome compared to the total as the dependent variable. When the game is an over, the dependent variable takes a value of one, when the game is an under, the dependent variable takes a value of zero. All pushes (cases where the total runs scored equal the total) are removed from the data set for this regression. The independent variables included in this regression are the same as the previous regression as shown in equation (2).

The results of the logit model show that betting strategies based upon air density and wind speed in a baseball could yield positive returns. Both air density and wind speed were found to have statistically significant effects on the number of overs in the baseball betting market with air density being significant at the 1% level and wind speed being significant at the 5% level. Higher air density was shown to make the under a more likely result in the baseball totals betting market, while greater wind speed was shown to make the over a more likely result. As for other independent variables in the logit model, the total was shown to have a negative and significant effect (1% level), meaning that higher totals result in more unders in the betting market, which has been shown previously in the literature in baseball (Paul et al., 2013). The dummy variable for clear days was again shown to have a negative impact the on probability of an over with statistical significance at the 10% level. All other independent variables, other than the intercept, were not shown to have a significant impact on the outcome of the game in the totals betting market.

Before applying the results seen above to betting simulations to calculate the returns to betting strategies based upon air density and wind speed, it is useful to attempt to understand why the sports book may not include these factors in the market price. One rationale could be that the book makers do not fully understand the impact of the subtleties of the atmospheric conditions, but another possibility is that the bettors could misinterpret the effects of certain weather-related factors in the marketplace. To address this possibility, we use data on betting percentages available from www.sportsinsights.com. Their data set includes information on the percentage of bets on the over and the under in the totals market for baseball. The use of this data provides the opportunity to investigate how bettors respond to the atmospheric conditions in a given city on the day a baseball game is played. The same independent variables are used as the previous two regressions (from equation (2)), but the dependent variable is now the percentage bet on the over by bettors in the wagering market. Regression results using ordinary least squares are shown in Table 4.

The percentage bet on the over was shown to rise with the total, as the total has a positive and significant effect on the percentage bet on the over at the 1% level. This has been shown previously in baseball (Paul et al., 2013), as bettors prefer wagering on the over compared to the under, especially when two high-scoring teams play each other, as betting on more scoring compared to less likely brings more consumption value to the bettor.

Air density was shown to have a positive and significant effect on the percentage bet on the over at the 1% level. As air density rises, more bettors place wagers on the over, despite the results seen in Bahill et al. (2009) and the previous regression results shown in this paper which illustrate that air density is inversely related to scoring in a baseball game. In short, the bettors appear to not understand the relationship between air density and scoring as they simply appear to get the relationship wrong. Bettors may not have access to (or consider using) detailed weather data, or may misinterpret or simply ignore this variable. This result could imply that sports books do not fully adjust the totals market to atmospheric conditions while the bettors ignore or misinterpret the relationship between these variables.

Wind speed was not shown to have a statistically significant effect on the percentage bet on the over. The remaining independent variables were also not shown to have statistically significant effects on the percentage bet on the over other than the intercept and the dummy variables for the weather conditions of cloudy and clear (both significant at the 10% level). Both cloudy and clear days were shown to increase the percentage bet on the over in the regression model.

Given the results in this section, it is likely that betting strategies based upon atmospheric conditions may yield positive results in the baseball totals betting market. To illustrate this possibility we construct a simple series of betting rules based on these variables in the next section.

Betting Simulations

Given that air density and wind speed were shown to have significant impacts on the total number of runs scored in baseball games, simple betting simulations are generated to illustrate the returns to wagering strategies based upon these atmospheric conditions. Air density was shown to have a negative impact on runs scored (increasing the likelihood of the under) and wind speed was shown to have a positive impact on runs scored (increasing the likelihood of the over). Given that the impact of these variables is more likely influence the betting market when they are considerably above or below the mean value of these variables, we have chosen a simple betting rule based on a value of a standard deviation above or below the mean of these values as the trigger point for placing a wager. Although simplistic, we believe that calculating the returns from this basic strategy will allow for the determination if it is possible to earn positive returns through knowledge of the atmospheric conditions for a baseball game.

When the air density is shown to be a standard deviation above its mean value, this would lead to a bet on the under (as air density is inversely related to runs scored), while if it is a standard deviation below its mean value it would lead to a bet on the under. The wagering strategy based on wind speed is similar, except that wind speed was shown to have a positive relationship with runs scored, so that when wind speed is a standard deviation above its mean, a bet on the over is placed, and when it is a standard deviation below its mean, an under wager is made. Given that clear days were also shown to have a negative impact on total runs scored and was also statistically significant in the logit model on the frequency of overs, the results for placing a wager on the under on clear days is also calculated. The results for these strategies are shown in Table 5 with the number of bets placed (N), the actual return to the betting strategy noted, the expected return of the betting strategy, and the z-test comparing the actual and expected returns based upon the test established by Gandar et al. (2002).

The simple betting simulations reveal that strategies based upon air density yielded positive returns during the 2012 baseball season. When air density is high (a standard deviation above the mean), a positive return of 0.0609 per dollar bet is shown when wagering on the under due to the inverse relationship between air density and total runs scored. In addition, when air density is low (a standard deviation below the mean), a bet on the over is shown to generate positive returns of 0.0434 per dollar bet. Combining these strategies together yields a large enough sample that the z-test is able to reject the null hypothesis of a fair bet at the 5% level. It appears that simple wagering strategies based upon air density yield positive and significant results for bettors.

Wind speed did not yield as straightforward results as air density. When wind speed was high, wagers on the over outperformed wagers on the under, yielding slight positive (statistically insignificant) returns. When wind speed was low, however, the bet on the under was outperformed by the over, as over wagers yielded positive but insignificant returns. It appears that wind speed and its impact on scoring may be a bit more nuanced than this simple betting rule captures.

In relation to clear days, where the regression results of the previous section revealed that fewer runs are scored on days that meet this weather condition, the betting simulation confirmed that result as wagers placed on the under on clear days earned returns of 0.0168 per dollar bet. This result, combined with the sample size, could not reject the null hypothesis of market efficiency.

Combinations of these atmospheric conditions were tried, but given the small sample size where these factors overlap, meaningful results could not be inferred. Perhaps future studies with more years of detailed data on baseball games will reveal interesting results with strategies involving a combination of these factors.

Conclusions

Based on the work on Bahill et al. (2009) on the physics of baseball, the impact of air density on total runs scored in a baseball game compared to the posted total on the game in the baseball betting market was examined. Air density, which consists of altitude, temperature, humidity, and barometric pressure, was shown to have a negative impact on runs scored beyond that suggested by the posted betting market total. In addition, air density was shown to have a negative and significant effect on the frequency of overs in a logit model related to outcome of wagers placed in the baseball totals market.

In addition to the effects of air density, we added the variables of wind speed, wind direction, and box score-listed weather conditions to the analysis of the effects of atmospheric conditions on the baseball totals market. Wind speed was shown to have a positive and significant effect on runs scored (beyond the posted total) and was shown to positively influence the frequency of overs. The wind direction variables did not reveal significant results and all but one of the weather condition variables were also statistically insignificant. The only weather condition variable that was shown to have a significant result in both the total runs scored regression and in the logit model of game outcomes was clear days. Clear days were shown to negatively impact scoring, likely due to clear days offering optimal weather conditions for fielders to perform defensively.

These factors were then included in simple betting simulations to determine if they could yield positive returns in the baseball totals market. For air density and wind speed, a simple betting rule based on the actual atmospheric conditions being a standard deviation above or below the mean value of these variables was the foundation for the betting rule, with returns to wagering on the over and the under calculated for these situations. For clear days, the returns to the days where this condition was noted in the box score of the baseball game were also calculated.

Wagering on the under when air density was high and wagering on the over when air density was low were both shown to generate positive returns. When combining both strategies related to air density, statistically significant results using the z-test of Gandar et al. (2002) were found at the 5% level. Although returns to wagering on the over when wind speed was high was shown to outperform wagering on the under, both strategies yielded negative returns. When the wind speed was low, however, the opposite of the expected result was found as the over outperformed the under, but statistically insignificant results were found. Wagering the under on clear days was also shown to generate positive returns, but they were not quite large enough to generate statistically significant results. It should be noted that we are using a single season of Major League Baseball data. Given the use of a single season of data, it is still important to note the comments of Osborne (2001) as it relates to market efficiency studies containing short samples. Longer time horizons are preferable to shorter time frames when studying market efficiency and betting market returns. However, a single season of baseball games does contain 2,430 observations and is equivalent to over nine NFL seasons when considering the number of games per year.

Overall, it appears that air density and other atmospheric conditions play a key role in the amount of scoring in a baseball game. Given the positive returns to these strategies, it is possible that the book makers do not fully appreciate the impact of these conditions on scoring, or they may understand that the bettors do not recognize the importance of these conditions. A regression using the percentage bet on the over in the baseball totals market revealed that air density was shown to have a positive and significant effect on the percentage bet on the over, which is the exact opposite result of what is seen with game outcomes. It appears the bettors may actually interpret the impact of atmospheric conditions in the wrong direction, preferring the over to the under when air density is high, even though these conditions make the under the more likely betting market outcome. This result may help in understanding why the sports book does not appear to fully incorporate air density into the total as the betting public seemingly ignores or misinterprets the impact of atmospheric conditions on the number of runs scored in a baseball game.

Appendix 1: Frequencies of Binary Variables-Wind Direction
and Weather Conditions

Sample of 2012 Major League Baseball Games (2430 observations)

Wind Direction   Frequency   Weather Condition   Frequency

In from CF          126            Sunny            259
In from RF          178           Cloudy            420
In from LF          162            Clear            516
Out to CF           317          Overcast           127
Out to RF           284        Partly Cloudy        743
Out to LF           204           Drizzle           16
Right to Left       344            Rain             11
Left to Right       408
Varies              55
None                271
Dome                81


References

Bahill, T. A., Baldwin, D. G., & Ramberg, J. S. (2009). Effects of altitude and atmospheric conditions on the flight of a baseball. International Journal of Sports Science and Engineering, 3(2), 109-128.

Borghesi, R. (2008). Weather biases in the NFL totals market. Applied Financial Economics, 18(12), 947-953.

Brown, K. H., & Abraham, J. F. (2002). Testing market efficiency in the Major League Baseball over-under betting market. Journal of Sports Economics, 3(4), 311-319.

Brown, K. H., & Abraham, J. F. (2004). Response to comment on testing market efficiency in the Major League Baseball over-under betting market. Journal of Sports Economics, 5(1), 96-99.

Brown, W., & Sauer, R. (1993). Does the basketball market believe in the hot hand? American Economic Review, 83(5), 1377-1386.

Figlewski, S. (1979). Subjective information and market efficiency in a betting market. Journal of Political Economy, 87(1), 75-88.

Gandar, J. M., Zuber, R. A., Johnson, R. S., & Dare, W. (2002). Re-examining the betting market on Major League Baseball games: Is there a reverse favorite-longshot bias? Applied Economics, 34, 1309-1317.

Gandar, J. M., & Zuber, R. A. (2004). An evaluation of the debate over testing market efficiency in the Major League Baseball over-under betting market. Journal of Sports Economics, 5(1), 100-105.

Hirshleifer, D., & Shumway, T. (2003). Good day sunshine: Stock returns and the weather. Journal of Finance, 58(3), 1009-1032.

Osborne, E. (2001). Efficient markets? Don't bet on it. Journal of Sports Economics, 2(1), 50-61.

Paul, R. J., & Weinbach, A. P. (2004). Comment on testing market efficiency in the Major League Baseball over-under betting market. Journal of Sports Economics, 5(1), 93-95.

Paul, R. J., Humphreys B. R., & Weinbach, A. P. (2013) The lure of the pitcher: How the baseball betting market is influenced by elite starting pitchers. In L. V. Williams & D. S. Siegel (Eds.), The Oxford handbook of the economics of gambling. Oxford, UK: Oxford University Press.

Roll, R. (1984). Orange juice and weather. American Economic Review, 74(5), 861-880.

Roll, R. (1988) R2. Journal of Finance, 43(3), 541-566.

Sauer, R., Brajer, V., Ferris, S. & Marr, M. (1988). Hold your bets: Another look at the efficiency of the gambling market for National Football League games. Journal of Political Economy, 96(1), 206-213.

Rodney J. Paul (1), Andrew P. Weinbach (2), and Chris Weinbach (3)

(1) Syracuse University

(2) Coastal Carolina University

(3) STEM Partners, LLC

Rodney J. Paul is a professor 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 sports.

Andrew P. Weinbach is an associate professor of economics and the Colonel Lindsey H. Vereen Endowed Business Professor at the E. Craig Wall Sr. College of Business Administration. His research interests include the economics and finance of sports, consumer behavior, and the economics of lotteries and gambling.

Chris Weinbach is a technology consultant for STEM Partners, LLC. He has a lifetime interest in the game of baseball and baseball statistics.

Table 1: Summary Statistics-2012 Major League Baseball
(2430 observations)

                     Total    Combined    Scored     Wind
                                Runs        Air      Speed
                                          Density

Mean                 8.1599    8.3687     1.1422     7.3681
Standard Deviation   1.1234    4.5822     0.0461     4.9832
Median               8.0000    8.0000     1.1466     7.0000

Table 2: Regression Model of Total Runs Scored in Baseball Games
based on Atmospheric Conditions

                            Dependent Variable: Total Runs Scored

                          Variable       Coefficient     Coefficient
                          (t-stat)        Variable        (t-stat)

Intercept               14.7510 ***     Right to Left      -0.3593
                          (4.0792)                        (-0.5044)
Total                    0.6940 ***     Left to Right      -0.8239
                          (6.7961)                        (-1.2197)
Over Probability Based    -0.3845          Varies          -0.1016
  on Odds Adjustment     (-0.2378)     (Swirling Wind)    (-0.1242)
Air Density             -10.1084 ***        Dome           -1.0379
                         (-4.0159)                        (-1.2690)
Wind Speed               0.0859 ***      Roof Closed       0.1976
                          (3.2119)                        (0.2932)
In from CF                -0.8325           Sunny          0.1842
                         (-1.1520)                        (0.5806)
In from RF                -0.7177          Cloudy          -0.0045
                         (-1.0005)                        (-0.0165)
In from LF                -0.3996           Clear         -0.4369 *
                         (-0.5715)                        (-1.6849)
Out to CF                 -0.7016         Overcast         -0.0366
                         (-1.0317)                        (-0.0910)
Out to RF                 -0.6003          Drizzle         -0.4309
                         (-0.8820)                        (-0.3825)
Out to LF                 -0.8452           Rain           1.2937
                         (-1.2266)                        (0.9214)
                                          R-squared         0.077

* -notation denotes statistical significance of t-test (* -10% level,
** -5% level, *** -1% level)

Table 3: Logit Model of Over Results in the Baseball Totals Betting
Market

                               Dependent Variable: Game Result is
                                  Over in Totals Market (Binary)

                        Variable        Coefficient       Coefficient
                        (t-stat)          Variable         (t-stat)

Intercept              5.6177 ***      Right to Left        -0.0650
                        (3.4317)                           (-0.1538)
Total                  -0.1219 ***     Left to Right        -0.0598
                        (-2.6785)                          (-0.1433)
Over Probability         0.2517            Varies           -0.0339
  Based
  on Odds Adjustment    (0.2973)      (Swirling Wind)      (-0.0704)
Air Density            -4.2534 ***          Dome            -0.4690
                        (-3.8207)                          (-0.9975)
Wind Speed              0.0253 **       Roof Closed         0.2202
                        (2.3197)                           (0.5276)
In from CF               -0.1102           Sunny            0.0183
                        (-0.2490)                          (0.1228)
In from RF               -0.2363           Cloudy           -0.0514
                        (-0.5464)                          (-0.4053)
In from LF               -0.0505           Clear           -0.2277 *
                        (-0.1154)                          (-1.9140)
Out to CF                -0.0378          Overcast          0.1113
                        (-0.0900)                          (0.5623)
Out to RF                -0.0536          Drizzle           -0.1359
                        (-0.1268)                          (-0.2637)
Out to LF                -0.0471            Rain            0.2867
                        (-0.1094)                          (0.4670)
                                     McFadden R-Squared      0.010

*-notation denotes statistical significance of t-test (* -10% level,
** -5% level, *** -1% level)

Table 4: Percentage Bet on the Over in the Baseball Totals Market

                         Dependent Variable: Percentage Bet on the Over

Variable                 Coefficient       Variable      Coefficient
                           (t-stat)                       (t-stat)

Intercept                -30.4803 ***   Right to Left      2.2730
                          (-1.8378)                       (0.6613)
Total                     2.4442 ***    Left to Right      5.1827
                           (5.3251)                       (1.5118)
Over Probability Based     12.6474          Varies         2.1846
  on Odds Adjustment       (1.5987)     (Swirling Wind)   (0.4516)
Air Density              53.6446 ***         Dome          2.1256
                           (4.6128)                       (0.8270)
Wind Speed                  0.0015       Roof Closed       -0.0069
                           (0.0168)                       (-0.0019)
In from CF                  3.5968          Sunny          -1.0662
                           (1.0041)                       (-0.9336)
In from RF                  0.9809          Cloudy        1.6965 *
                           (0.2716)                       (1.8378)
In from LF                  2.2092          Clear         1.8431 *
                           (0.6097)                       (1.8923)
Out to CF                   2.8023         Overcast        1.0794
                           (1.0651)                       (0.7503)
Out to RF                   4.5872         Drizzle         -2.5667
                           (1.3047)                       (-0.6169)
Out to LF                   2.7056           Rain          -1.0376
                           (0.7678)                       (-0.2052)
                                          R-squared         0.051

* -notation denotes statistical significance of t-test (* -10% level,
** -5% level, *** -1% level)

Table 5: Betting Simulations of Simple Strategies in Baseball
Totals Market

Betting Situation    N      Actual     Expected       Z
                            Return     Return

Air Density High-   239     0.0609     -0.0436      1.6248
  Wager on Under    228     0.0434     -0.0441      1.3402
Air Density Low -
  Wager on Over     467     0.0523     -0.0438     2.0995 **
Combined Bets on
  Air Density       372     0.0115     -0.0439      1.0788
Wind Speed High -
  Wager on Over     325    -0.0769     -0.0435      -0.6044
Wind Speed Low -
  Wager on Under    697    -0.0297     -0.0437      0.3762
Combined Bets on
  Wind Speed        257     0.0168     -0.0437      1.3412
Clear Day--Wager
  on Under

** -denotes statistical significance of z-test at the 5% level.
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Author:Paul, Rodney J.; Weinbach, Andrew P.; Weinbach, Chris
Publication:International Journal of Sport Finance
Article Type:Report
Geographic Code:1USA
Date:Aug 1, 2014
Words:6030
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