\nIn this paper we attempt to predict the total points scored in National Football League (NFL) games for the 2010-2011 season. Separate regression equations are identified for predicting points for the home and away teams in individual games based on information known prior to the games. The sum of the predictions for the home and away teams computed from the regression equations (updated weekly) are then compared to the over\/under line on individual NFL games in a wagering experiment to determine if a successful betting strategy can be identified. All predictions in this paper are out-of-sample\u2014meaning that all of the information necessary for the predictions was available before the games were played. Using this methodology, we find that several successful wagering strategies could have been applied to the 2010-2011 NFL season. We also estimate a single equation to predict the over\/under line for individual games. That is, we test to see if the variables we have collected and formulated are important in predicting the betting line for NFL games. These results can be used by either bettors or bookmakers wanting to increase their odds of success in the gaming industry.<\/p>\n
INTRODUCTION<\/strong> This research attempts to identify methods of predicting the total points scored in a particular game based on information available prior to that game. The primary research question is whether or not these methods can then be utilized to formulate a successful gambling strategy for the over\/under wager, with success requiring a winning percentage of at least 52.4%.<\/p>\n The remainder of this paper is organized as follows: in the next section we describe the efficient markets hypothesis as it applies to the NFL wagering market; we then offer a brief review of the literature; in the following section we describe the data and method; descriptive statistics and the main regression results are then presented; these are followed by the wagering simulations; we next discuss our investigation of the determinants of the over\/under line; and finally offer our conclusions.<\/p>\n NFL Betting as a Test of the Efficient Markets Hypothesis<\/strong> If the EMH holds, asset prices are formed on the basis of all information. If true, then the historical time series of such asset prices would not provide information that would allow investors to outperform the na\u00efve strategy of buy-and-hold (see, for example, Vergin 2001). As applied to NFL betting, if the use of past performance information on NFL teams cannot generate a betting strategy that would exceed the 52.4% win criterion, the EMH hypothesis holds for this market. Thus, the thrust of much of the research on the NFL has taken the form of attempts to find winning betting strategies, that is, strategies that violate the weak form of the EMH. <\/p>\n A Brief Review of the Recent Literature<\/strong> Based on NFL games from 1976 to 1994, Gray and Gray (1997) find some evidence that the betting spread is not an unbiased predictor of the actual point spread on NFL games. They argue that the spread underestimates home team advantage, and overstates the favorite\u2019s advantage. They further find that teams who have performed well against the spread in recent games are less likely to cover in the current game, and those teams that have performed poorly in recent games against the spread are more likely to cover in the current game. Further Gray and Gray find that teams with better season-long win percentages versus the spread (at a given point in the season) are more likely to beat the spread in the current game. In general, they conclude that bettors value current information too highly, and conversely place too little value on longer term performance. That conclusion is congruent with some stock market momentum\/contrarian views on stock performance. Gray and Gray then use the information to generate probit regression models to predict the probability that a team will cover the spread. Gray and Gray find several strategies that would beat the 52.4% win percentage in out-of-sample experiments (along with some inconsistencies). They also point out that some of the advantages in wagering strategies tend to dissipate over time.<\/p>\n Vergin (2001), using data from the 1981-1995 seasons, considers 11 different betting strategies based on presumed bettor overreaction to the most recent performance and outstanding positive performance. He finds that bettors do indeed overreact to outstanding positive performance and recent information, but that bettors do not overreact to outstanding negative performance. Vergin suggests that bettors can use such information to their advantage in making wagers, but warns that the market and therefore this pattern may not hold for the future.<\/p>\n A paper by Paul and Weinbach (2002) is a departure from the analysis of the spread in NFL games. They (as do we in this paper) target the over\/under wager, constructing simple betting rules in a search for profitable methods. These authors posit that rooting for high scores is more attractive than rooting for low scores. Ceteris paribus, then, bettors would be more likely to choose \u201cover\u201d bets. Paul and Weinbach show that from 1979-2000, the under bet won 51% of all games. When the over\/under line was high (exceeded the mean), the under bet won with increasing frequency. For example, when the line exceeded 47.5 points, the under bet was successful in 58.7% of the games. This result can be interpreted as a violation of the EMH at least with respect to the over\/under line.<\/p>\n Levitt (of Freakonomics fame) approaches the efficiency question from a different perspective. It is clear that if NFL bets are balanced, the bookmaker will profit by collecting $11 for each $10 paid out. As we suggested earlier, bookmakers at times take a \u201cposition\u201d on unbalanced bets, on the assumption that the bookmaker knows more about a particular wager than the bettors. Levitt presents evidence that the spread on games is not set according to market efficiency. For example, using data from the 2001-2002 seasons, he shows that home underdogs beat the spread in 58% of the games, and twice as much was bet on the visiting favorites. Bookmakers did not \u201cmove the line\u201d to balance these bets, thus increasing their profits as the visiting favorite failed to cover in 58% of the cases.<\/p>\n Dare and Holland (2004) re-specify work by Dare and MacDonald (1996) and Gray and Gray (1997) and find no evidence of the momentum effect suggested by Gray and Gray, and some, but less, evidence of the home underdog bias that has been consistently pointed out as a violation of the EMH. Dare and Holland ultimately conclude that the bias they find is too small to reject a null hypothesis of efficient markets, and also that the bias may be too small to exploit in a gambling framework.<\/p>\n Still more recently, Borghesi (2007) analyzes NFL spreads in terms of game day weather conditions. He finds that game day temperatures affect performance, especially for home teams playing in the coldest temperatures. These teams outperform expectations in part because the opponents were adversely acclimatized (for example, a warm weather team visiting a cold weather team). Borghesi shows this bias persists even after controlling for the home underdog advantage.<\/p>\n METHODS<\/strong> Match-ups Matter (we think)<\/strong> For example, suppose team \u201cA\u201d is averaging 325 yards (that\u2019s high) per game in passing offense and is playing team \u201cB\u201d which is giving up 330 yards (also, of course, high) per game in passing defense. The total of 655 would predict many passing yards will be gained by team \u201cA,\u201d and likely many points will be scored by team \u201cA.\u201d<\/p>\n Similarly, we theorize that if a team\u2019s offense that commits many turnovers plays a team whose defense causes many turnovers, points scored for the offensive team may be lower (and perhaps more points will be scored by the defensive team). For turnovers, we created variables similar to the passing and rushing yards in the previous paragraph: We also test to ascertain whether or not scoring is contagious. That is, if a given team scores many (or few) points, is the other team likely to score many (or few) points as well? We test for this by two-stage least squares regressions in which the predicted points scored by each team serve as explanatory variables in the companion equation.<\/p>\n General Regression Equations<\/strong> Equations such as 1 and 2 are estimated using data for weeks 5 through 17 of the 2010-11 season. We chose to wait until week five to begin the estimations so that statistics on offense, defense, turnovers, etc., are more reliable than would be the case for earlier weeks. <\/p>\n RESULTS AND DISCUSSION Regression Results<\/strong> The dome effect in a previous paper (see Pfitzner, Lang, & Rishel, 2009) found that teams scored approximately 5.4 more points when the game was played in a closed dome stadium for the 2005-2006 season. However, for the 2010-2011 season, games played in domes averaged 45.4 points and games played outdoors averaged 44.3. That difference is not statistically significant; the t-test for independent samples yields a calculated value of 0.54. The dome effect may be idiosyncratic in that, in some seasons, the high scoring teams may happen to be those who play home games in domed stadiums. <\/p>\n The representative estimated equations (at the end of the 16th week) are given in Table II. For the home points equation, the passing yardage and the rushing yardage are significant at \u03b1 < .01, and \u03b1 < .05 levels, respectively. The equation explains a modest 4.2% ( ) of the variance in home points scored. On the other hand, the F-statistic indicates that the overall equation meets the test of significance at \u03b1 < .01. The estimated coefficients for the variables have the anticipated signs. To interpret those coefficients, an additional 100 yards passing (recall that this is the sum of the home team\u2019s passing offense and the visitor\u2019s passing defense) implies approximately 4.3 additional points for the home team, whereas an additional 100 yards rushing implies approximately 4.2 additional points.\n\nTable II: Regression Results for Total Points Scored<\/strong> The visiting team estimation yields a similar equation in terms of the overall fit. The explanatory variables are statistically significant\u2014the passing yardage variable at \u03b1 < .05, and the rushing yardage variable is significant at \u03b1 < .01. The equation explains only 3.7% ( ) of the variance in visiting team points, and the F-statistic implies overall significance at \u03b1 < .05. The coefficients perhaps suggest a more important role for rushing than for passing in scoring for the visiting team. If the coefficients are to be believed, an additional 100 yards passing yields approximately 2.8 points for the visiting team, and an additional 100 yards rushing is worth 6.7 points. \n\nThe reader may find such low values to be of concern, but recognize that the variables for which we are attempting estimates are very difficult to predict and are subject to wide variation. As we show in a later section, the lines on the games are much easier to predict. The model is best judged by its prediction qualities\u2014here based on wagering success. \n\nOther Hypotheses<\/strong> As indicated above, we also find no evidence that teams are \u201cstreaky\u201d with respect to points scored. In short, we find that points scored in the immediately prior week do not contribute to the explanation of points scored in the current week. That conclusion holds up for the regressions in section VI as well. Wagering on the Over\/Under Line <\/strong> Out-of-Sample Method<\/em> Betting Strategies<\/em> As stated previously, a betting strategy on such games must predict correctly at least 52.4% of the time to be successful. If a given method cannot beat this 52.4% criterion, as a betting strategy it is deemed to be a failure.<\/p>\n Table III contains a summary of the results for the three betting strategies. The first betting strategy yields only ten \u201cplays\u201d over weeks 6 to 17. That betting strategy would have produced five wins, and five losses. For this (very) small sample, this strategy is, of course, not profitable, with only a 50% winning percentage. The second strategy (a differential greater than 5 points) yields 39 plays and a record of 17-10-0\u2014a winning percentage of 63%. Finally for every game played, the method produces a still profitable record of 97-78-5, with the winning percentage at 55.4%.<\/p>\n
\nBookmakers set over\/under lines for virtually all NFL games. Suppose the over\/under line for total points in a particular game is 40. Suppose further that a gambler wagers with the bookmaker that the actual points scored in the game will exceed 40, that is, he bets the \u201cover.\u201d If the teams then score more than 40 points, the gambler wins the wager. If the teams score under 40 points, the gambler loses the bet. If the teams score exactly 40 points, the wager is tied and no money changes hands. The process works symmetrically for bets that the teams will score fewer than 40 points, or betting the \u201cunder.\u201d The over\/under line differs, of course, on individual games. Since losing bets pay a premium (often called the \u201cvigorish,\u201d \u201cvig,\u201d or \u201cjuice\u201d and typically equal 10%), the bookmakers will profit as long the money bet on the \u201cover\u201d is approximately equal to the amount of money bet on the \u201cunder\u201d (bookmakers also sometimes \u201ctake a position,\u201d that is, they will welcome unbalanced bets from the public if the bookmaker has strong feelings regarding the outcome of the wager [see also the reference to Levitt\u2019s work in the literature review]). It is widely known a gambler must win 52.4% of the wagers to be successful. That particular calculation can be established simply. Let Pw = the proportion of winning bets and (1 \u2013 Pw ) = the proportion of losing bets. The equation for breaking even on such bets where every winning wager nets $10 and each losing wager represents a loss of $11 is:
\nPw ($10) = (1 \u2013 Pw ) ($11) , and solving for Pw
\nPw = 11\u221521 = .5238, or approximately 52.4%<\/p>\n
\nA number of important papers have treated wagering on NFL games as a test of the Efficient Market Hypothesis (EMH). This hypothesis has been widely studied in economics and finance, often with focus on either stock prices or foreign exchange markets. Because of the difficulties of capturing EMH conclusions given the complexities of those markets, some researchers have turned to the simpler betting markets, including sports (and the NFL), as a vehicle for such tests.<\/p>\n
\nNearly all of the extant literature on NFL betting uses the point \u201cspread\u201d as the wager of interest. The spread is the number of points by which one team (the favorite) is favored over the opponent (the underdog). Suppose team A is favored over team B by 7 points. A wager on team A is successful only if team A wins by more than 7 points (also known as \u201ccovering\u201d the spread). Symmetrically, a wager on team B is successful only if team B loses by fewer than 7 points or, of course, team B wins or ties the game\u2014in any of these cases, team B \u201ccovers.\u201d Vergin (2001) and Gray and Gray (1997) are examples of research that focus on the spread. <\/p>\n
\nWe focus on the total points scored in NFL games and the corresponding over\/under line for that game. With the objective of estimating regression equations for home and away team scoring, data were gathered for the 2010-11 season for the analysis. The variables include:
\nTP = total points scored for the home and visiting teams for each game played
\nPO = passing offense in yards per game
\nRO = rushing offense in yards per game
\nPD = passing defense in yards per game
\nRD = rushing defense in yards per game
\nGA = \u201cgive aways,\u201d offensive turnovers per game
\nTA = \u201ctake aways,\u201d defensive turnovers per game
\nD = a dummy variable equal to 1 if the game is played in a closed dome, 0 otherwise
\nPP = points scored by a given team in their prior game
\nL = the over\/under betting line on the game<\/p>\n
\nThe general regression format is based on the assumption that \u201cmatch ups\u201d are important in determining points scored in individual games. For example, if team \u201cA\u201d with the best passing offense is playing team \u201cB\u201d with the worst passing defense, ceteris paribus, team \u201cA\u201d would be expected to score many points. Similarly, a team with a very good rushing defense would be expected to allow relatively few points to a team with a poor rushing offense. In accord with this rationale, we formed the following variables:
\nPY = PO + PD = passing yards
\nRY = RO + RD = rushing yards<\/p>\n
\nTO = GA + TA, that is, turnovers = \u201cgive aways\u201d for a given team plus \u201ctake aways\u201d for the opposition team.
\nThe dome variable will be a check to see if teams score more (or fewer) points if the game is played indoors.
\nThe variable for points scored in the prior game (PP) is intended to check for streakiness in scoring. That is, if a team scores many (or few) points in a given game, are they likely to have a similar performance in the ensuing game?<\/p>\n
\nThe general sets of regressions attempted are of the form:
\n![\"Screen](\"https:\/\/i0.wp.com\/thesportjournal.org\/wp-content\/uploads\/2014\/02\/Screen-Shot-2014-02-14-at-4.10.13-PM.png\")
<\/a>where the subscripts h and v refer to the home and visiting teams respectively, and the i subscript indicates a particular game. <\/p>\n
\nDescriptive Statistics<\/strong>
\nTable I contains some summary statistics for the data set. Teams averaged approximately 223 yards passing per game (offense or defense, of course) for the season, and they averaged approximately 115 yards rushing. The statistics reported on the rushing and passing standard deviations without parentheses are for the offenses and the defensive standard deviations are (as you might guess) in parentheses. Interestingly, passing defense is less variable across teams than is passing offense (we hypothesize that teams must be more balanced on defense to keep other teams from exploiting an obvious defensive weakness, but teams may be relatively unbalanced offensively and still be successful [see the 2011 Packers, for example, who ranked near the top in passing offense and near the bottom in rushing defense]). Home teams scored approximately 23.2 points on average for the season and outscored the visitors by 1.7 points. Total points averaged 44.5 in 2010-2011 and the over\/under line averaged 42.8 (the difference between these means is statistically significant at \u03b1 < .10; the calculated value for the t-test of paired samples is approximately 1.92). Not surprisingly, the standard deviation was much smaller for the line than for total points.\n\nTable I: Summary Statistics<\/strong>
\n<\/a><\/p>\n
\nThough equations 1 and 2 from above represent our theoretical foundation, we did not find empirical support for the dome effect, points scored in the prior game, or for turnovers in predicting points for either the home or away teams. Thus we do not report regressions with those variables included (such estimations are available from the authors upon request). Since our objective is to produce predictions based on variables (and their effects) that are known prior to the games, we updated the equations weekly and checked for effects for those excluded variables. We did not find convincing evidence that any of the excluded variables should be included in the predictive equations.<\/p>\n
\n<\/a><\/p>\n
\nAnother hypothesis we wished to entertain is whether or not scoring is contagious. A priori, we surmised that points scored in given games for visiting and home teams would be positively related. In keeping with our earlier work, there is no evidence that such is the case. The estimated simple correlation coefficient between home team and visiting team points is -0.106, which is not statistically different from zero and \u201cwrong\u201d signed according to our intuition. Our initial thinking was that if team \u201cA\u201d scores and perhaps takes a lead, team \u201cB\u201d has greater incentive to score. An obvious complicating factor is that a given team may dominate time of possession, thus preventing the opposing team opportunities to score. We also experimented with two-stage least squares to test the hypotheses that scoring was contagious. In that formulation we developed a \u201cpredicted points\u201d variable for the home team, entered that variable as an independent variable in the visiting team equation, and reversed the procedure for the home team equation. Neither of the predicted points variables were statistically significant. The variable was positively signed for the home team equation, and negatively signed for the away team equation.<\/p>\n
\nFinally, though turnovers clearly matter in who wins or loses, there is no evidence from our work that measuring teams\u2019 turnovers per game prior to the current game aids in predicting points scored by the individual teams. <\/p>\n
\nIn this simulated wagering project we use the estimated equations to predict scores of the home and away teams for all of the games played over weeks 8 through week 17 (end of the regular season). The points predicted in this manner are then compared to the over\/under line for each game. We then simulate betting strategies on those games.<\/p>\n
\nSince it is widely known that betting strategies that yield profitable results \u201cin sample,\u201d are often failures in \u201cout-of-sample\u201d simulations, we use a sequentially updating regression technique for each week of games. Suppose, for example, we are predicting points for week 8. We then estimate equations TPhi and TPvi with the data from weeks 5, 6, and 7, then \u201cfeed\u201d those equations with the known data for each game through the end of week 7, generating predicted points for the visiting and home team for all individual games in week 8. The predicted points are then totaled and compared to the over\/under line for each game. Next we add the data from week 8, re-estimate equations TPhi and TPvi, and make predictions for week 9. The same updating procedure is then used to generate predictions for weeks 10 through 17. This method ensures that our results are not tainted with in-sample bias.<\/p>\n
\nWe entertain three betting strategies for the predicted points versus the over\/under line on the games. These strategies are:
\n1. Bet only games for which our predicted total points differ from the line by more than 7 points.
\n2. Bet only games for which our predicted total points differ from the line by more than 5 points.
\n3. Bet all games for which our predicted total points differ from the line by any amount\u2014in our case, all games.<\/p>\n