1
Determining The Value Of NFL Quarterbacks Economics Honors Thesis Steve Alexander 2008 2
Table of Contents ™ Variable Reference Guide ™ Introduction ™ Related Research ™ Results ™ Conclusion ™ Summary Statistics ™ Appendix ™ Works Cited 3
Variable Reference Guide (Every statistic is specific to a particular season/year) Ht Height in inches Wt Weight in pounds Age Age in years Exp Years of NFL experience (Equals 1 if player is a rookie) G Games attended (Quarterback did not necessarily play) GS Games where Quarterback was designated as a starting player Pcomp Number of pass completions Patt Number of passes attempted PcompPct Pcomp/Patt Pyds Number of passing yards PTD Number of passing touchdowns scored Interceptions Number of interceptions thrown Sck Number of times sacked SckY Number of yards lost as a result of being sacked Rate Passer Rating (this is explained elsewhere in the paper) Ratt Number of rushing attempts Ryds Number of rushing yards RTD Number of rushing touchdowns scored Fum Number of fumbles FumL Number of fumbles lost BaseSalary Annual base salary in dollars (there is a minimum salary level explained later) SignBonus Signing bonus in dollars as a lump sum in the year it was negotiated OtherBonus Roster, report, workout, and other bonuses in dollars TotalSalary BaseSalary + SignBonus + OtherBonus CapValue BaseSalary + OtherBonus + Pro‐Rated Signing Bonus 4
Introduction This study is comprised of several closely related goals. First and foremost, an attempt will be made to predict the future earnings of National Football League quarterbacks based on their performance in previous seasons, as judged by multiple key statistics. With these statistics, efforts will also be made to predict a team’s future record and to see exactly how vital these players are to their teams. The reader is assumed to have a basic understanding of the game of football. The quarterback position is unique. Historically, it has been regarded as the most critical role. The pay many of these players receive reflects that fact well. The quarterback is the first player to receive the football when it is put into play and the decisions he makes can alter the team’s odds of success or failure drastically. He must be hyper‐aware of every one of the other 21 players on the field—genetically gifted super‐athletes with lightning‐quick speed. He can throw the ball, run with the ball, or give the ball to another player. From memory, he can call and execute the complex plays the coach has chosen, or he can modify these calls after evaluating the opposing defense with audibles. Quarterbacks represent the majority of Most Valuable Player Award recipients and are remembered for years after the rest of their teammates have been forgotten. 5
One would therefore imagine that a quarterback’s pay would be highly sensitive to his on‐field performance, and perhaps the performance of his team would be affected likewise. The players whose perceived values are most directly related to measurable statistics are probably quarterbacks. For instance, this study would be difficult or impossible if it concerned offensive linemen because their performance is not easily quantifiable. Other positions can be similarly confounding, such as defensive backs. A great defensive back may not have outstanding statistics, such as a high amount of interceptions, because the ball will not often be thrown in their direction. For these reasons, I have chosen to study the quarterback position over all others. I compiled a list of every quarterback on the NFL’s payroll from the 2000 season to the end of the most recent (2007) season. I then gathered their individual statistics for each season from the official NFL records database. Next, using U.S.A Today’s database, I was able to find 6
figures for each player’s annual compensation and bonuses. Finally, using the STATA program, I attempted to create meaningful regressions. Primarily, four methods were used to regress these time‐series panel data. To control for such a high level of endogeneity and fluctuation of the variance of the error term, the fixed effects method was employed. Three dummy‐variable approaches were also utilized. The first used a dummy variable representing each year (season). The second involved 96 dummy variables, each assigned to a specific quarterback. The third used 32 dummy variables, each assigned to a specific team. Before the results of this study are shown, it is necessary for a quick primer on NFL pay schemes. Each player receives an annual salary, and there are minimum salary levels that vary with the number of years a player has been in the NFL. This minimum level is determined by the Collective Bargaining Agreement (CBA), which is negotiated by the NFL Players Association, a union of which every NFL player is a member. Years of Experience Minimum Salary 0
$285,000 1
$360,000 2
$435,000 3
$510,000 4‐6 $595,000 7‐9 $720,000 10+ $820,000 7
Players may also receive signing bonuses when they commit to a contract and other performance‐based bonuses. This makes a player’s total salary numbers fluctuate wildly from year to year, as signing bonuses can be very large and are counted as a lump‐sum payment in the year they were signed for. For this reason I have chosen to use the CapValue variable as my main salary statistic rather than the TotalSalary variable. The CapValue variable pro‐rates the amount of any signing bonuses over the years for which the player has contractually agreed to play. An NFL team may not pay more than a certain level of compensation to its players as a whole. This level is called the salary cap, and the CapValue statistic is the amount of compensation used for the purposes of calculating whether a team is at or under the cap. Because choosing to pay a quarterback more means choosing to pay other players less, the CapValue statistic gives an accurate measurement of the degree to which a team values their quarterback. Quarterbacks who have been drafted but have not yet played in their first season are allowed to negotiate their compensation only with the team that drafted them. For this reason, pay levels for rookies tends to be extremely low relative to more experienced quarterbacks who have negotiated another contract. 8
Related Research During the course of this study, some academic research that is directly relevant was found. Far more research was found that was tangentially relevant, but still interesting. When dealing with rookies, Hendricks, DeBrock, and Koenker found that the sooner draftees were allowed to negotiate with other teams, the lower the demand for athletes with more uncertain futures. These athletes might include players from lesser‐known schools or players with a history of injuries. These researchers also found that the visibility of the football program that a player came from was significant and positively correlated with their success in the draft. However, players from less visible programs seemed to have better careers over the long term, as their salaries were less likely to fall over time. These players received less pay initially, though. In my research I expected to find large returns to experience. Clark and Hall found that teams with a greater number of veteran players were of a better quality and had more success. The number of veterans was positively correlated with the level of competition between teammates. This is because the more veterans a team has, the less the likelihood of an open position. The increased competition for the spot drives the starter and the backups to perform better than they otherwise would. Clark and Hall also found that the preseason (which is not discussed in this study due to its nature) is a good thing, as it increases competition between teammates and therefore improves team quality. To 9
perform their research, these individuals used a team‐dummy method identical to the method performed in this study. To discuss the research done by Leeds and Kowaleski, it is necessary to again refer to the CBA. The current CBA went through a major overhaul in 1993. These researchers found that dramatic shifts in quarterback pay resulted. After this overhaul, the top quartile of quarterbacks (ranked according to pay) was rewarded more for starting games than for their performance in those games. In other words, performance was de‐emphasized relative to merely starting in games. For the lowest quartile of quarterbacks, however, the basis of pay stayed basically the same as before the new CBA. For these quarterbacks, there was a far greater emphasis on performance than for the higher‐ranked players. These low‐ranked quarterbacks could dramatically improve their pay by performing better. In my research I expected to find a large return to starting games, as this study might suggest. 10
Results Determining Cap Value as a Result of Passing Touchdowns Scored The primary way a quarterback scores points is by passing the football. Thus, the PTD statistic should prove especially relevant in determining the value of the quarterback. My first approach was via the fixed effects method. 1. CapValue Coefficient Std. Error P>|t| L1.PTD 108,527.1 15,302.3 0.000 R2=0.4692 #Obs=367 #Groups=84 I started simply by regressing the number of passing touchdowns scored in the previous season (L.PTD) on the cap value of the quarterback in the next season. With a high level of significance, it can be said that scoring one extra passing touchdown is predicted to increase a quarterback’s compensation in the next season by approximately $108,527. So a very large premium is placed on scoring an additional passing touchdown. The current record for the most passing touchdowns scored in a single regular season is 50, set in the 2007 season by New England Patriots quarterback Tom Brady. 11
2. CapValue Coefficient Std. Error P>|t| L1.PTD 64,066.31 17,442.06 0.000 L2.PTD 100,066.7 16,769.26 0.000 R2=0.5361 #Obs=281 #Groups=71 Here, I also control for the number of passing touchdowns scored two seasons ago (L2.PTD). As expected, the R‐squared value increases favorably. This regression indicates that an extra touchdown scored two seasons ago is predicted to add $100,066 to a quarterback’s pay today, while an extra touchdown scored one season ago is predicted to add only $64,066 to a quarterback’s pay today. Adding the passing touchdowns scored in even earlier seasons to the regression resulted in terribly insignificant p‐values or a lower adjusted coefficient of determination, so for the sake of brevity these regressions will not be shown. 3. CapValue Coefficient Std. Error P>|z| L1.PTD 181,995.5 12205.54 0.000 R2=0.4943 #Obs=367 #Groups=84 12
+6 Time Dummies In this regression, I abandoned the fixed‐effects approach for the time‐dummy approach. Again I started simply by using the passing touchdowns scored in the previous year and the quarterback’s cap value in the current year. An additional passing touchdown in the previous season was predicted to increase the quarterback’s compensation in the current season by $181,995. As expected, this is a very large figure. This regression shows a slightly higher r‐squared than the fixed effects method, but this is due to the inclusion of six more variables. The adjusted r‐squared actually decreased. So the time‐dummy approach is a less than optimal method in this case. This is fairly conclusive evidence that passing touchdowns play a large role in determining quarterback salary. Although this statement may seem intuitive or obvious, at least there is now evidence to back up that intuition. Determining Cap Value as a Result of Quarterback Passer Rating As stated before, passing ability is likely the greatest factor in judging a quarterback’s performance. To this end, statisticians have devised a system for ranking quarterbacks called the Passer Rating (Rate). Although the methods of calculating this number shall not be divulged here, suffice it to say that it includes the percentage of passes completed, the average yards gained per pass completion, the average number of touchdowns scored per pass attempt, and the average number of interceptions thrown per pass attempt. It is a very useful statistic, 13
yet it does not include sacks, rushing ability, or intangibles. Still, it should be a better measure of overall performance than passing touchdowns. The fixed effects method gave very poor results with low R‐squared values and high p‐
values. The fixed effects regressions will not be included in this section for this reason. The time‐dummy method gave a poor r‐squared value, and the player‐dummy method gave a poor adjusted r‐squared. The best method proved to be the team‐dummy approach. 4. CapValue Coefficient Std. Error P>|z| L1.Rate 20,254.68 3,819.688 0.000 R2=0.2873 #Obs=367 #Groups=84 +31 Team Dummies Here, each additional point of passer rating a quarterback earns is predicted to increase his value by $20,254.68 in the next season. A player who goes from an abysmal rating of 50 to an elite rating of 100 will receive an estimated $1,012,734 more in his next season. Determining Cap Value as a Result of Age and Experience One would predict that age and experience should increase a quarterback’s performance to a point, after which performance suffers. Rare players, like Brett Favre, who 14
played well into his late thirties, seem to defy this prediction. Nonetheless, since diminishing returns to age and experience are expected, the variables AgeSQ (age‐squared) and ExpSQ (experience‐squared) were added to the models in the following regressions. Using fixed effects, regressing age and age‐squared on cap value prove inconclusive. So experience and experience‐squared were substituted: 5. Log(CapValue) Coefficient Std. Error P>|t| L1.Exp 0.440962 .0392241 0.000 L1.ExpSQ ‐0.0239457 .0027393 0.000 R2=0.1752 #Obs=365 #Groups=84 This regression has a rather low R‐squared, but that is somewhat expected because experience is certainly not the biggest factor in determining quarterback salary. There are many seasoned veterans who are backups and do not receive compensation like the starting players receive. LCapValue is the log of CapValue. So we do observe a diminishing returns effect when dealing with experience. The massive return to experience is somewhat expected, based on the findings of previous researchers. It shows that rookies and inexperienced players (who cannot negotiate with teams other than the team that drafted them) do indeed make far less than more seasoned quarterbacks. 15
Using time‐dummies, experience was again superior to age. However, the fixed effects method provided a more meaningful regression. Using the team‐dummy and player‐dummy methods, age was mostly a worthless variable and the effect of experience was virtually identical to the time‐dummy and fixed effects methods. Every regression showed a prominent diminishing returns to experience effect. Determining Cap Value as a Result of Games and Games Started Being a starter should be fairly relevant to a quarterback’s salary. Backup quarterbacks are little‐known and receive, on average, far less pay than their superiors. As the number of games started increases (to a maximum of 16), I predict that expected compensation will rise. Being present for a game, regardless of starter status, should also have a positive correlation, but will likely not be as strong. 6. Log(CapValue) Coefficient Std. Error P>|t| L1.GS .0602965 .0081548 0.000 R2=0.5035 #Obs=365 #Groups=84 16
In this regression, the log of cap value was regressed on the number of games started in the previous season. Starting one additional game is predicted to increase the next year’s salary by about 6%. Using the time‐dummy approach, a 9% increase in cap value was predicted as a result of starting an additional game in the previous season, but adjusted r‐squared was low. Though the regressions will not be shown here, a 9% effect was also shown using the team‐dummy approach, but a 6% effect was shown using the player‐dummy approach. Perhaps the true figure is in between 6% and 9%. To account for games attended (G), I created a new variable: PctOfGamesStarted, which is simply GS divided by G: 7. Log(CapValue) Coefficient Std. Error P>|t| L1.PctOfGamesStarted 0.7105315 .137685 0.000 R2=0.3698 #Obs=317 #Groups=81 Here we see that a player who starts all 16 regular season games is predicted to have a 71% higher value than a player who is not a starter. This is a large and significant difference. 17
Determining Cap Value as a Result of Height and Weight Quarterback heights were very closely clustered with a mean of 74.9 inches. Weights had more variance and a mean of 224 pounds. Here I used normal regressions. R‐squared values for both height and weight were extremely miniscule. It appears as though height and weight have very little to do with being a good quarterback. Fifty percent of quarterbacks were between 74 and 75 inches tall, with no one shorter than 71 inches. It seems as though there is a certain minimum height required to see over the heads of the massive linemen. The low R‐
squared associated with weight is also not surprising, since traditionally the quarterback position has not been a brutally physical one. Quarterbacks do not usually take large amounts of physical punishment relative to their teammates so high weight is not that important. Conversely, light weight (normally associated with speed) is not extremely valuable either, since speed is not as vital for a quarterback as it is for a defensive back or running back. Here are the results of the regressions: 8. Log(CapValue) Coefficient Std. Error P>|t| Ht .0617184 .0375106 0.101 Log(CapValue) Coefficient Std. Error P>|t| Wt .0118269 .004162 0.005 R2=0.0058 #Obs=464 9. R2=0.0172 18
#Obs=464 Determining Cap Value as a Result of Interceptions Until this point, the independent variables used in the regressions have added value to a quarterback. Now the independent variable, Interceptions, is expected to decrease value. Throwing an interception is certainly a very bad thing as it gives possession of the football back to the opposing team. Surprisingly, however, using the Interceptions variable almost always showed a positive correlation to cap value. This was likely due to the fact that the highest paid quarterbacks threw many passes, a few of which were of course intercepted, while the lowest paid quarterbacks did not even receive any playing time and therefore never had the chance to throw an interception. I created a variable, InterceptionAvg, that I believed would suit the needs of this study better. This statistic is simply Interceptions divided by Patt. It shows how often, on average, a quarterback throws an interception when he throws a pass. This way, the quarterbacks who do not play and therefore do not throw passes are eliminated from the equation. The coefficient on this variable was negative in all regressions, but with unacceptable p‐values up to 0.6. I then restricted the regressions to include only quarterbacks who had started at least one game, and finally to include quarterbacks who had started at least half of the games during the regular season. The coefficient on Interceptions continued to be positive with high significance and the coefficient on InterceptionAvg continued to be negative with unacceptable significance. 19
10. Log(CapValue) Coefficient Std. Error P>|z| L1.Interceptions .0189017 .0101443 0.062 L1.PTD .0434145 .0071356 0.000 R2=0.4129 #Obs=173 #Groups=55 +6 Time Dummies Here is an example of one of the better regressions involving interceptions. It employs the time‐dummy method, includes only quarterbacks who started 8 games or more (half of the season), and also controls for passing touchdowns. I controlled for passing touchdowns because while the coefficient on Interceptions was consistently positive, it was less than half of the value associated with the “good” statistic, PTD. Perhaps there is a positive return for making risky passes which are sometimes intercepted, and this accounts for the way that the seemingly “bad” statistic adds value. Finally, using team‐dummy variables and controlling for the number of games started in the previous year, I was able to generate a regression that looks more like one might expect: 11. 20
Log(CapValue) Coefficient Std. Error P>|z| L1.Interceptions ‐.0303574 .0126393 0.016 L1.GS .1176726 .0138604 0.000 R2=0.5454 #Obs=365 #Groups=84 +31 Team Dummies Here it is shown that throwing an additional interception in the previous season is predicted to decrease cap value in the current season by about 3%. Multiple Regression Models with Cap Value as the Dependent Variable 12. Log(CapValue) Coefficient Std. Error P>|z| L1.Exp .3627602 .0426122 0.000 L1.ExpSQ ‐.0188704 .0028811 0.000 L1.PcompPct .3464153 .3267147 0.289 L1.Pyds .0002402 .000106 0.023 L1.PTD .0002593 .0112537 0.982 L1.Interceptions ‐.0088807 .010640 0.404 L1.Sck ‐.0017347 .0048105 0.718 R2=0.8063 #Obs=303 21
#Groups=78 +96 Player Dummies In these multiple regression analyses, I began to put together all of the more simple regressions I had done in order to find the absolute best model to predict quarterback compensation. The goal was to see how different statistics interacted with each other and to control for all relevant variables in hope of finding elasticities. The player‐dummy approach yielded the best results for the model specified above. I included all variables pertaining to the passing game from the previous year and also experience. The diminishing returns to experience were discovered once again, with high significance. The way in which passing yards contribute to compensation was also determined with accuracy far better than the desired 10% level. Every additional 100 yards a quarterback passed for during the previous season was estimated to increase cap value by about 2.6% in the current season. The p‐values associated with the other variables indicate that they are not statistically significant, but the coefficients associated with the other variables are what one might expect. For example, touchdowns add to compensation while throwing an interception and being sacked decrease compensation. 13. Log(CapValue) Coefficient Std. Error P>|t| L1.Exp .360661 .0424637 0.000 L1.ExpSQ ‐.0189453 .0028701 0.000 L1.PcompPct .3756269 .3259106 0.250 L1.Pyds .0000914 .0001388 0.511 L1.PTD ‐.0002735 .0112141 0.981 ‐.0132091 .0109181 0.228 L1.Interceptions 22
L1.Sck ‐.0048875 .0051582 0.344 L1.GS .0462967 .0280463 0.100 R2=0.3791 #Obs=303 #Groups=78 Here we have a fixed effects regression using the same variables as before but also controlling for the number of games started in the previous year. This statistic is almost significant at the 10% level, and so this regression has been included. It appears as if the estimated return to starting one additional game in the previous season is a 4.6% increase in cap value. Though the regressions will not be shown here, I began replacing the statistics from the previous year (the variables with the L.‐prefix) with statistics from two years ago, three years ago and so forth. I found that the R‐squared values as I went further back in time decreased drastically and the variable coefficients lost significance. This seems to indicate that when this many variables are accounted for, more recent performance outweighs past performance when determining compensation. The time‐dummy approach worked best when the rushing statistics were introduced into the model. Number of rushing touchdowns and number of rushing yards were added to the previous regression. Traditionally, rushing ability has not been extremely vital for the quarterback position. Some uncommon players, such as the Atlanta Falcons’ Michael Vick, can run with the football or pass it with equal skill. Players like this are a double threat, and this versatility has proven value—Michael Vick had the highest cap value of any player at any 23
position in the 2005 season. The results are somewhat confounded by the fact that possessing these dual abilities is extremely rare and by the fact that Michael Vick had a rather short stint in the NFL due to legal issues. Though the p‐values on the rushing statistics are high, one can see that, on average, scoring rushing touchdowns decreases cap value. This may be due to the fact that rushing touchdowns are usually better attempted by rushing specialists—not every quarterback is a Michael Vick. A quarterback who rushes and is then tackled has a higher risk of injury, which will hurt future performance and possibly cause him to miss games. 14. Log(CapValue) Coefficient Std. Error P>|z| L1.Exp .2837511 .0366806 0.000 L1.ExpSQ ‐.0158642 .0024127 0.000 L1.Rate .0021731 .001125 0.053 L1.Sck ‐.0041898 .0051741 0.418 L1.GS .0751387 .012307 0.000 L1.RTD ‐.0228148 .0350197 0.515 L1.Ryds .0003807 .0005437 0.484 .0245214 .0119926 0.041 L1.RegSeasonWins R2=0.5726 #Obs=365 #Groups=84 +6 Time Dummies Here I have also replaced the variables pertaining to the passing game with the quarterback passer rating, which was briefly described earlier. It includes elements of all the 24
variables that were removed. In the regression above, the number of wins the quarterback’s team had in the previous season are also accounted for. This is attempting to control for the team’s performance as a whole as well as the individual quarterbacks’ performances. The coefficient on RegSeasonWins is both positive and significant at the 5% level. If a quarterback’s team goes from winning half of the regular season games (8) to winning all of the regular season games (16), his cap value is estimated to increase by about 19.6%. This was the best model that I was able to specify for the regular season, after much deliberation. 15. Log(CapValue) Coefficient Std. Error P>|z| L1.Exp .2233226 .0435665 0.000 L1.ExpSQ ‐.0126701 .0027003 0.000 L1.Rate .0026633 .0013548 0.049 L1.Sck ‐.0085052 .0057719 0.141 L1.GS .0960592 .0132456 0.000 L1.RTD ‐.0232805 .0392978 0.554 L1.Ryds .000808 .0005548 0.145 L1.RegSeasonWins .0291631 .0137586 0.034 L2.WonSuperBowl .1422508 .2025955 0.483 R2=0.6035 #Obs=279 #Groups=71 +5 Time Dummies Here, the WonSuperbowl variable was added to control for post‐season performance. The best regression containing this variable used the time‐dummy approach and considered 25
only the effect of winning the Superbowl 2 years ago. Winning the Superbowl in any other year had inconclusive effects. It could be that there is a time‐delay effect in receiving extra compensation for being a Superbowl‐winning quarterback. The variable has low statistical significance but has a large positive coefficient. A quarterback who won the Superbowl 2 years ago is predicted to reap a 14.2% benefit today from doing so. Estimating Team Performance as a Result of Quarterback Performance It was stated before that quarterbacks are very important to their teams. But just how important is this position? While the primary goal of this study was to determine how a quarterback’s measurable qualities contribute to his monetary value, the secondary goal was to see how these qualities affect the performance of his team. 16. RegSeasonWins Coefficient Std. Error P>|z| PTD .1804051 .0270309 0.000 R2=0.4643 #Obs=200 #Groups=60 Restriction: GS greater than or equal to 8 +31 Team Dummies The best results were obtained by using the team‐dummy method and specifying that to be included in the model, a quarterback must have started at least half of the games during the regular season. This is an effective way to eliminate second‐ and third‐string quarterbacks who 26
confound the results. Though the coefficient on PTD appears small, it is significant, and it indicates that passing for an extra 11 touchdowns is estimated to increase an entire team’s record by about 2 wins. Two wins can make the difference between a chance at going to the playoffs and being uninvited to the post‐season tournament. 17. Log(RegSeasonWins) Coefficient Std. Error P>|z| Rate .0026498 0.000 .0185015 R2=0.4352 #Obs=200 #Groups=60 Restriction: GS greater than or equal to 8 +31 Team Dummies This regression demonstrates how powerfully the passer rating, as a measure of a quarterback’s ability, affects a team’s regular season wins. Here I used the log of RegSeasonWins, LRegSeasonWins. This shows that if a quarterback were to add 50 points to his passer rating (say for example, he doubles it from a score of 50 to a score of 100), the percent of games his team won would be predicted to almost double. This shows the difference between having a mediocre quarterback as a starter and having an elite quarterback as a starter. It is substantial. 18. Log(RegSeasonWins) Coefficient Std. Error P>|z| Exp .0284501 0.482 ‐.0200178 27
ExpSQ .000605 .0017967 0.736 Rate .0151807 .0025851 0.000 Sck ‐.011551 .0027757 0.000 RTD ‐.0213521 .0186355 0.252 Ryds .0003679 .0002519 0.144 GS .0669841 .0109636 0.000 R2=0.5645 #Obs=200 #Groups=60 Same restrictions apply +31 Team Dummies Here, many more variables are controlled for in order to isolate a ceteris paribus effect. The effect of passer rating decreases slightly but maintains high significance. There appears to be a strong positive return to having a quarterback who is a consistent starter, as indicated by the GS coefficient. Every other correlation matches with what one might expect except for the reversal of the diminishing returns effect discussed earlier with Exp and ExpSQ. The significance of these values, however, is very poor. 19. Log(RegSeasonWins) Coefficient Std. Error P>|z| Exp ‐.0978946 .0510495 0.055 ExpSQ .0042126 .0026727 0.115 Rate .0094647 .0036782 0.010 Sck ‐.0158034 .0041549 0.000 28
RTD .0219416 .0266615 0.411 Ryds .0002524 .0003816 0.508 GS .0673368 .0146347 0.000 .2685658 .192206 0.162 L3.WonSuperbowl R2=0.7553 #Obs=103 #Groups=40 Same restrictions apply +31 Team Dummies The only variable added here is WonSuperbowl. Except for winning the Superbowl 3 years ago, winning the Superbowl in every other year was extremely statistically insignificant. It appears as though winning the Superbowl 3 years prior has the most significant effect on regular season wins, with an estimated increase in wins of 26% if true. Estimating Team Performance as a Result of Compensation Surprisingly, none of the measures of compensation used in this study had any conclusive effect on team performance. All regressions were awful, with insignificant p‐values and r‐squared values lower than 1 percent. This indicates that paying a quarterback more does not necessarily increase team performance in the regular season. Perhaps quarterbacks at the professional level are in a position so advanced that their incentives to perform better (their internal drive to win) cannot be influenced by merely attempting to motivate them with 29
money. There are ten other players on the offensive side of the ball who must also perform well to win games, although the quarterbacks’ performance seems to matter to a disproportionately high degree. It is not possible to be a more successful team by giving your quarterback more money. 30
Conclusion Overall, the goals of this study have been achieved. With few surprises, the effects of quarterback statistics are what one would probably expect. With so many measurable factors, controlling for as many variables as possible without generating a meaningless regression proved difficult at times. Sometimes the results of each of the four methods were nearly identical, and other times they were drastically different. To write a somewhat concise paper, only the “best” regressions (as determined by adjusted r‐squared and F‐tests) were shown. This study has shown that the effects of “good” statistics on quarterback value are positive and of a magnitude consistent with their commonly accepted importance , while the effects of “bad” statistics on quarterback value are negative and of a magnitude consistent with their commonly accepted importance. While this may seem mundane, nothing should be assumed without empirical evidence, and that is what is provided here. For instance, there is some degree of evidence for the idea that throwing risky passes (thereby increasing interceptions) may increase quarterback value, although to a lesser degree than more commonly accepted “good” statistics. The fixed‐effects method seemed to work best for the regressions involving one independent variable of interest. The other methods tended to reduce the adjusted r‐squared value while inflating the r‐squared value. This was, of course, due to the drastically increased degrees of freedom when using the dummies. The team‐dummy method was noticeably 31
preferable to all other methods when there were many independent variables of interest. They usually provided the best estimate of the elasticities. The statistics that were most important and that were not direct measures of on‐field performance were experience and games started. Being a consistent starter was extremely valuable year after year. The huge effect of experience on compensation was mostly due to the better negotiating ability achieved after the first few years of professional play. For direct measures of on‐field performance, I had expected passing touchdowns to have the greatest effect. Although a large positive effect was confirmed using simple linear regression analysis with fixed effects, when controlling for more variables the significance of passing touchdowns became inferior to the passer rating statistic. So for the multiple regression models, which I believe were the best predictors of compensation, passer rating seemed to be the most important measure of direct on‐field performance. The results of this study were consistent with the findings of other researchers that dealt with related topics. Some of the regressions performed strongly reinforce the assertions made by these researchers. Both measurable and intangible qualities were controlled for, and I believe some good information was found. 32
Summary Statistics Variable
Obs
Mean
Std. Dev.
Min
Max
ht
wt
age
exp
466
466
466
466
74.87339
223.7639
27.88841
5.371245
1.419927
12.72809
4.239141
3.917091
71
196
21
1
78
285
44
21
g
gs
pcomp
patt
pyds
466
466
466
466
466
8.203863
6.729614
132.0773
219.0837
1512.7
6.046706
6.478074
126.9787
206.4902
1475.598
0
0
0
0
0
16
16
440
652
4830
ptd
intercepti~s
sck
scky
rate
466
466
466
466
466
9.17382
6.845494
14.60515
93.26609
64.71953
10.00719
6.682488
14.48376
92.13887
34.59824
0
0
0
0
0
50
29
76
424
156.9
ratt
ryds
rtd
fum
fuml
466
466
466
466
466
18.66524
69.79828
.6416309
4.375536
1.890558
21.37015
126.9199
1.317502
4.483969
2.10244
0
-13
0
0
0
123
1039
10
23
9
basesalary
signbonus
otherbonus
totalsalary
capvalue
466
466
466
466
466
1190329
1494023
538608.8
3027277
2597211
1562312
3311007
1643858
3926658
2832639
0
0
0
66176
0
1.10e+07
3.45e+07
1.23e+07
3.50e+07
1.54e+07
regseasonw~s
regseasonl~s
wonsuperbowl
466
466
466
8.081545
7.914163
.0321888
2.989192
2.991949
.176691
1
0
0
16
15
1
33
Appendix 1. . xtreg capvalue l.ptd ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
=
capvalue
Number of obs
=
Number of groups
0.1514
0.7170
0.4692
Obs per group: min =
avg =
max =
0.5536
F(1,282)
Prob > F
Coef.
ptd
L1.
_cons
108527.1
1914881
sigma_u
sigma_e
rho
1776989
1847200.7
.48063419
F test that all u_i=0:
Std. Err.
15302.3
176411.1
t
P>|t|
7.09
10.85
367
84
=
=
=
1
4.4
7
50.30
0.0000
[95% Conf. Interval]
0.000
0.000
78405.84
1567632
138648.3
2262131
(fraction of variance due to u_i)
F( 83, 282) =
2.73
Prob > F = 0.0000
2. . xtreg capvalue l.ptd l2.ptd ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
capvalue
=
Obs per group: min =
avg =
max =
0.4510
F(2,208)
Prob > F
Coef.
ptd
L1.
L2.
_cons
64066.31
100066.7
1631497
sigma_u
sigma_e
rho
1773286.1
1733970.4
.51120839
Std. Err.
17442.06
16769.26
256585.7
t
P>|t|
3.67
5.97
6.36
0.000
0.000
0.000
281
71
=
0.2141
0.6737
0.5361
F test that all u_i=0:
Number of obs
=
Number of groups
=
=
1
4.0
6
28.34
0.0000
[95% Conf. Interval]
29680.43
67007.18
1125655
98452.2
133126.2
2137339
(fraction of variance due to u_i)
F( 70, 208) =
2.91
Prob > F = 0.0000
34
3. . xtreg capvalue l.ptd y00 y01 y02 y03 y04 y05 y06 y07
note: y00 dropped because of collinearity
note: y02 dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
Number of obs
=
Number of groups
0.2294
0.6959
0.4943
Random effects u_i ~
corr(u_i, X)
=
capvalue
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
ptd
L1.
y01
y03
y04
y05
y06
y07
_cons
181995.5
-522170
798669
190079.5
439813.4
1094537
1025555
575559.4
sigma_u
sigma_e
rho
789524.41
1751372.5
.16889926
367
84
=
Std. Err.
12205.54
469627.4
425670.2
414823.9
402574.6
394331
392056.8
355044.3
Wald chi2( 7)
Prob > chi2
z
P>|z|
14.91
-1.11
1.88
0.46
1.09
2.78
2.62
1.62
0.000
0.266
0.061
0.647
0.275
0.006
0.009
0.105
=
=
1
4.4
7
235.85
0.0000
[95% Conf. Interval]
158073.1
-1442623
-35629.28
-622960.4
-349218.3
321662.2
257137.6
-120314.7
205917.9
398282.8
1632967
1003120
1228845
1867411
1793972
1271433
(fraction of variance due to u_i)
4. . xtreg capvalue l.rate dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions dpack
> ers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers dcharger
> s d49ers dseahawks drams dbuccaneers dtitans dredskins
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.0835
0.3124
0.2873
Random effects u_i ~
corr(u_i, X)
=
capvalue
Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
rate
L1.
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
_cons
20254.68
-3839118
-4270060
-2596233
-4339563
-5444537
-4731928
-2411496
-3757052
-3884920
-3411288
-4544148
-2514947
-1512382
-5775319
-4154680
-4843109
-4079960
-3679358
-4651285
-3146108
-5003556
-3043044
-3703831
-3988432
-2186909
-4660108
-3900727
-4268885
-4415544
-3429250
-3731496
5238504
sigma_u
sigma_e
rho
1545848.9
1954847.2
.38473986
367
84
Std. Err.
3819.688
1391916
1457252
1687118
1758252
1673851
1685671
1490864
1450133
1561670
1469921
1439423
1857192
1644751
1498789
1457836
1424244
1363923
1506940
1515097
1472795
1445665
1607036
1507980
1579748
1499619
1442309
1341221
1402118
1434426
1497973
1393326
1206533
Wald chi2( 32)
Prob > chi2
z
5.30
-2.76
-2.93
-1.54
-2.47
-3.25
-2.81
-1.62
-2.59
-2.49
-2.32
-3.16
-1.35
-0.92
-3.85
-2.85
-3.40
-2.99
-2.44
-3.07
-2.14
-3.46
-1.89
-2.46
-2.52
-1.46
-3.23
-2.91
-3.04
-3.08
-2.29
-2.68
4.34
P>|z|
0.000
0.006
0.003
0.124
0.014
0.001
0.005
0.106
0.010
0.013
0.020
0.002
0.176
0.358
0.000
0.004
0.001
0.003
0.015
0.002
0.033
0.001
0.058
0.014
0.012
0.145
0.001
0.004
0.002
0.002
0.022
0.007
0.000
=
=
1
4.4
7
64.12
0.0006
[95% Conf. Interval]
12768.22
-6567224
-7126221
-5902923
-7785673
-8725225
-8035783
-5333536
-6599260
-6945737
-6292280
-7365366
-6154976
-4736034
-8712891
-7011985
-7634576
-6753200
-6632906
-7620820
-6032734
-7837007
-6192777
-6659418
-7084682
-5126108
-7486981
-6529471
-7016986
-7226968
-6365224
-6462365
2873742
(fraction of variance due to u_i)
27741.13
-1111012
-1413899
710457.3
-893452.2
-2163850
-1428073
510543.7
-914842.8
-824103.3
-530296.8
-1722931
1125082
1711271
-2837747
-1297374
-2051642
-1406719
-725810.3
-1681750
-259482.4
-2170104
106689.9
-748244.1
-892181.7
752290.9
-1833235
-1271983
-1520784
-1604120
-493276.4
-1000627
7603266
35
5. . xtreg lcapvalue l.exp l.expsq ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
=
lcapvalue
Number of obs
=
Number of groups
0.3249
0.1681
0.1752
Obs per group: min =
avg =
max =
-0.1298
F(2,279)
Prob > F
Coef.
Std. Err.
t
P>|t|
365
84
=
=
=
1
4.3
7
67.13
0.0000
[95% Conf. Interval]
exp
L1.
expsq
L1.
_cons
.440962
.0392241
11.24
0.000
.3637493
.5181748
-.0239457
13.07829
.0027393
.1191112
-8.74
109.80
0.000
0.000
-.029338
12.84382
-.0185533
13.31277
sigma_u
sigma_e
rho
.91683304
.56348094
.72583324
(fraction of variance due to u_i)
F test that all u_i=0:
F( 83, 279) =
9.96
Prob > F = 0.0000
6. . xtreg lcapvalue l.gs ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
=
lcapvalue
Number of obs
=
Number of groups
0.1634
0.6849
0.5035
Obs per group: min =
avg =
max =
0.5721
F(1,280)
Prob > F
Coef.
gs
L1.
_cons
.0602965
13.95662
sigma_u
sigma_e
rho
.75915108
.62614982
.59513203
F test that all u_i=0:
Std. Err.
.0081548
.0666776
t
P>|t|
7.39
209.31
365
84
=
=
=
1
4.3
7
54.67
0.0000
[95% Conf. Interval]
0.000
0.000
.0442439
13.82536
.076349
14.08787
(fraction of variance due to u_i)
F( 83, 280) =
3.14
Prob > F = 0.0000
7.
. xtreg lcapvalue l.pctofgamesstarted ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
lcapvalue
=
Number of obs
=
Number of groups
0.1018
0.4602
0.3698
Obs per group: min =
avg =
max =
0.4728
F(1,235)
Prob > F
Coef.
pctofgames~d
L1.
_cons
.7105315
14.07673
sigma_u
sigma_e
rho
.79062173
.62573563
.61485897
F test that all u_i=0:
Std. Err.
.137685
.0987915
t
P>|t|
5.16
142.49
0.000
0.000
317
81
=
=
=
1
3.9
7
26.63
0.0000
[95% Conf. Interval]
.4392769
13.8821
.981786
14.27136
(fraction of variance due to u_i)
F( 80, 235) =
3.66
Prob > F = 0.0000
36
8. . reg lcapvalue ht
Source
SS
df
MS
Model
Residual
3.53660431
603.541125
1
462
3.53660431
1.30636607
Total
607.077729
463
1.311183
lcapvalue
Coef.
Std. Err.
.0617184
9.563875
ht
_cons
t
.0375106
2.808704
1.65
3.41
Number of obs =
F( 1,
462)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
=
=
=
=
=
464
2.71
0.1006
0.0058
0.0037
1.143
[95% Conf. Interval]
0.101
0.001
-.0119941
4.044457
.135431
15.08329
9. . reg lcapvalue wt
Source
df
MS
Model
Residual
10.4285052
596.649224
SS
1
462
10.4285052
1.29144854
Total
607.077729
463
1.311183
lcapvalue
wt
_cons
Coef.
Std. Err.
.0118269
11.53885
.004162
.9324726
t
2.84
12.37
Number of obs =
F( 1,
462)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.005
0.000
=
=
=
=
=
464
8.08
0.0047
0.0172
0.0151
1.1364
[95% Conf. Interval]
.0036482
9.706436
.0200056
13.37126
10. . xtreg lcapvalue l.interceptions l.ptd y00 y01 y02 y03 y04 y05 y06 y07 if gs>=8
note: y00 dropped because of collinearity
note: y02 dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.3775
0.4995
0.4129
Random effects u_i ~
corr(u_i, X)
=
lcapvalue
Number of obs
=
Number of groups
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
173
55
=
Std. Err.
Wald chi2( 8)
Prob > chi2
z
P>|z|
=
=
1
3.1
7
108.79
0.0000
[95% Conf. Interval]
intercepti~s
L1.
ptd
L1.
y01
y03
y04
y05
y06
y07
_cons
.0189017
.0101443
1.86
0.062
-.0009808
.0387841
.0434145
-.2847873
.1016172
.0153469
.1705334
.4587249
.3985979
13.89946
.0071356
.1875889
.1819525
.1751459
.1789033
.1782349
.1844948
.1780126
6.08
-1.52
0.56
0.09
0.95
2.57
2.16
78.08
0.000
0.129
0.577
0.930
0.340
0.010
0.031
0.000
.0294291
-.6524548
-.255003
-.3279327
-.1801105
.109391
.0369946
13.55057
.0574
.0828803
.4582375
.3586266
.5211774
.8080588
.7602011
14.24836
sigma_u
sigma_e
rho
.41590058
.5519512
.36215413
(fraction of variance due to u_i)
11. 37
. xtreg lcapvalue l.interceptions l.gs dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbronc
> os dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dst
> eelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.2221
0.6854
0.5454
Random effects u_i ~
corr(u_i, X)
=
lcapvalue
intercepti~s
L1.
gs
L1.
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
_cons
Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
365
84
Std. Err.
Wald chi2( 33)
Prob > chi2
z
P>|z|
=
1
4.3
7
198.97
0.0000
=
[95% Conf. Interval]
-.0303574
.0126393
-2.40
0.016
-.05513
-.0055847
.1176726
-.9756889
-.9939535
-.7104435
-1.04008
-.9680404
-1.202484
-.4961707
-.7727529
-.4087923
-.610266
-1.203488
-1.158997
-.5702867
-.9119642
-.9744763
-1.174528
-.9855204
-1.079872
-.8983575
-.9373878
-1.217716
-.9411509
-1.085783
-1.585512
-.5936432
-.9936043
-.8032136
-.9228794
-1.271332
-.6604002
-.7226087
14.60054
.0138604
.431719
.4521004
.523095
.5480824
.5222009
.5251186
.4645388
.4515109
.488364
.456446
.4475691
.5763073
.5102558
.4696097
.4526173
.4444627
.4239892
.4670408
.4723382
.4563711
.4516811
.498073
.4693304
.487937
.467485
.4480642
.4181844
.4372712
.4453714
.4663019
.4333973
.3730729
8.49
-2.26
-2.20
-1.36
-1.90
-1.85
-2.29
-1.07
-1.71
-0.84
-1.34
-2.69
-2.01
-1.12
-1.94
-2.15
-2.64
-2.32
-2.31
-1.90
-2.05
-2.70
-1.89
-2.31
-3.25
-1.27
-2.22
-1.92
-2.11
-2.85
-1.42
-1.67
39.14
0.000
0.024
0.028
0.174
0.058
0.064
0.022
0.285
0.087
0.403
0.181
0.007
0.044
0.264
0.052
0.031
0.008
0.020
0.021
0.057
0.040
0.007
0.059
0.021
0.001
0.204
0.027
0.055
0.035
0.004
0.157
0.095
0.000
.0905068
-1.821843
-1.880054
-1.735691
-2.114302
-1.991535
-2.231697
-1.40665
-1.657698
-1.365968
-1.504884
-2.080708
-2.288538
-1.57037
-1.832382
-1.86159
-2.045658
-1.816524
-1.995255
-1.824123
-1.831859
-2.102995
-1.917356
-2.005654
-2.541851
-1.509897
-1.871794
-1.62284
-1.779915
-2.144244
-1.574335
-1.572052
13.86933
.1448385
-.1295351
-.107853
.3148037
.0341416
.0554547
-.17327
.4143086
.1121922
.5483835
.2843517
-.3262689
-.0294552
.4297963
.0084538
-.0873626
-.3033968
-.1545169
-.1644888
.0274083
-.0429169
-.3324377
.0350542
-.1659123
-.6291728
.3226105
-.1154146
.0164128
-.0658435
-.3984203
.2535348
.1268345
15.33175
sigma_u
.4700485
12. .
>
>
>
xtreg lcapvalue
d14 d15 d16 d17
d44 d45 d46 d47
d74 d75 d76 d77
l.exp l.expsq l.pcomppct l.pyds
d18 d19 d20 d21 d22 d23 d24 d25
d48 d49 d50 d51 d52 d53 d54 d55
d78 d79 d80 d81 d82 d83 d84 d85
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
lcapvalue
exp
L1.
expsq
L1.
pcomppct
L1.
pyds
L1.
ptd
L1.
intercepti~s
L1.
sck
L1.
Number of obs
=
Number of groups
0.3957
1.0000
0.8063
Random effects u_i ~
corr(u_i, X)
=
Std. Err.
Wald chi2( 84)
Prob > chi2
z
P>|z|
=
=
303
78
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
l.ptd l.interceptions l.sck d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13
d26 d27 d28 d29 d30 d31 d32 d33 d34 d35 d36 d37 d38 d39 d40 d41 d42 d43
d56 d57 d58 d59 d60 d61 d62 d63 d64 d65 d66 d67 d68 d69 d70 d71 d72 d73
d86 d87 d88 d89 d90 d91 d92 d93 d94 d95 d96
1
3.9
7
907.21
0.0000
[95% Conf. Interval]
.3627602
.0426122
8.51
0.000
.2792418
.4462786
-.0188704
.0028811
-6.55
0.000
-.0245173
-.0132236
.3464153
.3267147
1.06
0.289
-.2939338
.9867644
.0002402
.000106
2.27
0.023
.0000326
.0004479
.0002593
.0112537
0.02
0.982
-.0217977
.0223162
-.0088807
.0106404
-0.83
0.404
-.0297355
.011974
-.0017347
.0048105
-0.36
0.718
-.0111631
.0076937
38
_cons
13.21509
.5233163
sigma_u
sigma_e
rho
0
.5214263
0
25.25
0.000
12.18941
(fraction of variance due to u_i)
14.24077
13. . xtreg lcapvalue l.exp l.expsq l.pcomppct l.pyds l.ptd l.interceptions l.sck l.gs ,fe
Fixed-effects (within) regression
Group variable: obs
R-sq:
within =
between =
overall =
corr(u_i, Xb)
lcapvalue
exp
L1.
expsq
L1.
pcomppct
L1.
pyds
L1.
ptd
L1.
intercepti~s
L1.
sck
L1.
gs
L1.
_cons
sigma_u
sigma_e
rho
=
Obs per group: min =
avg =
max =
-0.0041
F(8,217)
Prob > F
Coef.
14. Std. Err.
t
P>|t|
303
78
=
0.4032
0.4053
0.3791
=
=
1
3.9
7
18.33
0.0000
[95% Conf. Interval]
.360661
.0424637
8.49
0.000
.2769669
.444355
-.0189453
.0028701
-6.60
0.000
-.0246021
-.0132884
.3756269
.3259106
1.15
0.250
-.2667287
1.017982
.0000914
.0001388
0.66
0.511
-.0001822
.000365
-.0002735
.0112141
-0.02
0.981
-.0223761
.0218291
-.0132091
.0109181
-1.21
0.228
-.0347281
.00831
-.0048875
.0051582
-0.95
0.344
-.0150541
.0052791
.0462967
12.87554
.0280463
.2167161
1.65
59.41
0.100
0.000
-.0089814
12.4484
.1015748
13.30267
.72287905
.51937562
.65953644
F test that all u_i=0:
Number of obs
=
Number of groups
(fraction of variance due to u_i)
F( 77, 217) =
4.15
Prob > F = 0.0000
39
. xtreg lcapvalue l.exp l.expsq l.rate l.sck l.gs l.rtd l.ryds l.regseasonwins y00 y01 y02 y03 y04 y05 y06 y07
note: y00 dropped because of collinearity
note: y02 dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.4259
0.6561
0.5726
Random effects u_i ~
corr(u_i, X)
=
lcapvalue
exp
L1.
expsq
L1.
rate
L1.
sck
L1.
gs
L1.
rtd
L1.
ryds
L1.
regseasonw~s
L1.
y01
y03
y04
y05
y06
y07
_cons
sigma_u
sigma_e
rho
15. Number of obs
=
Number of groups
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
365
84
=
Std. Err.
Wald chi2( 14)
Prob > chi2
z
P>|z|
=
=
1
4.3
7
324.84
0.0000
[95% Conf. Interval]
.2837511
.0366806
7.74
0.000
.2118585
.3556436
-.0158642
.0024127
-6.58
0.000
-.0205929
-.0111355
.0021731
.001125
1.93
0.053
-.0000318
.004378
-.0041898
.0051741
-0.81
0.418
-.0143309
.0059513
.0751387
.012307
6.11
0.000
.0510173
.0992601
-.0228148
.0350197
-0.65
0.515
-.0914522
.0458226
.0003807
.0005437
0.70
0.484
-.0006849
.0014463
.0245214
-.0257068
.2530758
.0955666
.1503134
.1979179
.3330096
12.55499
.0119926
.1369846
.1246211
.124395
.1239032
.1258871
.1304998
.1674532
2.04
-0.19
2.03
0.77
1.21
1.57
2.55
74.98
0.041
0.851
0.042
0.442
0.225
0.116
0.011
0.000
.0010162
-.2941917
.0088229
-.1482432
-.0925325
-.0488163
.0772348
12.22679
.0480265
.2427781
.4973287
.3393764
.3931593
.4446521
.5887844
12.88319
.44833444
.50365711
.44208302
(fraction of variance due to u_i)
40
. xtreg lcapvalue
> y07
note: y00 dropped
note: y01 dropped
note: y02 dropped
l.exp l.expsq l.rate l.sck l.gs l.rtd l.ryds l.regseasonwins l2.wonsuperbowl y00 y01 y02 y03 y04 y05 y06
because of collinearity
because of collinearity
because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.2745
0.7747
0.6035
Random effects u_i ~
corr(u_i, X)
=
lcapvalue
exp
L1.
expsq
L1.
rate
L1.
sck
L1.
gs
L1.
rtd
L1.
ryds
L1.
regseasonw~s
L1.
wonsuperbowl
L2.
y03
y04
y05
y06
y07
_cons
sigma_u
sigma_e
rho
Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
279
71
Std. Err.
Wald chi2( 14)
Prob > chi2
z
P>|z|
=
=
1
3.9
6
271.23
0.0000
[95% Conf. Interval]
.2233226
.0435665
5.13
0.000
.1379338
.3087113
-.0126701
.0027003
-4.69
0.000
-.0179626
-.0073776
.0026633
.0013548
1.97
0.049
7.99e-06
.0053187
-.0085052
.0057719
-1.47
0.141
-.0198179
.0028074
.0960592
.0132456
7.25
0.000
.0700983
.1220202
-.0232805
.0392978
-0.59
0.554
-.1003027
.0537418
.000808
.0005548
1.46
0.145
-.0002794
.0018955
.0291631
.0137586
2.12
0.034
.0021968
.0561295
.1422508
.247729
.0671194
.1673362
.2352229
.3742909
12.56888
.2025955
.1424452
.1415351
.1369741
.1355663
.1371455
.2074093
0.70
1.74
0.47
1.22
1.74
2.73
60.60
0.483
0.082
0.635
0.222
0.083
0.006
0.000
-.2548291
-.0314584
-.2102842
-.1011282
-.0304822
.1054907
12.16237
.5393306
.5269165
.344523
.4358005
.500928
.6430912
12.9754
.26157212
.50391917
.21225072
(fraction of variance due to u_i)
16. 41
. xtreg regseasonwins ptd dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions d
> packers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers dc
> hargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.2493
0.5890
0.4643
Random effects u_i ~
corr(u_i, X)
=
regseasonw~s
Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
Std. Err.
Wald chi2( 32)
Prob > chi2
z
.0270309
1.341551
1.326908
1.398919
1.780537
1.482343
1.582736
1.326885
1.474692
1.736475
1.512685
1.397487
1.67217
1.411115
1.530055
1.495999
1.488615
1.416055
1.323021
1.742061
1.494993
1.334023
1.587116
1.349605
1.448989
1.388213
1.61154
1.338168
1.366791
1.370503
1.286229
1.612956
1.184958
sigma_u
sigma_e
rho
.58316445
2.3735634
.05692792
(fraction of variance due to u_i)
17. 0.000
0.367
0.680
0.753
0.686
0.206
0.132
0.105
0.487
0.599
0.103
0.409
0.179
0.798
0.475
0.868
0.049
0.926
0.104
0.787
0.293
0.391
0.465
0.230
0.599
0.673
0.531
0.735
0.392
0.679
0.613
0.585
0.000
=
=
1
3.3
8
127.10
0.0000
[95% Conf. Interval]
.1804051
-1.210322
.5468973
-.4393028
-.7204439
-1.873512
2.384208
-2.15358
1.024919
.9138654
-2.468439
-1.152624
-2.246461
.3609777
1.093331
.2492065
-2.931395
-.1314502
2.150056
.4716759
-1.573288
-1.143425
-1.158669
1.620261
.7614854
.586182
1.009167
.453696
-1.16893
-.5671619
.6501697
-.88162
5.036159
6.67
-0.90
0.41
-0.31
-0.40
-1.26
1.51
-1.62
0.70
0.53
-1.63
-0.82
-1.34
0.26
0.71
0.17
-1.97
-0.09
1.63
0.27
-1.05
-0.86
-0.73
1.20
0.53
0.42
0.63
0.34
-0.86
-0.41
0.51
-0.55
4.25
P>|z|
ptd
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
_cons
200
60
.1274255
-3.839714
-2.053795
-3.181134
-4.210233
-4.77885
-.717898
-4.754226
-1.865424
-2.489564
-5.433247
-3.891649
-5.523855
-2.404757
-1.905523
-2.682898
-5.849026
-2.906867
-.4430176
-2.9427
-4.503421
-3.758063
-4.269359
-1.024916
-2.078481
-2.134665
-2.149394
-2.169065
-3.847791
-3.253299
-1.870793
-4.042955
2.713683
.2333848
1.419069
3.147589
2.302529
2.769345
1.031826
5.486313
.4470664
3.915262
4.317295
.4963684
1.586401
1.030932
3.126712
4.092184
3.181311
-.0137636
2.643967
4.74313
3.886052
1.356844
1.471213
1.952021
4.265439
3.601452
3.307029
4.167728
3.076457
1.509931
2.118975
3.171132
2.279715
7.358635
42
. xtreg lregseasonwins rate dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions
> dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers
> dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.3033
0.5142
0.4352
Random effects u_i ~
corr(u_i, X)
=
lregseason~s
Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
Std. Err.
Wald chi2( 32)
Prob > chi2
z
.0026498
.2531486
.2532998
.2641275
.3184502
.271264
.2836575
.2573132
.280479
.3100152
.284227
.2565305
.3190813
.3161445
.2831561
.3214248
.2626931
.2627777
.2697232
.3058804
.2842079
.254382
.2793882
.2833584
.286444
.2775081
.2764847
.2377384
.2711652
.2514342
.2491229
.2962498
.3054938
sigma_u
sigma_e
rho
.20511788
.3153457
.29730397
(fraction of variance due to u_i)
18. 0.000
0.411
0.821
0.494
0.692
0.158
0.266
0.133
0.769
0.952
0.151
0.888
0.086
0.840
0.876
0.851
0.119
0.733
0.789
0.940
0.496
0.129
0.352
0.668
0.476
0.867
0.968
0.883
0.165
0.474
0.975
0.684
0.043
=
=
1
3.3
8
101.57
0.0000
[95% Conf. Interval]
.0185015
-.2082741
-.0572861
-.180657
-.1261122
-.3832408
.3157724
-.387051
.0822455
-.0185914
-.4082978
-.0361522
-.5486389
-.0638885
.0443465
-.0601978
-.4095488
-.0895699
.0721761
-.0230991
-.1935594
-.3865581
-.2598571
.1215891
-.2042278
-.0464107
-.010978
-.0349606
-.3766757
-.180036
.0076742
-.12063
.6169718
6.98
-0.82
-0.23
-0.68
-0.40
-1.41
1.11
-1.50
0.29
-0.06
-1.44
-0.14
-1.72
-0.20
0.16
-0.19
-1.56
-0.34
0.27
-0.08
-0.68
-1.52
-0.93
0.43
-0.71
-0.17
-0.04
-0.15
-1.39
-0.72
0.03
-0.41
2.02
P>|z|
rate
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
_cons
200
60
.013308
-.7044362
-.5537447
-.6983374
-.7502632
-.9149085
-.240186
-.8913755
-.4674832
-.6262101
-.9653724
-.5389426
-1.174027
-.6835204
-.5106292
-.6901789
-.9244179
-.6046047
-.4564716
-.6226137
-.7505966
-.8851376
-.8074478
-.4337831
-.7656478
-.5903165
-.5528781
-.5009194
-.9081497
-.672838
-.4805978
-.7012688
.018215
.023695
.287888
.4391724
.3370235
.4980388
.1484269
.8717309
.1172735
.6319742
.5890273
.1487769
.4666383
.0767489
.5557433
.5993221
.5697833
.1053203
.4254648
.6008238
.5764155
.3634778
.1120214
.2877337
.6769613
.3571922
.4974951
.5309221
.4309982
.1547982
.3127659
.4959462
.4600089
1.215729
43
. xtreg lregseasonwins exp expsq rate sck rtd ryds gs
dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals drave
> ns dcowboys dbroncos dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets
> draiders deagles dsteelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.4615
0.6164
0.5645
Random effects u_i ~
corr(u_i, X)
=
lregseason~s
exp
expsq
rate
sck
rtd
ryds
gs
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
19. Number of obs
=
Number of groups
=
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
-.0200178
.000605
.0151807
-.011551
-.0213521
.0003679
.0669841
.0076087
.0307806
-.2008257
.0589324
-.3617107
.3019682
-.2778486
.1903495
.1409161
-.3917313
.0518478
-.4984856
-.1143498
.1663781
.0420877
-.2324319
.0484898
.1663909
-.026323
-.1065287
-.0945079
-.1204437
.2600247
.032353
-.0663288
.2460976
.1579454
-.0214787
-.05001
.0840737
-.0573071
200
60
Std. Err.
.0284501
.0017967
.0025851
.0027757
.0186355
.0002519
.0109636
.2324205
.228223
.2469289
.2901562
.2436637
.2573444
.227862
.2560168
.281334
.2579886
.235885
.2820265
.2696525
.2483173
.2676383
.2391436
.2405346
.2341795
.2722946
.2489577
.2289799
.2531647
.2547739
.2595417
.2466082
.2526856
.2131944
.2440969
.2241611
.2236808
.2734202
Wald chi2( 38)
Prob > chi2
z
-0.70
0.34
5.87
-4.16
-1.15
1.46
6.11
0.03
0.13
-0.81
0.20
-1.48
1.17
-1.22
0.74
0.50
-1.52
0.22
-1.77
-0.42
0.67
0.16
-0.97
0.20
0.71
-0.10
-0.43
-0.41
-0.48
1.02
0.12
-0.27
0.97
0.74
-0.09
-0.22
0.38
-0.21
P>|z|
0.482
0.736
0.000
0.000
0.252
0.144
0.000
0.974
0.893
0.416
0.839
0.138
0.241
0.223
0.457
0.616
0.129
0.826
0.077
0.672
0.503
0.875
0.331
0.840
0.477
0.923
0.669
0.680
0.634
0.307
0.901
0.788
0.330
0.459
0.930
0.823
0.707
0.834
=
=
1
3.3
8
180.26
0.0000
[95% Conf. Interval]
-.075779
-.0029164
.010114
-.0169912
-.0578771
-.0001258
.0454959
-.4479272
-.4165282
-.6847975
-.5097633
-.8392827
-.2024177
-.7244499
-.3114342
-.4104885
-.8973796
-.4104782
-1.051247
-.642859
-.3203148
-.4824737
-.7011448
-.4229493
-.2925925
-.5600106
-.5944768
-.5433004
-.6166375
-.2393231
-.4763395
-.5496719
-.2491571
-.2599079
-.4998997
-.4893577
-.3543327
-.5932008
.0357434
.0041264
.0202475
-.0061108
.0151729
.0008615
.0884723
.4631446
.4780894
.283146
.6276282
.1158613
.806354
.1687527
.6921332
.6923206
.113917
.5141738
.0542763
.4141594
.653071
.5666492
.2362811
.5199289
.6253743
.5073646
.3814194
.3542845
.37575
.7593724
.5410455
.4170144
.7413523
.5757987
.4569424
.3893377
.52248
.4785866
44
. xtreg lregseasonwins exp expsq rate sck rtd ryds gs l3.wonsuperbowl
dbrowns dcardinals dfalcons dbills dpanthers dbear
> s dbengals dravens dcowboys dbroncos dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaint
> s dgiants djets draiders deagles dsteelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8
note: dpackers dropped because of collinearity
Random-effects GLS regression
Group variable: obs
R-sq:
within =
between =
overall =
0.5025
0.8640
0.7553
Random effects u_i ~
corr(u_i, X)
=
lregseason~s
Number of obs
=
Number of groups
Obs per group: min =
avg =
max =
Gaussian
0 (assumed)
Coef.
103
40
=
Std. Err.
Wald chi2( 37)
Prob > chi2
z
P>|z|
=
=
1
2.6
5
154.37
0.0000
[95% Conf. Interval]
exp
expsq
rate
sck
rtd
ryds
gs
wonsuperbowl
L3.
dbrowns
dcardinals
dfalcons
dbills
dpanthers
dbears
dbengals
dravens
dcowboys
dbroncos
dlions
dtexans
dcolts
djaguars
dchiefs
ddolphins
dvikings
dpatriots
dsaints
dgiants
djets
draiders
deagles
dsteelers
dchargers
d49ers
dseahawks
drams
dbuccaneers
dtitans
dredskins
_cons
-.0978946
.0042126
.0094647
-.0158034
.0219416
.0002524
.0673368
.0510495
.0026727
.0036782
.0041549
.0266615
.0003816
.0146347
-1.92
1.58
2.57
-3.80
0.82
0.66
4.60
0.055
0.115
0.010
0.000
0.411
0.508
0.000
-.1979498
-.0010259
.0022556
-.0239468
-.0303139
-.0004955
.0386534
.0021605
.0094511
.0166738
-.00766
.0741971
.0010004
.0960203
.2685658
-.1216714
.0565413
-.0113941
(dropped)
.0228089
.3162462
-.09739
-.0274463
.1259591
.0797176
-.1918851
-.597166
.0662275
.1125682
.3092648
-.1456616
-.0847136
.1090676
.2019396
-.1137503
-.0393572
-.2780264
.2693371
.1513948
-.1095333
-.0148283
.2517392
.0281761
.0235807
-.048114
(dropped)
1.127414
.192206
.2501946
.2368391
.3054718
1.40
-0.49
0.24
-0.04
0.162
0.627
0.811
0.970
-.108151
-.6120439
-.4076548
-.6101077
.6452826
.3687011
.5207374
.5873196
.2782492
.3405066
.2148152
.3464091
.2848702
.3449221
.2699186
.3069655
.2375632
.2401711
.2285346
.2489925
.3150534
.2570223
.2729145
.2403596
.2159081
.3272844
.2517683
.3570334
.230542
.2759735
.1985793
.2301507
.2397807
.2266213
0.08
0.93
-0.45
-0.08
0.44
0.23
-0.71
-1.95
0.28
0.47
1.35
-0.59
-0.27
0.42
0.74
-0.47
-0.18
-0.85
1.07
0.42
-0.48
-0.05
1.27
0.12
0.10
-0.21
0.935
0.353
0.650
0.937
0.658
0.817
0.477
0.052
0.780
0.639
0.176
0.559
0.788
0.671
0.459
0.636
0.855
0.396
0.285
0.672
0.635
0.957
0.205
0.903
0.922
0.832
-.5225495
-.3511345
-.5184201
-.7063957
-.4323763
-.5963172
-.7209159
-1.198807
-.3993879
-.3581585
-.1386549
-.6336779
-.7022068
-.3946867
-.332963
-.5848464
-.4625293
-.9194921
-.2241196
-.5483779
-.5613873
-.5557265
-.137469
-.422911
-.4463809
-.4922837
.5681674
.9836268
.32364
.6515031
.6842945
.7557524
.3371457
.0044753
.5318428
.5832949
.7571845
.3423547
.5327797
.612822
.7368423
.3573457
.383815
.3634392
.7627938
.8511675
.3423208
.5260698
.6409474
.4792633
.4935422
.3960556
.4297942
2.62
0.009
.2850325
1.969795
sigma_u
sigma_e
rho
.11536654
.25628941
.16848734
(fraction of variance due to u_i)
Works Cited USATODAY Player Salaries Database. Retrieved throughout February 2008. (http://content.usatoday.com/sports/football/nfl/salaries/) Official NFL Players Database. Retrieved throughout February 2008. (http://www.nfl.com/players/) NFL Team History Database. Retrieved throughout February 2008. (http://www.nflteamhistory.com/nfl_team_history.html) Salary Cap Frequently Asked Questions. Retrieved throughout March 2008. (http://askthecommish.com/salarycap/faq.asp) Hendricks, Wallace; DeBrock, Lawrence; Koenker, Roger. “Uncertainty, Hiring, and Subsequent Performance: The NFL Draft.” Journal of Labor Economics, October 2003, v. 21, iss. 4, pp. 857‐
86 Craig, Lee A.; Hall, Alastair R. “Trying Out for the Team: Do Exhibitions Matter? Evidence from the National Football League.” Journal of the American Statistical Association, September 1994, v. 89, iss. 427, pp. 1091‐99 Leeds, Michael A.; Kowalewski, Sandra. “Winner Take All in the NFL: The Effect of the Salary Cap and Free Agency on the Compensation of Skill Position Players.” Journal of Sports Economics, August 2001, v. 2, iss. 3, pp. 244‐56 45