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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
R2=0.0058
#Obs=464
9.
Log(CapValue) Coefficient Std. Error P>|t|
Wt .0118269 .004162 0.005
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
L1.Interceptions ‐.0132091 .0109181 0.228
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
L1.RegSeasonWins .0245214 .0119926 0.041
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 .0185015 .0026498 0.000
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 ‐.0200178 .0284501 0.482
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
L3.WonSuperbowl .2685658 .192206 0.162
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
fuml 466 1.890558 2.10244 0 9 fum 466 4.375536 4.483969 0 23 rtd 466 .6416309 1.317502 0 10 ryds 466 69.79828 126.9199 -13 1039 ratt 466 18.66524 21.37015 0 123 rate 466 64.71953 34.59824 0 156.9 scky 466 93.26609 92.13887 0 424 sck 466 14.60515 14.48376 0 76intercepti~s 466 6.845494 6.682488 0 29 ptd 466 9.17382 10.00719 0 50 pyds 466 1512.7 1475.598 0 4830 patt 466 219.0837 206.4902 0 652 pcomp 466 132.0773 126.9787 0 440 gs 466 6.729614 6.478074 0 16 g 466 8.203863 6.046706 0 16 exp 466 5.371245 3.917091 1 21 age 466 27.88841 4.239141 21 44 wt 466 223.7639 12.72809 196 285 ht 466 74.87339 1.419927 71 78
wonsuperbowl 466 .0321888 .176691 0 1regseasonl~s 466 7.914163 2.991949 0 15regseasonw~s 466 8.081545 2.989192 1 16 capvalue 466 2597211 2832639 0 1.54e+07 totalsalary 466 3027277 3926658 66176 3.50e+07 otherbonus 466 538608.8 1643858 0 1.23e+07 signbonus 466 1494023 3311007 0 3.45e+07 basesalary 466 1190329 1562312 0 1.10e+07
33
Appendix
1.
F test that all u_i=0: F( 83, 282) = 2.73 Prob > F = 0.0000 rho .48063419 (fraction of variance due to u_i) sigma_e 1847200.7 sigma_u 1776989 _cons 1914881 176411.1 10.85 0.000 1567632 2262131 L1. 108527.1 15302.3 7.09 0.000 78405.84 138648.3 ptd capvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = 0.5536 Prob > F = 0.0000 F(1,282) = 50.30
overall = 0.4692 max = 7 between = 0.7170 avg = 4.4R-sq: within = 0.1514 Obs per group: min = 1
Group variable: obs Number of groups = 84Fixed-effects (within) regression Number of obs = 367
. xtreg capvalue l.ptd ,fe
2.
F test that all u_i=0: F( 70, 208) = 2.91 Prob > F = 0.0000 rho .51120839 (fraction of variance due to u_i) sigma_e 1733970.4 sigma_u 1773286.1 _cons 1631497 256585.7 6.36 0.000 1125655 2137339 L2. 100066.7 16769.26 5.97 0.000 67007.18 133126.2 L1. 64066.31 17442.06 3.67 0.000 29680.43 98452.2 ptd capvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = 0.4510 Prob > F = 0.0000 F(2,208) = 28.34
overall = 0.5361 max = 6 between = 0.6737 avg = 4.0R-sq: within = 0.2141 Obs per group: min = 1
Group variable: obs Number of groups = 71Fixed-effects (within) regression Number of obs = 281
. xtreg capvalue l.ptd l2.ptd ,fe
34
3.
rho .16889926 (fraction of variance due to u_i) sigma_e 1751372.5 sigma_u 789524.41 _cons 575559.4 355044.3 1.62 0.105 -120314.7 1271433 y07 1025555 392056.8 2.62 0.009 257137.6 1793972 y06 1094537 394331 2.78 0.006 321662.2 1867411 y05 439813.4 402574.6 1.09 0.275 -349218.3 1228845 y04 190079.5 414823.9 0.46 0.647 -622960.4 1003120 y03 798669 425670.2 1.88 0.061 -35629.28 1632967 y01 -522170 469627.4 -1.11 0.266 -1442623 398282.8 L1. 181995.5 12205.54 14.91 0.000 158073.1 205917.9 ptd capvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 7) = 235.85
overall = 0.4943 max = 7 between = 0.6959 avg = 4.4R-sq: within = 0.2294 Obs per group: min = 1
Group variable: obs Number of groups = 84Random-effects GLS regression Number of obs = 367
note: y02 dropped because of collinearitynote: y00 dropped because of collinearity. xtreg capvalue l.ptd y00 y01 y02 y03 y04 y05 y06 y07
4.
rho .38473986 (fraction of variance due to u_i) sigma_e 1954847.2 sigma_u 1545848.9 _cons 5238504 1206533 4.34 0.000 2873742 7603266 dredskins -3731496 1393326 -2.68 0.007 -6462365 -1000627 dtitans -3429250 1497973 -2.29 0.022 -6365224 -493276.4 dbuccaneers -4415544 1434426 -3.08 0.002 -7226968 -1604120 drams -4268885 1402118 -3.04 0.002 -7016986 -1520784 dseahawks -3900727 1341221 -2.91 0.004 -6529471 -1271983 d49ers -4660108 1442309 -3.23 0.001 -7486981 -1833235 dchargers -2186909 1499619 -1.46 0.145 -5126108 752290.9 dsteelers -3988432 1579748 -2.52 0.012 -7084682 -892181.7 deagles -3703831 1507980 -2.46 0.014 -6659418 -748244.1 draiders -3043044 1607036 -1.89 0.058 -6192777 106689.9 djets -5003556 1445665 -3.46 0.001 -7837007 -2170104 dgiants -3146108 1472795 -2.14 0.033 -6032734 -259482.4 dsaints -4651285 1515097 -3.07 0.002 -7620820 -1681750 dpatriots -3679358 1506940 -2.44 0.015 -6632906 -725810.3 dvikings -4079960 1363923 -2.99 0.003 -6753200 -1406719 ddolphins -4843109 1424244 -3.40 0.001 -7634576 -2051642 dchiefs -4154680 1457836 -2.85 0.004 -7011985 -1297374 djaguars -5775319 1498789 -3.85 0.000 -8712891 -2837747 dcolts -1512382 1644751 -0.92 0.358 -4736034 1711271 dtexans -2514947 1857192 -1.35 0.176 -6154976 1125082 dlions -4544148 1439423 -3.16 0.002 -7365366 -1722931 dbroncos -3411288 1469921 -2.32 0.020 -6292280 -530296.8 dcowboys -3884920 1561670 -2.49 0.013 -6945737 -824103.3 dravens -3757052 1450133 -2.59 0.010 -6599260 -914842.8 dbengals -2411496 1490864 -1.62 0.106 -5333536 510543.7 dbears -4731928 1685671 -2.81 0.005 -8035783 -1428073 dpanthers -5444537 1673851 -3.25 0.001 -8725225 -2163850 dbills -4339563 1758252 -2.47 0.014 -7785673 -893452.2 dfalcons -2596233 1687118 -1.54 0.124 -5902923 710457.3 dcardinals -4270060 1457252 -2.93 0.003 -7126221 -1413899 dbrowns -3839118 1391916 -2.76 0.006 -6567224 -1111012 L1. 20254.68 3819.688 5.30 0.000 12768.22 27741.13 rate capvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0006Random effects u_i ~ Gaussian Wald chi2( 32) = 64.12
overall = 0.2873 max = 7 between = 0.3124 avg = 4.4R-sq: within = 0.0835 Obs per group: min = 1
Group variable: obs Number of groups = 84Random-effects GLS regression Number of obs = 367
note: dpackers dropped because of collinearity> s d49ers dseahawks drams dbuccaneers dtitans dredskins> ers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers dcharger. xtreg capvalue l.rate dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions dpack
35
5.
F test that all u_i=0: F( 83, 279) = 9.96 Prob > F = 0.0000 rho .72583324 (fraction of variance due to u_i) sigma_e .56348094 sigma_u .91683304 _cons 13.07829 .1191112 109.80 0.000 12.84382 13.31277 L1. -.0239457 .0027393 -8.74 0.000 -.029338 -.0185533 expsq L1. .440962 .0392241 11.24 0.000 .3637493 .5181748 exp lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = -0.1298 Prob > F = 0.0000 F(2,279) = 67.13
overall = 0.1752 max = 7 between = 0.1681 avg = 4.3R-sq: within = 0.3249 Obs per group: min = 1
Group variable: obs Number of groups = 84Fixed-effects (within) regression Number of obs = 365
. xtreg lcapvalue l.exp l.expsq ,fe
6.
F test that all u_i=0: F( 83, 280) = 3.14 Prob > F = 0.0000 rho .59513203 (fraction of variance due to u_i) sigma_e .62614982 sigma_u .75915108 _cons 13.95662 .0666776 209.31 0.000 13.82536 14.08787 L1. .0602965 .0081548 7.39 0.000 .0442439 .076349 gs lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = 0.5721 Prob > F = 0.0000 F(1,280) = 54.67
overall = 0.5035 max = 7 between = 0.6849 avg = 4.3R-sq: within = 0.1634 Obs per group: min = 1
Group variable: obs Number of groups = 84Fixed-effects (within) regression Number of obs = 365
. xtreg lcapvalue l.gs ,fe
7.
F test that all u_i=0: F( 80, 235) = 3.66 Prob > F = 0.0000 rho .61485897 (fraction of variance due to u_i) sigma_e .62573563 sigma_u .79062173 _cons 14.07673 .0987915 142.49 0.000 13.8821 14.27136 L1. .7105315 .137685 5.16 0.000 .4392769 .981786pctofgames~d lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = 0.4728 Prob > F = 0.0000 F(1,235) = 26.63
overall = 0.3698 max = 7 between = 0.4602 avg = 3.9R-sq: within = 0.1018 Obs per group: min = 1
Group variable: obs Number of groups = 81Fixed-effects (within) regression Number of obs = 317
. xtreg lcapvalue l.pctofgamesstarted ,fe
36
8.
_cons 9.563875 2.808704 3.41 0.001 4.044457 15.08329 ht .0617184 .0375106 1.65 0.101 -.0119941 .135431 lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 607.077729 463 1.311183 Root MSE = 1.143 Adj R-squared = 0.0037 Residual 603.541125 462 1.30636607 R-squared = 0.0058 Model 3.53660431 1 3.53660431 Prob > F = 0.1006 F( 1, 462) = 2.71 Source SS df MS Number of obs = 464
. reg lcapvalue ht
9.
_cons 11.53885 .9324726 12.37 0.000 9.706436 13.37126 wt .0118269 .004162 2.84 0.005 .0036482 .0200056 lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 607.077729 463 1.311183 Root MSE = 1.1364 Adj R-squared = 0.0151 Residual 596.649224 462 1.29144854 R-squared = 0.0172 Model 10.4285052 1 10.4285052 Prob > F = 0.0047 F( 1, 462) = 8.08 Source SS df MS Number of obs = 464
. reg lcapvalue wt
10.
rho .36215413 (fraction of variance due to u_i) sigma_e .5519512 sigma_u .41590058 _cons 13.89946 .1780126 78.08 0.000 13.55057 14.24836 y07 .3985979 .1844948 2.16 0.031 .0369946 .7602011 y06 .4587249 .1782349 2.57 0.010 .109391 .8080588 y05 .1705334 .1789033 0.95 0.340 -.1801105 .5211774 y04 .0153469 .1751459 0.09 0.930 -.3279327 .3586266 y03 .1016172 .1819525 0.56 0.577 -.255003 .4582375 y01 -.2847873 .1875889 -1.52 0.129 -.6524548 .0828803 L1. .0434145 .0071356 6.08 0.000 .0294291 .0574 ptd L1. .0189017 .0101443 1.86 0.062 -.0009808 .0387841intercepti~s lcapvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 8) = 108.79
overall = 0.4129 max = 7 between = 0.4995 avg = 3.1R-sq: within = 0.3775 Obs per group: min = 1
Group variable: obs Number of groups = 55Random-effects GLS regression Number of obs = 173
note: y02 dropped because of collinearitynote: y00 dropped because of collinearity. xtreg lcapvalue l.interceptions l.ptd y00 y01 y02 y03 y04 y05 y06 y07 if gs>=8
11.
37
sigma_u .4700485 _cons 14.60054 .3730729 39.14 0.000 13.86933 15.33175 dredskins -.7226087 .4333973 -1.67 0.095 -1.572052 .1268345 dtitans -.6604002 .4663019 -1.42 0.157 -1.574335 .2535348 dbuccaneers -1.271332 .4453714 -2.85 0.004 -2.144244 -.3984203 drams -.9228794 .4372712 -2.11 0.035 -1.779915 -.0658435 dseahawks -.8032136 .4181844 -1.92 0.055 -1.62284 .0164128 d49ers -.9936043 .4480642 -2.22 0.027 -1.871794 -.1154146 dchargers -.5936432 .467485 -1.27 0.204 -1.509897 .3226105 dsteelers -1.585512 .487937 -3.25 0.001 -2.541851 -.6291728 deagles -1.085783 .4693304 -2.31 0.021 -2.005654 -.1659123 draiders -.9411509 .498073 -1.89 0.059 -1.917356 .0350542 djets -1.217716 .4516811 -2.70 0.007 -2.102995 -.3324377 dgiants -.9373878 .4563711 -2.05 0.040 -1.831859 -.0429169 dsaints -.8983575 .4723382 -1.90 0.057 -1.824123 .0274083 dpatriots -1.079872 .4670408 -2.31 0.021 -1.995255 -.1644888 dvikings -.9855204 .4239892 -2.32 0.020 -1.816524 -.1545169 ddolphins -1.174528 .4444627 -2.64 0.008 -2.045658 -.3033968 dchiefs -.9744763 .4526173 -2.15 0.031 -1.86159 -.0873626 djaguars -.9119642 .4696097 -1.94 0.052 -1.832382 .0084538 dcolts -.5702867 .5102558 -1.12 0.264 -1.57037 .4297963 dtexans -1.158997 .5763073 -2.01 0.044 -2.288538 -.0294552 dlions -1.203488 .4475691 -2.69 0.007 -2.080708 -.3262689 dbroncos -.610266 .456446 -1.34 0.181 -1.504884 .2843517 dcowboys -.4087923 .488364 -0.84 0.403 -1.365968 .5483835 dravens -.7727529 .4515109 -1.71 0.087 -1.657698 .1121922 dbengals -.4961707 .4645388 -1.07 0.285 -1.40665 .4143086 dbears -1.202484 .5251186 -2.29 0.022 -2.231697 -.17327 dpanthers -.9680404 .5222009 -1.85 0.064 -1.991535 .0554547 dbills -1.04008 .5480824 -1.90 0.058 -2.114302 .0341416 dfalcons -.7104435 .523095 -1.36 0.174 -1.735691 .3148037 dcardinals -.9939535 .4521004 -2.20 0.028 -1.880054 -.107853 dbrowns -.9756889 .431719 -2.26 0.024 -1.821843 -.1295351 L1. .1176726 .0138604 8.49 0.000 .0905068 .1448385 gs L1. -.0303574 .0126393 -2.40 0.016 -.05513 -.0055847intercepti~s lcapvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 33) = 198.97
overall = 0.5454 max = 7 between = 0.6854 avg = 4.3R-sq: within = 0.2221 Obs per group: min = 1
Group variable: obs Number of groups = 84Random-effects GLS regression Number of obs = 365
note: dpackers dropped because of collinearity> eelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins> os dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dst. xtreg lcapvalue l.interceptions l.gs dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbronc
12.
> d74 d75 d76 d77 d78 d79 d80 d81 d82 d83 d84 d85 d86 d87 d88 d89 d90 d91 d92 d93 d94 d95 d96> d44 d45 d46 d47 d48 d49 d50 d51 d52 d53 d54 d55 d56 d57 d58 d59 d60 d61 d62 d63 d64 d65 d66 d67 d68 d69 d70 d71 d72 d73 > d14 d15 d16 d17 d18 d19 d20 d21 d22 d23 d24 d25 d26 d27 d28 d29 d30 d31 d32 d33 d34 d35 d36 d37 d38 d39 d40 d41 d42 d43 . xtreg lcapvalue l.exp l.expsq l.pcomppct l.pyds l.ptd l.interceptions l.sck d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12 d13
L1. -.0017347 .0048105 -0.36 0.718 -.0111631 .0076937 sck L1. -.0088807 .0106404 -0.83 0.404 -.0297355 .011974intercepti~s L1. .0002593 .0112537 0.02 0.982 -.0217977 .0223162 ptd L1. .0002402 .000106 2.27 0.023 .0000326 .0004479 pyds L1. .3464153 .3267147 1.06 0.289 -.2939338 .9867644 pcomppct L1. -.0188704 .0028811 -6.55 0.000 -.0245173 -.0132236 expsq L1. .3627602 .0426122 8.51 0.000 .2792418 .4462786 exp lcapvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 84) = 907.21
overall = 0.8063 max = 7 between = 1.0000 avg = 3.9R-sq: within = 0.3957 Obs per group: min = 1
Group variable: obs Number of groups = 78Random-effects GLS regression Number of obs = 303
38
rho 0 (fraction of variance due to u_i) sigma_e .5214263 sigma_u 0 _cons 13.21509 .5233163 25.25 0.000 12.18941 14.24077
13.
F test that all u_i=0: F( 77, 217) = 4.15 Prob > F = 0.0000 rho .65953644 (fraction of variance due to u_i) sigma_e .51937562 sigma_u .72287905 _cons 12.87554 .2167161 59.41 0.000 12.4484 13.30267 L1. .0462967 .0280463 1.65 0.100 -.0089814 .1015748 gs L1. -.0048875 .0051582 -0.95 0.344 -.0150541 .0052791 sck L1. -.0132091 .0109181 -1.21 0.228 -.0347281 .00831intercepti~s L1. -.0002735 .0112141 -0.02 0.981 -.0223761 .0218291 ptd L1. .0000914 .0001388 0.66 0.511 -.0001822 .000365 pyds L1. .3756269 .3259106 1.15 0.250 -.2667287 1.017982 pcomppct L1. -.0189453 .0028701 -6.60 0.000 -.0246021 -.0132884 expsq L1. .360661 .0424637 8.49 0.000 .2769669 .444355 exp lcapvalue Coef. Std. Err. t P>|t| [95% Conf. Interval]
corr(u_i, Xb) = -0.0041 Prob > F = 0.0000 F(8,217) = 18.33
overall = 0.3791 max = 7 between = 0.4053 avg = 3.9R-sq: within = 0.4032 Obs per group: min = 1
Group variable: obs Number of groups = 78Fixed-effects (within) regression Number of obs = 303
. xtreg lcapvalue l.exp l.expsq l.pcomppct l.pyds l.ptd l.interceptions l.sck l.gs ,fe
14.
39
rho .44208302 (fraction of variance due to u_i) sigma_e .50365711 sigma_u .44833444 _cons 12.55499 .1674532 74.98 0.000 12.22679 12.88319 y07 .3330096 .1304998 2.55 0.011 .0772348 .5887844 y06 .1979179 .1258871 1.57 0.116 -.0488163 .4446521 y05 .1503134 .1239032 1.21 0.225 -.0925325 .3931593 y04 .0955666 .124395 0.77 0.442 -.1482432 .3393764 y03 .2530758 .1246211 2.03 0.042 .0088229 .4973287 y01 -.0257068 .1369846 -0.19 0.851 -.2941917 .2427781 L1. .0245214 .0119926 2.04 0.041 .0010162 .0480265regseasonw~s L1. .0003807 .0005437 0.70 0.484 -.0006849 .0014463 ryds L1. -.0228148 .0350197 -0.65 0.515 -.0914522 .0458226 rtd L1. .0751387 .012307 6.11 0.000 .0510173 .0992601 gs L1. -.0041898 .0051741 -0.81 0.418 -.0143309 .0059513 sck L1. .0021731 .001125 1.93 0.053 -.0000318 .004378 rate L1. -.0158642 .0024127 -6.58 0.000 -.0205929 -.0111355 expsq L1. .2837511 .0366806 7.74 0.000 .2118585 .3556436 exp lcapvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 14) = 324.84
overall = 0.5726 max = 7 between = 0.6561 avg = 4.3R-sq: within = 0.4259 Obs per group: min = 1
Group variable: obs Number of groups = 84Random-effects GLS regression Number of obs = 365
note: y02 dropped because of collinearitynote: y00 dropped because of collinearity. 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
15.
40
rho .21225072 (fraction of variance due to u_i) sigma_e .50391917 sigma_u .26157212 _cons 12.56888 .2074093 60.60 0.000 12.16237 12.9754 y07 .3742909 .1371455 2.73 0.006 .1054907 .6430912 y06 .2352229 .1355663 1.74 0.083 -.0304822 .500928 y05 .1673362 .1369741 1.22 0.222 -.1011282 .4358005 y04 .0671194 .1415351 0.47 0.635 -.2102842 .344523 y03 .247729 .1424452 1.74 0.082 -.0314584 .5269165 L2. .1422508 .2025955 0.70 0.483 -.2548291 .5393306wonsuperbowl L1. .0291631 .0137586 2.12 0.034 .0021968 .0561295regseasonw~s L1. .000808 .0005548 1.46 0.145 -.0002794 .0018955 ryds L1. -.0232805 .0392978 -0.59 0.554 -.1003027 .0537418 rtd L1. .0960592 .0132456 7.25 0.000 .0700983 .1220202 gs L1. -.0085052 .0057719 -1.47 0.141 -.0198179 .0028074 sck L1. .0026633 .0013548 1.97 0.049 7.99e-06 .0053187 rate L1. -.0126701 .0027003 -4.69 0.000 -.0179626 -.0073776 expsq L1. .2233226 .0435665 5.13 0.000 .1379338 .3087113 exp lcapvalue Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 14) = 271.23
overall = 0.6035 max = 6 between = 0.7747 avg = 3.9R-sq: within = 0.2745 Obs per group: min = 1
Group variable: obs Number of groups = 71Random-effects GLS regression Number of obs = 279
note: y02 dropped because of collinearitynote: y01 dropped because of collinearitynote: y00 dropped because of collinearity> y07. xtreg lcapvalue 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
16.
41
rho .05692792 (fraction of variance due to u_i) sigma_e 2.3735634 sigma_u .58316445 _cons 5.036159 1.184958 4.25 0.000 2.713683 7.358635 dredskins -.88162 1.612956 -0.55 0.585 -4.042955 2.279715 dtitans .6501697 1.286229 0.51 0.613 -1.870793 3.171132 dbuccaneers -.5671619 1.370503 -0.41 0.679 -3.253299 2.118975 drams -1.16893 1.366791 -0.86 0.392 -3.847791 1.509931 dseahawks .453696 1.338168 0.34 0.735 -2.169065 3.076457 d49ers 1.009167 1.61154 0.63 0.531 -2.149394 4.167728 dchargers .586182 1.388213 0.42 0.673 -2.134665 3.307029 dsteelers .7614854 1.448989 0.53 0.599 -2.078481 3.601452 deagles 1.620261 1.349605 1.20 0.230 -1.024916 4.265439 draiders -1.158669 1.587116 -0.73 0.465 -4.269359 1.952021 djets -1.143425 1.334023 -0.86 0.391 -3.758063 1.471213 dgiants -1.573288 1.494993 -1.05 0.293 -4.503421 1.356844 dsaints .4716759 1.742061 0.27 0.787 -2.9427 3.886052 dpatriots 2.150056 1.323021 1.63 0.104 -.4430176 4.74313 dvikings -.1314502 1.416055 -0.09 0.926 -2.906867 2.643967 ddolphins -2.931395 1.488615 -1.97 0.049 -5.849026 -.0137636 dchiefs .2492065 1.495999 0.17 0.868 -2.682898 3.181311 djaguars 1.093331 1.530055 0.71 0.475 -1.905523 4.092184 dcolts .3609777 1.411115 0.26 0.798 -2.404757 3.126712 dtexans -2.246461 1.67217 -1.34 0.179 -5.523855 1.030932 dlions -1.152624 1.397487 -0.82 0.409 -3.891649 1.586401 dbroncos -2.468439 1.512685 -1.63 0.103 -5.433247 .4963684 dcowboys .9138654 1.736475 0.53 0.599 -2.489564 4.317295 dravens 1.024919 1.474692 0.70 0.487 -1.865424 3.915262 dbengals -2.15358 1.326885 -1.62 0.105 -4.754226 .4470664 dbears 2.384208 1.582736 1.51 0.132 -.717898 5.486313 dpanthers -1.873512 1.482343 -1.26 0.206 -4.77885 1.031826 dbills -.7204439 1.780537 -0.40 0.686 -4.210233 2.769345 dfalcons -.4393028 1.398919 -0.31 0.753 -3.181134 2.302529 dcardinals .5468973 1.326908 0.41 0.680 -2.053795 3.147589 dbrowns -1.210322 1.341551 -0.90 0.367 -3.839714 1.419069 ptd .1804051 .0270309 6.67 0.000 .1274255 .2333848 regseasonw~s Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 32) = 127.10
overall = 0.4643 max = 8 between = 0.5890 avg = 3.3R-sq: within = 0.2493 Obs per group: min = 1
Group variable: obs Number of groups = 60Random-effects GLS regression Number of obs = 200
note: dpackers dropped because of collinearity> hargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8> packers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers dc. xtreg regseasonwins ptd dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions d
17.
42
rho .29730397 (fraction of variance due to u_i) sigma_e .3153457 sigma_u .20511788 _cons .6169718 .3054938 2.02 0.043 .018215 1.215729 dredskins -.12063 .2962498 -0.41 0.684 -.7012688 .4600089 dtitans .0076742 .2491229 0.03 0.975 -.4805978 .4959462 dbuccaneers -.180036 .2514342 -0.72 0.474 -.672838 .3127659 drams -.3766757 .2711652 -1.39 0.165 -.9081497 .1547982 dseahawks -.0349606 .2377384 -0.15 0.883 -.5009194 .4309982 d49ers -.010978 .2764847 -0.04 0.968 -.5528781 .5309221 dchargers -.0464107 .2775081 -0.17 0.867 -.5903165 .4974951 dsteelers -.2042278 .286444 -0.71 0.476 -.7656478 .3571922 deagles .1215891 .2833584 0.43 0.668 -.4337831 .6769613 draiders -.2598571 .2793882 -0.93 0.352 -.8074478 .2877337 djets -.3865581 .254382 -1.52 0.129 -.8851376 .1120214 dgiants -.1935594 .2842079 -0.68 0.496 -.7505966 .3634778 dsaints -.0230991 .3058804 -0.08 0.940 -.6226137 .5764155 dpatriots .0721761 .2697232 0.27 0.789 -.4564716 .6008238 dvikings -.0895699 .2627777 -0.34 0.733 -.6046047 .4254648 ddolphins -.4095488 .2626931 -1.56 0.119 -.9244179 .1053203 dchiefs -.0601978 .3214248 -0.19 0.851 -.6901789 .5697833 djaguars .0443465 .2831561 0.16 0.876 -.5106292 .5993221 dcolts -.0638885 .3161445 -0.20 0.840 -.6835204 .5557433 dtexans -.5486389 .3190813 -1.72 0.086 -1.174027 .0767489 dlions -.0361522 .2565305 -0.14 0.888 -.5389426 .4666383 dbroncos -.4082978 .284227 -1.44 0.151 -.9653724 .1487769 dcowboys -.0185914 .3100152 -0.06 0.952 -.6262101 .5890273 dravens .0822455 .280479 0.29 0.769 -.4674832 .6319742 dbengals -.387051 .2573132 -1.50 0.133 -.8913755 .1172735 dbears .3157724 .2836575 1.11 0.266 -.240186 .8717309 dpanthers -.3832408 .271264 -1.41 0.158 -.9149085 .1484269 dbills -.1261122 .3184502 -0.40 0.692 -.7502632 .4980388 dfalcons -.180657 .2641275 -0.68 0.494 -.6983374 .3370235 dcardinals -.0572861 .2532998 -0.23 0.821 -.5537447 .4391724 dbrowns -.2082741 .2531486 -0.82 0.411 -.7044362 .287888 rate .0185015 .0026498 6.98 0.000 .013308 .023695 lregseason~s Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 32) = 101.57
overall = 0.4352 max = 8 between = 0.5142 avg = 3.3R-sq: within = 0.3033 Obs per group: min = 1
Group variable: obs Number of groups = 60Random-effects GLS regression Number of obs = 200
note: dpackers dropped because of collinearity> dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8> dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets draiders deagles dsteelers . xtreg lregseasonwins rate dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals dravens dcowboys dbroncos dlions
18.
43
dredskins -.0573071 .2734202 -0.21 0.834 -.5932008 .4785866 dtitans .0840737 .2236808 0.38 0.707 -.3543327 .52248 dbuccaneers -.05001 .2241611 -0.22 0.823 -.4893577 .3893377 drams -.0214787 .2440969 -0.09 0.930 -.4998997 .4569424 dseahawks .1579454 .2131944 0.74 0.459 -.2599079 .5757987 d49ers .2460976 .2526856 0.97 0.330 -.2491571 .7413523 dchargers -.0663288 .2466082 -0.27 0.788 -.5496719 .4170144 dsteelers .032353 .2595417 0.12 0.901 -.4763395 .5410455 deagles .2600247 .2547739 1.02 0.307 -.2393231 .7593724 draiders -.1204437 .2531647 -0.48 0.634 -.6166375 .37575 djets -.0945079 .2289799 -0.41 0.680 -.5433004 .3542845 dgiants -.1065287 .2489577 -0.43 0.669 -.5944768 .3814194 dsaints -.026323 .2722946 -0.10 0.923 -.5600106 .5073646 dpatriots .1663909 .2341795 0.71 0.477 -.2925925 .6253743 dvikings .0484898 .2405346 0.20 0.840 -.4229493 .5199289 ddolphins -.2324319 .2391436 -0.97 0.331 -.7011448 .2362811 dchiefs .0420877 .2676383 0.16 0.875 -.4824737 .5666492 djaguars .1663781 .2483173 0.67 0.503 -.3203148 .653071 dcolts -.1143498 .2696525 -0.42 0.672 -.642859 .4141594 dtexans -.4984856 .2820265 -1.77 0.077 -1.051247 .0542763 dlions .0518478 .235885 0.22 0.826 -.4104782 .5141738 dbroncos -.3917313 .2579886 -1.52 0.129 -.8973796 .113917 dcowboys .1409161 .281334 0.50 0.616 -.4104885 .6923206 dravens .1903495 .2560168 0.74 0.457 -.3114342 .6921332 dbengals -.2778486 .227862 -1.22 0.223 -.7244499 .1687527 dbears .3019682 .2573444 1.17 0.241 -.2024177 .806354 dpanthers -.3617107 .2436637 -1.48 0.138 -.8392827 .1158613 dbills .0589324 .2901562 0.20 0.839 -.5097633 .6276282 dfalcons -.2008257 .2469289 -0.81 0.416 -.6847975 .283146 dcardinals .0307806 .228223 0.13 0.893 -.4165282 .4780894 dbrowns .0076087 .2324205 0.03 0.974 -.4479272 .4631446 gs .0669841 .0109636 6.11 0.000 .0454959 .0884723 ryds .0003679 .0002519 1.46 0.144 -.0001258 .0008615 rtd -.0213521 .0186355 -1.15 0.252 -.0578771 .0151729 sck -.011551 .0027757 -4.16 0.000 -.0169912 -.0061108 rate .0151807 .0025851 5.87 0.000 .010114 .0202475 expsq .000605 .0017967 0.34 0.736 -.0029164 .0041264 exp -.0200178 .0284501 -0.70 0.482 -.075779 .0357434 lregseason~s Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 38) = 180.26
overall = 0.5645 max = 8 between = 0.6164 avg = 3.3R-sq: within = 0.4615 Obs per group: min = 1
Group variable: obs Number of groups = 60Random-effects GLS regression Number of obs = 200
note: dpackers dropped because of collinearity> draiders deagles dsteelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8> ns dcowboys dbroncos dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaints dgiants djets . xtreg lregseasonwins exp expsq rate sck rtd ryds gs dbrowns dcardinals dfalcons dbills dpanthers dbears dbengals drave
19.
44
rho .16848734 (fraction of variance due to u_i) sigma_e .25628941 sigma_u .11536654 _cons 1.127414 .4297942 2.62 0.009 .2850325 1.969795 dredskins (dropped) dtitans -.048114 .2266213 -0.21 0.832 -.4922837 .3960556 dbuccaneers .0235807 .2397807 0.10 0.922 -.4463809 .4935422 drams .0281761 .2301507 0.12 0.903 -.422911 .4792633 dseahawks .2517392 .1985793 1.27 0.205 -.137469 .6409474 d49ers -.0148283 .2759735 -0.05 0.957 -.5557265 .5260698 dchargers -.1095333 .230542 -0.48 0.635 -.5613873 .3423208 dsteelers .1513948 .3570334 0.42 0.672 -.5483779 .8511675 deagles .2693371 .2517683 1.07 0.285 -.2241196 .7627938 draiders -.2780264 .3272844 -0.85 0.396 -.9194921 .3634392 djets -.0393572 .2159081 -0.18 0.855 -.4625293 .383815 dgiants -.1137503 .2403596 -0.47 0.636 -.5848464 .3573457 dsaints .2019396 .2729145 0.74 0.459 -.332963 .7368423 dpatriots .1090676 .2570223 0.42 0.671 -.3946867 .612822 dvikings -.0847136 .3150534 -0.27 0.788 -.7022068 .5327797 ddolphins -.1456616 .2489925 -0.59 0.559 -.6336779 .3423547 dchiefs .3092648 .2285346 1.35 0.176 -.1386549 .7571845 djaguars .1125682 .2401711 0.47 0.639 -.3581585 .5832949 dcolts .0662275 .2375632 0.28 0.780 -.3993879 .5318428 dtexans -.597166 .3069655 -1.95 0.052 -1.198807 .0044753 dlions -.1918851 .2699186 -0.71 0.477 -.7209159 .3371457 dbroncos .0797176 .3449221 0.23 0.817 -.5963172 .7557524 dcowboys .1259591 .2848702 0.44 0.658 -.4323763 .6842945 dravens -.0274463 .3464091 -0.08 0.937 -.7063957 .6515031 dbengals -.09739 .2148152 -0.45 0.650 -.5184201 .32364 dbears .3162462 .3405066 0.93 0.353 -.3511345 .9836268 dpanthers .0228089 .2782492 0.08 0.935 -.5225495 .5681674 dbills (dropped) dfalcons -.0113941 .3054718 -0.04 0.970 -.6101077 .5873196 dcardinals .0565413 .2368391 0.24 0.811 -.4076548 .5207374 dbrowns -.1216714 .2501946 -0.49 0.627 -.6120439 .3687011 L3. .2685658 .192206 1.40 0.162 -.108151 .6452826wonsuperbowl gs .0673368 .0146347 4.60 0.000 .0386534 .0960203 ryds .0002524 .0003816 0.66 0.508 -.0004955 .0010004 rtd .0219416 .0266615 0.82 0.411 -.0303139 .0741971 sck -.0158034 .0041549 -3.80 0.000 -.0239468 -.00766 rate .0094647 .0036782 2.57 0.010 .0022556 .0166738 expsq .0042126 .0026727 1.58 0.115 -.0010259 .0094511 exp -.0978946 .0510495 -1.92 0.055 -.1979498 .0021605 lregseason~s Coef. Std. Err. z P>|z| [95% Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000Random effects u_i ~ Gaussian Wald chi2( 37) = 154.37
overall = 0.7553 max = 5 between = 0.8640 avg = 2.6R-sq: within = 0.5025 Obs per group: min = 1
Group variable: obs Number of groups = 40Random-effects GLS regression Number of obs = 103
note: dpackers dropped because of collinearity> s dgiants djets draiders deagles dsteelers dchargers d49ers dseahawks drams dbuccaneers dtitans dredskins if gs>=8> s dbengals dravens dcowboys dbroncos dlions dpackers dtexans dcolts djaguars dchiefs ddolphins dvikings dpatriots dsaint. xtreg lregseasonwins exp expsq rate sck rtd ryds gs l3.wonsuperbowl dbrowns dcardinals dfalcons dbills dpanthers dbear
Works Cited
USATODAY Player Salaries Database. Retrieved throughout February 2008.
(http://content.usatoday.com/sports/football/nfl/salaries/)
45
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