Alex Brakey

4/08/16

ECO 315-B

Professor Tebaldi

Does it really Pay to Play?

Section I. Introduction

All over television and the internet, you will see the astronomically high salaries that star

athletes make in professional sports. The professional sports league that is most famous for huge

contracts is Major League Baseball. The average team payroll on opening day in 2000 was over

$77 million (in 2015 dollars). On opening day 2015, the average payroll was $122 million

according to usatoday.com (2015). There are a few different theories about why payrolls have

increased so much in the past 15 years. One theory, hypothesized by Stephen Hall, Stefan

Szymanski, and Andrew Zimbalist (2002), is that players on winning teams have experienced

expectancy theory and require higher yearly salaries to maintain their level of play. Another

theory is that there is a link between a team’s performance and their payroll. This is the theory

which I will be testing in this paper.

This topic is important because it will answer the question that has been posed in bars and

on sports shows across America for years which is whether or not these large payrolls are based

on performance or if there are other factors that influence payroll. Does it make sense that the

2015 New York Yankees won 87 games while hitting .251, had a team ERA of 4.03, and had a

payroll north of $197 million while the 2015 Pittsburgh Pirates won 98 games while hitting .260,

had a team ERA of 3.21, and a payroll of just $86 million? I have chosen to research this topic

because I am one of those sports fans that is genuinely curious about whether my team’s payroll

is linked to performance or if the ever-rising payroll is just the result of capitalism in its finest

form.

The information and data that I will be referencing throughout this paper will come from

various scholarly articles that have been found using economics databases and Google scholar. I

have identified multiple sources which will be helpful in answering my research question. Using

these sources, I will attempt to create a model that provides an accurate depiction of what

different factors have an effect on a team’s payroll.

Section II of this paper provides academic sources which explain the theories that support

my model. Section III provides a description of the data used for the analysis, including

descriptive statistics and graphical representations of different variables. Section IV contains the

final models used and the results of the analysis, as well as an in-depth discussion about the

forms of the model and interpretations of the models. Section V will answer the question “Does

the empirical work done provide support for the theory?” This section will also contain a

summary of the results and a reflection on the work that has been carried out.

Section II. Theories behind the Model

Yu-Li Tao, Hwei-Lin Chuang, and Eric Lin’s research (2015) has provided a lot of

insight for me. Their study, called “Compensation and Performance in Major League Baseball:

Evidence from salary dispersion and team performance,” focused on the relationship between

compensation and performance in Major League Baseball between 1985 and 2013 (2015). One

thing that makes the analysis done by Tao, Chuang, and Lin different from my analysis is that

they were measuring the effects of individual players on team performance while using two

different payroll variables, payroll level and payroll’s relative position (2015). The variable that I

found most interesting was the market variable which accounted for each team’s market area.

The rationale they discussed was that a larger market leads to higher team revenues which turns

into higher payroll and would then allow a team to go out and get better players which is likely

to increase the team’s production (2015). In both the Bloom and GMM estimates, the market

variable was insignificant even at the .10 significance level (2015). While Tao, Chuang, and

Lin’s results showed that the market variable was insignificant, I believe that it is something that

is important to control for because of the fact that bigger markets, such as New York and San

Francisco, will normally have higher payrolls than smaller market teams, such as the Kansas City

Royals.

Mikhail Averbukh, Scott Brown, and Brian Chase’s research (2015) is more closely

related to what I am attempting to prove using my model, although there are some notable

differences. The main difference is that this study was done to determine an individual player’s

salary. Different models were created for two positions: pitchers and batters. Their research

measured multiple statistics for both positions (ERA, Wins, Losses, and Strikeouts for Pitchers

and Hits, Homeruns, Batting Average, Runs Batted In, and On Base Percentage). The data that

they used was based on the 2000 through the 2007 season. In creating fitted line plots, the “Pay

and Performance” study found a relatively strong, positive correlation between wins and salary.

The correlation coefficient between the two variables was .657 (2015). There was also a

relatively strong positive correlation between strikeouts (for pitchers) and salary in their model

(2015).

Stephen Hall, Stefan Szymanski, and Andrew Zimbalist’s research paper titled “Testing

Causality between Team Performance and Payroll” (2002) didn’t contribute any variables to my

model however some important theories were discussed as to why salaries have been increasing.

Hall, Szymanski, and Zimbalist stated that one of the main reasons of higher salaries is weak

free-agent classes during the offseason. The less free-agents available for a team, the more

willing a team will be to spend more on free-agents (2002). There is also the idea that players on

winning teams tend to establish an expectancy theory with regard to wage. The more the team

wins and the better the player performs, the higher the wage that player will expect to get from

the team they currently play for (2002).

David Hoaglin and Paul Velleman, in their analysis titled “A Critical Look at some of the

Analyses of Major League Baseball Salaries” (1995), reviewed the most common methods of

working with a team’s salary that were used by 15 different groups as part of a data analysis

exposition. What they found was that the salary variable was skewed and that most of the

possible predictors, when graphed against salary, were not linearly related (1995). The reasons

for re-expressing the salary variable in log form included making the distribution more

symmetric (the graph for inflation adjusted payroll and log of inflation adjusted payroll in

section III of this paper shows this), obtaining a better fit, stabilizing salary variance, and

accounting for year-by-year increases and decreases in bonus salaries. Hoaglin and Velleman

also found that those who worked with the re-expression of salary in log form were more

successful in creating an accurate model (1995).

Section III. The Data

The data that I will be using comes from all 30 MLB teams from the 2000 season to the

2015 season. Because this data contains a time series (2000-2015) for each cross-sectional

member (each MLB team), I am working with panel/longitudinal data. Team payroll information

was obtained from usatoday.com and provides the dollar figure for each team’s payroll for each

year. One important thing to note is that $1 in 2000 was not worth the same in 2015 due to

inflation. Because of this, all payroll figures were inflated to the 2015 level (coefficients are

located below) so that all payroll figures used in this analysis represent the team payroll’s 2015

monetary value.

Table 1. Inflation CoefficientsYear Coefficients

2000 1.3764052001 1.3383232002 1.3174932003 1.2881362004 1.2547222005 1.2136052006 1.175682007 1.1431212008 1.1008532009 1.1047842010 1.0869552011 1.0536952012 1.0323312013 1.0174282014 1.0011872015 1

There were multiple transformations of the team variable between 2000 and 2015. From

the 2000 season until the end of 2004, the Washington Nationals were located in Montreal and

known as the Expos. To account for this, the names for both of these teams have been combined

into a singular “Washington Nats/Montreal Expos” cross-sectional member in the dataset to keep

every piece of data for the team together. The metropolitan population listed in the dataset

represents Montreal from 2000-2004 and Washington D.C from 2005-2015. A similar situation

occurred with the Miami Marlins who, until the end of the 2011 season, were known as the

Florida Marlins. Just like the case above, the team’s names were combined into a singular cross-

sectional member named the “Miami/Florida Marlins.” The values in the metropolitan

population variable are all from the greater Miami metropolitan area because the team was

located in that metropolitan area both before and after they moved.

The strength of the data being used is that it is not a sample of teams, it is all 30 MLB

teams over a 16 season period and, thus, a collection of data from the whole population of MLB

over those 16 seasons. This is a strength because all of the figures that are obtained in my

analysis will be representative of the entire population of MLB teams instead of just a fraction of

the teams. The one weakness with this data is that it is possible that heteroskedasticity is present.

This will have no effect on the slope of each variable however there will be an effect on the

variance in the standard errors which makes any t-test, f-test, or confidence intervals calculated

using normal estimators invalid. Because I have panel data, I cannot conduct the Breusch-Pagan

test or White test for heteroskedasticity. In order to account for possible heteroskedasticity, I will

work with the robust estimators for which ever model I end up using after conducting a Hausman

test.

Table 2. Descriptive StatisticsVariables: Mean St.

DeviationMin Max

Log of Inf Adj Payroll 18.2911 0.427 16.685 19.348Wins 80.97 11.42 43 116Runs Offensive 739.54 84.82 513 978Homeruns Offensive 166.67 33.56 91 260Slugging Percentage 0.4144 0.02687 0.335 0.491Batting Percentage 0.261 0.012 0.226 0.294ERA 4.236 0.536 2.94 5.71Log of Pitcher Strikeouts 7.01 0.113 6.64 7.28Log of Pitcher Walks 6.25 0.129 5.85 6.59Fielding Percentage 0.983 0.0027 0.976 0.991Log of Metro. Population 15.052 0.589 14.22 16.48

Section IV. Empirical Model and Results

Section IV. Part 1: Creating the Models

The following model will be estimated:

Log of inflation adj. payroll=

β0+β1 (wins )+β2 (runs offensive )+β3 (homeruns off )+ β4 ( slg percentage )+β5 (batting percentage )+β6 (era )+β7 (log pitcher strikeouts )+β8 ( log pitcher walks )+β9 (fielding percentage )+β10(log of met . pop ulation)

Deciding on the functional form of the variables was something that took some time to

figure out. One of the issues that I encountered was that my model contains some values that

represent an average (ex. Batting percentage, slugging percentage, and fielding percentage)

while others represent a certain numerical statistic which will be much greater than 1. I ran the

model originally with logs on just the inflation adjusted payroll and metropolitan population

because those numbers were in the millions and would have adversely effected my results. I

decided to put a log on inflation adjusted payroll, pitcher strikeouts, pitcher walks, and

metropolitan population because these numbers for each team tended to be relatively large. I did

not see the necessity of quadratics for any of the variables in my model.

Distribution of Metropolitan Population:

12500000100000007500000500000025000000-2500000

100

80

60

40

20

0

Mean 4187263StDev 3017073N 480

metpop

Freq

uenc

y

Histogram of metpopNormal

Distribution of Log of Metropolitan Population:

16.416.015.615.214.814.414.0

60

50

40

30

20

10

0

Mean 15.05StDev 0.5895N 480

lnmetpop

Freq

uenc

yHistogram of lnmetpop

Normal

Taking the log of metropolitan population leads to a more normal distribution. It is more

likely that log of metropolitan population would pass the normality assumption when compared

to metropolitan population although log of metro population still isn’t very normally distributed.

It is necessary to use the log of metro population because this will rescale the values for each

cross-sectional member to a number which is closer to the scales of the other variables in the

dataset.

Distribution of Inflation Adjusted Payroll:

240000000210000000180000000150000000120000000900000006000000030000000

70

60

50

40

30

20

10

0

Inf. Adj Payroll

Freq

uenc

y

Histogram of Inf. Adj Payroll

Distribution of Log of Inflation Adjusted Payroll:

19.218.818.418.017.617.216.8

60

50

40

30

20

10

0

lninfadjpayroll

Freq

uenc

yHistogram of lninfadjpayroll

Both the inflation adjusted payroll and log of inflation adjusted payroll are relatively

normally distributed and both would likely pass the normality assumption, however the log of

inflation adjusted payroll variable is more normally distributed. As Hoaglin and Velleman

(1995) noted, a team’s salary tends to be skewed when it is in its standard form but then re-

expressing salary with a log will make the distributions more symmetric. Using log of inflation

adjusted payroll is necessary for this model because of the values of payroll for each team. With

each data point being in the millions, these points needed to be rescaled so that they were closer

to the values of the other variables in the dataset.

Upon determining which variables, and in what form, would be included in the model,

three models were predicted. The first was a simple ordinary least squares model. The OLS

regression model acts as more of a benchmark than a model that will actually be considered. The

OLS results (listed in Column 1 of Table 3 below) aren’t particularly interesting in solving the

question I posed earlier. This is because OLS models, when dealing with panel data, suffer from

omitted variable bias. We have not accounted for unobserved heterogeneity in OLS which means

that we know, for sure, that our estimates are wrong. We must move on and test two other

models in order to get an accurate answer for my research question.

Table 3. Model Results:

(OLS) (FE) (RE)Log of Inf. Adj

PayrollLog of Inf. Adj

PayrollLog of Inf. Adj

Payroll Wins 0.00371 0.00100 0.00274

(0.91) (0.31) (0.85)

Runs (Off) -0.000612 -0.000546 -0.000575(-0.98) (-1.08) (-1.11)

Homeruns (Off)

0.00747*** 0.00374* 0.00438**

(4.05) (2.29) (2.67)

Slugging Pct. -11.98*** -7.148* -8.252**

(-3.42) (-2.31) (-2.66)

Batting Pct. 24.42*** 12.30** 15.00***

(5.53) (3.19) (3.88)

ERA 0.0393 0.0628 0.0776(0.48) (0.96) (1.17)

Log of pitcher strikeouts

0.872*** 0.828*** 0.770***

(4.60) (5.13) (4.77)

Log of pitcher walks

-0.485** -0.561*** -0.520***

(-3.09) (-4.01) (-3.73)

Fielding Pct. 14.24* 13.23* 11.12*

(2.10) (2.41) (2.02)

Log of the Metro. Pop

0.190*** -0.437 0.166**

(6.52) (-1.93) (2.62)

_cons -4.326 8.743 1.353(-0.64) (1.47) (0.24)

N 480 480 480t statistics in parentheses* p < 0.05, ** p < 0.01, *** p < 0.001

The second model I ran was a fixed effects model (Listed in Column 2 of Table 3). Fixed

effects estimators are more consistent because, as the sample size increases, β̂ will get closer to

the true β .One thing that was interesting was the p-value and sign of the log of the metropolitan

population variable. The p-value of .054 indicates that it is statistically significant at a .10

significance level. This variable can be interpreted as meaning that every additional 1% increase

in the population of the team’s metropolitan area will decrease the team’s inflation adjusted

payroll by .437%, ceteris paribus. This is the opposite of what Tao, Chuang, and Lin (2015)

hypothesized in their analysis. When they ran their models, the market variable was not

significant at the .10 level (2015). In general, this model is saying that a larger metropolitan area

will actually decrease the teams inflation adjusted payroll. Another interesting result from the

fixed effects model was the slope and p-value for era. The slope and p-value were .0628

and .337, respectively. The interesting point about this is that there is a positive relationship

between era and log of inflation adjusted payroll. In their pitchers equation, Averbukh, Brown,

and Chase (2015) found a negative correlation between era and salary which means that as era

increases, the salary of the team will decrease.

The third and final model that was estimated was a random effects model (Listed in

Column 3 of Table 3). Random effects estimators are more efficient because the estimators

provide smaller variance regressions, meaning the standard error for each coefficient will be

smallest in the random effects model. The coefficient for batting percentage had a p-value

of .000 which indicates that it is a very significant aspect when looking at a team’s inflation

adjusted payroll. This coefficient indicates that for each additional 1% increase in batting

percentage, the teams inflation adjusted payroll will increase by 15%, ceteris paribus. Another

variable that was significant in the random effects model was log of the pitcher walks. For each

additional 1% increase in walks by the pitching staff, inflation adjusted payroll will decrease by

about .52%, ceteris paribus. When comparing the random and fixed effects models, we can also

see that the p-values for wins and runs (off) decreased when going from the fixed effects

estimates to the random effects estimates.

To determine which model should be used, a Hausman test was conducted and a

significance level of .05 for rejection was set. The null hypothesis was that the random effects

model was better. The alternative hypothesis was that the fixed effects model was better. The test

resulted in a chi-square value of 16.04 and a p-value of .0661. With a p-value above our .05

significance level, we fail to reject the null hypothesis which means that we are 95% confident

that the random effects model is the best model to use for this analysis. As mentioned above,

while we know that we are using the right model after conducting the Hausman test, it is still

possible that our estimators are inaccurate due to heteroskedasticity. In order to account for this,

we need to look at the robust estimators of the random effects model. We will notice that the

coefficient itself will not change, but everything else will. One of the consequences of

heteroskedasticity is that the variance in the standard errors incorrect which will make any t-test,

f-test, or confidence interval invalid using those incorrect standard errors. This would make any

tests for significance inaccurate. Using the robust estimators is a way for us to correct for

heteroskedasticity and make all of our standard error estimates correct.

Section VI. Part 3: Interpreting the RE Model with Robust Estimators

The overall R2 of the model is .2997 which means that 29.97% of the total variation in

log of inflation adjusted payroll can be explained over time and across cross-sectional units using

this model. The between R2 is .4110 which means that 41.1% of the total variation in log of

inflation adjusted payroll between cross-sectional units can be explained using this model.

Table 4. Robust Estimators of the Random Effects Model:

(RE Robust)Log of Inf. Adj

PayrollWins 0.00274

(0.91)

Runs (Off) -0.000575(-1.20)

Homeruns (Off)

0.00438*

(2.33)

Slugging Pct. -8.252*

(-2.22)

Off. Bat Proportion

15.00***

(3.71)

ERA 0.0776(1.09)

Log of pitcher strikeouts

0.770***

(4.49)

Log of pitcher walks

-0.520***

(-3.67)

Fielding Pct. 11.12*

(2.01)

Log of the Metro Pop

0.166**

(2.62)

_cons 1.353(0.25)

N 480t statistics in parentheses* p < 0.05, ** p < 0.01, *** p < 0.001Standard errors are calculated using robust estimators

Both home runs (off) and slugging percentage were significant at the .01 level in the

model without robust errors. Both of these variables became less significant, though still

significant at the .05 level, when we calculated the robust estimators. The interpretation of home

runs (off) is that for each additional homerun a team hits, the team’s inflation adjusted payroll

will increase by .438%, ceteris paribus. In this model, slugging percentage is significant at

the .05 level. The variables coefficient means that for each additional 1% increase in slugging

percentage, a team’s inflation adjusted payroll will decrease by 8.25%, ceteris paribus. The

variable log of the metropolitan population is significant at the .01 level. In Tao, Chuang, and

Lin’s analysis (2015), their Market variable was insignificant at even the .10 level. The

relationship between log of the metropolitan population and log of inflation adjusted payroll is

what we would expect. For every 1% increase in metropolitan population, inflation adjusted

payroll will increase by .166%, ceteris paribus. This supports the theory presented in Tao,

Chuang, and Lin’s analysis (2015) that a higher market population will allow for teams to

increase payrolls and get better players.

Section V. Conclusion

The theory that was being tested in this analysis was that there was a link between a

team’s performance and their payroll. Upon finding which model would be the best to use for

this analysis, we can conclude that there is a link between the team’s performance and their

payroll, though other factors definitely play a role in determining a team’s payroll. So, to answer

the question posed in Section I, the empirical work done does provide some support for the

theory but there are other factors that should be controlled for to get the most accurate answer.

We know that there are other factors in play because of the between and overall R2 values. Apart

from the key performance indicators that were controlled for in the model, nearly 60% of the

variation in log of inflation adjusted payroll between cross-sectional units remains unexplained.

Almost every key performance variable included in this analysis was significant at the .05

level which indicates that these variables did have an effect on a team’s payroll. The log of

metropolitan population variable makes me hesitant to say that performance is the only

determinant of a team’s payroll. In Averbukh, Chase, and Brown’s analysis (2015), they

concluded that performance is only generally linked to pay and that there are definitely outside

factors that affect payrolls. Tao, Chuang, and Lin (2015) also found that there is a link between

performance and payroll. I believe that my conclusion aligns with what Averbukh, Chase, and

Brown (2015) found which is that there is only a general link between salary and performance in

Major League Baseball.

I believe that I could have accomplished quite a bit more with this project. The only issue

with everything that I wanted to do was time. One variable was age. I was going to find the

average age for each team for each year and then I was going to include both a linear age

variable and a quadratic age variable. “Pay, Productivity, and Aging in Major League Baseball”

(2011) authors Jahn Hakes and Chad Turner suggested that the age variable follows a quadratic

pattern. Up until the age 27, a player’s age has a positive return to performance. After 27 though,

a player’s performance will begin to decline (2011). I believe that including age, in both linear

and quadratic form, would have benefited my model because it would have accounted for what

Hakes and Turner were able to show which is that age is significant to both performance and

salary. Another variable that I would have added given more time would be Gini coefficients.

The Gini coefficient measures the inequality among values of a frequency distribution and is

commonly used to measure income inequality.

Work Cited

Averbukh, M., Brown, S., & Chase, B. (2015). Baseball Pay and Performance (PDF) [PDF].

Retrieved March 09, 2016, from https://ai.arizona.edu/sites/ai/files/MIS580/baseball.pdf

Hakes, J. K., & Turner, C. (2011). Pay, productivity and aging in Major League

Baseball. Journal of Productivity Analysis, 35(1), 61-74.

Hall, S., Szymanski, S., & Zimablist, A. (2002). Testing Causality between Team Performance

and Payroll. Journal of Sports Economics. Retrieved April 12, 2016, from

http://jse.sagepub.com/content/3/2/149.full.pdf html

Hoaglin, David C., and Paul F. Velleman. "A critical look at some analyses of major league

baseball salaries." The American Statistician 49.3 (1995): 277-285.

Tao, Y. L., Chuang, H. L., & Lin, E. S. (2015). Compensation and performance in Major League

Baseball: Evidence from salary dispersion and team performance. International Review

of Economics & Finance.

United States, U.S Census Bureau. (2010). Population Change for Metropolitan and

Micropolitan Statistical Areas in the United States and Puerto Rico: 2000 to 2010 (CPH-

T-2). DC.

United States, U.S Census Bureau. (2015). Annual Estimates of the Resident Population: April 1,

2010 to July 1, 2015 - United States – Metropolitan and Micropolitan Statistical Area;

and for Puerto Rico: 2015 Population Estimates. DC.

Data information:

For Payroll information: http://www.usatoday.com/sports/mlb/salaries/2000/team/all/

For Team statistics: http://mlb.mlb.com/stats/sortable.jsp#elem=%5Bobject+Object

%5D&tab_level=child&click_text=Sortable+Team+hitting&game_type='R'&season=201

5&season_type=ANY&league_code='MLB'&sectionType=st&statType=hitting&page=1

&ts=1462233385078&playerType=ALL&sportCode='mlb'&split=&team_id=&active_s

w=&position=&page_type=SortablePlayer&sortOrder='desc'&sortColumn=avg&results=

&perPage=50&timeframe=&last_x_days=&extended=0