Quantifying Market Inefficiencies in the Baseball Players’ Market

Among the central arguments of the bestselling book and movie Moneyball was the allegation that the labor market for baseball players was inefficient in 2002. At that time, Billy Beane and the Oakland Athletics used observations made by statistical analysts to exploit this market inefficiency, and acquire productive players on the cheap. Econometric analysis published in 2006 and 2007 confirmed the presence of an inefficient market for baseball players, but left open the question of to what extent, and how quickly, a market correction would occur. We find that this market had in fact already corrected by 2006, and moreover argue that the perceived market response to Moneyball in 2004 is properly viewed as part of a more gradual longer-term trend. In addition, we use official payroll data from Major League Baseball to refute a previous observation that the relationship between team payroll and performance has tightened since the publication of Moneyball.


INTRODUCTION
bestselling book Moneyball was published in June of 2003. 1 Stripped of its storytelling, Lewis presented two strong theses. The first was that baseball executives had been using the wrong metrics to value the productivity of players and that this tendency was reversed by general manager Billy Beane and the Oakland Athletics in the early 2000s. The second, as a consequence, was that certain skills were undervalued (market inefficiency) and intelligent executives at small market teams could overcome their competitive disadvantage by exploiting the skill undervaluation.
While the Athletics performed well above what their meager resources would lead one to predict, the notion that sabermetric smarts could undo baseball's competitive balance problems engendered significant skepticism. If Lewis was correct in his assertion of skill undervaluation then the world would know the secret shortly after his book was published and Billy Beane's advantage would soon disappear.
Central to Lewis's narrative was the importance of a player's walk rate. On-base percentage (OBP) 2 was more closely correlated with a team's win percentage and revealed higher consistency from 1 year to the next for individual players than did a player's simple batting average (BA). OBP was more closely correlated to win percentage because a walk not only put a runner on base and sometimes moved other runners up, but it also allowed an additional batter to reach the plate during an inning and wore down the arm of the opposing pitcher. BA did not capture the important skill of having a good batter's eye and being able to work a walk.
While the superior value of the OBP metric, relative to BA, is manifest upon a moment's reflection, sabermetricians have turned their attention to more ambitious metrics, such as quantifying a fielder's range or separating out the value of good defense from good pitching. The analytic work done in baseball front offices these days is guarded closely as proprietary with teams seeking to reap the benefits of their discoveries as long as possible before other teams catch on.
As the work of sabermetricians becomes more secretive, the speed of the market adjustment process may slow. In the case of OBP, however, once the Athletics' strategy was recognized, it was easy for other teams to emulate it. Indeed, it is possible that the appreciation of the skill of walking became overdeveloped and OBP became overvalued; that is, the return to the ability to work a walk may have outpaced the value of working a walk. Ironically, to the extent that the market overadjusts to OBP or to other skills, general managers who pursue this skill in the marketplace will find that their team can be disadvantaged by the application of sabermetric knowledge. In contrast, the laggard general manager who eschews analytics may be temporarily advantaged by his obscurantism.
In any event, now, with 10 years of market response since the publication of Moneyball, it is interesting to follow how the skill of plate discipline has changed in value over time.
To study this question, we began with two papers by Hakes and Sauer [2006;2007] In a 2006 article, Hakes and Sauer employ data from 2000 to 2004 to compare the relative salary returns to changes in slugging percentage (SLG) 3 and OBP to the relative impact of SLG and OBP on win percentage. The key distinction they make is that SLG is a traditional measure of batting prowess, whereas OBP is the sabermetric variable of choice. They find that OBP is undervalued relative to SLG between 2000 and 2003, but that this undervaluation is abruptly reversed in 2004, the year after Moneyball is published. While instructive, this first article is limited by (a) only including 5 years of data and 1 year of data after the book's publication; (b) the fact that both OBP and SLG include singles in the numerator and outs in the denominator and, hence, are correlated with each other; and (c) the likelihood that the valuation of SLG is a function both of its contribution to winning and, independently, its contribution to fan enjoyment of the games (extra base hits and home runs are more exciting to watch than walks and singles).
Hakes and Sauer published a second paper in 2007 that extended their data set to 1986-2006 and introduced a refined separation of different hitting skills. Hitting skills were now delineated as Bat (batting average), Eye ((walks+hit by pitch)/plate appearances), and Power (bases per hit). The basic results corroborated those of the 2006 study. Eye was relatively undervalued until 2004, when its valuation spiked. The authors also found that the returns to Eye diminished in 2005 and 2006, indicating a possible overcorrection in 2004. By 2006, the return to Eye or plate discipline was at the same level it had been in 2003.
In this paper, we seek to further refine the modeling of these relationships and to take advantage of the time passed to follow the pattern of market response through 2012.

DATA AND METHODOLOGY
Like Hakes and Sauer, our primary data source is the Lahman [2013] database that contains seasonal statistics for every player in Major League Baseball (MLB) history going back to 1876. We used the 2012 version of the database, and focused on the 798 teamseasons that occurred between 1985 and 2012.
The Lahman database also contains information about player salaries in each year. However, the structure that governs the salaries of Major League players is notoriously complex. There are three major categories into which each player falls at the end of each season: free agent, arbitration eligible, and pre-arbitration eligible (a.k.a. "0-3 player"). 4 If a player is a free agent then his salary in the next season is likely to closely reflect the market price for his services. Conversely, if he is pre-arbitration eligible then his salary is unilaterally determined by his club, and is likely to be very close to the league minimum (US$490,000 in 2013). If he is arbitration eligible then his salary will be determined through a two-party arbitration process, which will likely yield something approximating a market rate. (For free agent and arbitration-eligible players, the correspondence between output and salary will also be distorted by the presence of long-term contracts.) Thus, knowing into which of these three buckets each player falls is an important factor in estimating his salary. The central piece of data that determines into which bucket each player falls is his major league service time, which accrues daily for each day the player spends on a major league roster. These data can be hard to find, but our service time data, which does not come from the Lahman database, allows us to make more precise determinations on this subject than Hakes and Sauer.
What follows is an overview of our methodology: 1. We build a model for labor market productivity, using three key offensive performance variables. This gives us an understanding of how these skills translate into winning on the field. 2. We build a corresponding model for labor market valuation, using the same three key variables. This gives us an understanding of how these skills are compensated on the labor market. 3. We compare the values of the corresponding coefficients. This helps us to assess inefficiencies in the labor market for baseball players.

MODELING LABOR MARKET PRODUCTIVITY
First, we want to understand how certain skills translate into winning. Specifically, we are interested in the following three skills: 1. Eye: walks plus hit-by-pitches per plate appearance 2. Bat: hits per at-bat (a.k.a. batting average) 3. Power: total bases per hit (a.k.a. slugging percentage divided by batting average) These three skills are largely uncorrelated with each other, or at least far less correlated than more comprehensive statistics like OBP and SLG. While it may be the case that players with excellent plate discipline tend to hit for more power, there is no reason, a priori, to think that this might be the case. Moreover, unlike OBP and SLG, the pairwise relationships among these three variables contain no functional dependence.
Our dependent variable is team-winning percentage, but we consider this relative to 0.500, which defines an average team. Furthermore, because baseball has a beautiful (although perhaps often overlooked) duality between offense and defense, every statistic that measures something good for the offense measures something bad for the defense. Thus, our explanatory variables measure the difference between each team's offensive performance and its defensive performance. Accordingly, we fit, using least squares, the regression model: where EyeA, BatA, and PowerA represent the opposing team's offensive performance, when playing the team in question. This is equivalent to the model fit by Hakes and Sauer, and our coefficients agree with theirs over similar time intervals. A summary of the results is shown in Table 1.
The first column of Table 1 shows the regression results over the full 28-year period in our study. We note that while there is clearly more to winning games than the three variables we have measured here, our model explains 81 percent of the variation in teamwinning percentage. In the four rightmost columns in Table 1, we have broken the 28 years into four distinct periods. The first (1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997) is far longer, and can be thought of as something of a control. The period from 1998 to 2002 contains the 5 years immediately preceding the release of Moneyball, the book, along with the actual season about which the book was written (2002). The 5-year period immediately following (2003)(2004)(2005)(2006)(2007) is when we would expect to see the ideas from Moneyball implemented in the baseball industry. Finally, the 5 years from 2008 to 2012 provide hindsight to Hakes and Sauer, and reflect the pitching-dominant era into which baseball has been thrown since improved testing procedures and stricter penalties for performance-enhancing drugs were introduced.
We conclude from these data that although there is clearly some variation in the value of these coefficients, the relationships among them remain relatively stable. There seems to be little evidence of significant structural shifts over this time period. 5 This makes sense, since while the game of baseball changes continuously because of the revolving door of player talent, and small rule changes, there is little reason to believe that the elements that lead to runs being scored and games being won change dramatically, especially over relatively short periods of time. Indeed, this was the conclusion that Hakes and Sauer reached in 2007. In Figure 1, we plot the value of each of the three coefficients as a time series after running the regression on each individual season.

MODELING LABOR MARKET VALUATION
Now that we have a quantitative understanding of how Eye, Bat, and Power translate into winning, it remains to assess how individual players are compensated for these skills on the labor market. As we discussed above, baseball's labor market is complicated by its institutional constraints; thus, our task is harder, and we are less successful at it. Unlike in the previous case, where our units of observation were teams, we now build a data set of Table 1 Estimates of Labor Market Productivity: The dependent variable is team-winning percentage 1985-2012 1985-1997 1998-2002 2003-2007 2008-2012  individual players. Following Hakes and Sauer, we use the natural logarithm of player salary in season t as our dependent variable, and build a model for that as a function of that player's Eye, Bat, and Power in the previous season (t−1). Only position players with a recorded salary and at least 130 plate appearances in the previous season were included. 6 There were 8,824 such players during the 1985-2012 period. Control variables were added to the regression model for: • TPA: How many total plate appearances did the players have? Clearly, if a player is playing regularly, he is likely to be paid more, regardless of how well he hits. • Catcher: Was the player primarily a catcher? Catcher is primarily a defensive position, so it stands to reason that catchers will make more money than players of corresponding offensive value. • Infielder: Was the player primarily a second baseman, a shortstop, or a third baseman? A similar, albeit weaker, argument can be made for infielders. • ArbElig: Was the player arbitration eligible? • FreeAgent: Was the player a free agent? • Fixed effects for Year: in order to control for baseball inflation and other factors Thus, our model for labor market valuation is: Year is a vector of indicator variables for each year from 1986 to 2012, and γ is a vector of the corresponding coefficients. The results from this regression are shown in Table 2. As we are using the same model as Hakes and Sauer, we attribute our slightly higher R 2 values to the more accurate contract data that we are using.
Here, unlike our estimates of labor market productivity, we see dramatic changes in the way that players are compensated for certain skills on the labor market over time. In particular, we note that whereas a free agent outfielder of average ability in 2000 would expect to earn approximately $36,000 more had his walk rate been 10 points higher, a player with the same statistics could have expected approximately $125,000 in additional salary in 2010, more than a threefold increase. 7 In Figure 2, we illustrate the changes in the coefficients for Eye, Bat, and Power over time.
There appears to be greater fluctuation in the labor market valuations, as opposed to the labor market productivity, but the evidence in favor of structural shifts is weak. 8 As Hakes and Sauer noted, after reaching a relatively low point in 2001, the return to Eye begins to pick up in 2002 and 2003, before the publication of Moneyball. The return then spikes in 2004, the year following the book's publication. They interpret these changes as evidence that Moneyball changed the labor market in baseball. In the longer-run context of the baseball labor market, while the rapid increase in the value of Eye surrounding the publication of Moneyball may or may not be causal, a more robust story is that the assimilation of the value of OBP by general managers has been a gradual change that has evolved over decades. During the 28-year window explored, the post-Moneyball spike in the valuation of Eye is just a blip in a long-term change. 9  In their 2007 paper, Hakes and Sauer question whether the noted decline in the value of Eye from 2004 to 2006 is attributable to random variation, or evidence of a market correction among general managers who had come to see Eye as being overvalued by the market. Although the valuation of Eye was equally high in 2010, it does appear that the post-Moneyball valuation of Eye was abnormally high. Thus, while we see some evidence that the average valuation of Eye is higher in the post-Moneyball era, the immediate effect of the book's publication has been mitigated. Given the rigidities in the baseball players' market and the pile-on nature of market response, it is not surprising that the observed annual market correction process is neither gradual nor linear.

ELASTICITY MODELS
Since the three offensive performance variables have different scales, it is desirable to consider the elasticity of salary with respect to Eye, Bat, and Power, by taking logs on both sides of the regression model. This represents a departure from Hakes and Sauer, and allows us to interpret the coefficients as the percentage change in salary that is associated with a 1 percent increase in each of the explanatory variables. In Table 3, we provide results from this regression. We note that the salary return to Eye increased by 5.8 percent (from 0.138 to 0.146) between the 1985-1997 and the 1998-2002 periods, but then shot up by 63.7 percent (from 0.146 to 0.239) in the immediate post-Moneyball period, before retreating 11.7 percent (from 0.239 to 0.211) during the 2008-2012 period. It is also notable that the returns to Power and Bat increased over the extended period as well, 10 although they did not grow as rapidly as the return to Eye. This result is evident in the Eyeto-Bat and Eye-to-Power coefficient ratios.
Similarly, we can consider our labor market productivity model in terms of elasticities. Since we include only salaries for position players in our labor market valuation, here we include only offense, and use team runs scored as the dependent variable. The results are shown in Table 4. We note that these coefficients show considerable long-term stability, with little evidence of meaningful change over an extended period of time. This corroborates our earlier observation that the elements of the game that translate into winning are relatively static.
By considering the ratio of the elasticity of each statistic to salary over the elasticity of that statistic to runs scored, we gain an estimate of the relative return to each skill. We show these results in Table 5.  1985-2012 1985-1997 1998-2002 2003-2007 2008-2012 In It is interesting to note here that the ratio of these elasticities for BA has remained remarkably consistent over these time periods. This suggests that however fairly BA is being compensated on the labor market, that relationship has not changed much, if at all. Conversely, the return to Power has changed dramatically from sub-period to sub-period. Now, it appears to be only about half as high as it was during the home run boom of the 1998-2002 period. However, it is also notable that between the 1985-1997 and 2008-2012 periods the return to power has stayed relatively constant, increasing by only 1.6 percent. Conversely, the return to Eye increased by 61 percent between the first and last periods. Moreover, the relative return to Power was three times higher than it was for Eye in the pre-Moneyball era, but in the most recent period is only slightly higher. 11 Further, the pattern of the relative return to Eye corresponds to the market correction process described earlier. Namely, the relative return to Eye increases in the late 1990s and the first

PAYROLL EFFICIENCY
Next we consider the relationship between team-winning percentage and payroll, indexed as the share of league payroll. 12 Hakes and Sauer posit that with the improved metrics of the Moneyball era, teams are better able to discern the true productivity of the players, and therefore the statistical relationship between team payroll and performance should tighten.
Indeed, Hakes and Sauer found that "the explanatory power of team payroll in predicting winning percentage has improved over time," citing as evidence the increase in R 2 values for a simple linear regression model for team-winning percentage as a function Table 4 Estimated labor market productivity elasticities: The dependent variable is ln (Team Runs Scored) 1985-2012 1985-1997 1998-2002 2003-2007 2008-2012 1985-2012 1985-1997 1998-2002 2003-2007 2008-2012  of team payroll. Specifically, they found the R 2 in this payroll model for all 587 teamseasons between 1986 and 2006 to be 0.146. As noted above, the salary figures in the Lahman database are incomplete. Further, to derive team payroll data from individual players' salaries would require knowledge of the entire 25-man roster, the players on the disabled list, and the proportion of a year that traded players were on the team. We use official payroll data from the Labor Relations Department (LRD) of MLB, and find that the correct R 2 over this time period is 0.228, a substantial increase over the figure reported by Hakes and Sauer. In Table 6, we present the results from applying a regression model to team-winning percentage as a function of each team's LRD payroll. 13 More importantly, Hakes and Sauer cited an increase in the strength of the relationship between team-winning percentage and payroll over time as evidence that the window for exploiting market inefficiency in the manner described in Moneyball may be narrowing; or, stated differently, Hakes and Sauer suggest that the new, improved performance metrics and correction of market inefficiencies since Moneyball has tightened the relationship between team payroll and win percentage. We find no such evidence. While it is true that the predictive power of payroll upon winning percentage is higher than it was in the 1980s and early 1990s, it has been much lower in the 2000s than it was in the late 1990s, and shows few signs of a long-term upward trend (see Figure 3). 14 An elegant way to test for structural breaks in this relationship is to model winning percentage as a function of normalized payroll, along with interaction terms for normalized payroll and an indicator  1985-2012 1985-1997 1998-2002 2003-2007 2008-2012   variable for each one of the periods. None of the terms involving any period was statistically significant.

CONCLUSION
We employed data from 1985 through 2012, with periodization, to test the existence of, the timing of the recognition of, and the market's adjustment to an inefficiency in the baseball labor market (specifically, the hypothesized undervaluation of walks). We adapted the basic model proposed by Hakes and Sauer, extended the data set, integrated more accurate data on player service time, player salaries and team payrolls, and introduced a double-log model to properly correct for scale differences in the variables describing hitters' skills. Finally, we explored whether a core assertion of Hakes and Sauerthat there has been a linear trend for the R 2 between team payroll and win percentage to increase as a result of the use of better performance metrics and the disappearance or, at least, attenuation of the labor market inefficiencyholds up when more accurate data are used and the time period is extended. Hakes and Sauer found evidence that the market inefficiency surrounding OBP described in Moneyball existed in the period before the publication of the book and then closed in the few years following its publication. As evidence, they cite growing equality in the expected salary return to certain batting skills from 2004 to 2006. We address this same question with a double-log model, and compare the relative return to these same batting skills.
Hakes and Sauer left open the question of how the labor market would react going forward. We find that the sharp increase in the return to on-base skills in the period surrounding Moneyball is properly viewed in the context of a longer-term evolution in valuation on the part of general managers. While Moneyball may well have been a catalyst in accelerating the incorporation of value of OBP into the baseball labor market, these changes were part of a more modest, longer-term trend. Moreover, the relative return to walking has fallen since Hakes and Sauer's observation, confirming their suspicions that the market correction in the years following Moneyball would not be permanent.
Finally, using accurate payroll data from MLB, we find no evidence that the relationship between team payroll and winning percentage has strengthened over time. This runs counter to the observation made by Hakes and Sauer. Notes 7. Over the 1998-2002 period, the average values for TPA, Eye, Bat, and Power among the 1,748 players in the data set were 447, 0.091, 0.272, and 1.59, respectively. If such a player were a free agent outfielder then his salary for the year 2000 predicted by our model is $2,424,780. However, if that player's walk rate was 0.101 then his predicted salary becomes $2,460,973, an increase of $36,193. The corresponding difference for a free agent outfielder in 2010 with the same statistics would be $124,655. Since we are using year fixed effects, this result controls for baseball inflation over the period. 8. The P-values for the F-statistic of the simple linear model for Eye, Bat, or Power as a function of Year are 0.089, 0.098, and 0.062, respectively. The corresponding P-values for a nested F-test that includes periodized interaction terms are 0.27, 0.846, 0.007. In particular, there is evidence that Power was compensated more highly from the 1998-2007 periods than in the other years. 9. This point is elaborated in Baumer and Zimbalist [2014, Ch. 1]. 10. This growth in the return to all skills should not be surprising, given the sustained increase in average salaries between 1985 ($370,000) and 2012 ($3.2 million). 11. It is also interesting to observe that the return to power was highest during the apex of the steroid power boom.
While an economist might scratch her head at this outcome (thinking that the plethora of home runs would lower the relative value of the same), it appears that the steroid power boom produced both an ownership and fan fascination with the long ball only serving to increase its market value. 12. Equivalently, each team's payroll divided by the league's average payroll in that season. 13. LRD payroll stands for MLB's Labor Relations Department measurement of payroll. LRD payroll differs from the CBT (competitive balance tax) and players' association payroll in its treatment of benefits, deferred salary, and discount rates. 14. Again, a simple linear model for R 2 in the payroll model as a function of time is not statistically significant at the 5 percent level. The increase in R 2 between 1985-1997 and 1998-2002 may be a function of widening revenue inequality across teams, and the Yankees' remarkable string of success during the latter period.