Submitted by Scott J. Callan, Ph.D. and Janet M. Thomas, Ph.D.

Scott J. Callan is a professor in the Department of Economics at Bentley University. Janet M. Thomas is an Emeritus Professor of Economics at Bentley University.

**ABSTRACT**

In this research, we contribute to the literature on amateur sports competition by empirically estimating a production frontier for a sample of Texas high school football programs. Modeling a production frontier in this context allows us to empirically isolate the influence of offensive and defensive inputs on a team’s overall performance. In so doing, we are able to predict a measure of relative production efficiency for each team. Research-based estimates indicate that, on average, Texas high school football teams play well below potential, which in turn is linked to coaching inefficiency. Each team’s predicted efficiency level is then used in a salary regression and is found to be an indirect determinant of a head coach’s salary.

**INTRODUCTION**

Few would argue that competitive sports provide more than adequate opportunity for debate about which team is superior and which franchise works to its potential. For many fans, this debate is an integral part of the entertainment of sports. For sports economists, however, investigating the debate and attempting to settle it objectively is an empirical challenge, which can be addressed using production theory and frontier modeling techniques.

Proposing the application of production theory to professional sports is credited to Rottenberg (1956), who did so in an economic investigation of the labor market for Major League Baseball (MLB). The underlying motivation for this proposition is that the overall performance objective of team sports lends itself readily to the input-output relationship that underlies a classical production function.

Early empirical estimations of production functions for team sports include a study by Scully (1974), considered the seminal empirical paper in this genre. Scully’s (1974) main objective was to determine the marginal revenue product for individual MLB players, an effort that was facilitated by the estimation of a team production function using ordinary least squares (OLS) analysis. According to Dawson, Dobson, and Gerrard (2000), the research following Scully (1974) has evolved away from OLS and toward frontier analysis. Frontier analysis allows for the estimation of potential output, or success in a sports context, with which one can compare actual performance. This shift in estimation methodology reflects both scholarly and popular interest in whether a winning team is necessarily among those that work to their potential.

Applications of production theory to American football are relatively rare because of the intricacy of football plays and the complex interrelationships among player positions. These realities make it difficult to link the production of points to a particular player, when so many individuals and plays typically contribute to moving the ball down the field and scoring. In any case, most researchers who estimate American football production do so at the professional level.

One example is Hofler and Payne (1996), who were the first researchers to apply production frontier analysis to professional football. Focusing on the 1989 to 1993 seasons, they used a Cobb-Douglas function to estimate a production frontier for the offensive unit of all 28 National Football League (NFL) teams. Using the natural log of total points scored on offense as the dependent variable, their empirical results identified five regressors that are statistically significant.[i] They further found that the average efficiency level of these teams was 95.8%, implying that they operated close to their potential over the period under study. More importantly, they demonstrated that team efficiency and actual success, i.e., number of wins, were distinctly different.

A more recent research paper by Hadley, Poitras, Ruggierio, and Knowles (2000) used a production frontier estimation approach by applying a Poisson regression model to estimate an NFL team’s production of wins as a function of several offensive and defensive measures. They then calculated an efficiency performance measure for each team as a ratio of observed to predicted wins. Following Porter and Scully (1982), they examined the effect of coaching experience on team efficiency, which they estimated to be increasing at a decreasing rate.

Although the use of production frontier analysis in sports has been prevalent at the professional level, the same cannot be said about its application to amateur sports, despite the assertion of some, including Hofler and Payne (1997), that such an extension is important. Depken and Wilson (n.d.) is the only research paper discovered during this research estimated amateur football production and coaches’ relative efficiency at the college level. In their study, Depken and Wilson used production frontier analysis to examine the efficiency of Division IA college football coaches from 1990 to 2004. Specifying a Cobb-Douglas production function, they used four inputs – total points scored, total points scored against, a team rating measure, and coaching experience, to explain team output measured as total season wins. Depending on the assumptions of their model, they estimated that the average efficiency for an NCAA Division I coach is between 77.2 and 81.4%. They further found a positive and significant correlation between the team’s technical efficiency and the coach’s experience.

In part, the observation of relatively few production analyses of amateur sports is due to the limited data and recorded statistics available at that level of competition relative to professional sports. Nonetheless, extending the investigation of production and technical efficiency to the amateur level is important in order to determine if findings at the professional level can be generalized.

To that end, this research applied production frontier analysis to American high school football. It is believed that this context has not been used in any prior efficiency study of team sports. The empirical context is Texas Class 5A high school football teams for the 2004 and 2005 seasons. The coaches and teams in the sample are such that team performance is linked to only one head coach. Hence, the estimate of each team’s efficiency is coincident with the head coach’s efficiency while leading that particular team.

It is believed that conducting an efficiency examination in a high school context is important, not only to fill an empirical void in the amateur sports literature, but also to test the reasonable expectation that efficiency attainment is more challenging at the high school level relative to collegiate and professional competition. This expectation is based on the inherently limited recruiting opportunities in high school sports and by the budgetary constraints faced by public school systems.

Texas offers a particularly good perspective for this study for a number of reasons. One is that the huge popularity of high school football in Texas, particularly at the Class 5A level, means that public schools are willing to allocate significant resources from their overall budgets to their football programs. Consequently, there is broad-based interest in determining the extent of performance efficiency relative to the public sector’s large investment in high school athletics. Related to this assertion is growing grassroots interest in the salaries earned by head coaches at public schools, particularly in light of pervasive cutbacks of academic and non-academic programs during an economic slowdown. Also important is the national recognition given to large football programs, such as those in Texas, which gives this research broad appeal. Consequently, this study may stimulate further research. Lastly, from a practical perspective, data on certain elements of Texas’ high school football programs have recently become available, making empirical estimation possible at this level of competition.

Following convention, this study used a Cobb-Douglas production function to estimate each team’s frontier output for the 2004 and 2005 seasons. Once done, we determined the level of technical efficiency for each coach in both periods. This approach is analogous in motivation to Hofler and Payne (1996) who used a similar model to study 28 NFL teams and to Einolf (2004), whose production estimation found that NFL franchises are more efficient than MLB teams. The researchers believe that the results of this study’s production estimation are of interest in their own right. However, they also provide data to facilitate an investigation of coaches’ remuneration. Given the budgetary constraints and associated program cuts in many public school systems, such an examination is timely and relevant.

Ultimately, it was found that an efficiency-based prediction measure did help to explain the level of coaches’ salaries, although the quantitative effect is relatively small. Such an outcome suggests an interesting parallel between the evaluation process within a high school athletic department and assessment practices in a corporate context, which also considers efficiency in performance reviews. This observation follows Fizel and D’Itri (1996; 1999), who explicitly linked coaching to managing in their empirical investigations of college basketball. They argued that coaches must recruit, train, and motivate their players just as managers do for their direct reports in a business setting.

At a fundamental level the researchers believe that this research study makes two important contributions to the literature. First, it illustrates the applicability of frontier theory to the production of sports performance at the high school level of football competition, a relevant and unexplored context in which to test efficiency attainment under constrained conditions. Second, the empirical findings identify efficiency as a statistically significant, direct determinant of team success and as a statistically significant, indirect influence on head coaches’ salaries.

**METHODS**

To estimate the production function for Class 5A Texas high school football teams and the associated measure of technical efficiency, the researchers developed an empirical model based on the stochastic frontier analysis (SFA) introduced at essentially the same time by Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977).[ii] A frontier production function defines the maximum production, or number of wins, achievable from a given stock of inputs, which, in this context, was the skill and experience of the team and its coach. If a team operates *on* its production frontier, it achieves technical efficiency because the actual number of wins is equivalent to the maximum number possible. If it operates *below* the production frontier, it suffers some amount of technical inefficiency, because the team does not work to its full potential. This in turn suggests that a measure of technical efficiency can be defined as the ratio of actual wins to potential wins for each team in the sample.

The result is a production function, representing the number of wins that team i accumulates in a season, given available team inputs, as follows:

(1) Wins_{i} = f(**X**_{i}**α**) exp(e_{i}), i = 1,…,n,

where:

Wins_{i} is the number of realized wins by team i in the season,[iii]

**X**_{i} is a vector of inputs for team i

**α** is a vector of frontier parameters to be estimated,

e_{i} is a composite error term equal to (v_{i} – u_{i}), where:

v_{i} is the white noise error term linked to random events, which is assumed to be

independently and identically distributed with a mean of zero and a finite variance,

and

u_{i }is a non-negative error term that captures team i’s technical inefficiency, and

n is the number of high school football teams.

The researchers used realized wins as the dependent variable, which follows Hofler and Payne (1997), Hadley et al. (2000), and Depken and Wilson (n.d.). Doing so suggests that teams are motivated to maximize wins, given their talent and resources. Note that this specification is justifiable even when recognizing that teams have other goals, such as attracting and training new talent, because these secondary motives are pursued to ultimately improve team performance or output (Hadley et al., 2000).

Using the composite error term allows for the estimation of a frontier production function instead of an average production function. Moreover, the decomposition of the error term distinguishes between statistical noise in the system due to randomness and deviations from the frontier that are under the control of the coach.

TE_{i}, or equivalently exp(–u_{i}), represents team i’s technical efficiency measured as the ratio of its actual, or observed, wins to its potential wins, such that TE ≤ 1. Hence, if TE equals unity, team i is playing at its potential, or is operating *on* its production frontier. If TE is less than unity, its value represents the production deficit linked to inefficiency, or the extent to which team i underperforms and operates *below* its production frontier.

The researchers applied the general specification given by equation (1) to the amateur football context, using a Cobb-Douglas functional form. Doing so follows others in the literature, including Zak, Huang, and Siegfried (1979), Porter and Scully (1982), Hofler and Payne (1996), Lee and Berri (2008), Smart, Winfree, and Wolfe (2008), and Depken and Wilson (n.d.). The resulting equation is:

(2) Wins_{i} = α_{0 }(PointsFor_{i})^{α1}(InvPointsAgainst_{i})^{α2 }(FinalRating_{i})^{α3 }(TotalCareerWinPer_{i})^{α4} exp(v_{i}) exp(−u_{i}),

where:

PointsFor_{i} captures the team’s offensive performance, measured as the total number of points scored by team i in the season,

InvPointsAgainst_{i} represents defensive performance, measured as the reciprocal of the total number of points allowed by team i throughout the season,[iv]

FinalRating_{i} is a numerical value that measures a team’s overall standing based upon an average of 17 different rankings, including strength of schedule,[v] and

TotalCareerWinPer_{i} captures the cumulative winning percentage at the high school level for the coach of team i measured over his head coaching career.

Several points of clarification are in order. First, the use of PointsFor and InvPointsAgainst captures the overall offensive and defensive skills of the team and links them to realized wins. These input measures define competitive sports in the context of optimizing behavior, whereby teams have incentives to maximize points earned and to minimize points scored against. Doing so motivates the use of a production frontier model, as discussed in Hofler and Payne (1996).

The use of aggregate skill measures was necessitated by data limitations at this level of competition. As for precedence, this aggregation is similar to that employed by Scully (1974), who also was confined by data, and more recently by Berri (1999) and Carmichael, Thomas, and Ward (2001).[vi] Further support is offered by Scully (1994) who points out that across all sports, football has the least number of measurements of playing skill. Related in context to this research, Depken and Wilson (n.d.) used an analogous approach to model collegiate football output. They argued that such an approach was appropriate because, unlike baseball and other team sports, points earned in football are not easily attributable to a single offensive player. This reality is due in part to the series of plays and the complex interchange of players that are necessary to “produce” a touchdown, field goal, or other scoring result.

Second, the FinalRating variable is an *ex post* measure of a team’s performance. In that sense, it is somewhat analogous to indices used in studies of professional sports, such as Fizel and D’Itri (1996) and Lee and Berri (2008), among others. For example, in their study of Division I collegiate basketball, Fizel and D’Itri (1996) used a “power” rating devised by Wise Research Associates, which is a composite measure of the difficulty of a team’s schedule. Depken and Wilson (n.d.) used a team quality rating variable for collegiate football based on the Wilson Performance Rating System, which was formed from the win-loss records of each team and its opponents.[vii]

Worthy of note is that FinalRating includes a measure of the strength of schedule, which is an important ranking because it controls for the opposition faced by each team. Doing so is critical in order to avoid falsely crediting a team with a higher efficiency level solely because that team faces relatively weak opponents compared to the team’s own talent (Fizel & D’Itri, 1996).

Lastly, the model includes a measure of each coach’s high school career record, TotalCareerWinPer, to capture the influence of team coaching on overall team performance. Fizel and D’Itri (1996) discussed the importance of coaching in collegiate basketball and the link between coaching and team potential. Porter and Scully (1982) estimated a managerial efficiency learning curve that explicitly considers the importance of coaching experience in MLB.[viii] Others who used coaching experience in their respective production models include Hadley, et al. (2000) in their study of the NFL, Smart and Wolfe (2003), who examined performance in MLB, and Depken and Wilson (n.d.), who investigated collegiate football.

Rewriting the function in log-linear form yields the following empirical model used to estimate the stochastic production frontier for the 2004 and 2005 seasons:

(3) ln(Wins_{i}) = α_{0} + α_{1 }ln(PointsFor_{i}) + α_{2 }ln(InvPointsAgainst_{i}) + α_{3 }ln(FinalRating_{i}) + α_{4 }ln(TotalCareerWinPer_{i}) + v_{i} − u_{i}.

Logically, it was anticipated that a positive relationship exists between ln(PointsFor_{i}) and ln(Wins_{i}) and between ln(InvPointsAgainst_{i})and ln(Wins_{i}). Because FinalRating_{i} is a measure of the team’s overall success, it too should have a positive influence on ln(Wins_{i}). Lastly, because ln(TotalCareerWinPer_{i}) captures the accumulated knowledge and practical experience of the head coach, it should positively affect the number of wins achieved in a season.

Once the stochastic frontier analysis (SFA) was completed for both seasons, we used the associated post-estimation value of technical efficiency in 2004 and the frontier estimate in 2005 to construct a measure of expected wins to use in a subsequent regression of the coach’s salary. Specifically, the researchers constructed a new variable, EfficFrontier, as the product of the technical efficiency estimate for 2004 and the potential output (or frontier) estimate for 2005, which generated a measure of expected output for 2005. This constructed variable is similar to that used by Zak et al. (1979) to estimate expected output for NBA teams. This helps to determine if efficiency in team sports is rewarded through a coach’s remuneration at the amateur level.

Therefore, it was estimated that the following salary model applied for each head coach in the sample, using 2005 data for salary and 2004 data for the explanatory variables:

(4) CoachSalary_{i} = β_{0} + β_{1}EfficFrontier_{i}+ β_{2}TotalCareerGames_{i} + β_{3}TotalCareerGames_{i}^{2} β_{4}TotalCareerWinPer_{i} + β_{5}Region2_{ i} + β_{6}Region3_{ i} +β_{7}Region4_{ i} + μ_{i},

where:

CoachSalary_{i} is the annual remuneration awarded to the coach of team i in 2005,

EfficFrontier_{i} is the product of TE in 2004 and potential (or frontier) output for 2005, which is a proxy for expected wins in 2005,[ix]

TotalCareerGames_{i} captures coaching tenure, or longevity, as the total number of high school games supervised by the head coach of team i throughout his career as of 2004,

TotalCareerWinPer_{i} captures coaching performance, as defined previously,[x]

Region2_{i}, Region3_{i}, and Region4_{i} are geographical areas representing groups of the 32 districts identified by the University Interscholastic League (UIL) within which high school football teams compete, and

μ_{i} represents the random disturbance term.

As for predicting the influence of EfficFrontier on CoachSalary, a reasonable argument is that coaches who are able to lead their respective teams to play at their potential should be rewarded appropriately. Whether or not this is done in practice is among the popular debates in sport. Obviously, win-loss records are typically perceived as the ultimate measure of success for the team and its coach. However, both sports scholars and sports enthusiasts are interested in whether a team has worked to its potential, independent of its win-loss record (Hofler & Payne, 1997). Following this logic, a related question is whether coaches are recognized monetarily for leading their teams to play at their potential, irrespective of the number of wins accumulated in a given season. If so, the relationship between EfficFrontier and CoachSalary should be positive.

Logically, it was anticipated that TotalCareerGames would be positively related to CoachSalary, because higher remuneration is generally needed to attract and retain more seasoned coaches. The same expectation applies to TotalCareerWinPer to recognize that more successful coaches are paid higher salaries than less experienced or rookie coaches.

With regard to the Region variables, location-specific differences in salary are not uncommon across geographical areas, although intrastate differences might not be discernable. In any case, if such distinctions do exist among Texas high school football coaches, the algebraic sign of regionally defined salary differences cannot be predicted *a priori*, aside from the expectation that more populous urban areas generally are linked to higher living costs and hence higher levels of remuneration.

In terms of the overall context for the model, the following is offered. In the state of Texas, the University Interscholastic League (UIL) organizes and provides oversight for high school football and other athletic programs. The UIL is a non-profit organization that was created by the University of Texas at Austin in 1910, “…to provide educational extracurricular academic, athletic, and music contests.” (University Interscholastic League, 2010). Based on UIL guidelines, Texas schools are placed in class designations: 1A, 2A, 3A, 4A, and 5A, based on enrollment (with 5A being the largest), and geographic location.[xi] Geographically, the UIL assigns each Texas high school to one of four regions. These four regions are in turn divided into 32 districts.

For football, enrollment also determines whether the team competes in a 6-man or an 11-man conference. Schools with relatively low enrollments, i.e., less than 100 students, compete in 6-man or 11-man conferences, leaving the majority to participate in 11-man conferences. Because our sample includes solely Class 5A teams, all teams play in 11-man conferences.

With the exception of coach’s salary, data for the 2004 and 2005 seasons for Texas Class 5A high school football teams were drawn from Texas Football Ratings (2004; 2005). Coaches’ salary data are for 2005 and were sourced from Austin American-Statesman Research (2010) in Austin, Texas.

**RESULTS AND DISCUSSION**

The log-linear production frontier model in equation (3) was estimated using maximum likelihood estimation (MLE). When conducting this estimation, the distribution of the technical efficiency term must be specified. Therefore, following common practice, we assumed a half-normal distribution of the associated inefficiency term. Validating the assertion of others, such as Dawson, et al. (2000), we found similar results when assuming an exponential distribution.[xii]

Descriptive statistics for all variables used in the frontier estimation for 2004 and 2005 are given in Table 1. After deleting observations with missing values (or zero values for logged variables), the sample comprises 201 observations for the 2004 production frontier and for the salary regression and 224 observations for the 2005 frontier.

Note that the mean values are reasonable and fairly consistent across the two periods. For example, for both years, the average number of wins is just over 5.5. The mean values of PointsFor and PointsAgainst also are consistent over the two periods, and the mean of PointsFor exceeds that of PointsAgainst for both years, which might reflect that teams in the Class 5A conference have stronger offensive units than defensive units. Lastly, the mean value of TotalCareerWinPer is over 53% for both years, which likely reflects that more successful coaches are rehired in subsequent periods. The means for FinalRating represent average team quality, since FinalRating is an index measure of each team’s ranking on specific attributes, as discussed previously. The minimum values for 2004 and 2005 are 105.28 and 122.59, respectively, and the maximums are 196.37 and 208.62, respectively, which gives context to the means of 153.071 for 2004 and 165.491 for 2005.

The production frontier results for 2004 and 2005 are presented in Table 2. The reported statistics suggest that the production model fits the data well and that the use of frontier analysis is appropriate. According to the likelihood-ratio χ^{2} test statistics of 573.20 for the overall 2004 estimation and 989.97 for the 2005 estimation, the coefficients of the independent variables are jointly significant in each case.

It was also determined that the stochastic frontier production estimation is appropriate for each year, based on the respective likelihood-ratio tests of the inefficiency component of the error terms equaling zero. For 2004, this χ^{2} statistic is 25.58, and for 2005, the comparable magnitude is 36.83, both indicating that there is inefficiency in the error structure.

It was further observed that, with the exception of ln(FinalRating) in the 2004 results, all estimated parameters had the expected signs based on the priors, and all are statistically significant at the five percent level or better. More specific analysis is possible by examining the individual parameter values. Because the frontiers were specified as log-log functions, the parameters are aptly interpreted as elasticities.

To begin, consider the estimated parameter on ln(PointsFor) in 2004 of 1.014, which is significant at the one percent level. Its value implies that a 10% rise in points scored in the season, which would be an increase of 27.3 points evaluated at the mean, would bring about a 10.1% increase in games won, raising the average number of wins in the season to 6.39.

The analogous parameter on ln(PointsFor) for the 2005 production frontier was also significant at the one percent level, although smaller in magnitude at 0.738, and the same interpretation is applicable. Specifically, a 10% increase in total points scored would lead to a 7.4% increase in season wins.

The findings for ln(InvPointsAgainst) were also strongly significant both for 2004 and 2005, but they were smaller in magnitude than those for ln(PointsFor). The quantitative strength of the parameters on offensive scoring relative to defensive scoring is analogous to what Depken and Wilson (n.d.) found. As these authors pointed out, this set of comparative results points to the greater importance of offensive play in football. Simply put, team wins are produced by scoring more points than an opponent. And although defensive play can score points, the majority of a team’s points come from offensive performance. The estimates from this study are consistent with this observation.

The specific elasticity estimates for ln(Wins) with respect to ln(InvPointsAgainst) are 0.587 and 0.360 for 2004 and 2005, respectively. For the 2004 season, this means that if a team reduces by 10% the points scored by its opponents, it will enjoy a 5.87% increase in wins. For 2005, the comparable effect on wins of a 10% reduction in points scored against a team is 3.6%.

The parameter on the overall quality measure, FinalRating, was positive and significant at the one percent level for the 2005 frontier, although not statistically significant in the 2004 estimation. Because this rating measure is a composite of some 17 different rankings, it is difficult to tell why the results were not consistent year after year. However, it might be related to the strength of the parameters on other regressors relative to their counterparts for the 2005 estimation. It also could be the case that the insignificant parameter in 2004 might mean that teams within a given conference were more evenly matched in 2004 than in the subsequent year. In any case, the value of 1.109 in 2005 implies that a 10% increase in a team’s overall rating will be associated with an 11.09% increase in team wins, which, when assessed at the mean, this translates to 6.26 wins for the season instead of 5.63. At the margin, a 26-point increase in the FinalRating for 2005 is associated with an additional win in the season.

Lastly, the parameters on ln(TotalCareerWinPer) for both the 2004 and 2005 production frontiers are positive, as anticipated, and statistically significant at the five percent level, which validates the assumed importance of a coach’s expertise to his/her team’s performance. In 2004, the estimated parameter suggests that a 10% increase in the coach’s winning percentage should give rise to a 1.7% increase in team wins. For 2005, the comparable change in team wins was 1.4%.

At a broader level, the researchers estimated the frontier level of performance for both years. As shown in Table 3, the potential number of wins for the average team in 2004, given existing inputs, was 7.461, markedly higher than average wins in that season of 5.806. In 2005, the potential number of wins was 7.607 compared to actual average wins of 5.634. These relative values speak to the presence of technical inefficiency for both seasons, meaning that, on average, teams did not play at their potential.

Another way to assess inefficiency is to examine the direct estimates of TE from each model year. As shown in Table 3, average technical efficiency in 2004 is found to be 0.763, and in 2005, the measure is estimated at 0.717. This means that, on average, Class 5A high school football teams in Texas played at about 76% and 72% of their potential, in 2004 and 2005, respectively, or at a two-year average of about 74%.

In the absence of published frontier analyses done at the high school level, it is difficult to make a direct comparison of the efficiency results to that of other high school teams. However, the efficiency estimates are not dissimilar from analogous findings for collegiate football, estimated to be 79.5% across all teams examined by Depken and Wilson (n.d.).[xiii] At the professional level of football, published results suggest much higher efficiency levels, which is a sensible outcome. Among such reported findings were 95.8% for the NFL over the 1989 to 1993 period (Hofler & Payne, 1996), 89% for the NBA (Hofler & Payne, 1997), 99.87% for the NBA (Zak et al., 1979), and 87% for the NHL (Kahane, 2005).

Taken together, the production frontier estimates presented in Tables 2 and 3 are logical and in keeping with expectations. Overall, the collective findings communicate that amateur football performance is influenced by many of the same determinants found to be significant at the professional level. This research also indicates that teams and coaches suffer from some amount of technical inefficiency, meaning they do not perform at the level of their potential.

The efficiency estimates are of interest in their own right, but it was also expected that they might contribute to an understanding of how head coaches at the amateur level are remunerated. To that end, the researchers considered various salary models based on the literature, testing for the effect of coaching efficiency and other determinants on salary. The result of this effort was the specification given in equation (4). The estimated measure of technical efficiency was not found to significantly affect salary. Therefore, as noted previously, a variable was constructed that is dependent on efficiency but also integrates some measure of the team’s overall potential. The motivation behind this construction is that, if one assumes the same efficiency level in 2005 as was achieved in 2004, one can derive an estimate of the expected number of wins for each team in 2005. Logically, this proxy for expected wins can then be used to help explain coaches’ salaries in 2005.

Descriptive statistics for all variables used in the salary estimation are provided in Table 4. In this sample of 201 Class 5A high school football teams, the average salary for a head coach is just over $80,100. Reportedly, salaries of head coaches in Texas are considerably higher than that of Texas high school teachers, and, according to the Bureau of Labor Statistics (BLS), higher than the national median for coaches and scouts across industries (Associated Press, 2006; Bureau of Labor Statistics, 2010).[xiv] For this group of coaches, the total number of games supervised over their respective careers is about 103 on average, and they are employed in schools that are about equally distributed across the four UIL-designated regions. The mean value for EfficFrontier is 5.96, which represents a reasonable expectation of number of wins for the current season, given the mean values on realized wins.

The researchers used ordinary least squares (OLS) to estimate equation (4). The results of the regression are given in Table 5. Based on the reported F-statistic, the model fits the data well. The adjusted R-squared indicates that about 17% of the variance in coaches’ salaries is explained by the specified regressors.

Interestingly, it was found that Texas high school coaches were paid indirectly for how well they made their teams work to their potential, but only to the extent that efficiency was assumed to influence expected wins. Specifically, in this sample, the researchers found that the estimated parameter on EfficFrontier is positive and significant at the 10% level. The estimated value of 432.757 can be interpreted as the incremental salary earned in a season for every additional anticipated win based on the realized technical efficiency from the prior period. Although the quantitative value is relatively small, this is nonetheless an interesting outcome, because it suggests that team coaches are not recognized solely for their win-loss records or experience, but are remunerated in some way for their ability to motivate teams to work to their potential.

As anticipated, coaching experience is also recognized monetarily. The parameter on TotalCareerGames, representing high school coaching longevity, is positive and significant at the five percent level, and the parameter on TotalCareerGames^{2} is negative although significant only at the 15% level. These results imply that salary increases with experience at a decreasing rate, which in turn means that a coach’s experience positively influences salary only up to a point. In this case, this point occurs when TotalCareerGames reaches about 287, which is well above the sample mean of 103 and greater than 2 standard deviations above the mean. Hence, it can be reasonably assumed that a positive relationship exists for most of the coaches in this sample.[xv]

As for coaching performance, the parameter on TotalCareerWinPer, is positive and significant at the five percent level. The associated parameter suggests that a coach’s salary is higher by just over $13,000 annually for every additional percentage of the coach’s lifetime high school wins. Based on the reported mean values in this sample, achieving a one-unit increase in winning percentage translates to winning about six games in the regular season.[xvi] Not surprisingly, this finding supports the common view that winning is rewarded significantly for coaches of Texas 5A high school football.

Lastly, according to the regression results, coaches’ salaries vary by high school location within the identified regions. The parameters on all three regions are negative, and those associated with Region 2 and Region 4 are statistically significant. Coaches of teams located in Regions 2 and 4 earn significantly less per year, $3,333 and $4,311, respectively, than those in Region 1, the suppressed region. It is difficult to know why these differences exist. As discussed, these regions and their districts were created by the University Interscholastic League (UIL) based on enrollments, and they are realigned every two years. Consequently, although not imposed deliberately, some regions might comprise schools that are at a more intense level of competition than others, which may be linked to higher salaries for coaches. There also might be variation in the cost of living across the regions that explains these differences in remuneration.

**CONCLUSIONS**

Competition at all levels of team sports motivates lively debate about whether a team has worked to its potential, independent of its win-loss record. In the vernacular of economic theory, such a debate is equivalent to asking if a team performs *on* its production frontier, achieving technical efficiency, or if it operates *below* its frontier, implying some degree of technical inefficiency. For some time, researchers have studied team sports using production theory in an effort to explore team or coaching efficiency, and in some cases to compare this result to other measures of team performance. Although earlier studies use MLB as the context, subsequent work extended this research to other professional sports, including the NBA, English cricket, and, in a limited way, the NFL.

In this research, the authors used the parametric technique known as stochastic frontier analysis (SFA) to estimate coaching efficiency in amateur sports, specifically for Texas 5A high school football in the 2004 and 2005 seasons. Existing research on football performance and team efficiency is relatively uncommon, and of those studies that do exist, all but one discovered during this study conducted the analysis at the professional level. Hence, the researchers believe that this present study of amateur football contributes important, new information about coaching efficiency at the high school level.

Overall, the SFA was successful, based on the conventional statistical measures of fit. Moreover, all but one of the parameter estimates across both frontier models bear the expected signs and were statistically significant. Further, the estimates indicated that offensive skill has a stronger quantitative influence on wins than does defensive skill. This study also confirmed what others have found at higher levels of competition regarding the positive influence on wins of overall team quality and coaching experience. Of particular relevance are the empirical findings on efficiency. The results indicated that, on average, Class 5A Texas high school teams played at 76.3 and 71.7% of their potential for the 2004 and 2005 seasons, respectively. These estimates indicated a fair amount of coaching inefficiency for the teams in this sample. Supporting this set of findings are the estimates of potential wins for each season. In 2004, potential wins were estimated to be 7.461, which is considerably higher than realized wins of 5.806. The analogous values for the 2005 season were found to be 7.607 and 5.634 for potential and realized wins, respectively.

Quantitatively, these efficiency estimates were generally lower than what others have found for professional teams and coaches. To a large extent, lower efficiency at the amateur level is a logical outcome, given the emphasis placed on success in professional sports, the high financial stakes, and the intense degree of competition. That said, the estimated inefficiency found in this research might be cause for concern in the face of escalating high school coaching salaries in Texas.

This latter assertion links directly to the second part of this empirical study, which is the investigation of salaries for high school coaches in this sample. On this front, it was discovered that the estimated measure of coaching efficiency does not directly influence salary levels, but it does so indirectly. That is, a proxy for expected wins formed from the coach’s efficiency record from the prior period had a positive effect on coaching salaries.

Taken together, the researchers believe the results contribute positively to the evolving literature on sports production. Amateur football at the high school level provides an interesting context for frontier analysis because of the associated policy implications and the unique constraints on team management relative to collegiate and professional competition.

**APPLICATION IN SPORT**

From a sports management perspective, the production inefficiency estimates suggest that the coach’s tasks of recruiting, training, motivating, and assessing players are not being carried out in an efficient manner. This in turn could have negative implications for the high school sports program, which would be salient to the school’s athletic department, particularly in Texas where coaches’ salaries, which are high relative to teachers’ pay, have come under fire.

Germane to this study’s particular context and in keeping with expectations, it was also found that the measured inefficiency at the high school level is greater than that estimated at both the collegiate and professional levels, according to results reported in the literature. Likely, this relative finding reflects the comparatively short playing career and limited experience of high school players.

Overall, these results lend support to those of existing researchers, specifically that amateur coaches are unable to sufficiently motivate players to achieve their full potential. It was also found that, relative to the collegiate and professional coach, the average high school coach is a more inefficient manager. One explanation for this deficiency is the coach’s limited access to recruiting opportunities at the high school level. Another is the time constraint these coaches face as they attempt to train, motivate, and athletically develop student players within the restrictions of the academic calendar. Lastly, coaches are constrained by the relatively scarce resources of a public school’s athletic department and tight municipal budgets, which in turn are affected by economic conditions and policy decisions at the state and local levels.

The empirical results provide estimates of team potential and coaching efficiency, which should be of interest to researchers, sports enthusiasts, public school administrations, and other stakeholders. The subsequent use of these estimates to examine coaching salaries has real-world relevance, given existing concerns about high school football coaching salaries in Texas, particularly in the face of budget constraints and associated cutbacks of academic programs. Consequently, the researchers hope that this research will motivate further study of amateur team sports in other contexts to extend overall understanding of team potential, coaching inefficiency, and coaching remuneration beyond the professional and collegiate levels.

**ACKNOWLEDGEMENTS**

None

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**Table 1**

*Selected Descriptive Statistics for the Frontier Estimations*

**Variable N Mean Std Dev**

*2004 Data*

Wins 2004 201 5.806 2.996

PointsFor 2004 201 272.955 108.081

PointsAgainst 2004 201 251.910 62.253

InvPointsAgainst 2004 201 0.004 0.001

FinalRating 2004 201 153.071 20.482

TotalCareerWinPer 2004 201 0.546 0.142

____________________________________________________________________________

* **2005 Data*

Wins 2005 224 5.634 3.131

PointsFor 2005 224 261.670 111.344

PointsAgainst 2005 224 245.714 64.221

InvPointsAgainst 2005 224 0.004 0.001

FinalRating 2005 224 165.491 14.268

TotalCareerWinPer 2005 224 0.536 0.154

**Table 2**

** ***Regression Estimates for the 2004 and 2005 Stochastic Production Frontiers*

*Dependent Variable: ln(Wins) *

**2004 2005**

**VARIABLE Parameter Estimates (n = 201) Parameter Estimates (n = 224)**

** **

Intercept −0.184 5.723^{a
} (1.402) (1.815)

ln(PointsFor_{i}) 1.014^{a} 0.738^{a
} (0.079) (0.072)

ln(InvPointsAgainst_{i}) 0.587^{a} 0.360^{a
} (0.087) (0.084)

ln(FinalRating_{i}) –0.039 1.109^{a
} (0.274) (0.369)

ln(TotalCareerWinPer_{i}) 0.172^{b } 0.141^{b
} (0.073) (0.068)

σ2 0.164^{ } 0.241^{
} (0.023) (0.029)

γ 0.897 0.972

Log-likelihood −9.350 -29.262

χ^{2} 573.20^{a} 989.97^{a}

**NOTES: **

Standard errors are given in parentheses.

^{a} Significant at the 1 percent level.

^{b }Significant at the 5 percent level.

^{c }Significant at the10 percent level.** **

σ^{2} = σ^{2}_{u} + σ^{2}_{v}, which is the variance of the composite error.

γ = σ^{2}_{u} / σ^{2}, whichis the proportion of the error variance attributable to technical inefficiency

**Table 3**

*Mean Estimated Values of Production Frontier and Technical Efficiency for 2004 and 2005*

**Variable Mean Std Dev **

2004 Technical Efficiency 0.763 0.136

2004 Production Frontier 7.461 3.605

2005 Technical Efficiency 0.717 0.175

2005 Production Frontier 7.607 3.682

**Table 4**

*Descriptive Statistics for the Coach’s Salary Estimation*

** **

**Variable N Mean Std Dev **

** **

CoachSalary 2005 201 80,107.975 10,227.799

EfficFrontier 201 5.960 3.100

TotalCareerGames2004 201 102.965 89.681

TotalCareerWinPer2004 201 0.546 0.142

Region 1 201 0.264 0.442

Region 2 201 0.249 0.433

Region 3 201 0.244 0.430

Region 4 201 0.244 0.430

** **

**Table 5**

*Regression Estimates for Coach’s Salary Determination*

*Dependent Variable: CoachSalary*

**VARIABLE ****PARAMETER ESTIMATES**

Intercept 68,840.89^{a} (3,029.44)

EfficFrontier 432.757^{c} (249.053)

TotalCareerGames2004 54.474^{b} (23.154)

TotalCareerGames2004^{2 } −0.095 (0.065)

TotalCareerWinPer2004 13,061.950^{b} (5,799.880)

Region 2 −3,333.139^{c }(1855.797)

Region 3 −1,678.359 (1,840.923)

Region 4 −4,310.565^{b }(1,876.474)

F (7, 193) = 7.01^{a}

R-squared = 0.2028

Adjusted R-squared = 0.1739

n = 201

NOTES:

Standard errors are given in parentheses.

^{a} Significant at the 1 percent level.

^{b }Significant at the 5 percent level.

^{c }Significant at the 10 percent level.

**ENDNOTES**

- In their original model, Hofler and Payne (1996) included 13 independent variables. After joint testing the insignificant independent variables, their final model includes the following 5 independent variables: net yards gained by rushing, net yards gained by passing, third down efficiency, punt yard returns, and percent of successful field goals.
- The derivation of our empirical model also follows Battese and Coelli (1995) and Battese and Corra (1977).
- In this particular sample, the total number of possible wins is essentially the same across observations, which would make wins and winning percentage equivalent dependent variables. Therefore, we used total number of wins for simplicity and to facilitate parameter interpretation.
- Note that we define InvPointsAgainst as the reciprocal of total points scored against team i, so that the resulting variable has an effectively positive influence on the dependent variable. This is a common conversion used by others in the literature, including Fizel and D’Itri (1996) and Depken and Wilson (n.d.).
- This rating variable, which was constructed by Texas Football Ratings, is widely cited and used in
*Texas Football*, an annual publication published since the 1960s and considered to be the premier source of information on Texas collegiate and high school football. The 17 rankings used to construct the composite variable are: regional, state, and overall ranking; state and overall for the following: offensive points scored, defensive points allowed, margin of victory, movers, shakers, strength of schedules by record, and strength of schedules by opponents ratings. (Texas Football Ratings, December 25, 2004b.) - As noted previously, Berri (1999) and Carmichael, Thomas, and Ward (2001) employed aggregate offensive and defensive inputs as a baseline of comparison to production models that used more specific input measures. If more detailed high school data become available in the future, results from our aggregate specification and others like it can be compared to alternative models that evolve in much the same way that the professional sports literature has developed since Scully’s seminal production function estimation.
- At the professional level, Dawson, Dobson, and Gerrard (2000) and Dawson and Dobson (2002) pointed out the importance of using an overall measure of player talent in a sporting production function, which in their case is specified for English association football. In both research studies, the authors constructed their own index measure, because they were unable to use the Opta Index of player performance, which is available only for the Premier League in English association football.
- Both Kahn (1993) and Scully (1994) discussed the link between team management, (including the coaching staff), and team performance at some length.
- Because EfficFrontier is a proxy for expected wins, realized wins was not included in the model to avoid confounding the results.
- We did attempt to add a squared term for this variable, but it was not statistically significant.
- For our period of study, Class 5A schools have enrollments of at least 1,925 students.
- Results for the exponential distribution are available upon request.
- This estimate is associated with the assumption of a half-normal distribution for the efficiency measure.
- According to the 2006 article, the average football coach earned nearly $73.8 thousand annually, while the average teacher salary was $42.4 thousand. The BLS reported that nationally as of 2008, the median salary for coaches and scouts across industries was $28,340.
- Following Porter and Scully (1982), we did attempt to assess the existence of a learning curve effect by regressing coaching experience (proxied through TotalCareerGames and TotalCareerGames
^{2}), on the estimated efficiency measure, but the relationship was not found to be statistically significant. Although disappointing, this finding is similar to that presented by Fizel and D’Itri (1996). - At the end of a regular season, a high school coach adds 10 games to the total career games played, which would be 113 from the mean of 103. To raise the percentage of wins by one (from the mean of 55 percent, or 57 games, to 56 percent, or 63 games), the coach would have to win 6 games that season.

* *