TEAM EFFECTS ON COMPENSATION: AN APPLICATION TO SALARY DETERMINATION IN THE NATIONAL HOCKEY LEAGUE.

IDSON, TODD L. ; KAHANE, LEO H.

LEO H. KAHANE [*]

Studies of salary determination largely model pay as a function of the attributes of the individual and the workplace (i.e., employer size, job characteristics, and so forth). This article empirically investigates an additional factor that may influence individual pay, specifically coworker productivity. Data from professional sports are used to evaluate this question because both salary and teammate performance measures are readily available. We find that team attributes have both direct effects on an individual's pay, and indirect effects through altering the rates at which individual player productive characteristics are valued. (JEL J31, L83)

I. INTRODUCTION

When complementary exists between labor inputs, individual productivity may be poorly measured by treating the individual worker separately from the character of the organization, or team, within which he works. Namely, individual productivity may vary when working in different settings to the extent that coworkers offer different degrees of assistance. [1] If such complementary between human capital inputs is present, say a team dynamic, it suggests the possible existence of both team and individual effects on productivity, and hence compensation.

Empirical analysis of team effects has been hindered by the lack of data on individuals and their coworkers, including managers. One source, though, that contains much of this requisite information is data on professional sports teams. In this article, we exploit this unique attribute of professional sports data in order to empirically assess the effect of complementarities between labor inputs on individual compensation. Specifically, we use a team-sport setting to situate the worker in an organizational environment where we can observe the impact of coworkers on individual productivity, thereby allowing us to examine the separate effects of individual and team productivity on salary determination. The general question asked is whether individual attributes are valued, or rewarded, differently in different work environments--or, in our specific case, on different teams. This study will investigate these questions by empirically assessing the effects of coworker productivity, that is, the effects of the qual ity (productivity) of teammates, on individual compensation. Although a number of different highly team-oriented sports would serve our purpose, such as basketball, this particular study uses data from professional hockey to investigate team effects on compensation. [2]

The remainder of the article is organized as follows. Section II outlines the empirical framework used to analyze the posited team effects, and discusses some of the theoretical issues relating to team effects on compensation. Section III describes the data used in the analysis. Section IV reports the empirical results, and Section V reports our conclusions and directions for further work.

II. TEAM AND COWORKER EFFECTS

As noted above, the goal of this study is to evaluate empirically whether average coworker attributes (1) affect individual salaries, (2) affect the rate at which individual attributes are valued by the firm, and (3) are complementary inputs to the attributes of individual team members. The question arises, though, as to why we might expect, on theoretical grounds, that individual players would earn any part of the team's contribution. The classic model of Alchian and Demsetz [1972] suggests that the capitalist manager will reap the returns from any incremental team output, as the primary function of the manager is to discover complementarities between inputs and enhance organizational output by optimally combining resources, and to monitor effort so as to reduce shirking. As a result, the manager will capture all of the returns that are not specific to the inputs, in which case we would not expect to observe team effects on individual compensation. Yet, although a manager acting as the residual claimant is one solution to the problem of reducing shirking in the context of complementary inputs whose effort is difficult to monitor, other models suggest that it may be desirable to compensate team members based on the value of team output in order to provide optimal incentives for effort (for example, see Akerlof [1982]). [3] In fact, effort incentives derived from team output are not necessarily at variance with an Alchian and Demsetz managerial structure because managers may still find that they increase team output (and hence their compensation) through effort incentives linked to team productivity. To the extent that such incentive systems are seen as profitable, we may well observe that individual compensation is related to team output measures. As such, empirical evidence of the existence of team productivity effects on individual compensation may provide an empirical foundation for classes of models that suggest the efficacy of team output incentive mechanisms.

In order to empirically capture team effects on individual salaries we employ the following econometric model:

(1)

ln([Salary.sub.i]) = [[beta].sub.0] + [[beta].sub.1][X.sub.i] + [[beta].sub.2]t_[X.sub.i] + [[beta].sub.3][[X.sup.*].sub.i]t_[X.sub.i] + [[beta].sub.4][Z.sub.i] + [[varepsilon].sub.i],

where [X.sub.i] represents a vector of individual player performances measures, t_[X.sub.i] represents the corresponding team performance measures, [Z.sub.i] represents additional regressors, and [[varepsilon].sub.i] is an iid random error. The effects of an individual player's performance on his salary is given by

(1.1)

[partial]ln([Salary.sub.i])/[partial][X.sub.i] = [[beta].sub.1] + [[beta].sub.3]t_[X.sub.i].

Expression (1.1) has two components, (1) a direct productivity effect represented by [[beta].sub.1] and (2) an indirect effect, [[beta].sub.3], which measures the effect of average coworker productivity on the rate at which individual player productivity is valued. The multiplicative specification of the team effect captures the possibility that worker attributes will be differentially rewarded in different work environments if the cross-partial effect, [[beta].sub.3], is not zero.

Team effects on compensation will operate not only through altering the rates at which productivity-related attributes are rewarded but also directly through the [[beta].sub.2] term. Specifically, the effect of team attributes on individual player's salary is given by

(1.2)

[partial]ln([Salary.sub.i])/[partial]t_[X.sub.i] = [[beta].sub.2] + [[beta].sub.3][X.sub.i],

where [[beta].sub.2] represents the direct effect of team attributes on player salaries, and as above [[beta].sub.3] represents the effect of the interaction of team values and player attributes on player salaries.

We see from (1.1) and (1.2) that there are potentially three qualitatively distinct ways in which individual and coworker attributes may combine to affect player salaries. First, if [[beta].sub.3] = 0, then the effect of individual attributes on productivity, or salary, are invariant to the corresponding team value of this attribute, that is, production is strictly additive in its inputs. Second, if [[beta].sub.3] [greater than] 0, then individual attributes are rewarded more on teams with players who, on average, also have higher values for the corresponding attributes, that is, inputs are complementary. Third, if [[beta].sub.3] [less than] 0 then an individual's attribute is valued at a lower rate when other players on the team have, on average, higher values of this attribute. That is, on teams with high average values for a given productive attribute, the marginal contribution of another unit of this attribute is lower than on teams with players exhibiting lower average values for the attribute and hence is valued less, in other words, inputs are seen as substitutes along the particular dimension.

An example might clarify these points: say that the [[beta].sub.1] coefficient on player height is positive, so that taller players tend to be awarded with higher salaries. Furthermore, say that the [[beta].sub.2] coefficient on average team height is also positive, indicating that players on generally taller teams will receive higher pay. [4] Now, if the [[beta].sub.3] coefficient on the interaction of individual player height and average team height is positive, then this would imply that relatively tall players are complementary to each other, so teams place greater a value on an individual being relatively tall when other players on the team are also tall and as such will offer relatively higher wages in order to attract (and retain) these players. Alternatively, if the coefficient on the interaction is negative, then teams with many relatively tall players find that another relatively tall player is not especially valuable to them and as a result will not be willing to pay as high a salary for relativel y tall players. Thus, when assessing the overall influence of a player's height on his compensation, we need to evaluate both terms in (1.1), and similarly if we want to look at the effect of average coworker, or team, attributes on a player's compensation, we must evaluate both terms in (1.2).

Finally, it is important to recognize the potential importance of controlling for team effects when investigating the influence of player attributes per se on salary determination. Specifically, if [[beta].sub.2] [neq] 0 and cov(t_[X.sub.i],[X.sub.i]) [neq] 0, then failure to include t_[X.sub.i] in the regressions will bias the estimate of [[beta].sub.1] a standard example of potential omitted-variable misspecification bias.

III. SALARY STRUCTURE AND DATA DESCRIPTIONS

Salary Structure in the NHL

The institutional environment for salaries in the National Hockey League (NHL) is comparatively open. Unlike professional basketball and football, hockey has no salary cap. Management attempted to implement a graduated tax scheme wherein teams with large payrolls are taxed at a higher rates than those with small payrolls. This was an attempt to aid small market teams, but the tax scheme failed following the 1994-95 lockout. Salaries for players in the NHL have been low compared to those in other major league sports. This began to change somewhat in 1989, when superstar Wayne Gretzky signed an eight-year, $20 million contract. This was followed by a handful of other multimillion dollar contracts for talented players. [5] Contracts often contain base salaries and bonus incentives. In some cases, incentives are a function of the number of games a player appears in or whether the player appears in playoff games. [6]

There is a free agency agreement in place, which is rather complex. A 1986 agreement between players and owners describes a complicated free agency mechanism that provides a scheme of compensation to teams losing free agents. The compensation is based on the salary of the departing player and his age. The agreement has allowed for greater mobility of players, but the compensation scheme limits their movement somewhat. [7]

Data Sources

Our data set contains two years of salary and performance data and is drawn from two primary sources. First, the Hockey News, (February 8, 1991, and November 15, 1991], provides data supplied from the NI-IL Players Association on salary. The reported earnings for each player includes his base salary plus any signing bonus or deferred income allocated to that year. Second, the Hockey News Complete Hockey Book (various years) provides data on individual player performance. Our data set contains information on 930 players for the 1990-91 and 1991-92 seasons. Players are included if they played at least two years in the NHL and 26 or more games in at least one year prior to the 1990-91 season and if a salary is reported for the player. [8] All performance data are for regular season play.

One problem that arises is that some players played for more than one team in a single year (some, in fact, played for three teams in a single season). Since salary data were not generally available for the player for each team that he played with during the season, comparison of multiple movers is tenuous. Nevertheless, in our data set, a player is considered as a member of the team reporting his salary in a given year.

IV. EMPIRICAL ANALYSIS

The variables that comprise the vectors [X.sub.i] and t_[X.sub.i] are described in Table I. Career performance measures for individual players, as well as the team measures comprising t_[X.sub.i], use data up to and including the year prior to the year used for the dependent variable, the natural log of salary. We choose this specification because player performance variables in the prior year determine management's expectations about performance and hence enter into salary decisions for the subsequent year.

Individual Player Variables

The expected signs for the variables that make up the vector [X.sub.i], are shown in Table I. Following Jones and Walsh [1988], we employ a variety of variables designed to characterize a player's skill. Skills acquired through general occupational experience are captured by a quadratic in the number of games played over the player's career--following the general literature on wage profiles (Mincer [1974]) an inverted U-shaped experience effect is predicted.

The primary variable representing offensive ability (points) is expected to have a positive coefficient, since, all else equal, greater offensive contributions by a player increases the likelihood that the team will win a game and thus he should be rewarded with a greater salary. [9] Similarly, star players who demonstrate unusual skill that attracts fans should earn greater salaries, all else equal. In order to account for star status, we use a variable (star) that is calculated for each player by adding the number of his career all-star appearances and the number of major trophies won. We expect a positive coefficient for this variable.

The variable representing penalty minutes per game (penalties) is expected to capture a player's intensity of play and defensive skill. As noted in Jones and Walsh [1988], a more intense (perhaps intimidating) player demonstrates a willingness to make the sacrifices required for the team's success. This being the case, a positive coefficient is expected.

Another variable we use to represent both a player's offensive and defensive skill is the plus/minus statistic (plus/minus). The plus/minus statistic is calculated by assigning a player a + 1 if he is on the ice when his team scores a full-strength goal, and assigning him a - 1 if he is on the ice and his team gives up a full-strength goal. Career plus/minus statistics are not readily available for players; rather we calculate a game-weighted average over the previous three seasons for use as a proxy for a player's career plus/minus statistic. Ceteris paribus, we expect a positive coefficient for this variable.

In order to control for various physical attributes that may affect player performance, and that are not captured by other performance variables, we include measures for a player's height and weight. Other things equal, physically larger players may be more effective offensively and defensively, as they can use their size to gain strategic position during play. A larger player may also be able use his size to attract and "tie up" the play of opponents, thus "freeing up" his teammates for potential scoring opportunities. This being the case, we expect these variables to have a positive impact on a player's salary.

To the extent that initial playing skill is a reliable indicator of future performance, players who begin their NHL career with a greater stock of ability are expected to start their professional career with a larger salary. This initial salary differential may be reflected later in the player's career salary path. In order to control for differences in initial ability, we have constructed a dummy variable (draft) that takes the value of 1 if a player was selected in the first or second round of the rookie draft and 0 otherwise. As defined, we expect a positive coefficient for this variable.

In order to control for differences in compensation for player position, we use a dummy variable (forward) that takes the value of 1 if a player is a forward (center or winger) and 0 otherwise. All else equal, controlling for scoring ability, a defenseman is expected to earn a greater salary. That is, a defenseman with the same scoring ability as a forward would earn a higher salary because the defenseman has the added ability to prevent opponents from scoring. [10]

Finally, we include a dummy variable (free agent) that takes the value of 1 if the player was a free agent in the year previous to the current salary year. The role of this variable is to control for any performance difference the player may have demonstrated during the season of his free agency. The logic behind this variable is that a player who will become a free agent at the end of a season may play with a greater effort and intensity than they might otherwise in order to impress potential employers. Following this reasoning then, we expect a positive coefficient for this variable. [11] The free-agent status variable (free agent) and the plus/minus variable (plus/minus) have not to our knowledge been incorporated into previous empirical studies of pay and performance in the NHL.

Team Variables

As noted above, the vector t_[X.sub.i] contains team values of each [X.sub.i] performance measure, calculated so as to remove each individual ith player's contribution. For example, we calculate average points per game for the team as a whole, excluding the individual player i's points-per-game statistic. This provides us with an approximate measure of the quality of players around any individual player i. We follow the same procedure to construct team measures for the variables points, penalties, plus/minus, height, weight, and star.

Included in t_[X.sub.i], with no corresponding measure in [X.sub.i], are performance measures for the team's coach. This approach follows Kahn [1993] and hypothesizes that quality coaching can enhance player performance and hence salary. (See also Clement and McCormick [1989], Porter and Scully [1982].) Ice hockey coaches make numerous decisions that can affect team and player performance, including composing player lines, special team assignments and match ups with the opposing team's player lines. [12] It is hypothesized that coaches with greater experience and coaching talent will be able to enhance the individual player's (and team) performance by utilizing players in such a way that maximizes the team's likelihood of winning a game. We use two variables to control for differences in coaching quality. The first is the number of seasons the team's coach has coached in the NHL (coach years). It is assumed that greater experience in coaching would lead to greater coaching ability and as such we expect a pos itive sign for this variable. Second, we calculate the coach's career percentage of points won while coaching in the NHL (coach percentage). Coaches with a demonstrated ability to coach teams to victory should have a positive effect on player performance and hence salary. The two coaching-quality variables are constructed using data for the current-year coach. Current-year data is used because coaching strategy (and the resulting effectiveness of the strategy) depends on the composition of the specific team line up being coached. [13]

Finally, the vector [Z.sub.i] contains franchise variables that might exert independent effects on player salary. In particular, total franchise revenues are included in [Z.sub.i] in order to control for differences across teams in their available funds that can be used to compensate players. As with the coaching variables, current revenues are used, since they should best represent management's expected revenues generated by a team's current composition. We expect that greater revenues should be associated with greater player salaries, all else equal.

Estimation Results

Table II reports salary regressions for specifications with and without team variables. [14] Column 1 reports least-squares estimates of player salaries regressed on measures of their own productivity, [X.sub.i]. We next add team measures: first, coaching quality, and team revenues in column 2, and then additionally the average productivity of coworkers (i.e., teammates), t_[X.sub.i] in column 3. If the coefficients on own productivity, [[beta].sub.1], fall when team measures are added, then we interpret this result as indicating that part of the measured effect of player characteristics on their salary is due to teammate contributions to their productivity. [15] Further, if other player's productivity affects the wages of the individual player directly (i.e., [[beta].sub.2]), namely, not through the route of increasing the individual player's productivity per se (i.e., the [[beta].sub.3] effect), then this may be because the manager realizes that other player productivity is partly due to the contributions of the individual player in question. [16] We next include interactions of individual attributes and corresponding team averages in order to evaluate whether coworker productivity affects the valuation of individual attributes. [17] Finally, as a benchmark for comparing our results in Table II, we have also estimated a model that uses players' own-productivity, [X.sub.i], and simple team dummies (see Table A1 in the appendix). Overall, our model outperforms this benchmark model as the team dummies are not jointly significant and our model with team-productivity measures has a larger adjusted R-squared.

Looking now at the basic specification in column 1 we see that all of the regressors have the expected signs and that each is statistically significant, with the exception of player weight--results consistent with other studies (see, for example, Jones and Walsh [1987, 1988]; Lavoie, Grenier, and Coulombe [1987, 1992]; Mclean and Veall [1992]). In column 2, we add the first set of team variables: franchise revenue (revenues) and proxies for coaching quality (coach years, coach percentage). The variable revenues has the expected positive effect on player salaries and is significant at the 1% level. Coaching quality is seen to have a significant effect on player salaries; however, coach percentage has the incorrect sign. [18] Comparing changes in the magnitude of the player attribute effects, we see that seven of the 11 coefficients decline slightly between columns 1 and 2. This result is consistent with our hypothesis that coaching quality affects player salaries, that is, higher quality coaching results in h igher player salaries, and part of the return to individual player productive attributes is due to the positive correlation between player productivity and coaching quality. Furthermore, we note that the two coaching variables and total revenue are a jointly significant at the 1% level.

Specification (3) additionally includes a second vector of team variables that measure the average playing quality of teammates. Comparing specifications (2) and (3), we see that inclusion of team averages leads to a slight decline in the coefficients for games, height, and free agent and a sizeable (25%) decline for plus/minus. Examining the coefficients for direct team effect variables we see that, with the exception of t_weight, all are positive. This result suggests that a player who played on a team with prolific scorers, intense two-way play (as proxied by t_penalties and t_plus/minus, respectively), with greater height but not weight (perhaps implying swifter skaters), and a team with a considerable stock of talent (as proxied by star) earned higher salaries, other things equal. In addition, the team variables are jointly significant at the 5% level, and the coaching variables and total revenue remain jointly significant at the 1% level. These results further support our hypothesis that a player's salary is a function of the attributes of the players with whom he plays. [19]

Finally, specification (4) allows the productive attributes of players to have differential effects on player salaries based on the quality of their teammates. Our priors are that, other things equal, an individual player's productivity will be greater when he plays in a team environment with higher-quality teammates, that is, player inputs are complementary. This being the case, we expect that the team performance measures will have a positive effect on team valuation of a player's productive attributes. [20] Yet, as discussed above, diminishing returns to certain attributes may effectively make inputs substitutes along certain productive dimensions, yielding a negative interaction effect.

Looking now at specification (4), we see that when compared with specification (3), individual player coefficients have the same sign with the exception of weight whose sign is negative (as it was in specifications [1] and [2]) and is now significant at the 1% level. The variable points is no longer significant, probably due to its strong relationship to the interaction variable (t_points x points). [21] The coaching variables and total revenue continue to be jointly significant at the 1% level. We also see from the joint significance tests that allowing team values to have differential effects conditional on the characteristics of the players raises the joint significance of the direct team effects from 5% to 1% significance. Furthermore, the vector of interactions also achieves joint significance at the 5% level, indicating that, taken as a whole, individual player attributes are differentially valued based on the productive attributes of teammates.

Regarding the individual coefficients, we see that three of the six interactions are statistically significant at the 10% level or better. The interactions for points and plus/minus are positive, implying that individual productivity is rewarded at a higher rate on teams with better players, that is, these labor inputs appear to be complements. The interactions for penalties and height, however, are negative and significant. This may reflect diminishing returns to these attributes across a given team so that if teammates already have these attributes, then the management views additional units of them as relatively less valuable for the team and accordingly will pay relatively less for these qualities. In the case of own penalty minutes interacted with team penalty minutes, the minus sign likely means that teams that already have a great deal of penalty minutes do not highly value another player who commits many penalties as this would lead to more short-handed situations in games and reduce the probability of winning. [22]

Finally, in order to evaluate the total effect of an individual's productive attributes on his salary, both direct and indirect effects must be evaluated according to (1.1). We see in panel A of Table III that when we evaluate (1.1) at the mean value for the corresponding team attribute (t_[X.sub.i]), and using the estimated coefficient values even when insignificant, all of the variables (with the exception of weight) in column 4 have positive total (direct plus indirect) effects on player salaries, [23] except for player weight, which has a negative total effect. [24] A similar pattern is found in panel B when we evaluate the total effect of average coworker, or team, attributes on individual player salaries as given by expressions (1.2).

V. CONCLUSIONS

This article has investigated the general question of the effects of coworker attributes on the compensation of individuals in an organization. Our specific empirical focus has been on professional hockey because data are available for both players and coaches, thereby allowing for the construction of a vector of management productivity and coworker productivity variables. Our central findings are that team attributes have both direct effects on individual player compensation and indirect effects through altering the rates at which individual player productive characteristics are valued.

Furthermore, when team variables are incorporated into the regressions there occurs a decline in certain individual productivity effects. This result indicates that estimates of the effects of individual attributes on compensation are upwardly biased when team effects are not taken into account in standard salary regressions. In addition, it appears that, on average, players seem to be complements in the production process in professional hockey, though a significant pattern of complementarity is not uniformly found across all productive attributes. It might be the case, though, that certain combinations of positions are more strongly complements and that larger, more significantly positive interactions might follow if certain positions are paired. Such an analysis may be a fruitful subject for further research.

Given the above patterns, the question arises concerning the extent to which these results generalize to industrial settings and how these inferences might differ for different size firms. An industrial setting that is perhaps closest to NHL teams might be a small firm. If such a firm has, say, two high-quality engineers, management might feel that there will be relatively little gained by investing in another first-rate engineer, thereby leading to the possibility of negative interactions for some of the variables. [25] Yet, in a large firm, where larger teams are assembled (Idson [1995]) at different points in the production process, greater returns to complementarity in talent may be present as redundancy in skills become less important (see Kremer [1993]), leading to a prediction of uniformly positive interactions. As such, there may be distinct differences in the effects of coworker quality on compensation in different team production settings and in different size firms.

(*.) We would like to thank Larry Kahn, Joe Tracy, Finis Welch, and two anonymous reviewers for useful comments. All errors, though, remain the responsibility of the authors.

Idson: Associate Professor, Columbia University, New York, NY 10027.

Kahane: Associate Professor, California State University, Hayward, CA 94542.

(1.) This argument is similar to the effects of complementarity of physical capital with human capital (see Griliches [1969]).

(2.) See Kahn [1991], for a recent survey of the literature on pay differentials in professional sports.

(3.) See McLaughlin [1994], for another possible linkage between individual pay and team performance in the context of rent sharing between workers and firms. Compensation is shown to be equal to the worker's opportunity wage and a share of the rents from the match, where these rents will include the complementarities inherent in team production which were stressed by Alehian and Demsetz as the basis for managerial compensation. Also of interest is the work by Chapman and Southwick [1991].

(4.) Note that a positive value for the coefficient on an attribute in the t_[X.sub.i] vector does not necessarily imply complementarity between inputs along the dimension being evaluated. It would, though, indicate that players benefit from being on teams with certain attributes regardless of the level of this particular attribute that they possess.

(5.) See Staudohar [1996, 139-41].

(6.) There is no consolidated source for contract details that is made publicly available.

(7.) See Staudohar [1996, 154-58] for details.

(8.) This follows the methodology in Jones and Walsh [1988]. Rookie players (less than two years experience in the NHL) are excluded from our analysis, since there is no historical (career) data for them and as such we are unable to construct (lagged) career performance variables for them. One possibility we considered was to examine the rookies' data from their minor- or junior-league performance. This is not feasible, however, for several reasons: (1) there are several different and separate minor leagues of varying quality, and as such data would not be comparable for players coming from different leagues; (2) there are many different junior leagues, and the same problem arises; (3) some players come from foreign teams of differing quality, and, again, comparisons of players from different foreign leagues would not be valid. Part-time players are also excluded, since they arc not eligible for Alt-Star appearances and trophies (players that have not played at least 26 games in at least one NI-IL season are also excluded, since they do not qualify as full-time players). Goalies are also excluded from the analysis. This is because the statistics describing a goalie's performance are not comparable to that used for nongoalics.

(9.) In addition to improving a team's winning ability, skilled offensive players may also cause an increase in attendance by fans who wish to view skilled play, regardless of the outcome of the game.

(10.) Examples of defensemen with great scoring ability would include Paul Coffey (Detroit), Ray Bourque (Boston), and Brian Leetch (N.Y. Rangers). Arguably the greatest offensive defenseman ever was Bobby Orr (Boston), who twice won the Art Ross trophy for leading scorer among all players in the league (1969-70 and 1974-75 seasons). In principal, all (nongoalie) players are responsible for both offensive and defensive play. Forwards, however, have the primary task of scoring goals, whereas the primary role of defensemen is to prevent opponents from scoring.

(11.) Note that this variable is subject to other interpretations. For example, if wages are "sticky," then the variable free agent may capture salary adjustments for newly negotiated contracts.

(12.) A player "line" consists of a set of three players (a center and two wings) who play together. In addition, defensemen typically play in pairs that are determined by the coach.

(13.) Attempts were made to include interactions between the coaching variables and player performance measures. These interaction variables were dropped, however, because of severe multicollinearity between the player performance variables and the coach interaction variables. As an example, the correlation coefficient between player points per game (points) and a variable created by multiplying coach winning percentage by player points per game (coach percentage X points) is .965. Including both in a regression causes points to be insignificant and to have a negative coefficient in some regressions.

(14.) As reported in note a to Table I, mean salary for the 1990--91 and 1991--92 seasons (combined) was 277,369, with a standard deviation of 242,828. The dispersion of salaries within teams was somewhat lower--when we calculate the dispersion of salaries relative to team mean salary and then take the average of these values, the average standard deviation of salaries within teams is 233,808 for the 1990-91 and 1991-92 seasons combined. These values are consistent with spillover effects within teams, though other explanations cannot be ruled out, such as productivity considerations that may follow from a tighter dispersion of salaries across team members. Given we have two years of data, many players appear in both years, and we essentially have a "short" panel data set. As such, in order to estimate standard errors of coefficients correctly, we use White's [1980] robust variance-covariance matrix method with clustering on the individual. An alternative method would be to estimate a fixed-effects model. Att empts to do so by using a first-difference method, however, yielded poor results. This was primarily due to the fact that in taking first differences, our sample size decreased dramatically as the change in salaries for many players was zero.

(15.) This statement follows from recognition that if [[beta].sub.2] [greater than] and cov(t_[X.sub.i], [X.sub.i]) [greater than] 0, then omission of t_[X.sub.i] will upwardly bias the estimate of [[beta].sub.1].

(16.) Note that we cannot run own-productivity as a function of teammates productivity and simply interpret a positive coefficient as complementarity in labor inputs. This is because any observed relationship might reflect sorting, where the most productive players gravitate to other more productive players.

(17.) Ideally we would like to measure the individual's productivity as an individual specific output plus the contributions of the individual to productivity of other players on the team, minus the contribution of other players on the team to the output of the individual. Namely, some players may produce high yields in terms of helping other players in ways that traditional productivity measures do not capture, leading to an underestimate of their productivity to the statistician, but this may be known to the team and accordingly compensated (so that the player may look as though he is overpaid). Similarly, a player may have inflated measured productivity statistics, because much of his "output" is due to the contributions of other players and little is contributed to them (e.g., the player get lots of shots due to being setup by other players in ways that are too general to be counted as assists), so to the statistician he might look like he is underpaid. Data limitations, though, preclude identifying each of these separate effects for our NHL files.

(18.) This unexpected result is likely due to multicollinearity between coach percentage and the remaining independent variables. An auxiliary regression that regresses coach percentage on the remaining independent variables in specification (1) produced an F-statistic that was significant at the 1% level ([R.sup.2] = .21). Results are available on request.

(19.) We additionally evaluated team-level effects on own productivity and salary by estimating a "multilevel" model (sec Bryk and Raudenbush [1992]), which allows for a decomposition of the total variance of the dependent variable into the components that occur at the individual player level and at the team level. Although program parameters (we used the program MU, 1992) limit the number of team-level effects that can be estimated, we did estimate a multilevel model version of the specification in column 1 of Table II that allows for a team-level variance for points per game (points) as an explanatory variable. A test of significance for the inclusion of team-level variance in points" passed at better than the 1% significance level, supporting our main hypothesis that team effects are important in explaining individual performance and salary. (Results arc available on request.) An alternative explanation to our finding that team measures are important in explaining individual player salaries is that this r esult may reflect nonrandom sorting of players. An analysis of the correlation matrix for the explanatory variables and additional regressions that included player performance measures relative to team performance measures, however, do not support this alternative hypothesis. (Results are available on request.)

(20.) As an example of how individual performance is linked to the performance of teammates, we can consider Bernie Nicholls during his play with the Los Angeles Kings before and after the trade of Wayne Gretzky. In 1988 Wayne Gretzky, arguably the most talented hockey player in history, was traded by Edmonton to the Los Angeles Kings. He played together with Bernie Nicholls during the 1988-89 season in Los Angeles, and Nicholls had a career-high 150 points, 50 points more than he had scored in any of his previous six seasons in the NHL, with nearly two points per game. Nicholls was traded midway through the 1989-90 season to the New York Rangers, and his scoring dropped significantly. During the 1990-91 season with the Rangers, Nicholls scored 73 points, and his points per game dropped to approximately one. One can think of examples from other sports as well. For example, we can consider the impact that Magic Johnson had on the scoring ability of James Worthy when they played together in the NBA for the Los Angeles Lakers.

(21.) As evidence, the standard error of the estimate jumps over fivefold, and points and t_points x points are jointly significant at 1%.

(22.) Other potentially important interaction effects were tested. For example, management may be particularly willing to pay a premium for players who can fill particular gaps in their lineup. Thus, a team with, say, a pool of talented scorers might desire to hire a player who can act as an "enforcer" or "intimidator" to complete the lineup. To test for this effect, an interaction between team points per game (t_points) and penalty minutes per game (penalties) was tested. This effect, however, was not statistically significant and was consequently dropped from the model.

(23.) That is, even in the cases where the direct and indirect effects have different signs, the sum of these effects at mean team values is positive.

(24.) This may not be surprising, since after controlling for height and various performance measure, a heavier player may actually be a poorer player along certain productivity dimensions that are not directly measured in our regressions.

(25.) Similar considerations might apply within divisions of larger firms.

REFERENCES

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Alchian, Armen A., and Harold Demsetz. "Production, Information Costs, and Economic Organization." American Economic Review 67, December 1972, 777-95.

"Bottom Line on Salaries." Hockey News, 8 February 1991, 46-47.

Bryk, Anthony S., and Stephen Raudenbush. Hierarchical Linear Models: Applications and Data Analysis Methods. Newbury Park, Calif.: Sage Publications, 1992.

Chapman, Kenneth S., and Lawrence Southwick. "Testing the Matching Hypothesis: The Case of Major League Baseball." American Economic Review 81, December 1991, 1352-60.

Clement, Robert C., and Robert E. McCormick. "Coaching Team Production." Economic Inquiry 27, April 1989, 287-304

FW's Valuation Scoreboard. Financial World, July 1992, 50-51.

_____. Financial World, May 1993, 26-30.

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_____. "Salary Determination in the National Hockey League: The Effects of Skills, Franchise Characteristics, and Discrimination." Industrial and Labor Relations Review 41, July 1988, 592-604.

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_____. "Managerial Quality, Team Success, and Individual Player Performance in Major League Baseball." Industrial and Labor Relations Review 46, April 1993, 531-47.

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 Variable Definitions [a] (n = 930)
Variable Definition
A. Individual player variables:
 games Total number of games played over
 the player's NHL career
 points Career points per game (goals and
 assists)
 penalties Career penalty minutes per game
 plus / minus Game-weighted average plus/minus
 statistic
 height Players' height (in inches)
 weight Players' weight (in pounds)
 star The number of times the player was
 selected as an all star plus the
 number of NHL trophies he has won
 draft = 1 if the player was selected in the
 first or second draft round; 0 otherwise
 forward = 1 if the player is a forward;
 0 otherwise
 free agent = 1 if the player was a free-agent prior
 to the current salary year; 0 otherwise
B. Team/coach variables:
 revenues Total team revenues
 coach years Number of seasons the team's
 coach has coached in the NHL
 coach percentage Coach's career percentage of
 points won [b]
 Mean
Variable (Standard Deviation)
A. Individual player variables:
 games 336.524
 (256.834)
 points 0.522
 (.325)
 penalties 1.381
 (1.136)
 plus / minus 0.157
 (10.944)
 height 72.512
 (1.840)
 weight 190.011
 (13.011)
 star 0.366
 (2.193)
 draft 0.428
 (0.495)
 forward 0.633
 (0.482)
 free agent 0.116
 (0.320)
B. Team/coach variables:
 revenues 25.336
 (6.992)
 coach years 4.557
 (4.396)
 coach percentage 0.529
 (0.080)
Variable Expected Effect on Salary
A. Individual player variables:
 games +
 points +
 penalties +
 plus / minus +
 height +
 weight +
 star +
 draft +
 forward -
 free agent +
B. Team/coach variables:
 revenues +
 coach years +
 coach percentage +

(a.)The salary data are for the 1990--91 and 1991--92 seasons and were obtained from the Hockey News [8 February, 1991, 46--47; 15 November, 1991, 44]; mean (standard deviation) of player salaries are 277,369 (242,828). Performance data for players and coaches were taken from the Sporting News Complete Hockey Book. Data on team revenues was obtained from Financial World [1992, 1993]. Information on free-agents was obtained from the NHL.

(b.)In the NHL a win is worth two points, a tie one point, and a loss zero points. The variable coach percentage is calculated for each coach as the percent of points won out of total points possible over the coach's career.

 Salary Regressions
 Dependent Variable = log (salary), n = 930
Variable (1) (2) (3)
A. Individual player variables:
 games X [10.sup.2] 0.170 [*] 0.174 [*] 0.171 [*]
 (0.017) (0.017) (0.017)
 [games.sup.2] X [10.sup.5] -0.129 [*] -0.134 [*] -0.132 [*]
 (0.021) (0.020) (0.020)
 points 0.850 [*] 0.849 [*] 0.858 [*]
 (0.075) (0.072) (0.072)
 penalties X 10 0.521 [*] 0.500 [*] 0.500 [*]
 (0.108) (0.110) (0.111)
 plus / minus X [10.sup.2] 0.351 [*] 0.415 [*] 0.312 [*]
 (0.109) (0.116) (0.123)
 height x 10 0.215 [*] 0.212 [*] 0.205 [*]
 (0.079) (0.077) (0.078)
 weight x [10.sup.2] -0.011 -0.001 0.006
 (0.123) (0.124) (0.124)
 star x 10 0.416 [*] 0.398 [*] 0.403 [*]
 (0.126) (0.127) (0.132)
 draft X 10 0.517 [**] 0.540 [**] 0.540 [**]
 (0.250) (0.249) (0.248)
 forward -0.157 [*] -0.161 [*] -0.161 [*]
 (0.026) (0.026) (0.027)
 free agent 0.208 [*] 0.198 [*] 0.197 [*]
 (0.041) (0.040) (0.040)
Variable (4)
A. Individual player variables:
 games X [10.sup.2] 0.173 [*]
 (0.017)
 [games.sup.2] X [10.sup.5] -0.132 [*]
 (0.020)
 points 0.305
 (0.418)
 penalties X 10 1.794 [**]
 (0.797)
 plus / minus X [10.sup.2] 0.289 [**]
 (0.122)
 height x 10 24.930 [**]
 (12.827)
 weight x [10.sup.2] -17.376 [*]
 (6.073)
 star x 10 0.471 [*]
 (0.161)
 draft X 10 0.586 [**]
 (0.249)
 forward -0.157 [*]
 (0.027)
 free agent 0.200 [*]
 (0.039)
B. Team coach variables:
 revenues X [10.sup.2] ... 0.793 [*] 0.518 [*]
 (0.174) (0.189)
 coach years X [10.sup.2] ... 0.636 [**] 0.847 [*]
 (0.265) (0.315)
 coach percentage ... -0.302 [**] -0.583 [*]
 (0.144) (0.192)
C. Team averages:
 t_points ... ... 0.172
 (0.243)
 t_penalties ... ... 0.110 [***]
 (0.078)
 t_plus / minus X [10.sup.2] ... ... 0.507 [**]
 (0.266)
 t_height ... ... 0.005
 (0.043)
 t_weight ... ... -0.013 [*]
 (0.005)
 t_star ... ... 0.003
 (0.022)
D. Interactions:
 t_points X points ... ... ...
 t_penalties X penalties ... ... ...
 t_plus / minus ... ... ...
X plus / minus X [10.sup.3]
 t_height X height ... ... ...
 t_weight ... ... ...
X weight x [10.sup.3]
 t_star X star ... ... ...
Constant 10.000 [*] 9.925 [*] 12.027 [*]
 (0.478) (0.471) (2.857)
Adjusted [R.sup.2] 0.6100 0.6242 0.6280
F-statl 4.57 [*] 6.02 [*]
F-stat2 2.64 [*]
F-stat3
B. Team coach variables:
 revenues X [10.sup.2] 0.513 [*]
 (0.187)
 coach years X [10.sup.2] 0.759 [**]
 (0.327)
 coach percentage -0.583 [*]
 (0.189)
C. Team averages:
 t_points -0.348
 (0.353)
 t_penalties 0.258 [**]
 (0.127)
 t_plus / minus X [10.sup.2] 0.519 [**]
 (0.264)
 t_height 2.478 [**]
 (1.287)
 t_weight -0.186 [*]
 (0.061)
 t_star 0.011
 (0.023)
D. Interactions:
 t_points X points 1.051
 (0.779)
 t_penalties X penalties -0.096 [***]
 (0.059)
 t_plus / minus 0.027
X plus / minus X [10.sup.3] (0.121)
 t_height X height -0.034 [**]
 (0.018)
 t_weight 0.913 [*]
X weight x [10.sup.3] (0.318)
 t_star X star -0.013
 (0.019)
Constant -134.451 [+]
 (89.203)
Adjusted [R.sup.2] 0.6324
F-statl 5.59 [*]
F-stat2 3.34 [*]
F-stat3 2.26 [**]

Notes: Parameter estimates are reported with robust standard errors in parentheses. Significance at the 1%, 5%, 10% and 15% levels are denoted by superscripts (*.), (**.), (***.), and (+.) F-stat1 tests for the joint significance of the two coach variables and revenues; Fstat2 tests for the joint significance of the team averages for points through star; F-stat3 tests for the joint significance of the interactions of the team averages and the player values.

 Marginal Effects of Player and Team Attributes
 Effects of Individual Player Attributes on Salaries
Individual Performance Measures [partial]ln([Salary.sub.i])/
 [partial][X.sub.i] =
 [[beta].sub.1] +
 [[beta].sub.3]t_[X.sub.1]
points [*] 0.854
penalties [*] 0.468
plus / minus [*] 0.290
height [**] 0.276
weight [*] -0.028
star [*] 0.423
 B. Effects of Team Attributes on Salaries
Team Performance Measures [partial]([salary.sub.i])
 /[partial]t_[X.sub.i] =
 [[beta].sub.2] +
 [[beta].sub.3][X.sub.i]
t_points [**] -0.201
t_penalties [**] 0.125
t_plus / minus [**] 0.523
t_height [**] 0.013
t_weight [**] -0.012
t_star 0.006

Notes: Calculations are based on the parameter estimates reported in column 4 of Table II, evaluated at the mean values of the regressors in Table I. The marginal values reported in panel A are in the same scale as the coefficients values in panel A of Table II; the marginal values reported in panel B are in the same scale as in panel C of Table II.

Superscripts (*.)and (**.)indicate rejection (at 1% and 5%, respectively) of a test of the linear restriction that the additive and associated interaction term for each variable sums to zero.

ABBREVIATION

NHL: National Hockey League

 APPENDIX
 Salary Regressions
 Dependent Variable = log (salary), n = 930
Variable Coefficient
games X [10.sup.2] 0.176 [*]
 (0.017)
[games.sup.2] x [10.sup.5] -0.137 [*]
 (0.021)
points 0.854 [*]
 (0.073)
penalties X 10 0.476 [*]
 (0.113)
plus/minus X [10.sup.2] 0.311 [*]
 (0.124)
height X 10 0.204 [*]
 (0.077)
weight X [10.sup.2] 0.037
 (0.120)
star X 10 0.401 [*]
 (0.131)
draft X 10 0.552 [**]
 (0.247)
forward -0.160 [*]
 (0.026)
free agent 0.209 [*]
 (0.041)
revenues X [10.sup.2] 0.873 [*]
 (0.229)
team dummies [dum#s]
Constant 9.761 [*]
 (0.471)
Adjusted [R.sup.2] 0.6241

Notes: Parameter estimates are reported with robust standard errors in parentheses. Significance tests for the 1%, 5%, 10%, and 15% levels are denoted by superscripts (*.), (**.), (***.), and (+.). The joint significance test for team dummies produced an F-statistic of 1.31 (degrees of freedom of 20 and 541, where 541 is the number of clusters), which fails to achieve the 15% level of significance. (Parameter estimates and standard errors for the team dummies, "dum#s," are available on request.)