R.E. McCormick, R.D. Tollison J. of Economic Behavior Org. 44 2001 201–219 209
and white players are sometimes viewed as better by some fans. Thus, the source of the wage differential identified by Kahn and Sherer remains at issue. Indeed, according to the
model of racial discrimination, black players ought to make more money in certain NBA markets than comparable white players.
5. Playing time results
Our basic premise is that what may look like racialgenderage discrimination is some- times more properly characterized as simple price discrimination. To continue this investi-
gation, we now examine whether there is any systematic relation between minutes played and race.
Following other models of playing time, we regress average minutes played per game for each player on each team over the sample period 1980–1981 through 1987–1988 on a
vector of player skills Clement and McCormick, 1989. The dependent variable, average minutes played per game, is total minutes played in a season for each player divided by the
number of games that this player participated in during that season.
19
We include a dummy for race in this equation to assess whether this factor is relevant. These results, based on
2481 observations, are reported in Table 5. Basketball fans will not be surprised by the bulk of these results. The ceteris paribus
results are strong and straightforward. Older players play more, but at a decreasing rate. Playing time is maximized at 28 years old. Taller players play more. Playing time is max-
imized at 7
′
5
′′
which includes every person in the database with one exception.
20
Playing time increases with weight over all ranges of our data.
21
Better performance as measured in a variety of ways begets more playing time. All-stars play no more than regular players,
other things the same. All-defensive team players and those who make the all-rookie team play more than their skill levels suggest. Holding fouls per minute constant, players who foul
out more are given more minutes. There are numerous explanations of such a result; perhaps it reflects a strategic choice by coaches to use certain players to foul during the course of a
game. And perhaps oddly, good free throw shooters play less, other things the same.
The important result in Table 5 is that black players play more minutes than white players, all else equal. Adjusting for skills, a black player plays 1.47 more minutes per game than
an apparently comparable white player. Over an 82 game regular season, this amounts to about 120 more playing minutes. The result is statistically significant at the 1 percent level.
While it might be argued that our playing time result could be due to a correlation between race and some important omitted variable, we speculate that this is not the case. Given the
width and breadth of player data and the seemingly unquenchable thirst for data by fans, if
19
We excluded any player who did not play at least 100 min in any given season. This mutes the problem created by the fact that for any given team for any given season the error terms must sum to zero. This means that for
almost all teams for almost all years, there is at least one missing player. Pooling the data over many seasons and many teams minimizes any problems of this type.
20
Note that the quadratic term on height is statistically insignificant, implying a linear relation between playing time and height.
21
Data on playing position, e.g. guard or forward, are not available. Indeed these distinctions are blurry in the NBA. Moreover, the height and weight data proxy for playing position.
210 R.E. McCormick, R.D. Tollison J. of Economic Behavior Org. 44 2001 201–219
Table 5 OLS regression estimates of minutes played per game per player
sample size: 2481; F ratio: 219.3; R
2
: 0.6625; dependent mean: 22.23; root M.S.E.: 5.36 Variable
Parameter estimate
t Ratio Prob |t|
Intercept −
128.471 −
2.828 0.0047
0,1 Dummy for race, Black = 1 1.467
5.403 0.0001
Player’ age at the start of the season 3.247
7.197 0.0001
Player’s age squared −
0.058 −
7.060 0.0001
Player’s height 23.598
1.688 0.0915
Player’s height squared −
1.596 −
1.488 0.1368
Player’s weight 0.028
3.002 0.0027
Player’s weight squared 0.000
− 2.930
0.0034 0,1 Dummy if player selected to all-star team during season
0.079 0.183
0.8550 0,1 Dummy if player selected to all-defensive team during season
1.216 1.940
0.0525 0,1 Dummy if player selected to all-rookie team during season
4.414 5.021
0.0001 Free throws attempts per minute played
2.387 0.729
0.4660 Number of times player fouled out in season per minute played
1360.699 18.760
0.0001 Shooting percentage 2-point shots
25.127 11.219
0.0001 Shooting percentage 3-point shots
1.675 2.428
0.0152 Free throw shooting percentage
− 0.024
− 2.130
0.0333 Total points scored per minute played
16.345 12.232
0.0001 Fouls committed per minute played
− 140.159
− 37.765
0.0001 Offensive rebounds per minute played
− 8.476
− 1.326
0.1850 Defensive rebounds per minute played
38.016 9.928
0.0001 Shots blocked per minute played
27.939 4.047
0.0001 Assists per minute played
31.787 11.642
0.0001 Steals per minute played
9.815 1.454
0.1461
there was such a variable, it would be reported in the newspapers and data collection services. The absence of such a reporting boosts our confidence that there is not some important
missing variable. However, to investigate this question more carefully, we regressed each individual player skill on his age, his height, his weight, each of these squared, and a 0,1
race dummy. The race dummy is significant in most cases, but not in the same way in each case. Table 6 reports the player skills and the t ratio on the race dummy variable.
The results in Table 6 suggest that there are some basketball skills correlated with race. Therefore, our earlier finding that race is related to playing time is called into question. To
investigate this more, we reestimated the playing time equation omitting the age, height, and weight variables and included the residuals from the player skills equation where the
race dummy was excluded. We inserted each player skill separately and ran one regression per player skill. In all cases, the race dummy on playing time remains significant. In several
cases, the value of the coefficient declines, but never significantly. In sum, there appears to be some intangible player skill variable that we are not measuring which is correlated with
race, but not in a way that affects the basic finding that black players play more than white players, other things the same. Nevertheless, there seems to be room for additional work
here to attempt to quantify the absent player characteristics. In the end these may turn out to be subjective and unmeasurable, e.g. attitude, but without additional study, we are not
prepared to speculate.
R.E. McCormick, R.D. Tollison J. of Economic Behavior Org. 44 2001 201–219 211
Table 6 Player skills and race
a
Skill Regression
R
2
Number of observations
t Ratio on race dummy
Skills with a positive race coefficient Blocked shots per minute
0.498 2480
9.791 Offensive rebounds per minute
0.5395 2480
4.041 Defensive rebounds per minute
0.6302 2480
4.041 Foul Outs per minute
0.1169 2480
3.868 Points per minute
0.0592 2480
9.231 2-Point shooting percentage
0.0359 2480
3.869 Skills with a negative race coefficient
Fouls per minute 0.2108
2480 −
.3942 Assists per minute
0.5142 2480
− 4.871
3-Point shooting percentage 0.1188
2480 −
4.619 Skills with an insignificant race coefficient
Steals per minutes 0.2157
2480 0.686
a
In each case, we regressed the player skill on age, height, weight, and the square of each. We excluded all players with 100 min played per season. The race dummy takes the value 1 if the player is black.
We explored alternative specifications to determine whether the playing time relation was robust. For instance, we included the team whiteblack player ratio as a control variable.
We included a variable denoting whether a player was traded during the season. We deleted points scored per minute played from the regression. We reduced the sample to players who
played at least 500 min during a season. We also estimated the playing time equation using only those individuals who played in at least 50 games. In all cases, the coefficient on race
remains significant.
We divided teams into three classifications based on the team’s overall racial composition. The classifications are the middle half and the upper and lower quartiles of team racial
composition.
22
Then we reestimated the playing-time equation allowing for different race dummy variables in these three groups. The coefficients on the race dummy are reported in
Table 7, and they are all positive. The class is significant, and in pairwise tests of equality the coefficient in the blackest quartile is different from the coefficients in the other two
groups at the 10 percent level of significance. There is no evidence that the coefficient in the mixed half is different from the coefficient in the whitest quartile.
In addition, we allowed for different values of the race dummy variable in the minutes played equation depending on the racial composition of the underlying community see
Table 3. The results on the race dummies for this specification are reported in Table 8. We observe a positive and significant relation between playing time and race in the three
SMSA race classifications. The overall class is significant, but the individual coefficients are statistically the same. Overall, the evidence suggests that black players play more than
comparable white players in all cities and on all teams.
22
In our sample the top quartile of teams with the largest portion of white players has a player ratio equal to or 0.5. The blackest quartile of teams have a white to black player ratio of 0.2 or less. The remaining teams are in
the middle half.
212 R.E. McCormick, R.D. Tollison J. of Economic Behavior Org. 44 2001 201–219
Table 7 Coefficients on the race dummy in the playing time equation by different team racial classifications
a
Sample size: 2481; F ratio: 201.38; R
2
: 0.663 Team racial composition
Parameter estimate t Ratio
Prob |t| Blackest quartile
1.44 4.32
0.0001 Mixed half
1.64 5.54
0.0001 Whitest quartile
0.94 2.48
0.0134
a
Team racial composition is the ratio of total white players on the team to total number of black players on the team for the year in question. The blackest quartile of teams has a white to black ratio of players of 0.2 or less.
The whitest quartile has a white to black player ratio of 0.5 or greater. The mixed middle captures the remaining half of the teams. The F statistic on the class is 5.14 which is significant at the 1 percent level.
Table 8 Coefficients on the race dummy in the playing time equation across different SMSAs
a
Sample size: 2481; F ratio: 200.89; R
2
: 0.663 SMSA
Parameter estimate t Ratio
Prob |t| Blackest quartile
1.54 4.27
0.0001 Mixed half
1.49 5.14
0.0001 White quartile
1.31 1.31
0.0005
a
See Table 3 for a description of the SMSA variable. The F ratio on the SMSA class variable is 9.85 which is significant at the 1 percent level. None of the pairwise tests of equality is significant.
In the attendance results Table 2, we find no significant relation between the race of the coach and attendance. We interpret this as additional evidence that fans are color blind.
To explore the race issue further in the context of coaches, we investigated whether there was an impact on playing time which is related to the race of the player and the coach. We
created a match variable which is one if the race of the coach and the player match and zero otherwise. In our player sample, 13.2 percent of the players had coaches that were black,
and the rest had coaches that were white. Of the players, 73.8 were black. Table 9 reports the breakdown by race of player and coach.
The χ
2
statistic of independence is 3.723 which is not significant at the 10 percent level. A black player is as likely to have a black coach as is a white player, and vice-versa. In
total, we have 799 occasions where the race of the coach and the race of the player match,
Table 9 Frequency distribution of player race by race of coach
a
Black players White players
Total Coach is black
235 101
336 Coach is white
1635 564
2199 Coach is mixed
11 2
13 Total
1881 667
2548
a
There is one case where there was a coaching change during the season, and one of the coaches was white and one was black.
R.E. McCormick, R.D. Tollison J. of Economic Behavior Org. 44 2001 201–219 213
discarding the mixed situation. We estimated the playing time equation to determine if there was a propensity for white coaches to favor white players or black coaches to favor black
players by including a dummy for the race match variable. In this specification, a player actually plays less, if he is the same race as his coach, but this result is not significant
the t ratio is −1.193 and the coefficient is small, less than half-a-minute per game. This result could be driven by the large portion of black players who play for white coaches.
Consequently we estimated this effect separately for black and white coaches. The basic result persists that black players play more, but there is no significant difference between
coaches in this regard. Both black and white coaches play black players more, other things the same.
6. A model of wage determination with racial discrimination by customers
