Two African-American coaches, Doc Rivers of the Boston Celtics and Maurice Cheeks of the Philadelphia 76ers, declined to comment on the paper’s claims. Rod Thorn, the president of the New Jersey Nets and formerly the N.B.A.’s executive vice president for basketball operations, said: “I don’t believe it. I think officials get the vast majority of calls right. They don’t get them all right. The vast majority of our players are black.”

Image Commissioner David Stern says a league study demonstrates there is no bias. Credit... Lawrence Jackson/Associated Press

Mr. Wolfers and Mr. Price spend 41 pages accounting for such population disparities and more than a dozen other complicating factors.

For the 1991-92 through 2003-4 seasons, the authors analyzed every player’s box-score performance — minutes played, rebounds, shots made and missed, fouls and the like — in the context of the racial composition of the three-person crew refereeing that game. (The N.B.A. did not release its record of calls by specific officials to either Mr. Wolfers, Mr. Price or The Times, claiming it is kept for referee training purposes only.)

Mr. Wolfers said that he and Mr. Price classified each N.B.A. player and referee as either black or not black by assessing photographs and speaking with an anonymous former referee, and then using that information to predict how an official would view the player. About a dozen players could reasonably be placed in either category, but Mr. Wolfers said the classification of those players did not materially change the study’s findings.

During the 13-season period studied, black players played 83 percent of the minutes on the floor. With 68 percent of officials being white, three-person crews were either entirely white (30 percent of the time), had two white officials (47 percent), had two black officials (20 percent) or were entirely black (3 percent).

Mr. Stern said that the race of referees had never been considered when assembling crews for games.

With their database of almost 600,000 foul calls, Mr. Wolfers and Mr. Price used a common statistical technique called multivariable regression analysis, which can identify correlations between different variables. The economists accounted for a wide range of factors: that centers, who tend to draw more fouls, were disproportionately white; that veteran players and All-Stars tended to draw foul calls at different rates than rookies and non-stars; whether the players were at home or on the road, as officials can be influenced by crowd noise; particular coaches on the sidelines; the players’ assertiveness on the court, as defined by their established rates of assists, steals, turnovers and other statistics; and more subtle factors like how some substitute players enter games specifically to commit fouls.