Sequential judgment effects in the workplace: evidence from the national basketball association.
Many areas of economic activity involve an evaluator who must make subjective judgments. Often these judgments are made sequentially toward different individuals, with the intent that each judgment be made independently. Business managers complete annual evaluations regarding employee performance, raises, and promotions. Judges score gymnasts as they compete in a sequence and boxers as they fight round-by-round. Referees call fouls and violations as they occur play-by-play throughout a match. Might these economic agents be affected by the sequential nature of their decisions? For instance, might a tough raise or promotion decision between two subordinates this year affect a manager's evaluation of either or both the next year?
I study this aspect of decision-making using data on the offensive foul and violation decisions of National Basketball Association (NBA) referees. These calls always result in a turnover and loss of possession. Thus, when a tough call is made on one team, a referee crew may consciously or subconsciously increase scrutiny on the opposing team, potentially returning possession to the originally aggrieved team. This situation contains the anecdotally popular sequential judgment known as the "make-up call," but I do not claim to explicitly study make-up calls as the quality of each call is not known. However, if make-up calls exist, they would likely reveal themselves in this situation.
Identification is achieved by exploiting the fact that some offensive fouls and violations regularly involve substantial referee judgment while others do not. For example, charging and traveling calls can be relatively subjective, allowing room for referee discretion, while very little judgment is required when a player steps out of bounds or passes the ball out of bounds. There is scope for referee decisions to be sequentially biased following more subjective calls that should not exist following more objective calls. Using three assumptions, I examine referee decisions following both types of calls. Relatively objective calls serve in a control capacity to test for equilibrium changes in player behavior. If player behavior hypotheses are rejected, sequential bias among referee decisions will be identified.
The behavioral concept of changing one's scrutiny of others following a tough decision is not unique to sports. It can apply to professor grading decisions, supervisor promotion decisions, voter award decisions (e.g., Oscars, Grammys, Most Valuable Player awards), police investigative decisions, and so on. The sports world happens to be an excellent setting to empirically investigate these situations because of the wide availability of clean and precise data on actual decisions and because such allegations toward referees are quite common in a number of sports. These may involve the judgment calls of pass interference or roughing the passer in football, strikes and balls in baseball, cross-checking or two-man advantages in hockey, red/yellow cards or offside in soccer, point deductions in boxing, and offensive fouls and violations in basketball.
Prior research provides some supporting evidence of sequential bias. Plessner and Betsch (2001) conduct an experimental study of videotaped scenes of potential penalties from soccer matches. They find a negative correlation between participants' successive penalty decisions regarding the same team and a positive correlation between successive penalty decisions regarding opposing teams. Damisch, Mussweiler, and Plessner (2006) study actual gymnastics scores from the 2004 Olympics and find that evaluations of consecutively performing gymnasts are correlated in a positive way (an assimilation effect). However, Morgan and Rotthoff (2012) examine gymnastics scores at the 2009 World Artistic Gymnastics Championships and find no evidence that a previous gymnast's score affects the score of the next gymnast. In nonsports contexts, assimilation effects have been found by Page and Page (2010) in rankings of sequential performances in the Idol singing competition and by Attali (2011) in sequential grading of essays in standardized tests.
In the remainder of the paper, I analyze sequential situations in the NBA using play-by-play data to investigate the changes in probability of particular referee judgment calls following a recent judgment call on the opposing team. Findings are consistent with a hypothesis in which referees increase scrutiny on one team following a potentially difficult decision against the opposing team. I also investigate the changes in probability of particular referee judgment calls following a recent judgment call on the same team. Results initially appear to support a hypothesis of decreased referee scrutiny, but a more thorough analysis suggests that they are likely explained by changes in player awareness of the recently committed violation.
II. NBA REFEREEING PHILOSOPHIES
The NBA may well have a latent history with changes in referee behavior following tough decisions. Shortly after referee Tim Donaghy's gambling scandal in the summer of 2007, (1) the league hired former federal prosecutor Lawrence Pedowitz to conduct an investigation of its refereeing structure. Among other things, Donaghy alleged that an incorrect call had been made in a Minnesota/New Orleans game and he was later told by another referee that they "could have made something up at the other end ... calling a traveling violation on Kevin Garnett" (Donaghy 2009).
After interviewing every referee as well as other team and league personnel, Pedowitz found that there were two historical refereeing philosophies, pre- and post-2003. Under the pre-2003 philosophy, "... if a referee recognized that he or his crew had made an incorrect call, a referee might whistle a 'make-up call' soon thereafter" (Pedowitz 2008). The league changed its officiating philosophy in 2003, establishing 16 performance standards where referees were to "strive for the unattainable goal of perfection," "get the calls right," and make "accurate calls, regardless of the circumstances of the game" (Pedowitz 2008). The dataset I analyze is from 2006 to 2011, firmly within the post-2003 refereeing time period. During this time, the league imposed strict performance standards discouraging, among other things, make-up calls and sequential bias. Thus, not only is this a nonexperimental study of sequential bias by elite, highly trained employees, but the employees are also operating under instructions from their employer not to consider prior decisions when making current ones.
Play-by-play data were obtained from Basketballvalue.com for five NBA regular and post seasons from 2006-07 to 2010-11. (2) The data contained complete information for 6,538 games. (3) For each play, I observed the date, the two teams competing, the current score, the quarter and time remaining, which team has possession of the ball, and the outcome of the play. Figure 1 shows a sample of the raw data for the game between the Houston Rockets and Portland Trailblazers on October 27,2009. The play-by-play data were then aggregated to the possession-by-possession level. (4) The outcome of interest for each possession was whether it ends in a turnover, and, if so, which type. Possessions were dropped if there were less than 24 seconds remaining in the period or less than 2 minutes remaining in the fourth quarter or overtime. This was done in an effort to exclude situations where the offense may not execute a typical full possession or the defense is likely to intentionally foul when trailing toward the end of a game. The final dataset contained over 1.1 million possessions.
The specific sequential situations I examined are those involving turnovers, that is, offensive fouls and violations. Table 1 shows a description of the turnovers examined in this study as well as their number of occurrences during the sample period. Referee scrutiny can also enter other facets of the game including defensive foul and technical foul calls. However, these are beyond the scope of this study, which examined one particular type of sequential bias. (5) I examined sequential bias in the context of offensive fouls or violations because they always result in a turnover and loss of possession. If there is any doubt about the accuracy of one of these calls, it could consciously or subconsciously be remedied by whistling an offensive foul or violation on the opposing team, thus returning possession to the initially aggrieved team. As mentioned above, such a scenario was described by the disgraced former NBA referee Tim Donaghy. He writes about a situation where it was suggested that a traveling violation (a judgment call) could have been whistled on Keven Garnett to "even things out" after a bad call on the other end of the floor (Donaghy 2009).
Bertrand, Chugh, and Mullainathan (2005) note that time pressure and ambiguity are two conditions under which implicit attitudes may arise. NBA referees are under enormous time pressure as they have to make split-second decisions, but the ambiguity involved in their calls can vary. Calls involving more ambiguity should also tend to involve a greater use of judgment. Of the nine turnover types with the highest frequency of occurrence in the data, three arguably involve a substantial amount of judgment. Offensive foul calls include the judgment decisions of charging, illegal screens, and hooking/pushing with the off arm. Traveling violations include the judgment decisions of sliding/switching pivot feet, same foot hops, spin move steps, and illegal jump stops. Three-second violations are called at the referee's discretion and frequently take place away from the ball where less attention is received. Thus, I classify offensive fouls, traveling violations, and 3-second violations as judgment calls (JCs). The remaining six turnover types, 24-second violations, step out-of-bounds turnovers, bad pass turnovers, lost ball turnovers, bad pass steals, and lost ball steals are classified as non-judgment calls (NJCs). With 24-second violations, there is an audible buzzer to aid the referee. Step out-of-bounds turnovers involve the relatively objective determination of a player's foot crossing the out-of-bounds line. Bad pass and lost ball turnovers and steals are primarily player-generated turnovers. These classifications do not imply that NJCs involve a complete lack of judgment, just significantly less than JCs and not so much as to influence referee behavior. (6) Anecdotal evidence of the minimal judgment involved in NJCs is seen in the relative lack of media coverage and fan uproar over questionable NJCs relative to that of questionable offensive fouls, traveling, and 3-second violations.
Table 2 shows summary statistics for JCs and NJCs for all five seasons combined as well as for each season individually. If referee behavior impacts JCs more than NJCs, then we might see more inter-season variability in the mean number of JCs relative to NJCs because referee judgment and discretion could change across seasons as league points of emphasis, training materials, and protocols change, or as the mixture of games officiated by different referees changes. (7)
This is explored in the far right column of Table 2, showing t test results of 10 inter-season mean comparisons. (8) For each JC, seven or more of the inter-season differences are significant. Only one of the six NJCs has more than three significant inter-season differences and half have no significant differences at all. While only descriptive, this provides elementary support for the notion that referee behavior has more of an impact on JCs than NJCs.
IV. HYPOTHESES AND METHOD
Two types of sequential judgments are examined in this study. The first examines how a judgment call on one team affects referee scrutiny on the opposing team. The second examines how a judgment call on one team affects referee scrutiny on that same team. I examined changes in the probability of judgment calls in these two situations following a recent prior judgment call and tested for changes in player behavior that may confound the analysis such as changes in player effort and awareness of certain situations.
HYPOTHESIS 1: The probability of a judgment call turnover on one team increases following a recent judgment call turnover on the opposing team.
HYPOTHESIS 2: The probability of a judgment cal! turnover on one team decreases following a recent judgment call turnover on the same team.
Hypothesis 1 is examined in a rudimentary manner in Table 3. This table shows differences in mean occurrences of individual turnover types immediately following a particular JC on the opposing team. It does not control for confounding factors, but does appear to suggest that some JCs may increase in likelihood immediately following certain JCs on the opposing team, while no NJCs show probability increases.
B. Method and Assumptions
To analyze the effects of sequential judgments, I estimated the impact of past decisions on current ones. Specifically, the following regression specification is estimated
(1) [Y.sub.ij] = [[alpha].sub.i] + [Z.sub.ij][[beta].sub.1] + [X'.sub.ij][[beta].sub.2] + [[epsilon].sub.ij]
where Y is an indicator variable equal to one if there is a particular turnover in the current possession, X is a vector of game/possession characteristics, and Z is an indicator variable equal to one if there was a particular turnover in the opponent's previous possession, for i = 1 ... 4,350 offense-defense-season combinations and j = 1 ... [J.sub.i] possessions. (9) The offense-defense-season fixed effect accounts for all invariant characteristics (observable or unobservable) relevant to one team's offense against another team's defense in a season, for example, during possessions of the Lakers' offense against the Warriors' defense during the 2010-11 season. The parameter of interest is [[beta].sub.1], the marginal impact of a particular recent turnover by the opposing team (Z) on the probability of a particular turnover by the current team (Y). (10)
This model may not identify the true effect of sequential judgments if, in addition to the referees, players change their behavior between the past and present decisions. In other words, the probability of a particular turnover is not solely dependent on referee behavior. It is also affected by team characteristics such as the players, their motivation to play the opposing team, and the coaching styles; game/possession characteristics such as the score differential, time remaining, home/away possession, and if it is a playoff game; the current effort level of the players; and the current awareness level of the players. Considering all of these factors, the probability of a turnover in the current possession is
(2) Pr (7) = [f.sub.Y] (team, ref (X, Z), oef (X, Z), [oaw.sub.Y] (X, Z), def (X, Z), [daw.sub.Y] (X, Z))
where team is a vector of observable and unobservable team characteristics; ref is a referee behavior function; oef and def are offensive and defensive player effort functions, respectively; and oaw and daw are offensive and defensive player awareness functions, respectively, and all functions satisfy suitable regularity conditions.
I defined player effort as the energy and care taken to avoid committing any turnover by the offense or to obtain any turnover by the defense. Player awareness is the attention paid by the offense (defense) to avoid (obtain) a particular turnover as a result of the extra knowledge of the same turnover following a recent occurrence. Thus, player awareness affects specific turnovers while player effort is an energy or hustle factor that affects all turnovers.
Changes in player effort and awareness will have different impacts on the game depending on whether they are made by the offense or defense. Increases in offensive (defensive) player effort should decrease (increase) the probability of any turnover being committed. Likewise, increases in offensive (defensive) player awareness should decrease (increase) the probability of the particular turnover of interest being committed. While awareness tunes the offense into particular turnovers in order to avoid them, defensive awareness could have varying effects. It may increase the likelihood of offensive fouls by causing defenders to think more about putting themselves in the proper position to take a charge. Similar logic could apply to steals. However, it is unlikely that defenders' thoughts about traveling, 3-second violations, 24-second violations, stepping out of bounds, bad pass turnovers, or lost ball turnovers will lead to increases in their likelihood of occurrence. Offensive players have more direct control over these turnovers through their decisions whereas the defense impacts them in a more indirect manner through its structure and pressure.
Three basic assumptions are required in order for [[beta].sub.1], to identify the effect of changes in referee behavior on foul calls. These assumptions allow multiple tests to be conducted to examine whether changes in player effort or awareness are likely biasing the [[beta].sub.1] estimates.
ASSUMPTION 1. Referee behavior is not affected by non-judgment calls. ([[partial derivative]ref/[partial derivative]Z] = 0 [for all] NJC Z)
ASSUMPTION 2. Recent turnovers of type i do not affect player awareness of turnovers of type j [not equal to] i. ([[partial derivative][oaw.sub.y]/[partial derivative]Z] = [[partial derivative][daw.sub.Y]/[partial derivative]Z] = 0 [for all] Y [not equal to] Z)
ASSUMPTION 3. Increases in offensive (defensive) effort reduce (increase) the probability of a turnover. Increases in offensive (defensive) awareness reduce (nonmonotonically increase) the probability of the specific turnover type, ([[partial derivative][f.sub.y]/[partial derivative]oef] <0, [[partial derivative][f.sub.y]/[partial derivative]def] > 0, [[partial derivative][f.sub.Y]/[partial derivative][oaw.sub.Y]] < 0, [[partial derivative][f.sub.Y]/[partial derivative][daw.sub.Y]] [greater than or equal to] 0 [for all] Y)
Assumption 1 implies that equilibrium adjustments in player behavior can be tested following NJCs without the confounding influence of changes in referee behavior. Because NJCs do not involve judgment (or involve very little), there is no scope for changes in referee behavior to affect current outcomes. Thus, observed changes in current outcomes would only serve as evidence of equilibrium changes in player behavior.
Assumption 2 makes explicit that player awareness of a particular turnover is affected only by that specific turnover. It implies that a bad pass turnover might make players think about avoiding future bad passes, but it does not make them think about traveling. Suppose previous and current calls are displayed in a matrix with the previous calls along the columns and the current calls down the rows. Assumptions 1 and 2 imply that we can test for changes in player effort following NJCs along the non-diagonal matrix elements. If these changes are not supported, the assumptions further imply that we can test for changes in player awareness following NJCs along the diagonal elements.
Assumption 3 makes explicit that player effort impacts all turnover types. It also addresses the directional impact of offensive and defensive effort and awareness on the probability of a turnover's occurrence. The "[greater than or equal to]" reflects the previous discussion that defensive awareness is likely benign for everything other than steals and offensive fouls. Assumption 3 is used in the final step of the identification process to address any lingering concerns about changes in player behavior following JCs.
Assumptions 1-3 are not strong, yet they allow for a process by which sequential changes in referee behavior may potentially be identified using a series of tests. Assumptions 1 and 2 allow for examination and testing of equilibrium changes in player effort. With satisfactory results, they next allow for examination and testing of equilibrium changes in player awareness. With satisfactory results, Assumption 3 then allows for testing any remaining concerns that equilibrium player behavior may change only following JCs.
The complete identification process is as follows: (1) Test whether players change their effort levels following NJCs. If the evidence is not supportive, (2) test whether players change their awareness levels following NJCs. If the evidence is not supportive, (3) test whether players change their behavior following JCs.
V. EMPIRICAL RESULTS
I estimated Equation (1) with a fixed effects logit (FE Logit) model. The FE Logit allows for the inclusion of a detailed set of fixed effects, but its conditional logit estimation procedure does not allow for subsequent estimation of the average partial effect (APE) of [[beta].sub.1]. Thus, the value of a parameter estimate is the information contained in its sign and significance. Once general conclusions are reached from this process, I used a fixed effects linear probability model (FELPM) to provide point estimates of the APEs of some key statistics.
Table 4 presents FE Logit regression results for various specifications when both Z and Y represent the most common JC occurrence of an offensive foul (Z in the previous possession by the opposing team and Y in the current possession). (11) The variable of interest is Recent Z. Its parameter estimates are all positive and significant at the 1% level for all specifications. The Likelihood Ratio Test supports selection of Specification 3 where all game and possession characteristics are included but interactions of Z with those characteristics, such as point differential or time remaining, are excluded. Thus, the interactions do not have enough explanatory value to justify their additional cost to efficiency. In the analyses that follow, I report the sign and significance of [[beta].sub.1] from the FE Logit using Specification 3 and offense-defense-season fixed effects.
A. Changes in Scrutiny on the Opposing Team
Hypothesis 1 is examined in Tables 5-7. In Table 5, each estimate is the sign and significance of[[beta].sub.1] from a regression of the current outcome in each row (T) on the JC outcome in the previous possession of the opposing team in each column (Z). The upper panel shows the probability changes of current JCs associated with various JCs in the opponent's previous possession while the lower panel shows the probability changes of current NJCs. Two main points are evident from examination of the upper panel. (1) Each JC in the previous possession is associated with an increased probability of the exact same JC and a 3-second violation in the next possession of the opposing team. (2) An offensive foul in the previous possession is associated with an increased probability of all three JCs in the next possession of the opposing team.
Are the increases in likelihood of JCs the result of referee sequential bias? Before I can reasonably conclude that these findings are because of changes in referee behavior, I tested whether equilibrium changes in player behavior may be impacting the results. I used a treatment and control-style analysis in the spirit of Price et al. (2012), exploiting the differences in judgment involved between JCs and NJCs.
Table 6 presents estimates of the change in likelihood of current outcomes (rows) following a particular NJC in the previous possession (column). As per Assumptions 1 and 2, changes in player effort are the sole factors affecting the non-diagonal elements. Overall, 44 of 48 estimates are insignificant and only one significant estimate is positive. This general lack of significance suggests that offensive and defensive players do not change their net effort levels in a meaningful way following NJCs (Identification Step 1). (12) Because changes in player effort are not supported, player awareness is examined along the highlighted diagonal of the bottom panel. This shows how the probability of an NJC changes immediately following the exact same NJC. All results are insignificant, suggesting that changes in player awareness do not play a meaningful role (Identification Step 2).
The first two steps of the identification process suggest that players do not change their effort and awareness levels immediately following a turnover by the opposing team. However, it could be argued that while players might not change their behavior following NJC turnovers, they may do so following JC turnovers. This could confound any findings regarding referee behavior and is addressed via Assumption 3.
Assumption 3 makes explicit the directional impact of offensive and defensive effort and awareness on the probability of a turnover's occurrence. If offensive player effort or awareness were to increase following a JC, this should decrease current turnover probabilities and therefore strengthen the positive and significant findings in the top panel of Table 5. The point estimates would simply serve as lower bounds. However, if defensive player effort or awareness were to increase following a JC, this would increase turnover probabilities and mean that the positive and significant findings in the top panel of Table 5 may not be entirely because of referee behavior.
Equilibrium increases in defensive player effort following a JC should be evident through an increase in the probability of all turnovers in each JC column of Table 5 because defensive effort is the energy and care taken to obtain any turnover. This is clearly not supported in the data (Identification Step 3). Following a JC in the previous possession, the probability of a current NJC significantly decreases or is benign while the probability of a current JC significantly increases or is benign. This would not be the case if changes in defensive player effort were driving the results and, thus, findings are not supportive of this particular player behavioral change. (13)
Instead, there is a clear distinction between the impact on JCs and NJCs. Because defensive player effort is not changing and, by definition, referee behavior can affect current JCs but not current NJCs, it is reasonable to conclude that the significant probability increases in the non-diagonal estimates of the upper panel of Table 5 are attributable to changes in referee scrutiny following a JC in the previous possession.
Changes in defensive player awareness could potentially affect estimates along the diagonal. Logically, defensive awareness should not affect traveling and 3-second violations because defender decisions do not play a direct role in causing these violations. (14) However, defensive awareness could play a direct role in offensive fouls through player positioning to take charges. While the results in Table 6 suggest that player awareness is not likely changing in a meaningful way because it does not change following NJCs (especially following steals), it is still possible that defensive awareness of charges could increase following a charge in the previous possession. While there is no supporting evidence, this possibility is a limitation of the non-experimental nature of the study. However, even if defensive awareness were to behave in such a manner, all other findings would remain valid.
When examining changes in the likelihood of current JCs following previous JCs on the opposing team, six of nine estimates are positive and significant. There are 72 other estimates presented in Tables 5 and 6 and only one of them is positive and significant. (15) While controlling for equilibrium adjustments in player behavior cannot be explicitly performed, testing for these adjustments can. The three tests of the identification process along with the strong findings when referee judgment is involved in both decisions (6 of 9) and the weak findings when it is not (1 of 72) provide robust evidence of increases in referee scrutiny on one team following prior decisions involving judgment on the opposing team.
Point estimates of the key findings for Hypothesis 1 are presented in Table 7 using the FELPM. Each number is the APE percent change of [[beta].sub.1]. Estimates range from a 16% to 66% change in probability of current JCs following a JC on the opposing team. Results suggest that elite, NBA referees working under a performance standard to make "accurate calls, regardless of the circumstances of the game," have sequential bias in certain decisions in what fans may describe as a make-up call situation. It appears that referees increase scrutiny on one team immediately following a judgment call, with the potential for error, on the opposing team. Interestingly, this finding does not seem to be substantially influenced by situational factors in the game or whether it is the regular season or the playoffs. (16)
B. Changes in Scrutiny on the Same Team
Table 8 presents initial estimates useful for testing Hypothesis 2. In this table, each estimate is the sign and significance of [[beta].sub.1] from a regression of the current outcome in each row (Y) on the outcome
in the previous possession of the same team in each column (Z). (17) While most of the results are benign, the most evident ones are that each JC in a team's previous possession is associated with a decreased probability of the exact same JC in their very next possession. But, is referee behavior driving this result or is it player behavior?
Similar to the analyses for Hypothesis 1, if changes in player effort are affecting current turnovers, this should be apparent upon examination of the columns of Table 9. The weak results in each column suggest that changes in player effort are not meaningful. Results are thus far consistent with the idea that referees scrutinize a team less for the same JC on two possessions in a row. However, evidence from the highlighted diagonal in the lower panel strongly suggests that players change their behavior after their team commits a turnover. In particular, the offense becomes more aware of that exact same turnover type. A team is much less likely to commit the same NJC turnover twice in consecutive possessions. While this result makes intuitive sense for 24-second violations, it is particularly noteworthy for other NJCs such as lost ball turnovers or stepping out of bounds. The distinct diagonal nature of the findings is striking. Coupled with the results from earlier studies, it suggests that NBA players are very aware of turnovers committed by themselves and their teammates and respond accordingly, but they are not similarly affected by turnovers committed by their opponents. In other words, it is more powerful for an event to happen to a team than to observe the same thing happening to the opponent.
Because changes in offensive awareness lead to a reduced likelihood of repeating the same turnover in two consecutive possessions, this is also a very plausible explanation for the observed JC results in the upper panel of Table 8. Thus, one cannot disentangle the effects of possible changes in referee behavior from changes in player awareness. The results in this table do not support Hypothesis 2 and suggest that changes in offensive awareness play a substantial role in explaining the observed JC probability declines.
VI. EXTENSIONS AND ROBUSTNESS
A logical extension is to examine if the sequential bias of Hypothesis 1 is transitory or if it lasts for a meaningful period of time during a game. To assess this issue, I employed the same regression model, but where Z may now represent the occurrence of a particular JC turnover in the opponent's most recent possession (1), second most recent possession (2), third most recent possession (3), and so forth. The model examines how a JC turnover in a particular recent opponent possession is associated with probability changes for various outcomes in a team's current possession. (18) In order to minimize the impact of confounding events, observations were included only if no JCs have been called on either team in the possessions between the occurrence of Z and Y.
Figure 2 shows the changes in probability of current outcomes associated with an offensive foul call on the opposing team in the seven most recent possessions. The results suggest that increased scrutiny of offensive fouls on the other team lasts for up to their next six possessions. Increased scrutiny of traveling and 3-second violations lasts for up to three possessions. NJCs are trivially impacted as would be expected. Figure 3 shows that a traveling call on the opposing team can lead to increased scrutiny of traveling and 3-second violations on the other team for their next two possessions. Figure 4 shows that the impact of a 3-second violation on the opposing team is transitory.
A 3-second violation is often away from the ball, so the initial call may be less questionable and therefore results in less referee desire (implicit or explicit) to increase scrutiny. Offensive fouls and traveling violations occur mostly on or near the ball where time and social pressures may be the greatest. In particular, when a block/charge event occurs, everyone in the arena knows that something just happened, but there might be differences of opinion as to what it was. The motivation to subsequently increase scrutiny may be greater in these situations and may therefore explain the longer-lasting increases in probability of JCs on one team following an offensive foul call on the opposing team.
A. Robustness Checks
Results of five different regression models are shown in Table 10, examining the most common JC occurrence of an offensive foul in the previous and current possession. There are no differences in sign or significance of the Recent Z parameter of interest and there is only one meaningful difference in sign and significance of any other parameter estimates across the models. The estimate for Playoffs is positive and significant (or close to it) in the three specifications without a fixed effect for defense, while it is negative and insignificant in the offense-defense-season fixed effect model. (19) The former results seem to imply that offensive fouls occur or are called more frequently in the playoffs. However, once one controls not only for what team is on offense, but also for what team is on defense, the increased frequency disappears. In other words, better defensive teams tend to be playing in the playoffs and therefore generate more offensive foul calls.
This same analysis was performed for every estimate in Tables 5 and 6 to see if the findings for Recent Z substantially varied across three key models: logit with offense-defense-season fixed effects, logit with offense-game fixed effects, and a multinomial logit with offense-season fixed effects. (20) There was only one substantive difference. When offense-game fixed effects are used, the estimate of the change in probability of a current 3-second violation following a 3-second violation on the opposing team becomes marginally insignificant. (21) While using offense-game fixed effects better controls for possible differences between referee crews, over 811,000 observations (73% of the data) must be dropped because there is no within-group variation of the 3-second violation outcome variable, that is, a team is not called for a 3-second violation in a particular game. Thus, while this particular estimate has marginally better control features, it ignores three-fourths of the data. While this is a potential limitation of the findings, the parameter estimate is only marginally insignificant using 28% of the data, and is solidly significant when a majority of the data is employed.
VII. SUMMARY AND CONCLUSIONS
Sequential bias is a phenomenon that may be prevalent in many aspects of life and economic activity, one of which is sports. This study takes a nonexperimental approach to examining sequential judgment effects not only in the sports environment of NBA referees but also in a business setting examining elite employee behavior for a global business enterprise.
I find evidence consistent with sequential bias at the highest level of professional basketball following a potentially tough decision on the opposing team but not supportive of sequential bias following a potentially tough decision on the same team. On- or near-the-ball judgment calls such as offensive fouls and traveling violations are associated with non-transitory increases in scrutiny on the opposing team, with offensive fouls leading to up to six possessions of increased scrutiny. Decreased scrutiny on the same team following a judgment call is partially unsupported and partially inconclusive because players become more aware of turnovers recently committed by themselves or their teammates.
Immediately after an offensive foul on one team, the likelihood of a judgment call on the next possession of the opposing team increases by 16%-66%, depending on the type of call. Increases of this magnitude may lend credence to the position that the sequential bias is subconscious. For example, a 3-second violation occurs roughly one time every 300 possessions. (22) A conscious attempt to increase scrutiny of this violation on the opposing team seems likely to increase the probability of this event by substantially more than 66% because of its infrequency of occurrence. On the other hand, the statistics in this study use information on all judgment call turnovers. It can be argued that they are therefore a lower bound to the probability increases that would be associated with knowledge of truly erroneous or questionable judgment calls.
The analysis of sequential judgment effects among NBA referees has important implications not only for the league's training and monitoring programs, but also for its reputation and perceived quality to consumers. There are obvious strategic implications for coaches and players, as well as potential business implications for casinos offering real-time micro betting on near-future events. While more bias-targeted referee training may help reduce its occurrence, early research suggests that simple awareness of bias may lead to behavioral changes (Pope, Price, and Wolfers 2013).
NBA referees are among the most skilled workers in their labor market, that is, they are "the best of the best." Even with their high level of skill, stringent performance standards from their employer, strict monitoring, and post-game assessment, referee workplace decisions are still impacted by sequential factors in certain situations. Thus, it stands to reason that sequential judgments likely play a more extensive role in decision-making in other areas of economic activity, where performance guidelines are less structured and individual decisions less scrutinized.
APE: Average Partial Effect
FE Logit: Fixed Effects Logit
FELPM: Fixed Effects Linear Probability
JC: Judgment Call
NBA: National Basketball Association
NJC: Non-Judgment Call
Attali, Y. "Sequential Effects in Essay Ratings." Educational and Psychological Measurement, 71(1), 2011, 68-79.
Basketballvalue.com. "Basketballvalue.com Data Files." Accessed June 13, 2011. http://basketballvalue.com/ downloads.php.
Berri, D. "A Simple Measure of Worker Productivity in the National Basketball Association," in The Business of Sport, edited by B. Humphreys and D. Howard. Westport, CT: Praeger, 2008, 1-40.
Bertrand, M., D. Chugh, and S. Mullainathan. "Implicit Discrimination." American Economic Review, 95(2), 2005, 94-98.
Damisch, L., T. Mussweiler, and H. Plessner. "Olympic Medals as Fruits of Comparison? Assimilation and Contrast in Sequential Performance Judgments." Journal of Experimental Psychology: Applied, 12(3), 2006, 166-178.
Donaghy, T. Personal Foul: A First-Person Account of the Scandal that Rocked the NBA. Kingston, PA: Four Daughters, 2009.
Morgan, H., and K. Rotthoff. "Sequential Order Judging: Findings of a Difficulty Bias." Social Science Research Network Working Paper, 2012.
National Basketball Association. "Official Rules 20102011." New York: National Basketball Association, 2010.
Page, L., and K. Page. "Last Shall Be First: A Field Study of Biases in Sequential Performance Evaluation on the Idol Series." Journal of Economic Behavior & Organization, 73(2), 2010, 186-98.
Pedowitz, L. "Report to the Board of Governors of the National Basketball Association." Wachtell, Lipton, Rosen & Katz, 2008. Accessed January 4, 2012. http:// www.nba.com/media/PedowitzReport.pdf.
Plessner, H., and T. Betsch. "Sequential Effects in Important Referee Decisions: The Case of Penalties in Soccer." Journal of Sport and Exercise Psychology, 23(3), 2001, 254-259.
Pope, D., J. Price, and J. Wolfers. "Awareness Reduces Racial Bias." NBER Working Paper No. 19765, 2013. Accessed September 18, 2014. http://www.nber.org/ papers/w19765.
Price, J., M. Remer, and D. Stone. "Subperfect Game: Profitable Biases of NBA Referees." Journal of Economics and Management Strategy, 21(1), 2012, 271-300.
Washington Post. "Donaghy Sentenced to 15 Months in Prison," 30 July 2008. Sec. E, p. 3.
(1.) See Washington Post (2008) for more information.
(2.) Accessed June 13, 2011. http://basketballvalue.com/ downloads.php.
(3.) Eighteen games were missing and seven were dropped because of incomplete data.
(4.) I use a precise version of Possessions Employed (Berri 2008) where a team's possession ends with a made basket or free throw, a turnover, a defensive rebound, or a missed shot out of bounds to the opposing team. This differs from the NBA's definition of a team possession, which "ends when the defensive team gains possession or there is a field goal attempt which hits the rim" (National Basketball Association, 2010).
(5.) No claims are made that this is the only type of sequential bias.
(6.) This classification of judgment and non-judgment calls is consistent with that of Price, Renter, and Stone (2012). They classify the three JCs as calls involving more referee discretion and the six NJCs as calls/events involving less discretion.
(7.) This inter-season variability is in addition to that which affects both JCs and NJCs such as player behavior, randomness, and rule changes. There were no significant rule changes during the sample period.
(8.) The ten comparisons are 2006-07 compared to the four subsequent seasons, 2007-08 compared to the three subsequent seasons, 2008-09 compared to the two subsequent seasons, and 2009-10 compared to 2010-11.
(9.) 30 teams x 29 other teams x 5 seasons = 4,350. Robustness checks are performed using team-season and team-game fixed effects and also using a multinomial logit system model recognizing the potential interdependencies between current turnovers (Y).
(10.) Z and Y can represent the same or different turnovers. Interactions of Z and X were tested and are discussed later in Section V.
(11.) This situation also contains the difficult block/charge referee decision.
(12.) It is possible that the offense and defense could both increase or decrease their effort levels at the same time where the effects offset. Thus, 44 of 48 estimates suggest that there is no meaningful net change in player effort following NJCs.
(13.) This also addresses similar concerns that differences in game intensity may be driving the results. While the control variables can account for general changes in intensity, it may be the case that JCs in the previous possession occur during more intense periods of play and this intensity leads to increased JCs in the subsequent possession. If such changes in game intensity, or defensive effort, were driving the results of Table 5, we should see a significant, positive effect on both NJCs and JCs. The opposing results for NJCs and JCs strongly rebut this notion.
(14.) It is possible that defensive awareness of 3-second violations and traveling could be verbalized to referees, letting them know they believe such violations have occurred. Such vocalization may increase when the same (or some other questionable) call is made against a particular team in the previous possession.
(15.) The 72 other estimates are composed of 18 estimates of current NJCs following previous JCs (Table 5) and 54 estimates of current outcomes (JC or NJC) following previous NJCs (Table 6).
(16.) There is some evidence to suggest that sequential bias in referee judgment decisions may contain an element of opportunism. Most interactions of Z with the game/possession characteristics of score differential, time remaining, home/away possession, and regular season/playoff game (X) are insignificant. Notable exceptions are that the increased scrutiny of an offensive foul following a prior offensive foul decreases at the 5% level in the second quarter, at the 10% level at the end of the first quarter, and at the beginning of the third quarter. However, this interaction is not powerful enough to pass the Likelihood Ratio Test. The increased scrutiny of a traveling violation following a prior offensive foul increases yet even more in the playoffs. This interaction is significant at the 10% level and passes Likelihood Ratio Test at 10% level. These results could suggest that a referee's bias in sequential judgments is strongest in the very beginning of the game, where neither team has a large advantage, and at the end of the game, where players may feel more critical. In playoff games, a referee could feel more pressure to give the appearance of calling violations both ways. Yet the majority of interaction effects are insignificant, leaving only weak evidence that time remaining and the playoff status of a game may impact referee behavior.
(17.) This analysis excludes observations where a judgment call was made on an opposing team in between a team's previous possession and its current one. Including this type of situation could bias the results.
(18.) When Z represents the opponent's most recent possession, the estimates coincide with those of Table 6. When Z represents more distant possessions, it is smoothed to include the possession of interest as well as those immediately prior and following, that is, it has a three-possession window.
(19.) The playoff variable is time invariant in the offense-game fixed effect model so a parameter estimate cannot be obtained.
(20.) The multinomial logit accounts for possible interdependence between current turnovers because only one can occur on any given possession. Offense-season fixed effects were used because they provided the finest level of detail without being too computationally burdensome. Offense-defense-season and offense-game fixed effects numbered 39,150 (9 x 30 x 29 x 5) and 110,700 (9 x 30 x 82 x 5), respectively. Unlike the FE Logit regressions, the conditional logit procedure could not be used. In all the analyses performed, multinomial logit estimates were qualitatively similar to the respective estimates from the FE Logit with the same fixed effects. Hausman tests of the Independence of Irrelevant Alternatives assumption were performed and the lowest obtained p value was .484.
(21.) There was one other statistical difference between the models but it had no substantive impact on the findings regarding Hypothesis 1. In the multinomial logit model, the estimate of the change in probability of a current lost ball steal following an offensive foul on the opposing team becomes insignificant.
(22.) Each team has approximately 93 possessions in a typical game during the sample period.
Gift: Associate Professor, Pepperdine University, Graziadio School of Business and Management, Los Angeles, CA 90045. Phone (310) 568-2319, Fax (310) 568-5778, E-mail firstname.lastname@example.org
TABLE 1 Description of Turnover Categories Number of Category Occurrences Description Bad Pass Steal 57,001 Bad pass turnover caused by opposing player's positive, aggressive action Lost Ball Steal 37,505 Lost ball turnover caused by opposing player's positive, aggressive action Offensive Foul 25,225 Illegal contact, committed by an offensive player, after the ball is live and there is team control Bad Pass 19,226 Bad pass turnover that is Turnover not a steal Traveling 15,536 Player takes too many steps without dribbling the basketball Lost Ball 11,395 Lost ball turnover that Turnover is not a steal 24-Second 7,766 Team possessing the ball Violation does not attempt a shot within 24 seconds 3-Second 4,402 Offensive player remains Violation in the lane for more than 3 seconds Step Out of 3,421 Player possessing the Bounds ball steps out of bounds 15 Other 5,084 Turnover Types Total Turnovers 186,561 Notes: Descriptions are from the 2010-11 NBA Official Rulebook, nba.com, and the 2009 FIBA Basketball Statisticians' Manual. Bad pass and lost ball turnovers occur when the ball is not stolen but instead goes out of bounds, thus stopping play. TABLE 2 Summary Statistics by Turnover Type All Seasons 2006-07 Mean SD Mean SD Any Turnover 15.37 36.07 16.22 36.87 Judgment Calls (JC) Offensive Foul 2.08 14.27 2.36 15.19 Traveling 1.28 11.24 1.70 12.92 3-Second Violation .36 6.01 .47 6.85 Non-Judgment Calls (NJC) 24-Second Violation JC) .64 7.97 .62 7.88 Step Out of Bounds .28 5.30 .27 5.22 Bad Pass Turnover 1.58 12.49 1.57 12.43 Lost Ball Turnover .94 9.64 .92 9.53 Bad Pass Steal 4.70 21.16 4.66 21.09 Lost Ball Steal 3.09 17.30 3.11 17.36 Regular Season Total Games 6,126 1,229 1,230 1,212 Total Possessions 1,139,909 228,387 229,486 224,597 Playoffs Total Games 412 79 86 84 Total Possessions 73,844 14,256 15,308 15,115 Overall Total Games 6,538 1,308 1,316 1,296 Total Possessions 1,213,753 242,643 244,794 239,712 2007-08 2008-09 Mean SD Mean SD Any Turnover 15.06 35.76 15.08 35.78 Judgment Calls (JC) Offensive Foul 1.96 13.87 1.92 13.72 Traveling 1.19 10.82 1.20 10.90 3-Second Violation .34 5.85 .29 5.37 Non-Judgment Calls (NJC) 24-Second Violation .60 7.73 .62 7.83 Step Out of Bounds .27 5.23 .29 5.34 Bad Pass Turnover 1.57 12.44 1.58 12.47 Lost Ball Turnover .93 9.60 .97 9.78 Bad Pass Steal 4.68 21.12 4.71 21.18 Lost Ball Steal 3.09 17.29 3.12 17.39 Regular Season Total Games 1,225 1,230 - Total Possessions 228,975 228,464 - Playoffs Total Games 82 81 - Total Possessions 14,770 14,395 - Overall Total Games 1,307 1,311 - Total Possessions 243,745 242,859 - 2009-10 2010-11 Mean SD Mean SD Any Turnover 15.17 35.87 15.33 36.03 Judgment Calls (JC) Offensive Foul 2.01 14.03 2.14 14.47 Traveling 1.20 10.89 1.11 10.49 3-Second Violation .34 5.85 .37 6.04 Non-Judgment Calls (NJC) 24-Second Violation .68 8.19 .68 8.22 Step Out of Bounds .29 5.40 .28 5.32 Bad Pass Turnover 1.64 12.69 1.56 12.40 Lost Ball Turnover .93 9.60 .95 9.71 Bad Pass Steal 4.68 21.11 4.76 21.29 Lost Ball Steal 3.00 17.07 3.13 17.41 Regular Season Total Games Total Possessions Playoffs Total Games Total Possessions Overall Total Games Total Possessions Inter-Season Significant Differences Any Turnover 6 Judgment Calls (JC) Offensive Foul 8 Traveling 7 3-Second Violation 7 Non-Judgment Calls (NJC) 24-Second Violation 6 Step Out of Bounds 0 Bad Pass Turnover 1 Lost Ball Turnover 0 Bad Pass Steal 0 Lost Ball Steal 3 Regular Season Total Games -- Total Possessions -- Playoffs Total Games -- Total Possessions -- Overall Total Games -- Total Possessions -- Notes: Mean is the average number of occurrences per 100 possessions. Inter-season significant differences are at the 5% level. TABLE 3 Frequency of Current Outcomes by Previous Possession of Opposing Team Previous Possession of Opposing Team (Z) Offensive Foul Current Outcome (Y) Yes No Difference Any Turnover 15.77 15.36 .41 Judgment Calls (JC) Offensive Foul 2.64 2.07 .57 ** Traveling 1.53 1.27 .25 ** 3-Second Violation .61 .36 .26 ** Non-Judgment Calls (NJC) 24-Second Violation .52 .64 -.12 ** Step Out of Bounds .29 .28 .01 Bad Pass Turnover 1.55 1.58 -.03 Lost Ball Turnover .91 .94 -.03 Bad Pass Steal 4.39 4.70 -.32 * Lost Ball Steal 2.91 3.09 -.19 Total Possessions 25,124 1,188,629 - Traveling Current Outcome (Y) Yes No Difference Any Turnover 15.54 15.37 .17 Judgment Calls (JC) Offensive Foul 2.21 2.08 .13 Traveling 1.72 1.27 .45 ** 3-Second Violation .58 .36 22 ** Non-Judgment Calls (NJC) 24-Second Violation .60 .64 -.04 Step Out of Bounds .24 .28 -.05 Bad Pass Turnover 1.28 1.59 -.31 ** Lost Ball Turnover .99 .94 .05 Bad Pass Steal 4.50 4.70 -.19 Lost Ball Steal 2.98 3.09 -.11 Total Possessions 15,229 1,198,524 -- 3-Second Violation Current Outcome (Y) Yes No Difference Any Turnover 15.78 15.37 .41 Judgment Calls (JC) Offensive Foul 2.42 2.08 .34 Traveling 1.65 1.28 .37 3-Second Violation .59 .36 .23 Non-Judgment Calls (NJC) 24-Second Violation .54 .64 -.10 Step Out of Bounds .19 .28 -.09 Bad Pass Turnover 1.76 1.58 .18 Lost Ball Turnover .75 .94 -.19 Bad Pass Steal 4.23 4.70 -.47 Lost Ball Steal 3.22 3.09 .13 Total Possessions 4,253 1,209,500 - Note: Frequency is per 100 possessions. * Significant at 5%; **significant at 1%. TABLE 4 Fixed Effects Logit Regression Estimates (1) (2) Beta SE Beta SE Recent Z .2231 ** .0411 .2244 ** .0411 Homeposs -.0724 ** .0130 Playoffs -.0091 .0422 PtsBehind 20+ PtsBehind 10-19 PtsBehind 4-9 PtsAhead 4-9 Pts Ahead 10-19 PtsAhead 20+ Q1 Last 5 Min Q2 First 7 Min Q2 Last 5 Min Q3 First 7 Min Q3 Last 5 Min Q4 First 7 Min Q4 Last 5 Min Overtime Interactions No No with Recent Z (3) (4) Beta SE Beta SE Recent Z .1952 ** .0411 .4713 ** .1454 Homeposs -.1077 ** .0134 -.1062 ** .0136 Playoffs -.0059 .0422 -.0087 .0426 PtsBehind 20+ -.3477 ** .0431 -.3451 ** .0439 PtsBehind 10-19 -.1104 ** .0241 -.1109 ** .0244 PtsBehind 4-9 -.0366 .0197 -.0388 .0199 PtsAhead 4-9 .1051 ** .0198 .1039 ** .0200 Pts Ahead 10-19 .0919" .0247 .0848 ** .0250 PtsAhead 20+ .1652 ** .0427 .1610 ** .0434 Q1 Last 5 Min .2652 ** .0284 .2735 ** .0288 Q2 First 7 Min .5083 ** .0263 .5187 ** .0265 Q2 Last 5 Min .3272 ** .0286 .3361 ** .0290 Q3 First 7 Min .4181 ** .0266 .4258 ** .0270 Q3 Last 5 Min .4562 ** .0291 .4593 ** .0295 Q4 First 7 Min .5574 ** .0270 .5633 ** .0274 Q4 Last 5 Min .3966 ** .0335 .4040 ** .0341 Overtime .2183 .1139 .2339 * .1151 Interactions No Yes with Recent Z Notes: All specifications use offense-defense-season fixed effects. Standard errors are White-corrected. Dependent Variable Y: Offensive Foul in the Current Possession. Recent Z: Offensive Foul in the Previous Possession of Opposing Team. * Significant at 5%; ** significant at 1%. TABLE 5 Sign and Significance of [[beta].sub.1] Following Judgment Call Z Previous Possession of Opposing Team (Z) Current Outcome (Y) Offensive Traveling 3--Second Foul Violation Panel A: Judgment Calls (JC) Offensive Foul + ** + + Traveling + ** + ** + 3-Second Violation + ** + ** + * Panel B: Non-Judgment Calls (NJC) 24-Second Violation -- * + -- Step Out of Bounds -- -- -- Bad Pass Turnover -- _ ** + Lost Ball Turnover -- + -- Bad Pass Steal _ ** -- -- Lost Ball Steal _ * -- + Notes: The model [mathematical not reproducible in ascii] is estimated using a logit with offense--defense--season fixed effects. X includes variables for home possession, playoff game, point differential, and time remaining. Standard errors are White--corrected. Significant at 5%; ** significant at 1%. TABLE 6 Sign and Significance of Following Non-Judgment Call Z Previous Possession of Opposing Team (Z) 24 Second Step Out Bad Pass Current Outcome (Y) Violation of Bounds Turnover Panel A: Judgment Calls (JC) Offensive Foul + - - Traveling + + + 3-Second Violation + - - Panel B: Non-Judgment Calls (NJC) 24-Second Violation + - Step Out of Bounds - + - Bad Pass Turnover - - + Lost Ball Turnover - - - * Bad Pass Steal - + + Lost Ball Steal - - - ** Lost Ball Bad Pass Lost Ball Current Outcome (Y) Turnover Steal Steal Panel A: Judgment Calls (JC) Offensive Foul + + + Traveling + - - 3-Second Violation - - Panel B: Non-Judgment Calls (NJC) 24-Second Violation + + + Step Out of Bounds + - - Bad Pass Turnover - * - + Lost Ball Turnover - + - Bad Pass Steal + + - Lost Ball Steal - - - Notes: The model [mathematical expression not reproducible in ascii] is estimated using a logit with offense-defense- season fixed effects. X includes variables for home possession, playoff game, point differential, and time remaining. The second most recent possession by the opposing team is used for both steal categories because these turnovers are often immediately followed by fastbreaks. Standard errors are White-corrected. * Significant at 5%; "significant at 1%. TABLE 7 Changes in Probability Following Judgment Call Z Previous Possession of Opposing Team (Z) Current Offensive Traveling 3-Second Outcome (Y) Foul (%) (%) Violation (%) Offensive Foul 21.7 ** 5.2 11.5 Traveling 16.4 ** 30.2 ** 23.6 3-Second Violation 66.5 ** 49.2 ** 52.3 * Notes: The model is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Each point estimate is the average partial effect percent change for [[beta].sub.1] from a linear probability model with offense-defense- season fixed effects. X includes variables for home posses- sion, playoff game, point differential, and time remaining. Standard errors are White-corrected. * Significant at 5%; ** significant at 1%. TABLE 8 Sign and Significance of Following Judgment Call Z on the Same Team Previous Possession of Same Team (Z) Current Offensive 3-Second Outcome (Y) Foul Traveling Violation Judgment Calls (JC) Offensive Foul -- ** - + Traveling - -- ** + 3-Second Violation - - _ ** Non-Judgment Calls (NJC) 24-Second Violation - + Step Out of Bounds - - + Bad Pass Turnover - - - Lost Ball Turnover + - + Bad Pass Steal + - + Lost Ball Steal + + - Notes: The model [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is esti-mated using a logit with offense-defense- season fixed effects. X includes variables for home possession, playoff game, point differential, and time remaining. Standard errors are White-corrected. * Significant at 5%; **significant at 1%. TABLE 9 Sign and Significance of [[beta].sub.1] Following Non-Judgment Call Z on the Same Team Previous Possession of Same Team (Z) 24-Second Step Out Bad Pass Current Outcome (Y) Violation of Bounds Turnover Panel A: Judgment Calls (JC) Offensive Foul + - + Traveling - - - 3-Second Violation + - + Panel B: Non-Judgment Calls (NJC) 24-Second Violation - ** - + Step Out of Bounds + + Bad Pass Turnover + + - ** Lost Ball Turnover - + - Bad Pass Steal + - + Lost Ball Steal + - + * Previous Possession of Same Team (Z) Current Outcome (Y) Turnover Steal Steal Panel A: Judgment Calls (JC) Offensive Foul + + + Traveling - + - 3-Second Violation _ - _ Panel B: Non-Judgment Calls (NJC) 24-Second Violation - + - ** Step Out of Bounds + - - Bad Pass Turnover + - * - Lost Ball Turnover - - Bad Pass Steal - - ** + Lost Ball Steal - - - Notes: The model [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is estimated using a logit with offense-defense- season fixed effects. X includes variables for home possession, playoff game, point differential, and time remaining. Standard errors are White-corrected. * Significant at 5%; "significant at 1%. TABLE 10 Regression Estimates Robustness Check Logit Logit Beta SE Beta SE Recent Z .2087 ** .0408 .1952 ** .0411 Homeposs -.0914 ** .0132 -.1077 ** .0134 Playoffs .0536 * .0267 -.0059 .0422 PtsBehind 20+ -.2867 ** .0405 -.3477 ** .0431 PtsBehind 10-19 -.0559 * .0225 -.1104 ** .0241 PtsBehind 4-9 -.0050 .0192 -.0366 .0197 PtsAhead 4-9 .0833 ** .0195 .1051 ** .0198 PtsAhead 10-19 .0449 .0230 .0919 ** .0247 PtsAhead 20+ .0792 * .0382 .1652 ** .0427 Q1 Last 5 Min .2644 ** .0288 .2652 ** .0284 Q2 First 7 Min .5072 ** .0258 .5083 ** .0263 Q2 Last 5 Min .3275 ** .0287 .3272 ** .0286 Q3 First 7 Min .4179 ** .0266 .4181 ** .0266 Q3 Last 5 Min .4559 ** .0284 .4562 ** .0291 Q4 First 7 Min .5579 ** .0266 .5574 ** .0270 Q4 Last 5 Min .3956 ** .0335 .3966 ** .0335 Overtime .2457 .1139 .2183 .1139 Constant -4.16 ** .0213 Fixed Effects No Offense-defense-season Logit Multinomial Logit Beta SE Beta SE Recent Z .1513 ** .0415 .2050 ** .0409 Homeposs -.0988 ** .0133 Playoffs .0492 .0268 PtsBehind 20+ -.5007 ** .0515 -.2907 ** .0406 PtsBehind 10-19 -.2294 ** .0284 -.0606 ** .0225 PtsBehind 4-9 -.0848 ** .0211 -.0084 .0192 PtsAhead 4-9 .1666 ** .0219 .0907 ** .0195 PtsAhead 10-19 .2319 ** .0295 .0610 ** .0231 PtsAhead 20+ .4114 ** .0530 .1062 ** .0383 Q1 Last 5 Min .2674 ** .0288 .2672 ** .0288 Q2 First 7 Min .5101 ** .0260 .5190 ** .0259 Q2 Last 5 Min .3291 ** .0291 .3296 ** .0287 Q3 First 7 Min .4190 ** .0269 .4243 *' .0267 Q3 Last 5 Min .4569 ** .0292 .4590 ** .0285 Q4 First 7 Min .5573 ** .0273 .5629 ** .0266 Q4 Last 5 Min .3948 ** .0338 .3869 ** .0336 Overtime .1748 .1139 .2255 * .1141 Constant -4.02 ** .0214 Fixed Effects Offense-game No Multinomial Logit Beta SE Recent Z .2037 ** .0409 Homeposs -.1015 ** .0133 Playoffs .0692 * .0282 PtsBehind 20+ -.2931 ** .0408 PtsBehind 10-19 -.0695 ** .0228 PtsBehind 4-9 -.0133 .0192 PtsAhead 4-9 .0981 ** .0196 PtsAhead 10-19 .0772 ** .0233 PtsAhead 20+ .1244 ** .0387 Q1 Last 5 Min .2664 ** .0288 Q2 First 7 Min .5183 ** .0259 Q2 Last 5 Min .3275 ** .0288 Q3 First 7 Min .4223 ** .0267 Q3 Last 5 Min .4572 ** .0285 Q4 First 7 Min .5606 ** .0266 Q4 Last 5 Min .3842 ** .0336 Overtime .2012 .1142 Constant Fixed Effects Offense-season Notes: The multinomial logit model involves the selection of one outcome from 10 alternatives (three JCs, six NJCs, and the base alternative). Offense-season fixed effects were estimated because more detailed fixed effects were too computationally intensive. Standard errors are White-corrected. Dependent Variable Y: Offensive Foul in the Current Possession, Recent Z: Offensive Foul in the Previous Possession of Opposing Team. * Significant at 5%; **significant at 1%. FIGURE 1 Sample Play-By-Play Data for the Houston Rockets versus Portland Trailblazers 20091027HOUPOR 44 00:43:10 [POR 11-10] Roy lump shot: Made (4 pts) Assist: Webster (1 AST) 20091027HOUPOR 45 00:42:52 [HOU 12-11] Brooks Running Hook Shot: Made (2 PTS) 20091027HOUPOR 46 00:42:30 [POR 14-12] Outlaw 3pt Shot: Made (3 PTS) Assist: Roy (1 AST) 20091027H0UP0R 47 00:42:05 [HOU] Battier 3pt Shot: Missed 20091027HOUPOR 48 00:42:05 [POR] Outlaw Rebound (off:0 Def:l) 20091027H0UP0R 49 00:41:56 [POR] Blake Turnover : Lost Ball (1 TO) steal:Brooks (1 ST) 20091027H0UP0R 50 00:41:50 [HOU] Brooks Foul : Offensive (1 PF) 20091027H0UP0R 51 00:41:50 [HOU] Brooks Turnover : Foul (1 TO)
|Printer friendly Cite/link Email Feedback|
|Date:||Apr 1, 2015|
|Previous Article:||Capacity constraints and information revelation in procurement auctions.|
|Next Article:||Profit-maximizing gate revenue sharing in sports leagues.|