Skill, Luck and Hot Hands on the PGA Tour
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Skill, Luck and Hot Hands on the PGA Tour June 21, 2005 Robert A. Connolly Richard J. Rendleman, Jr. Kenan-Flagler Business School Kenan-Flagler Business School CB3490, McColl Building CB3490, McColl Building UNC - Chapel Hill UNC - Chapel Hill Chapel Hill, NC 27599-3490 Chapel Hill, NC 27599-3490 (919) 962-0053 (phone) (919) 962-3188 (phone) (919) 962-5539 (fax) (919) 962-2068 (fax) [email protected] [email protected] We thank Tom Gresik and seminar participants at the University of Notre Dame for helpful comments. We provide special thanks to Carl Ackermann and David Ravenscraft who provided significant input and assistance during the early stages of this study. We also thank Mustafa Gultekin for computational assistance and Yuedong Wang and Douglas Bates for assistance in formulating, programming and testing portions of our estimation procedures. Please direct all correspondence to Richard Rendleman. Skill, Luck and Hot Hands on the PGA Tour 1. INTRODUCTION Like all sports, outcomes in golf involve elements of both skill and luck. Perhaps the highest level of skill in golf is displayed on the PGA Tour. Even among these highly skilled players, however, a small portion of each 18-hole score can be attributed to luck, or what players and commentators often refer to as “good and bad breaks.” The purpose of our work is to determine the extent to which skill and luck combine to determine 18-hole scores in PGA Tour events. We are also interested in the question of whether PGA players experience “hot or cold hands,” or runs of exceptionally good or bad scores, in relation to those predicted by their statistically-estimated skill levels. From a psychological standpoint, understanding the extent to which luck plays a role in determining 18-hole golf scores is important. Clearly, a player would not want to make swing changes to correct abnormally high scores that were due, primarily, to bad luck. Similarly, a player who shoots a low round should not get discouraged if he cannot sustain that level of play, especially if good luck was the primary reason for his score. At the same time, it is important for a player to know whether his general (mean) skill level is improving or deteriorating over time and to understand that deviations from past scoring patterns may not be due to luck alone. From a policy standpoint, it seems reasonable that the PGA Tour and other professional golf tournament organizations should attempt to minimize the role of luck in determining tournament outcomes and qualification for play in Tour events. Ideally, the PGA Tour should be comprised of the most highly-skilled players. Also, the Tour should strive to conduct tournaments that reward the most highly skilled players rather than those who experience the greatest luck. In some cases, luck in a round of golf can easily be identified. In the final round of the 2002 Bay Hill Invitational, David Duval’s approach shot to the 16th hole hit the pin and bounced back into a water hazard fronting the green. Duval took a nine on the hole. Few would argue that Duval’s score of nine was due to bad judgment or a sudden change in his skill level, and it is highly unlikely that Duval made swing changes or changes to his general approach to the game to correct the type of problem he incurred on the 16th hole. In contrast, consider the good fortune experienced by Craig Perks in the final round of the 2002 Players Championship, when he chipped in on holes 16 and 18 and sunk a 28-foot putt for 2 birdie on the 17th hole en route to victory. Was Perks’ victory, his first on the PGA Tour, a reflection of exceptional ability or four consecutive days of good luck? The fact that his best season since 2002 placed him 146th on the PGA Tour’s Official Money List suggests that luck may have been the overriding factor. Similarly, as of June, 2005, Ben Curtis has had only has had only one top-10 finish since winning the 2003 British Open. At this point, few would argue that Curtis’s victory was anything other than a fluke. In many situations, specific occurrences of good and bad luck may not be as obvious as in the examples above. Luck simply occurs in small quantities as part of the game. Even players as highly skilled as Tiger Woods and Vijay Singh cannot produce perfect swings on every shot. As a result, a certain human element of luck is introduced to a shot even before contact with the ball is made. (An excellent article on the role of luck in golf, with a focus on how the ultimate outcome of the dual between Tiger Woods and Chris DiMarco in the 2005 Masters was determined as much by luck as by skill, is provided in Jenkins (2005).) Our work draws on the rich literature in sports statistics. Klaassen and Magnus (2001) model the probability of winning a tennis point as a function of player quality, context (situational) variables, and a random component. They note that failing to account for quality differences will create pseudo-dependence in scoring outcomes, because winning reflects, in part, player quality, and player quality is generally persistent. Parameter estimates from their dynamic, panel random effects model using four years of data from Wimbledon matches suggest that the iid hypothesis for tennis points is a good approximation, provided there are controls for player quality. The most important advantage of their approach from our perspective is that it supports a decomposition of actual performance into two parts: an expected score based on skill and an unexpected residual portion. In this paper, we decompose individual golfer’s scores into skill-based and unexpected components by using Wang’s (1998) smoothing spline model to estimate each player’s mean skill level as a function of time while simultaneously estimating the correlation in the random error structure of fitted player scores and the relative difficulty of each round. The fitted spline values of this model provide estimates of expected player scores. We define luck as deviations from these estimated scores, whether positive or negative, and explore its statistical properties. Our tests show that after adjusting a player’s score for his general skill level and the relative difficulty of the course on the day a round is played, the average (and median) standard deviation of residual golf scores on the PGA Tour is approximately 2.7 strokes per 18-hole round, ranging 3 between 2.1 and 3.5 strokes per round per player. We find clear evidence of positive first-order autocorrelation in the error structure for over 13 percent of the golfers and, therefore, conclude that a significant number of PGA Tour players experience hot and cold hands. We also apply some traditional hot hands tests to the autocorrelated residuals and come to similar conclusions. However, after removing the effects of first-order autocorrelation from the residual scores, we find little additional evidence of hot hands. This suggests that any statistical study of sporting performance should estimate skill dynamics and deviations from normal performance while simultaneously accounting for the relative difficulty of the task and the potential autocorrelation in unexpected performance outcomes. The remainder of the paper is organized as follows. In Section 2 we describe our data and criteria for including players in our statistical samples. In Section 3 we present the results of a number of fixed-effects model specifications to identify the variables that are important in predicting player performance. In Section 4 we formulate and test the cubic spline-based model for estimating player skill. This model becomes the basis for our analysis of player skill, luck and hot hands summarized in Section 5. A final section provides a summary and concluding comments. 2. DATA We have collected individual 18-hole scores for every player in every stroke-play PGA Tour event for years 1998-2001 for a total of 76,456 scores distributed among 1,405 players. Our data include all stroke play events for which participants receive credit for earning official PGA Tour money, even though some of the events, including all four “majors,” are not actually run by the PGA Tour. The data were collected, primarily, from www.pgatour.com, www.golfweek.com, www.golfonline.com, www.golfnews.augustachronicle.com, www.insidetheropes.com, and www.golftoday.com. When we were unable to obtain all necessary data from these sources, we checked national and local newspapers, and in some instances, contacted tournament headquarters directly. Our data cover scores of players who made and missed cuts. (Although there are a few exceptions, after the second round of a typical PGA Tour event, the field is reduced to the 70 players, including ties, with the lowest total scores after the first two rounds.) The data also include scores for players who withdrew from tournaments and who were disqualified; as long as we have a score, we use it. We also gathered data on where each round was played. This is especially important for 4 tournaments such as the Bob Hope Chrysler Classic and ATT Pebble Beach National Pro-Am played on more than one course. The great majority of the players represented in the sample are not regular members of the PGA Tour. Nine players in the sample recorded only one 18-hole score over the 1998-2001 period, and 565 players recorded only two scores. Most of these 565 players qualified to play in a single US Open, British Open or PGA Championship, missed the cut after the first two rounds and subsequently played in no other PGA Tour events.