Reference Points, Prospect Theory, and Momentum on the PGA Tour

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Reference Points, Prospect Theory, and Momentum on the PGA Reference Points, Prospect Theory, and Momentum on the PGA

Tour Tour

Daniel F. Stone

Bowdoin College

Jeremy Arkes

Naval Postgraduate School

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Article

Reference Points,

Prospect Theory,

and Momentum

on the PGA Tour

Daniel F. Stone

and Jeremy Arkes

Abstract

Pope and Schweitzer (2011) study predictions of prospect theory for the reference

point of par on the current hole in professional golf. We study prospect-theory

predictions of three other plausible reference points: par for recent holes, for the

round, and for the tournament. A potentially competing force is momentum in

quality of play, that is, the hot or cold hand. While prospect theory predicts negative

serial correlation in better (worse)-than-average performance across holes, the hot

(cold) hand implies the opposite. We find evidence that, for each of the reference

points we study, when scores are better than par, hot-hand effects are dominated by

prospect-theory effects. These effects can occur via two mechanisms: greater

conservatism or less effort. We find evidence that the former (latter) dominates for

scores closer to (further from) the reference point. We also find evidence of

prospect theory effects (greater risk seeking) when scores are worse than par for

the round in Round 1 and of cold-hand effects for scores worse than par for the

tournament in Round 3. The magnitudes of some of the joint effects are comparable

to those found by Pope and Schweitzer and other related papers. We conclude by

discussing how, rather than compete, prospect-theory and cold-hand forces might

also cause one another.

Bowdoin College, Brunswick, ME, USA

Naval Postgraduate School, Monterey, CA, USA

Corresponding Author:

Daniel F. Stone, Bowdoin College, 9700 College Station, Brunswick, ME 04011, USA.

Email: [email protected]

Journal of Sports Economics

2016, Vol. 17(5) 453-482

ª The Author(s) 2016

Reprints and permission:

sagepub.com/journalsPermissions.nav

DOI: 10.1177/1527002516641167

jse.sagepub.com

Keywords

golf, reference points, prospect theory, hot hand, momentum

You get to like the 12th hole and I’m three under par and I don’t want to have one hole hurt

a round so I end up laying up

–Phil Mickelson, in an interview after the first round of the 2015 U.S. Open

Introduction

Pope and Schweitzer (2011; hereafter PS) find that professional golfers make putts

for a score of par around three percentage points more often than putts of the same

difficulty for a score one stroke better than par (birdie). Since nearly all strokes have

roughly the same expected effect on final tournament standing and earnings, golfers

should, normatively, treat otherwise equivalent par and birdie putts equally and thus

perform equally well on them. PS’s finding violates this normative standard but is

consistent with predictions of prospect theory (Kahneman & Tversky, 1979) that

individuals are often influenced by arbitrary reference points, in this case par for the

current hole, and have greater motivation to avoid a loss versus the reference point

than to attain a gain. While prospect theory had been studied extensively in the lab,

PS’s paper was one of the first to study the topic in a context with experienced

decision makers and high stakes.

In this article, we develop the analysis of PS further by studying three other types

of reference points: par for very recent holes (the combined score for the current and

last holes), par for the current round, and par for the tournament. The Phil Mickelson

quote above reflects prospect-theory-type thinking with respect to the round-level

reference point in particular, but the others we analyze are plausible a priori as well.

Our analysis of reference points is motivated by several goals. First, we hope our

results contribute to the literature on the determinants of reference points that most

affect behavior. As discussed in Barberis (2013; an excellent review of the elements of

prospect theory and follow-up theoretical and empirical work), one of the challenges in

applying prospect theory has been that the reference point’s definition is often ambig-

uous. Our work addresses the importance of expectations for determining reference

points, and the relevance of multiple reference points in a given context. There has been

some recent experimental work on these topics (Abeler, Falk, Goette, & Huffman,

2011; Baucells, Weber, & Welfens, 2011), but such research using nonlab data is highly

limited. Second, we dig deeper into the possible mechanisms underlying reference-

point effects, effort versus risk attitudes. PS discuss both of these mechanisms driving

their results—that players may exert more effort and/or become more risk seeking in

putts for par, causing the improvement in average performance. But PS do not analyze

which factor is dominant, or how this may depend on context. Third, we hope to enhance

the understanding of the general prevalence and magnitudes of prospect-theory effects.

454 Journal of Sports Economics 17(5)

This analysis is made more complex, however, by the fact that golfers may

experience periods of momentum in quality of play. That is, being in what prospect

theory refers to as ‘‘the domain of gains’’ (a position preferred to the reference point,

i.e., having a score below par for holes relevant to a given reference point) could be

indicative of the golfer having the ‘‘hot hand’’ and therefore being likely to play

better on subsequent holes. The ‘‘domain of losses’’ (a score above par) is analogous

and may imply a ‘‘cold hand.’’ While the conventional wisdom in behavioral eco-

nomics as of just a few years ago was that ‘‘the hot hand is a widespread cognitive

illusion’’ (Kahneman, 2011, p. 117), the hot hand is now recognized to exist in a

variety of settings.

Hot (cold) hand theories predict positive serial correlation in the

chance of outcomes being better (worse) than average, and prospect theory typically

predicts the opposite. The hot hand—or the hot-hand bias, that is, overestimation of

one’s own momentum—could thus be viewed as a confound to the analysis of

prospect theory. But the hot and cold hands, and the bias, are of interest also.

Thus,

most of our results do not purely capture either prospect-theory or momentum effects

but instead can be viewed as the outcome of a potential horse race between these

competing forces. We also conduct a limited analysis in which the two types of

effects are better separated; the results from this support our interpretation of the

results from the main analysis.

Our main findings are as follows. For each of the three types of reference points,

we find evidence that when in the domain of gains, prospect-theory effects dominate

hot-hand effects: Recent success predicts a decline in quality of subsequent perfor-

mance. The evidence on eff ort versus risk mechanisms is somewhat murky, but

overall the results suggest that the prospect-theory effects are driven more by risk

aversion when scores are closer to the reference point, and more by lower effort for

scores further from the reference point. There is some evidence of a hot-hand bias

(that golfers become overconfident), which could also account for this decline in

performance in the domain of gains.

We also find several types of evidence indicating that the actual reference points

golfers use are in fluenced by both salience and expectations that players adjust

expectations, and thus reference points, based on their own overall ability, how play

is going in a particular round, and the difficulty of the relevant holes. Since these

factors diminish the observability of reference-point effects, these results imply our

more aggregated results are, in general, likely attenuated.

When golfers enter a hole in the domain of losses, results are different. We find

some evidence of greater risk-seeking behavior (as predicted by prospect theory), as

being in the domain of losses for the round in Round 1 is associated with greater

chances of both below- and above-par scores on the current hole. We also find a

general decline in quality of play when scores are above par for the tournament in

Round 3, consistent with the cold hand (we restrict analysis to Rounds 1 and 3 for

reasons we explain in the Other C ontrols section). This effect occurs for both

relatively high- and low-ranked players, indicating it is not just driven by lack of

experience, skill, or lower stakes. While the exact mechanism behind the effect is

Stone and Arkes 455

somewhat unclear, the stronger result in Round 3 indicates it is unlikely to be caused

entirely by a few plausible alternative factors we discuss.

We also estimate the joint effects of the reference points across sets of holes

(rounds and half-rounds). Our estimates for these joint effects are precise and mostly

small (less than 0.1 strokes) for round-level effects. However, the joint effect for all

of Round 3 starting with a tournament score substantially above or below par is

around 0.2 strokes, which is similar to the round-level effects found by PS and

related papers. PS discuss how such a magnitude extrapolated for an entire tourna-

ment could imply annual losses of hundreds of thousands of dollars. Some of our

estimated half-round effects are larger on a per-hole basis, and all of these estimates

should be conservative due to the attenuation issue, and so while these estimates are

certainly not huge, they seem economically significant.

In summary, our results confirm the importance of reference points in real-world

behavior, as their effects are large enough to be observable and substantial despite

heterogeneity, unobserved expectations, and competing forces. Our results also pro-

vide evidence consistent with the existence and importance of momentum in perfor-

mance and the asymmetry of hot and cold momentum. In the final section, we discuss

reconciling our seemingly inconsistent results—the variation in dominance of risk,

effort, and cold-hand effects. Rather than being competing forces, cold-hand and

prospect-theory effects may actually, at least in part, cause one another.

Data and Prior Literature

Data

We use data fro m t he Professional Golfers’ Association (PGA)’s ShotLink data -

base available to the public online. The database inclu des inf ormatio n on every

shot at PGA tournament events from 2003 to the present, excluding the four

major tournaments. We use data t hrough the 2014 seaso n. Variables include the

tournament, course, round, hole, player, the score on the given hole, and, based

on laser-determined location, the location of the ball and distance to the hole for

each shot.

The unit of observation used for our analysis is player-year-tournament-round-

hole. We start with 3,549,186 observations. We deleted 627 rounds that were labeled

the fifth round (which occurs rarely in certain tournaments) and dropped 109 rounds

for which there were missing scores. We considered dropping the five tournaments

that comprise the golf play-offs, the FedEx Cup (the Tour Championships, Deutsche

Bank Championship, The Barclays, and the BMW Championship), but found results

were similar and prefer to keep these observations to maximize sample size. We

code hole number in the order in which holes are played by the player, so if a player

starts a round on Hole 10, which happens somewhat regularly, this hole is coded as

Hole 1, Hole 11 as 2, and so on.

456 Journal of Sports Economics 17(5)

Related Literature

Other papers (in addition to PS) that analyze prospect theory in golf are Sachau,

Simmering, and Adler (2012) and McFall (2015). The former finds that, consistent

with prospect theory, amateur golfers are more risk seeking after a bogey versus after

a par or birdie. In addition to analyzing the behavior of amateurs and not pros, their

paper also differs from ours by using survey data and not data on performance. The

latter finds that PGA players penalized a shot for hitting a par-5 tee shot out of

bounds are more likely to ‘‘go for the green’’ on the shot after the do-over drive,

indicating more risk-seeking behavior. This finding supports the importance of par

on the current hole as a reference point.

Another paper particularly closely related to ours is Smith et al. (2009) as they also

study the competition between prospect-theory and competing forces, including per-

ceived hot or cold streaks, but for a different context, (online) poker. Their main

finding is that play becomes more risky and aggressive after big losses, as predicted

by prospect theory.

Other economics papers using PGA tour data include the following. Brown

(2011) finds that the presence of Tiger Woods demotivated other top players, caus-

ing them to perform worse by 0.8 strokes per tournament. Kali, Pastoriza, and Plante

(2015) study the effects of nonmonetary incentives on performance and find that

players perform worse when these incentives (Ryder Cup points) are higher. They

find effects that range from 0.29 to 0.81 strokes per tournament. The study by

Rosenqvist and Skans (2015) is similar to our paper in that they study the effects

of confidence, closely related to the hot/cold hand, and indeed find evidence of

positive correlation in performance; their paper differs from ours in their focus on

performance variation across, and not within, tournaments.

Other papers in this genre differ from ours by analyzing effects for specific holes

or small sets of holes, and not entire rounds of play. Hickman and Metz (2015) show

that players choke on putts on the final holes of tournaments when monetary stakes

are higher, and Balsdon (2013) shows that risk strategies are affected by tournament

standing, more so when players are trying to make the cut rather than at the end of

tournaments. Ozbeklik and Smith (2014) show that risk strategies do vary wi th

tournaments but only analyze match play (a nonstandard format of tournament);

as the authors discuss, the incentives to change risk strategy are much stronger in the

match play format, which supports our assumption that risk attitudes, normatively,

should not change for the large majority of holes we analyze.

Theory and Identification

Prospect theory predicts that decisions are made based on salient changes or differ-

ences in a variable of interest, as compared to the so-called reference point. The

reference point is a psychologically plausible baseline and is typically arbitrary from

Stone and Arkes 457

a normative perspective. By contrast, standard economics of course assumes choices

are made based on the levels of relevant variables. Fo r example, if an agent is

proposed a ‘‘50–50 win US$150, lose US$100 gamble,’’ prospect theory assumes

the agent makes his choice ignoring the value of his initial wealth, since this value

is the natural reference point, and only the possible changes in wealth are relevant to

the decision. That is, prospect theory predicts the same choice whether initial wealth

is US$100 or US$1 million. Obviously, this factor could be normatively significant.

Prospect theory includes three other key elements, in addition to reference points:

(1) Agents evaluate a ‘‘value’’ (contra utility) function of outcomes versus the

reference point, whose slope is 2–3 times as steep in the domain of losses (negative

values of the difference/change variable) versus that of gains (positive values),

implying loss aversion; (2) there are diminishing marginal effects of both gains and

losses on the value function; and (3) ‘‘probability weighting,’’ in particular, very low

probabilities are weighted upward in calculating expectations. Since most of the

decisions we analyze do not involve very low probabilities, and in the interest of

parsimony, we do not consider probability weighting in our analysis (however, this

may be worth studying in future work).

We present an illustrative value function, applied to the context of golf for the

reference points that we consider, in Figure 1. Discussion of this figure is sufficient

x = -score relative to reference point (positive values = scores below par)

210–1–2

v(x)

–4

–3

–2

–1

Start hole in domain of losses:

Convex value function.

Highest returns to effort.

Start hole in domain of gains:

Concave value function.

Lowest returns to effort.

Start hole at reference pt:

Convex-concave value function.

Moderate returns to effort.

Figure 1. A prospect-theory value function for golfers (vðxÞ¼x

0:7

if x  0, ¼2:3ðxÞ

0:7

x < 0).

458 Journal of Sports Economics 17(5)

for understanding the key forces at play in our empirical setting; a detailed formal

analysis, which could include dynamic effects a cross holes, is beyond our scope.

Notethat,fornow,scoresbelow par are gains and thus imply positive va lue s of x

(score relative to par for a given reference point) and vice versa for scores above

par. For the moment, assume that x represents the score on just the current hole

being pla yed. The v alue function is steeper in the domain of losses due to loss

aversion, which implies the return to making a shot to avoid a loss (x ¼1) is

greater than the return to attaining a gain of the same size (x ¼ 1). PS pointed out

that this could cause golfers to put more effort into putts for par than for birdie (x ¼ 1)

and focus on this explanation for their result that par putts are indeed made more often.

Effort, in this context, refers to time and focus, which does plausibly vary at least to

some extent from hole to hole. The figure also shows that, since the value function is

convex to the left and concave to the right, golfers may be more risk seeking for shots

to avoid losses (or further losses) and more risk averse on shots to attain gains.

As discussed above, our work is motivated, first and foremost, by the existence of

several other plausible reference points in golf, beyond par on the current hole.

When a golfer steps up to the tee, he is not in the domain of gains or losses for that

particular hole. If he performed well on the previous hole, he might still feel the

emotional benefit from the gain and thus continue to act as if he is in the domain of

gains. And he might also consider to score relative to par for the round and/or for the

tournament. That is, x could represent the combined score from the current hole and

any of a few different sets of relevant previous holes.

If x > 0 at the start of a hole (the golfer is in the domain of gains), this could make

him exert less effort than otherwise, since the returns to further gains and costs of

losses are both relatively low. This would make him less likely to attain birdie, more

likely to attain bogey, and have a higher mean score on the current hole. At the same

time, x > 0 could also cause conservatism due to the concavity of the value function

(making the golfer less likely to attain both birdie and bogey, possibly also increas-

ing the mean score). If x < 0, the value function is convex and steep, which would

cause relatively high risk seeking and/or effort.

However, performance versus each of these reference points could also be indi-

cative of the extent to which a player is currently hot or cold. If a player performed

better than par on the last hole, his ability level might be temporarily elevated (due to

confidence, physical factors, etc.), implying he is likely to perform better on the

current hole as well (higher chance of birdie, lower chance of a bogey, and lower

mean score). The reverse would be true for players who performed poorly on recent

holes. We note that results consistent with the cold hand could be caused by injuries

or other physical problems.

Table 1 summarizes the predicted effects for various outcomes for the different

forces. Since each of the three mechanisms—hot/cold hands, effort, and risk pre-

ferences—makes a unique prediction of the combination of three types of outcomes

(above par, below par, and mean performance), we can, loosely speaking, identify

which of the three forces is dominant, by examining the effects of position versus the

Stone and Arkes 459

reference point for each of the three outcomes.

In the next section, we discuss the

methods we use for estimating these effects.

Empirical Methods

Model and Left-Hand Side (LHS) Variables

To test these predictions, it might be ideal to estimate a model that jointly analyzes

more than one of the outcomes, such as multinomial logit (ordered logit would not be

appropriate since the signs of the predicted effects are ambiguous). However, a

nonlinear model like this is computationally infeasible, given the size of our data

set and the need to include large sets of fixed effects (FEs). Instead, we use linear

probability models for each of the probability outcomes (bp and ap, binary variables

for below and above par on the current hole, respectively), and a linear model also

for the outcome of score (s).

Right-Hand Side (RHS) Variables

To capture the effects of par for the round and tournament as reference points, we

include, in all regressions, shots above par for the current round, rounda, shots below

par for round, roundb, and (after Round 1) shots above par and below par for the

tournament, tourna and tournb. Each of these variables thus takes a value of 0 or a

positive integer. While we focus on linear specifications, we also examine nonlinear

ones to examine ho w marginal effects and mechanisms may change as position

versus the reference-point changes.

To capture the effects of score versus par for the last hole (to address the reference

point of par for the rolling pair of the current and last holes), we include, in all

regressions, lastb and lasta (strokes below and above par on the last hole, respec-

tively). These variables are again weakly positive and integer valued. We explored

including analogous variables for the last two and last three holes, but they do not

qualitatively change the results and make the results harder to interpret and the

presentation much more complex.

Table 1. Hot Hand and Prospect-Theory Predictions.

Score at Start of Hole

Versus Reference Point Current Hole

Hot/Cold

Hand

Prospect

Theory: Effort

Prospect

Theory: Risk

Below par Pr (below par) þ 

Below par Pr (above par) þ 

Below par E (score) þþor 0

Above par Pr (below par) þ þ

Above par Pr (above par) þ þ

Above par E (score) þ ?

460 Journal of Sports Economics 17(5)

We include the last, round, and tourn variables in all models because they are

obviously correlated and act as controls for one another, and it would be difficult to

interpret results with these variables included in separate models. A natural alter-

native specification would be to include the scores for individual lagged holes as

separate regressors. We think our specification is preferable because it maps directly

to reference-point theory. It would be very difficult to interpret round and tourna-

ment reference-point effects with a lagged-hole score specification. However, one

should note that our specification does imply that, in general, the coefficients cannot

be interpreted as marginal effects, and it is more appropriate to consider the joint

marginal effects over a set of holes such as a round, as we discuss in Magnitudes

section.

Other Controls

In addition to the last=round=tourn variables acting as controls for one another,

there are many factors that could confound our analysis. An important one is tourna-

ment standing. Players may sometimes have strategic incentives to go for riskier or

more conservative shots to out-compete players with similar scores, and these

incentives could be correlated with our regressors of interest. In particular, players

likely vie to make the cut in the late holes of Round 2 (finish in the top half in order

to proceed to Rounds 3 and 4) and vie to out-compete players with similar scores in

the final holes of Round 4. Controlling for these incentives is very difficult and could

create a ‘‘bad control’’ problem, since standing at the start of a hole would be

affected by the regressors of interest. Thus, we limit our analysis to the rounds in

which strategic incentives such as these should be minimal, Rounds 1 and 3. Since

we have such a large sample, we can still obtain precise results for this limited scope.

Moreover, in auxiliary analyses reported in the working paper version of this article

(available on www.ssrn.com), we show results for Rounds 1 and 2 are similar, as are

results for 3 and 4.

Other important confounding variables are player and course heterogeneity. FEs

are clearly the ideal way to control for these factors. We include FEs for each hole-

day, which accounts for variation in difficulty of cours es, holes within courses,

placement of the pin, and weather across rounds (we cannot control for weather

changes within a round). Accounting for player heterogeneity is more difficult. We

consider several types of FEs: player-year, player-year-par value, player-year and

player-par value combined, and play er-course.

Each of these, except the latter,

accounts for player ability changing over time; player-course-year FEs would be

equivalent to player-tournament FEs, which would be very collinear with the tourn

variables. Accounting for par value is important since some players may be rela-

tively good at longer or shorter holes, and par values are not independent across

holes.

Including any of these FEs could cause a dynamic panel endogeneity bias, also

known as Nickell bias (Nickell, 1981). The lagged dependent variable is endogenous

Stone and Arkes 461

in FEs panel data models, and the last=round=tourn variables are highly correlated

with the lagged dependent variable in our models.

A standard way to address this

problem is to use other lags as instruments (e.g., the Arellano–Bond estimator), but

we cannot do this since we do not know which, if any, lags are exogenous. However,

this bias disappears as T (number of observations per FE group) grows large.

Thus,

we can minimize this issue by dropping FE groups with insufficiently large T.We

determine what cutoff to use for T empirically. We do this by checking results for

each type of FEs with progressively higher (minimum) T t hresh olds. If res ults

change substantially from one threshold to another, this means the results for the

lower threshold are very likely biased. If results are similar across thresholds, this

means the bias has likely become small.

We report a large set of these results in the working paper but only summarize

these results here in the interest of brevity. We find the Nickell bias is severe for

player-course FEs, as results change sharply as the threshold grows until a large

majority of the sample is lost. But the bias appears fairly mild for other FEs; results

are roughly stable for a range of thresholds that maintain the large majority of the

sample. Using our best judgment, we choose to proceed using player-year-par FEs

with a threshold of 50 observations per FE group. These FEs are more conservative

than both player-year and the player-year and player-par combination (since player-

year-par allows player ability to vary by par-value across years), but more aggressive

than player-year-par with higher thresholds, which lead to substantial sample losses

(over 30% for a threshold of 100) but largely similar results. Still, we should keep in

mind that we do not control for player-course effects.

The final (baseline) regression equation we use is thus:

¼ b

lastb

þ b

lasta

þ b

roundb

þ b

rounda

þ b

tournb

þ b

tourna

þ a

þ g

þ e

ð1Þ

The subscript i denotes the player-year-par FE group, and h denotes the course-

hole-day FE group. The dependent variable, y, is below par (bp, 0/1), above par (ap,

0/1), or the score (s, ..., 2, 1, 0, 1, 2, ... ). Standard errors are clustered by

player tournament. Summary statistics for our final sample, and variable definitions,

are provided in Table 2. Scores tend to be better for last and round variables in

Round 3, likely due to the better golfers making the cut to continue to Round 3.

Results

Main Results

Table 3 presents the various results for the baseline model. We interpret these results

using the predictions of Table 1 and focus our discussion on the significant results.

We do not present tests of joint significance for any of the variables because joint

462 Journal of Sports Economics 17(5)

Table 2. Summary Statistics for Final Sample.

Round 1 Round 3

Var. Definition Mean Min Max Mean Min Max

bp Dummy for s < 0 0.189 0 1 0.184 0 1

ap Dummy for s > 0 0.175 0 1 0.171 0 1

s Score versus par on current hole 0.006 3 12 0.006 39

lastb Strokes below par on last hole 0.198 0 3 0.215 0 3

lasta Strokes above par on last hole 0.193 0 12 0.178 0 9

roundb Strokes below par for round at start

of current hole

0.793 0 12 0.947 0 11

rounda Strokes above par for round at start

of current hole

0.760 0 18 0.606 0 15

tournb Strokes below par for tourn. at start

of current hole

n/a n/a n/a 3.839 0 27

tourna Strokes above par for tourn. at start

of current hole

n/a n/a n/a 0.613 0 25

Note. n ¼ 943,575 and 473,451 for all variables in Rounds 1 and 3, respectively.

Table 3. Main Results.

Round 1 Round 3

bp ap s bp ap s

lastb 0.0038***

(0.0010)

0.0008

(0.0010)

0.0040**

(0.0018)

0.0039***

(0.0014)

0.0031**

(0.0014)

0.0005

(0.0024)

roundb 0.0012***

(0.0004)

0.0002

(0.0003)

0.0009

(0.0006)

0.0023***

(0.0005)

0.0008

(0.0005)

0.0034***

(0.0009)

tournb 0.0013***

(0.0002)

0.0000

(0.0002)

0.0011***

(0.0004)

lasta 0.0004

(0.0009)

0.0021**

(0.0009)

0.0019

(0.0016)

0.0018

(0.0013)

0.0013

(0.0014)

0.0037

(0.0023)

rounda 0.0012***

(0.0003)

0.0013***

(0.0004)

0.0014**

(0.0006)

0.0008

(0.0006)

0.0004

(0.0007)

0.0002

(0.0011)

tourna 0.0007

(0.0004)

0.0015***

(0.0005)

0.0033***

(0.0008)

Adj R

.099 .052 .102 .087 .048 .093

n 943,575 943,575 943,575 473,451 473,451 473,451

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player-tournament.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

Stone and Arkes 463

effects are analyzed in the Magnitudes section. We first discuss the variables mea-

suring gains and then those for losses.

All five of the variables increasing in gains versus reference points (lastb, roundb,

and tournb, with the first two included in the models for both Rounds 1 and 3) have

significant negative effects on the probability of scoring below par on the current

hole. This is quite strong evidence that prospect-theory effects (effort and/or risk)

dominate hot-hand effects for this outcome. As players move further into the domain

of gains for each reference point, they do not exert as much effort or play more

conservatively on subsequent holes.

In Round 3, an increase in lastb is also associated with a lower probability of

scoring above par on the current hole. This result, together with the lower probability

of birdie, is consistent with the prospect-theory risk prediction that golfers become

more conservative when they have gains they wish to preserve. The evidence for a

lower above-par probability in the domain of gains is insignificant for the other gains

variables but usually directionally consistent with conservatism.

Turning to the outcome of average score, three of the five gains variables have

significant po sitive coefficients, and there are no significant negative ones. The

positive sign is consistent with both (prospect-theory) risk and effort predictions

and inconsistent with the hot-hand prediction.

In t he domain of losses, three of the five variables have signi ficant positive

effects on the probability o f scori ng above par on the current hole: lasta, rounda in

the first round and tourna in Round 3. There are no significant negative effects for

this outcome. In Round 1, being above par for the round also is associated with a

higher probability of a score below par. The combination of these two effects

(higher chances of scores below and above par) i s consiste nt with the risk predic-

tion (greater risk seeking). In Round 3, tourna is also associated with higher

average scores (and no increase in the chance of birdie). The combination of

effects for this variable is most consistent with the cold hand. It is unlikely that

this result is driven by within (current) round weather changes, since tournament

scores during Round 3 are mostly based on performance in the previous two days.

Player-course effects should affect performance in both rounds in a similar way, so

these effects likely do not explain the stronger cold-hand results in Round 3. I t is

possible that these results are influenced by injuries (or other factors), but these

likely should also cause similar results in Round 1. Therefore, we interpret our

results as implying that cold-hand forces are at least rela tively strong in Round 3

when tournament scores are high.

We explore these results further in several ways. First, we examine nonlinear

specifications. As discussed above, diminishing marginal sensitivity is an important

part of prospect theory. Thus, the benefit of an additional gain declines as scores fall

further below par. But this means the cost of an additional loss also declines. This

could cause golfers to actually reduce effort as scores rise above par (for the round or

tournament). On the other hand, if golfers integrate the value they could receive by

eliminating losses on future holes, effort could continue to grow as scores rise further

464 Journal of Sports Economics 17(5)

above par. Moreover, changes in the curvature of the value function at different

values of x could cause changes in risk attitudes.

We use two nonlinear specifications; first, a relatively flexible quadratic, allow-

ing marginal effects to grow, shrink, or neither. These results are in Table 4. The

roundb linear terms for outcomes bp and ap are both negative in Round 1, indicating

risk effects that were less clear in Table 3. However, the squared term is positive and

significant at the 10% leve l, indicating that the ap (bogey) effect disappears as

roundb increases. Thus, the combination of these results implies risk (effort) effects

are stronger for scores closer to (further from) the reference point. Moreover, in

Round 3 the roundb-square term is significantly positive for ap and s,further

indicating that effort effects are relatively dominant for higher values of roundb.

Regarding the domain of losses, for Round 1 the rounda-square terms for the LHS

variables ap and s are both positive, indicating the marginal cold-hand and/or risk

effects grow as a golfer moves further into the domain of losses and that effort

effects may be dominant for scores just above par. That is, golfers may successfully

Table 4. Quadratics.

Round 1 Round 3

bp ap s bp ap s

lastb 0.0037***

(0.0010)

0.0009

(0.0010)

0.0040**

(0.0018)

0.0038***

(0.0014)

0.0028**

(0.0014)

0.0002

(0.0024)

roundb 0.0015*

(0.0008)

0.0017**

(0.0008)

0.0008

(0.0014)

0.0029**

(0.0011)

0.0016

(0.0011)

0.0001

(0.0019)

roundbsq 0.0001

(0.0002)

0.0003*

(0.0002)

0.0003

(0.0003)

0.0001

(0.0002)

0.0005**

(0.0002)

0.0007*

(0.0004)

tournb 0.0019***

(0.0005)

0.0002

(0.0005)

0.0019**

(0.0009)

tournbsq 0.0000

(0.0000)

0.0000

(0.0000)

0.0001

(0.0001)

lasta 0.0003

(0.0009)

0.0022**

(0.0009)

0.0022

(0.0016)

0.0019

(0.0013)

0.0014

(0.0014)

0.0038

(0.0023)

rounda 0.0021***

(0.0007)

0.0007

(0.0007)

0.0021*

(0.0013)

0.0013

(0.0012)

0.0018

(0.0013)

0.0032

(0.0022)

roundasq

0.0002

(0.0001)

0.0004***

(0.0001)

0.0007***

(0.0002)

0.0002

(0.0002)

0.0004*

(0.0002)

0.0006

(0.0004)

tourna 0.0012

(0.0009)

0.0028***

(0.0009)

0.0057***

(0.0016)

tournasq 0.0001

(0.0001)

0.0002*

(0.0001)

0.0003*

(0.0002)

Adj R

.099 .052 .102 .087 .048 .093

n 943,575 943,575 943,575 473,451 473,451 473,451

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

Stone and Arkes 465

‘‘try harder’’ when scores are just above par in Round 1, but this extra effort does not

occur, or is less effective, as scores go further over par. For Round 3, again there is

no evidence of any type of increase in the chance of birdie. The cold-hand effect

indicated by higher scores for the tournament incre ases as scores grow, but the

marginal effect declines (the tourna-square terms are negative and significant for

outcomes of above par and average score).

Next, we examine a specification in which the RHS variables are recoded as

dummies equal to 1 if the original regressors took strictly positive values. This of

course is a much less flexible specification but is appropriate if golfers focus on

whether or not they are in the domain of gains or losses, and not ‘‘where’’ they are

in either of these domains. These results are presented in Table 5. There are some

intriguing results here that did not emerge in Table 3. In particular, being below par for

the round (roundbd ¼ 1) decreases the chance of ap ¼ 1inRound1anddoesnot

increase the chance of ap ¼ 1 in Round 3. These results differ from analogous results

for roundb reported in Tables 3 and 4. These new (Table 5) results are more indicative

of risk effects than effort effects. Since the Table 6 results are relatively highly

influenced by lower values of roundb, and the Tables 3 and 4 results more by higher

values, this comparison supports the conclusion that risk effects are stronger for lower

values of roundb, and effort effects stronger for higher values. For the domain of

losses, the results are also supportive of the interpretation of Table 5 discussed above.

A potential critique of our analysis is that par is not always the most relevant

reference point that players expect better scores on relatively easy holes and worse

Table 5. Score Above/Below Reference Point Dummy Variables.

Round 1 Round 3

bp ap s bp ap s

lastbd 0.0038***

(0.0011)

0.0009

(0.0011)

0.0042**

(0.0018)

0.0040***

(0.0015)

0.0027*

(0.0015)

0.0002

(0.0025)

roundbd 0.0031***

(0.0011)

0.0023**

(0.0010)

0.0001

(0.0018)

0.0070***

(0.0015)

0.0000

(0.0015)

0.0060**

(0.0025)

tournbd 0.0036

(0.0022)

0.0010

(0.0023)

0.0017

(0.0040)

lastad 0.0002

(0.0011)

0.0023**

(0.0011)

0.0027

(0.0019)

0.0009

(0.0016)

0.0021

(0.0016)

0.0038

(0.0028)

roundad 0.0024**

(0.0011)

0.0005

(0.0011)

0.0001

(0.0019)

0.0019

(0.0016)

0.0018

(0.0016)

0.0035

(0.0028)

tournad 0.0019

(0.0024)

0.0055**

(0.0026)

0.0103**

(0.0044)

Adj R

.099 .052 .102 .086 .048 .093

n 943,575 943,575 943,575 473,451 473,451 473,451

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

466 Journal of Sports Economics 17(5)

scores for tougher holes.

Consequently, a bogey on a difficult hole may be per-

ceived as a relatively small loss and a birdie on an easy hole a small gain. This theory

of expectations-dependent reference points was first developed formally by Koszegi

and Rabin (2006). PS analyze this theory in their context and find support for it.

We conduct a similar analysis, subtracting the mean score for each hole–tourna-

ment–round from each player’s score and then reconstruct the last, round, and tourn

variables with these difficulty-adjusted scores. The new variables are referred to as

adjlast, adjround, and adjtourn. For example, if a golfer birdied Hole 1 in Round 1,

but every golfer birdied that hole, then adjlastb ¼ 0 on Hole 2, while lastb ¼ 1. The

results are in Table 6. The point estimates are indeed generally a bit stronger than

their analogs in Table 3, supporting the theory, and largely consistent with the

interpretation of the original results discussed above. A difference worth noting is

that adjroundb has a significant positive effect on ap in Round 3 (while roundb did

not), supporting the effort prediction of that situation.

Hole and Player Heterogeneity

In order to better understand these results, we disaggregate the sample in several

ways. First, we split it into ‘‘front nine’’ holes (first nine holes played in the current

round, which sometimes are the course’s Holes 10–18) and ‘‘back nine’’ holes and

estimate the baseline models for each of these subsamples.

The results, in Table 7,

indicate that lastb effects are larger on the back nine, and roundb effects greater on

Table 6. Koszegi and Rabin (2006) Reference Points.

Round 1 Round 3

bp ap s bp ap s

adjlastb 0.0038***

(0.0012)

0.0016

(0.0012)

0.0053***

(0.0020)

0.0053***

(0.0017)

0.0038**

(0.0017)

0.0002

(0.0029)

adjroundb 0.0014***

(0.0004)

0.0003

(0.0004)

0.0010

(0.0007)

0.0025***

(0.0006)

0.0013**

(0.0006)

0.0041***

(0.0010)

adjtournb 0.0015***

(0.0002)

0.0002

(0.0002)

0.0013***

(0.0004)

adjlasta 0.0003

(0.0010)

0.0026**

(0.0010)

0.0029

(0.0018)

0.0026*

(0.0015)

0.0006

(0.0015)

0.0037

(0.0025)

adjrounda 0.0012***

(0.0004)

0.0013***

(0.0004)

0.0015**

(0.0007)

0.0006

(0.0006)

0.0011*

(0.0006)

0.0009

(0.0011)

adjtourna 0.0014***

(0.0005)

0.0021***

(0.0005)

0.0043***

(0.0009)

Adj R

.099 .052 .102 .087 .048 .093

n 943,575 943,575 943,575 473,451 473,451 473,451

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament. See text for definition of adjusted variables.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

Stone and Arkes 467

the front nine. This is consistent with the idea that score relative to par for the round

could be especially salient early in the round, when one has recently started from the

reference-point score of 0. Later in the round, golfers may become accustomed to

being below par and adjust their reference point (and focus more on the most recent

holes). Thus, this is additional evidence of expectations-adjusted reference points.

Table 7 results also suggest that cold-hand effects (for the round and tournament)

accelerate on the back nine.

Table 7. Front Nine/Back Nine.

Round 1 Round 3

bp ap s bp ap s

Front nine

lastb 0.0026

(0.0016)

0.0013

(0.0016)

0.0041

(0.0027)

0.0044**

(0.0022)

0.0026

(0.0022)

0.0007

(0.0037)

roundb 0.0034***

(0.0008)

0.0001

(0.0008)

0.0028**

(0.0014)

0.0043***

(0.0012)

0.0009

(0.0011)

0.0035*

(0.0020)

tournb 0.0015***

(0.0004)

0.0004

(0.0004)

0.0019***

(0.0007)

lasta 0.0007

(0.0014)

0.0033**

(0.0015)

0.0032

(0.0025)

0.0002

(0.0021)

0.0013

(0.0022)

0.0007

(0.0037)

rounda 0.0008

(0.0008)

0.0004

(0.0008)

0.0008

(0.0014)

0.0007

(0.0013)

0.0030**

(0.0013)

0.0040*

(0.0022)

tourna 0.0001

(0.0008)

0.0015*

(0.0008)

0.0024*

(0.0014)

Adj R

.100 .051 .103 .087 .049 .095

n 443,410 443,410 443,410 221,447 221,447 221,447

Back nine

lastb 0.0062***

(0.0014)

0.0007

(0.0014)

0.0061**

(0.0024)

0.0057***

(0.0019)

0.0026

(0.0020)

0.0014

(0.0033)

roundb 0.0011**

(0.0004)

0.0003

(0.0004)

0.0008

(0.0007)

0.0008

(0.0007)

0.0002

(0.0007)

0.0010

(0.0011)

tournb 0.0015***

(0.0004)

0.0002

(0.0004)

0.0014**

(0.0006)

lasta 0.0004

(0.0013)

0.0007

(0.0013)

0.0016

(0.0022)

0.0028

(0.0018)

0.0012

(0.0018)

0.0023

(0.0032)

rounda 0.0006

(0.0004)

0.0027***

(0.0004)

0.0039***

(0.0007)

0.0003

(0.0007)

0.0022***

(0.0008)

0.0022

(0.0014)

tourna 0.0005

(0.0006)

0.0011*

(0.0007)

0.0029***

(0.0011)

Adj R

.101 .052 .104 .088 .047 .092

n 500,165 500,165 500,165 252,004 252,004 252,004

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament.

*, **, and *** denote 10%, 5%, and 1% significance.

468 Journal of Sports Economics 17(5)

In Table 8, we present results for an analysis split by par value per hole. Effects

are generally stronger for the longer holes, consistent with the effects compounding

across shots. In particular, in Round 3, the roundb and tourna positive effects on

score increase for larger par values. We might expect, a priori, the risk effects to be

relatively visible for par-5 holes, as players then face the strategic choice of whether

to ‘‘go for the green’’ on the second shot. And indeed, the rounda effects in Round 1,

consistent with greater risk seeking, are strongest for par-5 holes. However, the

separate par-5 analysis does not reveal changes in other risk attitudes obscured by

the more aggregated analysis.

We next split our sample by official world golf ranking (for the end of the prior

calendar year) and present results in Table 9.

Results are fairly similar for higher and

lower ranked players. Important differences are that roundb effects in Round 1, and

tournb effects in Round 3, are stronger for the worse ranked players, consistent with

worse ranked players having expectations-based reference points closer to par in Round 1

and for the tournament in Round 3. However, we also find roundb effects for better ranked

players in Round 3. Perhaps this is because, after making the cut, their initial expectations/

goals have been met, and the reference point of par for the current round becomes more

relevant. The tourna effects in Round 3 are similar for both groups of players.

Drives, Approaches, and Putts

There are three main types of golf shots: drives, approaches, and putts. We analyze

the effects of reference points on these different shots to gain further insight into the

various types of behavior.

For drives, we use two LHS variables: distance and a dummy for whether or not

the drive landed on the fairway (0/1). Improvements in both distance and fairway

accuracy would indicate greater effort, while different signs for these effects would

indicate a change in risk attitude (longer drives and lower accuracy would indicate a

riskier strategy and vice versa for a more conservative strategy). We exclude holes

that have an average drive distance of less than 260 yards, as these are likely holes in

which drive distance is shortened by the layout of the hole.

For approaches, we use a sample of the first shot on a given hole for which the ball

is between 100 and 200 yards from the hole and on the fairway. We examine three

LHS variables, average distance from the hole after the shot, a dummy for ‘‘close’’

distance (within 8 feet of the hole), and another dummy, ‘‘far’’ (20 feet or more from

the hole). Results are similar with different cutoff values. These cutoff values defin-

ing close and far are based on the probability of making a putt from a given distance:

8 feet is a cutoff for which anything closer has, on average, a 50% or higher prob-

ability of having the putt made, while putts beyond 20 feet have less than a 15%

probability of being made. These three outcomes map to the three outcomes used for

the analysis of hole-level results, representing mean performance, and probabilities

of above- and below-average performance, respectively. We include a seventh-order

polynomial for distance from hole to account for difficulty of the shot.

Stone and Arkes 469

Table 8. Par 3/4/5.

Round 1 Round 3

bp ap s bp ap s

Par 3

lastb 0.0026

(0.0019)

0.0048**

(0.0021)

0.0058*

(0.0033)

0.0035

(0.0028)

0.0058*

(0.0030)

0.0043

(0.0048)

roundb 0.0008

(0.0007)

0.0005

(0.0007)

0.0003

(0.0012)

0.0005

(0.0011)

0.0010

(0.0011)

0.0001

(0.0018)

tournb 0.0022***

(0.0005)

0.0009*

(0.0005)

0.0027***

(0.0008)

lasta 0.0018

(0.0017)

0.0061***

(0.0020)

0.0047

(0.0032)

0.0003

(0.0027)

0.0037

(0.0030)

0.0032

(0.0048)

rounda 0.0010

(0.0006)

0.0014*

(0.0008)

0.0024*

(0.0013)

0.0003

(0.0012)

0.0023

(0.0015)

0.0028

(0.0023)

tourna 0.0011

(0.0009)

0.0006

(0.0011)

0.0006

(0.0018)

Adj R

.024 .039 .049 .026 .038 .049

n 216,353 216,353 216,353 99,016 99,016 99,016

Par 4

lastb 0.0033**

(0.0013)

0.0001

(0.0014)

0.0028

(0.0023)

0.0044***

(0.0017)

0.0026

(0.0017)

0.0007

(0.0029)

roundb 0.0011**

(0.0004)

0.0001

(0.0005)

0.0009

(0.0008)

0.0028***

(0.0006)

0.0012*

(0.0006)

0.0040***

(0.0011)

tournb 0.0008***

(0.0003)

0.0002

(0.0003)

0.0003

(0.0005)

lasta 0.0004

(0.0011)

0.0011

(0.0013)

0.0007

(0.0021)

0.0014

(0.0016)

0.0019

(0.0017)

0.0044

(0.0028)

rounda 0.0009**

(0.0004)

0.0012**

(0.0005)

0.0014*

(0.0008)

0.0011

(0.0007)

0.0002

(0.0008)

0.0014

(0.0013)

tourna 0.0009*

(0.0005)

0.0015***

(0.0006)

0.0034***

(0.0009)

Adj R

.041 .049 .066 .048 .047 .070

n 577,792 577,792 577,792 332,538 332,538 332,538

Par 5

lastb 0.0082**

(0.0036)

0.0024

(0.0022)

0.0064

(0.0051)

0.0007

(0.0067)

0.0002

(0.0039)

0.0007

(0.0097)

roundb 0.0020*

(0.0012)

0.0004

(0.0007)

0.0018

(0.0017)

0.0028

(0.0024)

0.0016

(0.0014)

0.0073**

(0.0034)

tournb 0.0032***

(0.0010)

0.0000

(0.0006)

0.0031**

(0.0015)

lasta 0.0015

(0.0028)

0.0003

(0.0018)

0.0028

(0.0042)

0.0084

(0.0057)

0.0079**

(0.0037)

0.0136

(0.0084)

(continued)

470 Journal of Sports Economics 17(5)

For putts, we use a binary dependent variable (miss/ make) and controls of

seventh-order polynomial for distance and an interaction of distance groups and

elevation-change-to-hole decile; (own) putt number for the hole; dummies for

whether the putt is for eagle, birdie, bogey, or double-bogey or greater. Our results

for these variables, which are not reported, are similar to those of PS but slightly

smaller in magnitude.

These results are in Table 10. In general, results are stronger for drives and

approaches, indicating reference points based on scores from past holes are more

important for initial shots on the hole, and the reference point of par on the current

hole is most important for the final shots on the hole (putts), supporting the idea that

more salient reference points have greater impacts. lastb is associated with a shorter

drive in both rounds, and lasta a longer drive in Round 3. These results could be

evidence of risk or effort effects. In Round 3, being above par for the round also

leads to riskier (longer and less accurate) drives. Being below par for the round in

Round 3 leads to shorter drives and worse all-around approach shots, consistent with

effort effects.

Scores below par for the round in Round 1, and for the tournament in Round 3, are

both associated with riskier drives (longer distance and lower accuracy). The com-

bination of these effects (greater risk but worse ‘‘quality’’) is not consistent with any

of the predictions from Table 1 but could result from the hot-hand bias, that is, a

misperception of being hot leading to riskier shots. These same variables, lastb and

tournb, are also associated with lower probabilities of making putts, and (all-around)

inferior approach shots in Round 3. Since our approach shot sample is limited to

shots taken from the fairway, this means the drive leading to the approach must have

been accurate. This suggests an interpretation of the results in which golfers who are

playing well for the tournament take aggressive drives, then ‘‘shirk’’ on the approach

shot and perhaps the putt as well (the putt effects could also be due to conservatism).

This contrast in behavior across types of shots for a given reference-point effect is

somewhat puzzling. Perhaps taking aggressive drives is simply highly appealing or

Table 8. (continued)

Round 1 Round 3

bp ap s bp ap s

rounda 0.0028***

(0.0010)

0.0018**

(0.0007)

0.0001

(0.0016)

0.0006

(0.0026)

0.0016

(0.0017)

0.0028

(0.0039)

tourna 0.0033

(0.0020)

0.0040***

(0.0014)

0.0095***

(0.0031)

Adj R

.074 .027 .074 .074 .029 .074

n 149,430 149,430 149,430 41,897 41,897 41,897

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

Stone and Arkes 471

fun (there is a saying that you ‘‘drive for show, putt for dough’’). It is also possible

that the decline in all-around performance on approach shots and putts when in the

domain of gains could be caused by the hot-hand bias causing golfers to exert less

effort due to overconfidence.

In the final analysis of this section, we exploit the disaggregation of shots within a

hole to better separately identify momentum and prospect-theory effects. We do this

by controlling for lagged performance for the particular shot type, in addition to the

Table 9. Player Rank.

Round 1 Round 3

bp ap s bp ap s

Rank < 200

lastb 0.0027*

(0.0015)

0.0011

(0.0014)

0.0013

(0.0025)

0.0045**

(0.0018)

0.0036**

(0.0018)

0.0016

(0.0031)

roundb 0.0011**

(0.0005)

0.0006

(0.0005)

0.0002

(0.0008)

0.0016**

(0.0007)

0.0013*

(0.0006)

0.0034***

(0.0012)

tournb 0.0012***

(0.0003)

0.0004

(0.0003)

0.0002

(0.0005)

lasta 0.0014

(0.0014)

0.0023*

(0.0014)

0.0014

(0.0024)

0.0001

(0.0018)

0.0013

(0.0018)

0.0013

(0.0030)

rounda 0.0007

(0.0005)

0.0011**

(0.0005)

0.0013

(0.0009)

0.0011

(0.0008)

0.0006

(0.0009)

0.0002

(0.0015)

tourna 0.0011**

(0.0005)

0.0016***

(0.0006)

0.0034***

(0.0010)

Adj R

.102 .049 .103 .091 .044 .093

n 466,410 466,410 466,410 288,930 288,930 288,930

Rank  200

lastb 0.0047***

(0.0015)

0.0029*

(0.0015)

0.0066***

(0.0026)

0.0030

(0.0023)

0.0027

(0.0024)

0.0010

(0.0040)

roundb 0.0014***

(0.0005)

0.0001

(0.0005)

0.0018*

(0.0009)

0.0036***

(0.0009)

0.0001

(0.0009)

0.0034**

(0.0015)

tournb 0.0016***

(0.0004)

0.0011**

(0.0004)

0.0028***

(0.0007)

lasta 0.0008

(0.0012)

0.0021

(0.0013)

0.0029

(0.0023)

0.0048**

(0.0021)

0.0019

(0.0022)

0.0078**

(0.0038)

rounda 0.0017***

(0.0004)

0.0013***

(0.0005)

0.0012

(0.0008)

0.0005

(0.0009)

0.0006

(0.0010)

0.0001

(0.0018)

tourna 0.0004

(0.0007)

0.0010

(0.0008)

0.0033**

(0.0014)

Adj R

.097 .053 .100 .083 .049 .091

n 477,165 477,165 477,165 184,521 184,521 184,521

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

472 Journal of Sports Economics 17(5)

standard last=round=tourn variables. Th e lagged shot-type performance controls

provide a relatively direct measure of recent quality of play for that particular shot

type and should thus capture momentum effects in a more pure way. Consequently,

the last=round=tourn should now more directly capture the more decision-theoretic

prospect-theory effects. We conduct this analysis as a final extension rather than as

the pr imary analysi s of the paper for two reasons. First, and foremost, we are

ultimately interested in total effects on scores. If the phenomena interact, exacerbate,

or mitigate one another with respect to these effects, so be it. The net effects are still

most important, and the hole-level score-based analysis implicitly addresses this

bottom-line issue of s core effect s most directly. Second, controlling for past

Table 10. Drives/Approaches/Putts.

Drive

Distance

Fairway

(0/1)

Distance

After

Approach

Close (0/1)

Approach

Far (0/1)

Make Putt

(0/1)

Round 1

lastb 0.0921

(0.0575)

0.0025

(0.0016)

1.8758*

(1.0523)

0.0036*

(0.0021)

0.0021

(0.0022)

0.0009

(0.0007)

roundb 0.0623***

(0.0214)

0.0014***

(0.0005)

0.0710

(0.3748)

0.0013*

(0.0007)

0.0003

(0.0007)

0.0011***

(0.0002)

lasta 0.0465

(0.0500)

0.0019

(0.0013)

0.4200

(0.9568)

0.0003

(0.0018)

0.0003

(0.0019)

0.0005

(0.0006)

rounda 0.1660***

(0.0203)

0.0006

(0.0005)

0.2675

(0.3764)

0.0001

(0.0007)

0.0008

(0.0007)

0.0002

(0.0002)

.554 .065 .131 .078 .112 .603

n 683,906 683,906 327,089 327,089 327,089 1430,394

Round 3

lastb 0.1708**

(0.0731)

0.0018

(0.0020)

1.1477

(1.2882)

0.0021

(0.0027)

0.0000

(0.0028)

0.0010

(0.0009)

roundb 0.2246***

(0.0301)

0.0014*

(0.0007)

1.5876***

(0.4868)

0.0033***

(0.0010)

0.0033***

(0.0010)

0.0005

(0.0003)

tournb 0.1582***

(0.0144)

0.0011***

(0.0003)

0.5946***

(0.2205)

0.0011**

(0.0005)

0.0017***

(0.0005)

0.0011***

(0.0002)

lasta 0.1608**

(0.0686)

0.0011

(0.0018)

3.0356**

(1.2737)

0.0063**

(0.0025)

0.0034

(0.0026)

0.0015*

(0.0008)

rounda 0.2674***

(0.0347)

0.0015*

(0.0008)

1.1888**

(0.6000)

0.0016

(0.0012)

0.0013

(0.0012)

0.0001

(0.0004)

tourna 0.1162***

(0.0282)

0.0005

(0.0006)

0.7779*

(0.4525)

0.0009

(0.0009)

0.0012

(0.0009)

0.0005

(0.0003)

.580 .071 .143 .083 .113 .613

n 396,249 396,249 191,519 191,519 191,519 820,068

Note. All models estimated by ordinary least squares with player-year-par and hole-day fixed effects, and

standard errors clustered by player tournament. See text for other controls.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

Stone and Arkes 473

performance for a particular shot type opens up a new can of methodological worms.

In fact, we limit this new analysis to just one shot type, drives, since measuring past

performance for other shot types is more difficult. Despite these issues, however, we

feel this additional analysis clarifies and complements the main results.

We control for recent performance on drives in two ways: (1) with lagged distance

(lagdist)andfairway(lagfairway) and (2) with a moving average of distance on the

last three holes (MAdist) and fairway accuracy on the last three holes (MAfairway)

calculated using just one or two values when they are the only ones available (for holes

2–3 of a round and when one or more of the last three holes was a par 3 or dropped par

4 or 5). Results are in Table 11. We include the results for drives reproduced from

Table 10 for ease of direct comparison. The coefficients for the last=round=tourn

variables with either type of lagged drive controls are typically stronger and direc-

tionally the same as the original coefficients. This supports the interpretations dis-

cussed above, in particular, the point that the estimated effects were mitigated

somewhat by countervailing momentum effects. Moreover, the coefficients for the

lagged drive performance variables are generally consistent with momentum. How-

ever, we cannot tell to what extent they are driven by the hot versus cold hands.

Table 11. Additional Analysis of Drives (Round 3 only).

LHS ¼ Distance LHS ¼ Fairway (0/1)

lastb 0.1506*

(0.0768)

0.2267**

(0.0896)

0.2116***

(0.0770)

0.0023

(0.0021)

0.0001

(0.0024)

0.0021

(0.0021)

roundb 0.2190***

(0.0316)

0.2339***

(0.0370)

0.2191***

(0.0304)

0.0016**

(0.0008)

0.0015

(0.0010)

0.0016**

(0.0008)

tournb 0.1415***

(0.0151)

0.1449***

(0.0172)

0.1322***

(0.0144)

0.0012***

(0.0004)

0.0013***

(0.0004)

0.0011***

(0.0004)

lasta 0.1630**

(0.0728)

0.2699***

(0.0916)

0.2162***

(0.0731)

0.0008

(0.0019)

0.0027

(0.0024)

0.0005

(0.0019)

rounda 0.2691***

(0.0365)

0.2673***

(0.0421)

0.2709***

(0.0351)

0.0024***

(0.0009)

0.0029***

(0.0011)

0.0023**

(0.0009)

tourna 0.1025***

(0.0296)

0.1116***

(0.0330)

0.0961***

(0.0283)

0.0004

(0.0007)

0.0006

(0.0008)



0.0004

(0.0007)

lagdist 0.0351***

(0.0023)

0.0000

(0.0001)

lagfairway 0.2997***

(0.0789)

0.0065***

(0.0021)

MAdist 0.0593***

(0.0026)

0.0001

(0.0001)

MAfairway 0.4458***

(0.0886)

0.0053**

(0.0024)

Adj R

.577 .588 .578 .071 .070 .071

n 356,728 245,341 356,104 356,728 245,341 356,104

Note. LHS ¼ left hand side. All models estimated by ordinary least squares with player-year-par and hole-

day fixed effects, and standard errors clustered by player tournament. See text for other controls.

*, **, and *** denote 10%, 5%, and 1% significance, respectively.

474 Journal of Sports Economics 17(5)

Magnitudes

Thus far we have omitted discussion of the magnitudes of effects due to the com-

plexity caused by interactions and dynamic effects across holes. When a golfer gets,

say, a bogey on one hole, this certainly increases the value of lasta for the next hole,

but may increase, decrease, or have no effect on all of the round and tourn regres-

sors. Moreover, the bogey also affects the values of these regressors for future holes.

In general, it is misleading to interpret the magnitude of any individual estimate

alone as a marginal effect and is more appropriate to estimate the reference-points’

joint effect for a larger set of holes, such as a round. Thus, in this section we estimate

the expected total score for various sets of holes, for example, E½

i¼1

ðs

; ...; s

i1

Þ

for the expected total score for a particular round with reference-point effects and

compare this to the expected score under the null of scores across holes.

We use the following simulation-based p rocedure. For simplicity, and to be

consistent with the empirical models used for the LHS variables ap and bp in Results

section, we restrict the score for each hole to be 1, 0, or 1. We assume

Prðs

¼ 1Þ¼0:165 and Prðs

¼1Þ¼0:195, the approximate empirical probabil-

ities of scores below and above par on the first hole of a tournament. Results are

similar when we use small variants on these numbers. After drawing s

for Round 1

from this distribution, we then draw s

with distribution Prðs

¼ 1Þ¼

and

Prðs

¼1 Þ¼

, where

and

are calculated using the baseline

results (those reported in Table 3), replacing the FEs with the ‘‘unconditio nal’’

bogey and birdie probabilities, 0.165 and 0.195, respectively. The distributions for

later holes are calculated in the same way, conditioning on scores from all prior

holes.

We estimate the effects for various scenarios for all of Round 1, Round 3,

and the back nine for each of those rounds (we analyze the back nine separately

because only the back nine’s starting round score can vary).

This procedure yields a distribution across simulation runs of round- or half-

round-level effects for a g iven set of point estimates for the last=round=tourn

coefficients. The mean of this distribution should be an unbiased estimate of the

expected (half)-round-level joint effect (of the last=round=ts variables). We find by

‘‘guess and check’’ that 100,000 simulation runs are necessary to get the mean of this

distribution to stabilize within 0.01 strokes across nearly all simulation runs, and we

therefore use this number of runs for this procedure.

This distribution across simulation runs is driven by randomness in hole-level

score draws and does not yield a confidence interval for the round-level effect. To

obtain such a confidence interval, we need to use the estimated sampling distribution

of the coefficient estimates. A naive way to do this would be to just use lower and

upper bounds of the (analogous) confidence intervals for each of the coefficients, but

this would ignore the joint distribution of the coefficient estimates. Thus, instead we

take draws from a multivariate normal distribution with means equal to the coeffi-

cient estimates, and covariance of the estimated covariance matrix, at the start of

Stone and Arkes 475

each simulation. We conduct a different 100,000 run (half)-round-level simulation

for each of these draws, storing each mean effect. We repeat this procedure 1,000

times, giving us a distribution of 1,000 estimates of (half)-round-level effects reflect-

ing sampling uncertainty. We then use the 2.5th and 97.5th percentiles from this

distribution to construct a 95% confidence interval for the average (half) round

effect.

To summarize, the procedure for a round-level effect is as follows (the half-round

procedure is analogous):

1. draw one coefficient vector from two multivariate normal distributions, one

with parameters equal to the bp model estimates, and one using the ap model

estimates,

2. draw a score of 1, 0, or þ1 for Hole 1 using probabilities 0.195, 0.640, and

0.165, respectively,

3. use this score and coefficients from Step 1 to calculate predicted probabilities

for Hole 2,

and

, and use these to draw a score for Hole 2,

4. continue this procedure for Holes 3–18; sum scores for Holes 1–18 to get a

round score,

5. repeat Steps 2–4 100,000 times and store the mean to obtain a precise esti-

mate of the mean round score for the coefficients from Step 1, and

6. repeat Steps 1–5 1,000 times to obtain an estimated sampling distribution of

this mean round score. Use the mean of this distribution as the point estimate,

and the 2.5th and 97.5th percentiles as the 95% confidence interval, for the

expected round-level score.

Results are presented in Table 12. We report the implied joint effects of reference

points compared to the null; these are equal to the differences between our estimated

half round/round scores and the corresponding expected scores under the null of

i.i.d. scores across holes, 18*(0.1650.195) ¼0.54 strokes under par per round,

and 0.27 per half-round.

There is only a small effect for Round 1, less than 0.1%

Table 12. Joint Reference-Point Effects for Sets of Holes.

Full Round

Back 9, roundb ¼ 5

(at start) Back 9, rounda ¼ 5

Round 1 0.033, [0.006, 0.057] 0.044, [0.014, 0.099] 0.104, [0.051, 0.157]

Rd 3, tsb ¼ 5 (at start) 0.169, [0.111, 0.226] 0.128, [0.057, 0.203] 0.166, [0.049, 0.281]

Rd 3, tsb ¼ tsa ¼ 0 0.088, [0.054, 0.119] 0.057, [0.017, 0.129] 0.104, [0.015, 0.204]

Rd 3, tsa ¼ 5 0.231, [0.133, 0.33] 0.123, [0.006, 0.241] 0.164, [0.078, 0.252]

Note. Table reports expected score for round or half-round (estimated by si mulation procedure

explained in text) with reference-point effects minus expected score under null hypothesis of i.i.d. hole

scores. last vars ¼ 0 at start of each back nine.

Values are point estimates [95% confidence interval].

476 Journal of Sports Economics 17(5)

of a typical score of around 70 strokes per round, but the confidence interval is so

small the effect is significant at the 5% level. This interval is likely especially small

due to the large sample for this analysis and negative covariances of many of the

coefficient estimates. Results on the back nine, however, are larger; a point estimate

of 0.044 strokes that narrowly fails to be significant at 5% when starting the back

nine five strokes below par, and a significant effect of 0.104 strokes when starting

five strokes above par. Prospect theory causes golfers to ‘‘take it easy’’ and/or ‘‘play

it safe’’ to protect gains after starting the round well, both of which cause perfor-

mance to decline on the back nine; the cold hand/risk seeking causes a larger effect.

For Round 3, the results are stronger, both for the back nine and for the entire

round, especially when the tournament score at the start of the round is nonzero.

Round-level effects are higher (worse), with increases in scores of 0.09 (starting at

par), 0.17 (starting five under), and 0.23 strokes (starting five over), all of which are

significant. Back-nine effects rang e from 0. 06 to 0. 17 strokes. T he round-level

effects, per hole, are smaller because effects in the domain of gains cause negative

feedback. When golfers play well early in a round, pushing scores below par,

prospect-theory forces cause future scores to be higher, causing average scores to

be closer to what they would be under the null.

By comparison, the mean round-level effects found by Brown (2011) and Kali

et al. (2015) were both around 0.2 strokes, and PS’s estimate was 0.25 strokes per

round. Our estimates are in the same ballpark (somewhat smaller for round effects

but larger, per hole, for half-round effects). Our estimated effects are, however,

likely lower bounds, since we impose the same reference points on all players,

whereas we know reference points actually vary depending on expectations. We are

essentially measuring players’ reference po ints with error, which attenuates our

estimated effects. And again, to the extent that the hot-/cold-hand effects mitigate

prospect-theory effects, we are understating the effects of the reference points; we

are (again) just estimating net effects.

Conclusion

We provide new evidence that highly experienced agents acting in a high-stakes

environment are influenced by a variety of reference points. The results indicate that

the reference points actually used vary considerably across and w ithin players

depending on subjective expectations and context. When putting, the reference point

of par on the current hole is (by far) the most powerful. Reference points based on

past holes are more important for initial shots on a hole (drives and approaches) and

have some influence on putts as well. Past-hole reference points appear to affect both

risk attitudes and effort, with risk effects (greater conservatism) dominating after

small gains, and effort effects dominating as gains grow. There is some evidence of

golfers becoming more risk-seeking after losses, and evidence that overall perfor-

mance worsens as losses grow (the cold hand).

Stone and Arkes 477

Our results reinforce the conclusion that prospect theory yields insight into

behavior beyo nd the stan dard mod el . The effec ts of indivi dua l past-h ole ref eren c e

points are substantially smaller than the current-hole reference-point effect on

putting found by PS. But past-hole reference-point effects interact within and

across holes and can yield a total effect over a set of holes of a similar magnitude

to that found by PS.

How can we reconcile the differing results—prospect-theory effort effects ‘‘far’’

into the domain of gains, risk effects after small gains, and cold-hand effects in the

domain of losses? A careful theoretical analysis is beyond our scope and could be a

good topic for future work, but we provide a few speculative comments here. The

variation within the domain of gains is perhaps relatively straightforward and con-

sistent with Figure 1—the decline in effort after large gains could be due to the

flattening slope of the value function; the steeper, more concave, value function

closer to the reference point may cause the greater conservatism there. An alterna-

tive explanation for the decline in effort is overconfidence due to the hot-hand bias.

The cold-hand dominance in the domain of losses is more puzzling. This could be

due to a decline in confidence or some other psychological problem greater than the

corresponding increase (if any) in the domain of gains. The cold hand could also be

due to lower effort in the domain of losses, since the value function does flatten far to

the left (in the domain of losses) as well as to the right. Thus, prospect-theory and

momentum effects may not be in conflict as much as we initially suggest.

Another potential reason for the cold hand in the domain of losses is that golfers

do try harder (as we initially thought prospect theory predicted), but that greater

effort in golf can, at some point, actually harm performance. See, for example, Cao,

Price, and Stone (2011) for a discussion of how too much effort can hurt perfor-

mance for skill tasks, that is, performance can be an inverse-U-shaped function of

effort. Perhaps good scores versus reference points push golfers toward the left end

of the hill, and poor scores toward the right end. This would again mean the cold

hand is actually the result of a prospect-theory effect, again implying the forces may

cause, rather than compete with, one another.

Last, we note the causality between prospect theory and the cold hand could go in

the other direction as well. The cold hand that seems to occur when scores go above

par may help to rationalize the emphasis that players put on avoiding a bogey on the

current hole found by PS, and the conservatism we find after small gains. Similarly,

a subconscious attempt to mitigate the hot-hand bias could push players to play more

conservatively when in the domain of gains. Hence, some aspects of prospect theory

may actually be adaptive or ‘‘ecologically rational’’ (Smith, 2003), given psycho-

logical factors causing the potential for cold momentum.

Acknowledgment

We thank Peter von Allmen (the editor), two anonymous referees, Neil Metz, Josh Price, Dan

Sachau, Skip Sauer, Yao Tang, Dan W ood, and participants at the 2015 NAASE/WEAI

478 Journal of Sports Economics 17(5)

meeting, and seminars at Clemson Universit y and the University of Virginia for helpful

comments.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, author-

ship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of

this article.

Notes

1. From video on http://www.foxsports.com/golf/usga/story/us-open-phil-mickelson-1-

under-69-first-round-061815?vid¼467701827998

2. See Iso-Ahola and Dotson (2014) for a good recent review. An (even more recent) paper

omitted is Miller and Sanjurjo (2014) who emphasize the important distinction between

hot and cold hands and present striking new experimental results. Arkes (In Press) found

evidence of the cold hand in golf across rounds but no evidence of a hot hand. However,

Stone (2012) and Arkes (2013) both show that typical tests of the hot/cold hand have low

power and may be biased toward null effects. It is important to note that the bias to infer

the hot hand prematurely (Falk & Konold, 1997; also known as the ‘‘hot-hand bias’’) is

still well established and unquestioned.

3. See, for example, Arkes and Stone (2014) for discussion of psychological and economic

implications.

4. Admittedly it is unclear from the figure whether the value function is ‘‘more concave’’

when x > 0 versus x ¼ 0, or ‘‘steeper’’ when x < 0 versus x ¼ 0, that is, some predictions

could depend on the parameterization of the value function and/or level of x. Moreover,

since the value function flattens to both the left and right, returns to effort could decrease

as x declines for x < 0. These returns could also increase if golfers consider the effects of

current and future holes on the value function when deciding on how to approach the

current shot. For example, if a golfer begins a hole at x ¼2, his benefit of a birdie and

moving to x ¼1 is less than that of a birdie pushing him from x ¼1tox ¼ 0,

suggesting his effort incentive when starting at x ¼2 is lower than at x ¼1. However,

if he also thinks about the potential future benefit of moving from x ¼1tox ¼ 0, he

might be more motivated at x ¼2 than x ¼1. In summary, prospect-theory predic-

tions are somewhat ambiguous, but the ones we discuss are those we feel ar e most

plausible, a priori, and are consistent with previous literature. Moreover, some of our

empirical analysis is flexible and allows for varying effects, and we also discuss different

types of effects in the Conclusion section.

5. We write ‘‘þ or 0’’ for the effect of an increase in conservatism on average score because

while the chances of extreme outcomes should clearly decline, and this may be associated

with an increa se in average scores, it is also possible that average scores could be

Stone and Arkes 479

unaffected. But it seems reasonable to rule out the case of conservatism causing average

scores to decline. We write ‘‘?’’ for the analogous increase in aggressiveness because this

could plausibly increase or decrease mean score (decrease if golfers were previously

failing to minimize score in order to avoid too much risk, and increase if golfers sacrifice

mean score for a chance of making big gains).

6. We use the procedure developed in Correia (2014) to account for the high dimensionality

of the fixed effects (FEs).

7. For intuition, consider a situation in which each player only played two holes. If the

player-level mean was controlled for, then lastb variables would mechanically be highly

negatively correlated with S

. For example, if a player shot par on average, the coefficient

on lastb would be 1 and the coefficient on lasta þ1, for a dependent variable of S

. The

bias referred to by Miller and Sanjurjo (2014), and studied explicitly by Miller, Sanjurjo,

et al. (2015), is very similar, as shown in appendix B3 of the latter paper, which refers to

the analogous nonpanel case. Note that this bias is not caused by time effects (in our

context, hole-day effects), since for these FE groups the lagged regressor is not in the

same group as the current observation.

8. See, for example, Flannery and Hankins (2013).

9. Scores for the last and round variables are relatively low compared to the mean of s in

Round 3 because a relatively high number of Par 5 observations are dropped from the

final sample in Round 3 (but not from the construction of last and round variables), and

scores on Par 5 holes are much lower on average than those of Par 3 and 4 holes. We

provide the minimum and maximum value for each variable, rather than the standard

deviation, because the latter provides less immediate information both on how the vari-

able is defined and the range of values it takes.

10. We also consider a robustness check in which we focus on just the third round and split

the sample into two, players in the top half of their tournament at the start of the round and

those out of the top half, to assess the effects of tournament standing. Results are largely

similar for the two groups (and unreported in the interest of brevity); the most interesting

difference is that the lower half group plays better in every dimension as rounda

increases, providing evidence of effort effects in the domain of losses. Perhaps most

importantly, the results confirm that the cold-hand effect in Round 3 associated with

high tournament scores is not driven by players rationally ‘‘giving up’’ due to being out of

contention as the cold-hand effects are similar for both groups of players.

11. Note that Nickell bias is likely more severe for this analysis since there will be player-

year-par groups with less than 50 observations.

12. Ranking data were obtained from http://www.pgatour.com/stats/stat.186.html site. The

number of observations per player-year-par group is very similar for both higher and

lower ranked players, so Nickell bias is not a greater threat for either group.

13. We obtain estimates of 0.033 and 0.019 for birdie effects for Rounds 1 and 3,

respectively, versus 0. 038 and 0.024, which PS reports in their Table 5. Results

may differ both due to our using a slightly different specification, and sample, both

with respect to the time frame and criteria for dropping observations. The groups we

use for the distance group-elevation decile interaction are: 0–5 feet, 5–10 feet, 10–15

480 Journal of Sports Economics 17(5)

feet, 15–30 feet, 30–45 feet, 45 –60 feet, an d >60 feet. In PS’s baseline model

(columns 1–3 of their Table 3), they do not control for e levation, and in other models

they control for elevation in a slightly different, but roughly equivalent, way (FEs for

regions of the green).

14. In the actual simulations, the predicted probabilities never fell outside of [0,1].

15. We have also compared the entire distributions of scores, and not just the means, and find

results are similar to those we discuss here; these results are available on request.

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Author Biographies

Daniel F. Stone is an assistant professor of economics at Bowdoin College. His research

interests include sports economics, behavioral economics, information, politics, and media.

Jeremy Arkes is an associate professor of economics in the Graduate School of Business and

Public Policy at the Naval Postgraduate School. His research interests are in military man-

power, substance abuse, the effects of divorces, behavioral economics, and sports economics.

482 Journal of Sports Economics 17(5)