Abstract:

Past studies have found a positive correlation between the use of
the designated hitter hi baseball and hit hatters, hut the reason for
this is debatable. Using a new micro-level data set of individual plate
appearances, we control for detailed cost-benefit attributes that affect
the decision calculus of the pitcher to isolate the deterrent impact of
requiring the pitcher to bat. We find that pitchers hit batters
strategically, and the deterrent effect of requiring pitchers to bat
explains 60%-80% of the difference in hit batsmen between leagues. We
also identify evidence of direct retaliation against plunking pitchers.
(JEL D81, KC42, L83)

Subject:

Pitchers (Baseball)
(Analysis)

Sports associations (Analysis)

Sports associations (Analysis)

Authors:

Bradbury, John Charles

Drinen, Douglas J.

Drinen, Douglas J.

Pub Date:

01/01/2007

Publication:

Name: Economic Inquiry Publisher: Western Economic Association International Audience: Academic Format: Magazine/Journal Subject: Business, general; Economics Copyright: COPYRIGHT 2007 Western Economic Association International ISSN: 0095-2583

Issue:

Date: Jan, 2007 Source Volume: 45 Source Issue: 1

Product:

Product Code: 8691000 Athletic Associations NAICS Code: 81399 Other Similar Organizations (except Business, Professional, Labor,
and Political Organizations) SIC Code: 8699 Membership organizations, not elsewhere classified

Organization:

Organization: Major League Baseball

Geographic:

Geographic Scope: United States Geographic Code: 1USA United States

Accession Number:

171399957

Full Text:

I. INTRODUCTION

Are you seriously going to throw at somebody when you're facing Randy Johnson?

--Curt Schilling (1)

In 1973, amid waning fan interest in baseball, the American League (AL) of Major League Baseball (MLB) instituted the designated hitter rule as an experiment. The stated goal of this rule change was to boost offensive output by increasing the talent pool of batters in the lineup. Traditionally, the competing teams field nine players who must play defense in the field and bat. Because of the importance of the pitcher, who is responsible for putting every ball in play, teams rely on pitching ability and ignore the hitting ability when choosing the pitcher in the lineup. Therefore, pitchers tend to be very poor hitters. By allowing teams to substitute a player of greater hitting ability known as the designated hitter (DH)--to bat for the pitcher the total offensive output increases. The experiment has since grown into an institution in the AL that differentiates it from the National League (NL), where all players must bat and play in the field.

The DH succeeded in turning the AL into the "power league" as intended, but an unintended consequence of the rule change is that the AL now has more batters hit by pitches than the NL. Traditional baseball lore holds that the lack of retaliatory punishment in the AL for hitting batters is the cause of this phenomenon. Veteran NL manager Dusty Baker describes the deterrent impact from a pitcher's point of view, "You can be bold in (the American) League and get away with [hitting batters]. It's different in our league where you have to hit." (2) Pitchers who do not have to bat (where they might face retaliation) are more willing to risk hitting batters than pitchers who do bat.

Given that the rules of the game in both leagues are identical
except for the use of the DH, MLB created ideal conditions for a natural
experiment to examine the impact of the DH on hit batters in a
controlled setting. (3) Several economists have looked at the issue and
found that there is a statistically significant relationship between the
DH and hit batters. However, the reason for this difference is subject
to much debate. It is possible for the hit batter differential to exist
without the deterrent effect of retaliation. The rulebook punishment for
hitting a batter--awarding the hit batter first base makes retaliatory
enforcement costly for teams to employ. Pitchers are typically poor
hitters who rarely reach base via hitting, while batters who are DHs are
typically good hitters. The fewer hit batters in the NL may reflect the
lineup composition, in which the pitcher must bat. NL teams will try to
avoid plunking pitchers because they are relatively less likely to reach
base than non-pitchers, while AL teams do not have this easy out in
their batting lineups.

A problem with previous studies of the subject is their reliance on yearly aggregate data. Hit batters are rare events--approximately 1% of all plate appearances result in a hit batter--that occur in the course of game where other incentives are quite relevant. This means that identifying small changes in pitcher behavior from specific factors will be difficult to identify with aggregate data. Differentiating between the two competing explanations for the hit batter differential between leagues deterrence and lineup composition--requires micro-level data where we can control for in-game strategic incentives for hitting batters.

Using a unique play-by-play data set that allows us to control for specific factors that affect hit batters, we find that pitchers are still more likely to hit batters with a DH in the lineup than without. And though pitchers are hit more rarely than other players, pitchers do experience retaliation for hitting batters. Retaliation against pitchers who plunk batters, which was not found in earlier studies, is needed to generate an effective deterrent for pitchers in the NL. These findings provide support for the deterrence hypothesis that the threat of retaliation against pitchers who hit batters does impact the probability that a pitcher will hit a batter and demonstrate the importance of informal enforcement mechanisms in deterring criminal behavior. Furthermore, we show that it is likely that the addition of formal administrative punishments for hitting batters by MLB may, in fact, have increased the incidence of hit batters in both leagues. The remainder of the paper is organized as follows. Section II reviews the previous literature, concentrating on the empirical problems with these studies. Section III discusses the strategic incentives for hitting batters. Section IV presents the empirical model with section V discussing the findings. Section VI discusses a peculiar deviation in the pattern of hit batters in the 1990s. Section VII concludes the paper.

II. A REVIEW OF PREVIOUS STUDIES

Goff, Shughart, and Tollison (1997, hereafter, GST) first examined the hypothesis that the cost differential to pitchers for hitting batters induced by the DH is the cause of the interleague difference in hit batsmen. Using annual league time series data from 1901 to 1990, GST analyzes changes in the difference in hit batsmen between leagues before and after the implementation of the DH. The results indicate that, controlling for several factors, the introduction of the DH raised the level of hit batsmen in the AL 15% higher than the NL, consistent with the hypothesis that the DH removes an effective criminal deterrent. However, though the evidence is clear that the DH is associated with an increase in hit batsmen, it is not necessarily a consequence of the lesser deterrent to hitting batters in the AL. Trandel, White, and Klein (1998, hereafter, TWK) and Levitt (1998) are skeptical of the interpretation of GST on two grounds. First, adding the DH raises the marginal benefit of hitting batters due to the existence of an additional good batter in the lineup. Because pitchers are poor hitters, few teams want to risk hitting the other pitcher. Replacing the pitcher with the DH places a batter more worthy of plunking in the lineup. As TWK state, "the difference [in hit batsmen] is largely because the AL batters are (on average) better hitters, and are thus less costly and more beneficial to hit." Therefore, the aggregate increase in hit batsmen in the AL over the NL may be attributed to increased rewards rather than the lowered punishment for hitting batters. (4) Second, the deterrence story seems implausible because pitchers are rarely hit, and therefore, the probabilistic penalty of retaliation for hitting batters is too small to have any real effect in a pitcher's decision calculus. Hitting the pitcher in retaliation may actually reward the instigator by putting the weak-hitting pitcher on base. Given that pitchers know this, it is unlikely that pitchers consider reciprocal retaliation to be a cost. As evidence, Levitt (1998) finds that there is no correlation between pitchers hitting batters and pitchers being hit themselves, and pitchers are hit so rarely that they would have to be hypersensitive to the relative price change induced by the DH. (5) Trandel (2004) finds similar results for the 1974-1977 seasons using the same test and finds little evidence of retaliation against any pitcher. All of these studies conclude that the increase in the frequency of hit batsmen in the AL likely reflects changes in the batting lineup composition and not a deterrent response by pitchers. (6)

The crux of the problem in identifying what factors influence the difference in hit batters across leagues lies with aggregate data, with which researchers can observe only the increase in the frequency of hit batsmen with the DH, a result that both competing hypotheses predict. Aggregate data, though informative, only reveal a quantity hit without regard to different situations where strategic incentives vary. A retaliatory plunking war in one game may be hidden within yearly statistics from a 162-game season, even though such an event may signal a credible threat of retaliation that persists into the future. As a result, the differing hypothesized effects are hard to disentangle from aggregate data. To remedy this we examine the costs and benefits of hitting any particular batter at the time the pitcher chooses a pitch. This allows us to examine pitcher decisions as the strategic incentives for hitting batters vary.

Readers familiar with the previous economics literature on this topic will recall that the earlier studies debate the cause of the DH-hit batter relationship as a moral hazard problem. In the AL, pitchers are actors who are improperly "monitored" by the opposition and therefore are more likely to engage the undesirable behavior of hitting batters than NL pitchers. The manger of the pitcher, in addition to the opposing team, may suffer as a consequence of their shirking agent, creating the potential for moral hazard. In this paper, we focus on the strategic incentives of pitchers as they impact pitcher behavior and shift the debate away from relying on moral hazard as the disputed explanation. The central debate has to do with a behavioral response to the cost of hitting batters. Is requiring pitchers to bat a real deterrent against hitting batters, or is the observed league difference in hit batsman a product of differing lineup compositions in the leagues? That pitchers might shirk in their duty to the manager is a possibility, but it is largely irrelevant to the debate and nearly impossible to identify empirically. Pitchers can still be perfect agents of their managers and act in accordance with their managers' wishes to hit more batters with a DH and fewer batters without a DH. Managers certainly have reason to fear that a pitcher who hits batters may end up injured, suspended, etc. Therefore, even without moral hazard, the cost differential of hitting batters in the leagues is sufficient to explain the league difference in hit batters.

III. THE COSTS AND BENEFITS OF HITTING BATTERS

Getting hit by a pitch is a painful, and sometimes dangerous, event in baseball. (7) To deter the pitcher from accidentally or purposely hitting batters the hit batter is awarded first base. Given this statutory constraint, we know the benefits and costs of hitting a batter. The benefits include preventing a batter from hitting the ball, which might generate more runs than simple base advancement; inflicting injury on a competitor; or decreasing the opposing team's willingness to stand close to the plate to hit outside pitches. The costs include putting a runner on base, which increases the chance of the opposing team scoring; ejections, fines, and suspensions to the pitcher or manager; physical retaliation via assault; retribution against the teammates of the pitcher by plunking them when they bat; and direct reciprocal retribution in the form of the current pitcher being intentionally plunked by the opposing pitcher.

It is this last cost which is the main subject of discussion. AL and NL pitchers bear identical costs and benefits when facing batters except for the reciprocal retribution. AL pitchers do not bat and therefore bear a lower cost for hitting a batter than NL pitchers. GST hypothesizes that this relative price difference induces AL pitchers to hit more batters than NL pitchers hit. (8) This is not to say that any pitcher actively seeks to consume hit batsmen purposely (at least not most of the time). Instead pitchers engage in riskier behavior (e.g., throwing inside, faster) when DHs bat in their place, which results in a greater number of hit batsmen. Cy Young award--winning pitcher Randy Johnson, who has played in both leagues, explains, "If you're the pitcher and you're playing in the American League, then you may have a tendency to throw inside a little bit more knowing when that ninth hole comes up, you won't be hitting. You're protected in that regard." (9) This is a behavioral response similar to automobile drivers in Peltzman (1975), which documents seat belt laws inducing motorists to engage in riskier behavior that leads to car accidents. Lowering the negative consequences of an action leads people to take more risks that result in the negative outcome.

However, the critics of the deterrent hypothesis propose that the expected punishment for plunking is too minor to have much, if any, effect on an NL pitcher's decision when he pitches to batters. On average, hitting the pitcher is not a good idea. If retaliation does provide a deterrent, pitchers would have to be extremely sensitive to that threat. However, circumstances arise where the benefits of hitting a pitcher exceed the costs. This may occur in the course of the same game in which a pitcher hits a batter, or it may occur in some future game. (10) Though pitchers are hit less than other position players, it is possible that pitchers may be hit more than they ought to be, given their hitting abilities. Ultimately, this is an empirical question. Using the data we describe in the following section, we can answer the questions central to the analysis.

IV. EMPIRICAL MODEL

A data-gathering project known as Retrosheet has carefully reconstructed baseball play-by-play data for many seasons. (11) Using a computer program, we extracted data by plate appearance from the Retrosheet data. From this we can observe the exact game situation when a player is hit or not hit; thereby, we can control for the costs and benefits of hitting any particular player during a plate appearance. (12) We use a predictive model to estimate the likelihood that a batter is hit given five categories of influences on hitting batters: deterrence (DH), batter quality (BQ), pitcher quality (PQ), retaliation (R), and game situation (GS). Thus,

[HBP.sub.j] = [[alpha].sub.j] + [beta][DH.sub.j] + [lambda][BQ.sub.j] + [gamma][PQ.sub.j] + [phi][R.sub.j] + [psi][GS.sub.j] + [eta][YR.sub.j] + [[epsilon].sub.j],

where HBP is a dummy variable equal to one when a pitcher hits a batter and zero otherwise. Subscript j represents a plate appearance, [alpha] is the constant, [epsilon] is the error term, and YR is the year. We estimate the equation using both logit and probit estimation techniques.

The variable DH is a dummy equal to one when the DH rule is in effect. According to baseball's conventional wisdom, it is the lower deterrence induced by the DH that explains the hit batsmen differential between leagues. If, after accounting for the many factors we include, the DH coefficient remains positive, it is likely that the deterrent effect of requiring the pitcher to bat explains some of the difference. If it is insignificant when controlling for other factors, it is likely that the deterrence hypothesis is incorrect.

The BQ vector includes two batter quality variables that effect the cost-benefit calculation of hitting batters. Better hitting batters that replace pitchers in the batting order ought to be hit more frequently than pitchers; therefore, we include the season OPS of the batter and a dummy equal to one if a pitcher is hitting. OPS is the sum of the slugging average and on-base percentage. (13) A batter with a higher OPS is more likely to generate offense and is therefore more beneficial to hit than a player with a lower OPS. Even when controlling for OPS, pitchers may be less desirable batters to hit relative to teammates, which is why we also include the pitcher dummy. These variables are important in determining whether composition of the lineup explains the hit batsmen differential.

Though it has not been discussed much in this literature, hit batsmen are largely accidental. Even pitchers who do not mean to brush a player back or pitch on the inside of the plate may accidentally hit batters. The variables in PQ control for the pitcher's propensity to hit batters accidentally. Pitchers who are more likely to hit batters mistakenly are also likely to make mistakes in the form of placing hittable balls in the strike zone when compared to other pitchers. These pitchers will generate more offense for opposing teams relative to more skilled pitchers. We use the pitcher's OPS allowed to hitters as a proxy for pitcher skill. Also, pitchers with less control are more likely to issue walks than pitchers with good control. Therefore, we also include the walk rate of the pitcher to control for pitcher quality. Both of these measures ought to be positively related to hit batsmen.

At the heart of this paper is the issue of using the pitcher to retaliate against the other team for hitting batters. This retaliation may be reciprocal, pitcher hitting pitcher, or pitchers may hit teammates of the pitcher. Trandel (2004) notes that observing retaliation in the data is of fundamental importance to this debate. (14) We have the ability to detect retaliation using the variables in vector R. There are several reasons pitchers retaliate against the other team through plunking, but we will focus on two common reasons that we can capture in the data. If a batter on the current pitcher's team was hit in the previous half-inning, this may provoke a retaliatory plunking of any player on the opposing team. We include a dummy variable equal to one if this occurs. Also, we test specifically for reciprocal retaliation against pitchers. We create a dummy for a pitcher appearing at the plate when a batter was hit in the previous half-inning. If retaliation does occur through pitchers hitting pitchers, this variable should be positive. Also, it is a baseball tradition to hit a batter following a home run, on occasion. We include a dummy equal to one if the previous batter hit a home run.

Finally, we include vector GS of many possible game situations affecting the cost-benefit calculation. The score, inning, number of outs, and base runners threatening to score all affect the costs and benefits of hitting a batter at the particular moment. The following example from an altercation between the Houston Astros and Chicago Cubs in 2004 demonstrates the role of the game situation in the decision to plunk a batter.

Both benches cleared briefly in the second inning of Friday's game between the Houston Astros and the Chicago Cubs when Cubs catcher Michael Barrett began jawing with Astros pitcher Roy Oswalt, 5 days after Oswalt hit him in the back with a pitch. Barrett said after that game that he "thought he [Oswalt] was better than that" and warned: "We're going to see him again." With Wood on the mound Friday in another matchup with Oswalt, Barrett and Oswalt exchanged words when the Houston pitcher came to the plate to lead off the second with the Astros leading 4-2. After the brief altercation, Oswalt grounded out and the two again exchanged a few words after Barrett had run down the first-base line to back up the throw. In the top of the sixth, Oswalt again came up as the leadoff hitter, with the Astros leading 8-4, and he was hit in the thigh by the first pitch from Cubs reliever Kent Mercker. (15)

Score differential and whether a team is winning or losing certainly impact the strategic incentives. A large absolute score difference means there is a low marginal impact of runs. When the score differential becomes large both teams may be more willing to engage in a plunking war than when the game is tight. However, because the costs of retaliation fall as the differential rises, pitchers may be extra careful as the likelihood of retaliation increases. Also, it may be the case that a big loser is more willing to plunk batters. As the score differential grows the losing team may feel the opponent is "rubbing it in," and thus may start plunking the winning team. It is unclear if batters are more likely to be hit early as opposed to later in a game. Plunking early may send a message to players not to "crowd" the inside of the plate throughout the game. However, if retaliation is an ex post enforcement mechanism to prevent plunking, later in the game teams have more incentive to renege on tacit agreements not to hit each others' players. The number of outs affects the probability that any base runner will score. A batter put on base with no outs has a better chance of scoring than a batter who reaches first with two outs. Finally, the number and positioning of base runners can also affect the costs and benefits. The specific predictions between differing runner configurations is a bit complicated and largely irrelevant to the discussion here. Generally, the more runners on base and the further the runners are around the bases, the greater the offensive damage a batter can do by putting a ball in play. Therefore, the seven of eight possible runner configurations included (we exclude empty bases for comparison) should be positively related to hit batsmen. We also include year effects to control for unobservable differences in hit batsmen that may change from year to year. Table 1 lists the summary statistics for the variables in both of our data sets.

V. RESULTS AND DISCUSSION

Retrosheet provides play-by-play data for both leagues from 1972 to 1992 and for the 1969 season. From this data, we extract two four-season data sets--the earliest and most recent available years--to estimate the empirical model. For each plate appearance, we extract data on the specific game situation and the seasonal performance of the players participating in the event. Initially, we focus on the four most recent years of available data, 1989-1992.

Table 2 lists the results of the logit and probit estimations of coefficients and z statistics. For ease of interpretation, we also include the odds ratio (logit) and marginal probability (probit) calculations for each variable. Most of the variables are significant and of the expected sign. Of particular interest, the DH variable is positive and significant when controlling for all other factors, including the lineup composition batter quality variables. When the DH is in effect pitchers are 15%-17% more likely to hit batters than when the DH is not in effect. Consistent with the lineup composition hypothesis, batter quality is positively related with hit batsmen, and pitchers are about 55% less likely to be hit than other batters. Also, pitcher quality is negatively related to hit batsmen. Both proxies for lack of pitch control are positively associated with a greater likelihood of hitting a batter; however, pitcher OPS is not statistically significant.

The retaliation variables tell an interesting story. Preceding events likely to provoke retaliation all tend to increase the likelihood that a pitcher hits a batter. A batter who appears at the plate following a home run is 32% more likely to be hit than when the preceding batter did not hit a home run. In response to hit batters, pitchers are more likely to hit batters the half-inning after the opposing pitcher hits a current pitcher's teammate. Though the results are not significant for players in general, a pitcher is four times more likely to be hit when an opposing player was hit in the previous half-inning. This variable, which is significant, is very important for the deterrence hypothesis. We now observe retaliation against pitchers who hit batters. This is a phenomenon previously unidentified in the aggregate data. Though pitchers are hit less frequently than other players on average, pitchers are more likely to be hit after plunking an opposing player. Therefore, pitchers do bear a very real cost to hitting batters, which is necessary to provide a deterrent to pitchers.

For the game situation variables, outs and innings seem to be relatively unimportant. Though the absolute score differential is unimportant, the relative score differential from the offense to the defense has a small but significant effect on the incidence of hit batsmen. The base runner composition variables appear to be important and consistent with our predictions.

Over this sample, the AL hit batsmen rate is 26% higher than the NL. Controlling for all of the above factors, including batter quality, the DH rule dummy explains approximately 60% of this difference. It is now clear that something other than the lineup composition created by the DH is responsible for the increase in hit batsmen differential between leagues. The most plausible explanation for this difference is the removal of a credible threat of ex post punishment of pitchers in the AL. However, it is possible that some yet unidentified factors unique to the AL may explain the sign and significance of the DH dummy since the DH occurs only in this league. For example, differences in strike zones, stadium configurations, league traditions, etc. between leagues are competing but less satisfying explanations. To be thorough, we examine a different data set that includes the 1969 and 1972-1974 seasons. (16) This is the earliest available data in the Retrosheet archives, plus this period includes some observations of the AL without the DH rule. We are slightly less confident in this data than our more recent data for a few reasons. First, the only two seasons available prior to the introduction of the DH are the noncontiguous years 1969 and 1972. Second, the late 1960s and early 1970s were low offense years for baseball. The DH was the last of several rules instituted to "fix" baseball, which was losing fans. The league responded in 1969 by lowering the pitchers mound, shrinking the strike zone to make it more hitter friendly, and adding two new teams to each league. Luckily, the exogenous events other than the DH apply to both leagues; therefore, we feel the data are useful in isolating the effect of the DH rule on hit batsmen, given that we acknowledge the deficiencies.

We apply the same model for the previous sample to this data, except that the DH variable is now coded zero for some observations in the AL, when the DH was not in effect. Therefore, the DH should now reflect the change in the costs of hitting batters induced by the DH rule itself. The results in Table 3 are quite similar to estimates of the more recent data. For this sample, when the DH is in effect pitchers are 11%-12% more likely to hit batters than without the DH rule, ceteris paribus. These results differ only slightly in magnitude from our previous estimates. We also observe pitchers, as well as position players, being hit in retaliation for hit batters in this sample. Most of the other factors are of similar magnitude and size, though some measures are no longer statistically significant. In the years following the DH, AL pitchers hit batters at a rate 14% higher than NL pitchers; thus, the DH rule dummy explains approximately 80% of the interleague difference in hit batsmen in this sample. Therefore, the results indicate that the removal of the threat of retaliation by the DH rule, not unobserved league-specific factors, is the likely cause.

VI. THE PROBLEM OF THE 1990s

Figure 1 shows the fluctuation in the rate of hit batsmen from 1921 to 2003. It is evident from the figure that the introduction of the DH in 1973 coincides with the AL's continued higher rate of hit batsmen as compared to the NL. From the introduction of the DH until the present, the AL's average yearly hit batsmen rate has exceeded the NL rate by 15%. From the beginning of the modern era of baseball until the 1970s, the hit batsmen rates varied closely, yet randomly, across leagues. For the next 21 years the AL rate exceeded the NL rate every year, which is consistent with both the deterrence and lineup composition hypotheses. But in the mid-1990s, just as economists began to study this issue and the public availability of play-by-play data ceased, a deviation in this pattern emerged. In 1994, the NL hit batsmen rate rose above the AL rate for the first time since the implementation of the DH, and the leagues' hit batter rates began to fluctuate more closely than in the past. Why was there such a dramatic shift and how does this shift impact the perceived relationship between the DH and hit batsmen? There are two important factors that contributed to this change: league expansion and a rule change in the punishment for pitchers hitting batters.

In response to criticism over the recent events of the 1990s, Goff, Shughart, and Tollison (1998) postulates that the NL expansion in 1993 diluted the talent pool of players increasing the number of accidental hit batsmen. We believe the evidence supports this contention. Fringe pitchers and hitters are more apt to make mistakes by hitting batters or getting hit by pitches. (17) In 1993, the NL hit batsmen rate fell to only 10% of the AL rate, while the average difference over the preceding 2 decades was 20%. Additionally, the mid-1990s corresponded with a dramatic increase in hit batsmen in both leagues, doubling over the decade. The expansion hypothesis is consistent with both the change in the difference between leagues and the rise in hit batters overall. In the two rounds of expansion, three of the four expansion teams joined the NL with one existing AL team (Milwaukee) shifting to the NL. But more importantly, the asymmetric expansion draft rules in 1993 favored expansion players coming from NL rosters. Therefore, the expansion diluted the talent pool of both leagues but affected NL rosters to a greater degree than AL rosters. (18) From 1993 to 1997, as the expansion players eased into the league, the NL rate exceeded the AL rate in 3 of the 5 years, with the AL rate being 2% lower than the NL rate, on average. But in 1998, when both leagues added a team, there was no asymmetry between leagues in drafting expansion players. Interestingly, the disruption to the previous pattern was noticeably smaller. Since the last round of expansion in 1998, only once (2000) has the NL rate exceeded the AL rate, with the AL rate exceeding the NL rate by an average of 6.5%.

[FIGURE 1 OMITTED]

The second factor was an important rule change in the punishment for hitting batters that significantly altered the cost of retaliating against pitchers who hit batters. In 1994, MLB adopted the "double-warning" rule, which authorized umpires to warn both teams if an umpire judges that a pitcher intentionally hits or attempts to hit a batter. (19) Once the umpire issues a warning, retaliation results in the immediate expulsion of the current pitcher and manager of the offending team, accompanied by monetary fines. This statutory constraint significantly raised the cost of retaliation, which, in turn, lowered the probability of punishment to pitchers for initially hitting batters. With the double-warning rule, pitchers can take risks that are more likely to result in hit batters than without the rule because pitchers know the opposition will be less likely to retaliate due to the increased penalty after the umpire issues a warning. The relative rise in the NL rate of hit batsmen in the 1990s is consistent with the deterrence hypothesis and therefore is expected. NL pitchers are now more protected from retaliation than before the double-warning rule, thereby inducing NL pitchers to behave more like their counterparts in the AL. A similar phenomenon occurred in Japanese professional baseball. The Nippon Professional Baseball League, which also has the DH in only one league, experienced a similar convergence of hit batsmen after several years of excessive hit batsmen in the DH league. Kawaura and La Croix (2002) find that this convergence correlates with a rule change that increased the penalty for intentionally hitting batters. Thus, the statutory constraint designed to limit hit batters likely had the unintended consequence of doing the opposite.

Finally, using game-level (not play by play) data from 1973 to 2003, Bradbury and Drinen (forthcoming) find that games in which the DH is allowed are associated with more hit batters than non-DH games, controlling for many relevant factors. Over the entire 31-year history of the DH, the 4 years in which the NL rate exceeded the NL rate appear to be exceptions rather than the rule. When relevant historical facts are included in the analysis, it is clear that the 1990s actually lend support to, rather than cast doubt upon, the deterrence explanation for differences in hit batsmen between leagues. The most recent history of hit batsmen rates is indicative of a return to the pre-1990s league differences in hit batsmen, with a tempering of the difference by the double-warning rule.

VII. CONCLUSION

Competing economic theories predict that DH rule is responsible for raising the rate of hit batsmen in the AL above the NE. As Figure 1 illustrates, for the 3 decades following the introduction of the DH the AL hit batsmen rate remained higher than the NL rate for all but 4 years. However, there is less agreement regarding the cause of this difference. While the lower cost of hitting batters for AL pitchers, who do not have to bat and face reciprocal retaliation, is a likely explanation, it is not the only explanation. An alternate hypothesis is that the DH increases the incidence of hit batters by altering the composition of the batting lineup. Replacing a poor-hitting pitcher with a good-hitting DH raises the marginal benefit to hitting an additional batter in the lineup in the AL. Additionally, the retaliation threat to pitchers may be too small to be relevant. If retaliation is not likely, there is no cost differential to pitchers between leagues to explain the hit batsmen difference.

Past empirical analyses of the competing hypotheses to explain the hit batsmen differential have relied on yearly aggregate data. However, the explicit cost-benefit analysis of the pitcher at the time of a pitch is not observable with aggregate data. Using two new unique data sets we have been able to observe the costs and benefits of hitting batters during individual plate appearances. Therefore, we isolate the predicted effects of the competing hypotheses and determine the impact of many relevant factors that influence the pitcher's consumption of risk in hitting a batter every time a hitter steps to the plate. Controlling for variables that proxy batter quality, pitcher quality, retaliation, and game situation, we find that the DH rule increases the likelihood that any batter will be hit during a plate appearance between 11% and 17%. This explains approximately 60%-80% of the hit batsmen rate differential between leagues. Furthermore, we observe retaliation against plunking pitchers, which is necessary for the deterrent explanation. In total, the findings support the fact that requiring pitchers to bat deters them from hitting batters. Though other factors are important in the economics of plunking, the threat of retaliation plays a nontrivial role. Players really do "take care of it" themselves through an informal enforcement mechanism, while official enforcement penalties may have, in fact, exacerbated the problem of hit batters.

ABBREVIATIONS

AL: American League

DH: Designated Hitter

MLB: Major League Baseball

NL: National League

OPS: On-Base Percentage Plus Slugging percentage

doi:10.1093/ei/cb1014

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Levitt, S. D. "The Hazards of Moral Hazard: Comment on Goff, Shughart, and Tollison." Economic Inquiry, 36(4), 1998, 685-7.

McCormick, R. E., and R. D. Tollison. "Crime on the Court." Journal of Political Economy, 92(2), 1984, 223-35.

Morgan, J. 2002. Attention Estes: throw at Clemens. June 14. 2002.

N.A.P.B.L. Umpire Manual. Chicago: Triumph Books. 1996.

Pappas, D. 1997. Terms of major league expansion. Outside the Lines, Fall, 3. http://roadsidephotos. com/baseball/expansion.htm

Peltzman, S. "The Effects of Automobile Safety Regulation." Journal of Political Economy, 83(4), 1975, 677-725.

Retrosheet. www.retrosheet.org (accessed June 12, 2003).

Rosenthal, K. 2002. Mets get shot with mighty Clemens at the bat. The Sporting News, June 13.

Rupert, M. E., and E. F. Stephenson, "On Moral Hazard and Hit Batsmen: A Comment on Goff et al. and Trandel et al." Manuscript, Berry College, 2002.

Schmidt, M. B., and D. J. Berri. "On the Evolution of Competitive Balance: The Impact of Increasing Global Search." Economic Inquiry, 41(4), 2003, 692-704.

Sobel, R. S.. and T. M. Nesbitt. "Automobile Safety Regulation and the Incentive to Drive Recklessly: Evidence from NASCAR." Manuscript, West Virginia University, 2004.

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Trandel, G. A. "Hit By Pitches: Moral Hazard, Cost-Benefit. Retaliation. or Lack of Evidence?" Journal of Sports Economics. 5(1). 2004, 87-92.

Trandel, G. A., L. H. White, and P. G. Klein. "The Effect of the Designated Hitter Rule on Hit Batsmen: Pitcher's Moral Hazard or the Team's Cost-Benefit Calculation." Economic Inquiry, 36(4), 1998, 679-84.

Turocy, T. L. "Offensive Performance, Omitted Variables, and the Value of Speed in Baseball." Economics Letters, 89(3), 2005, 283-6.

(1.) The quotation is from Rosenthat (2002).

(2.) The quotation is from Morgan (2002).

(3.) The controlled environment of sports has become a popular arena for testing economic hypotheses. Some examples include Duggan and Levitt (2002) on identifying corruption in sumo wrestling: Goff, McCormick, and Tollison (2002) on racial integration and innovation as measured by winning in baseball and basketball: McCormick and Tollison (1984) and Heckelman and Yates (2003) on crime and enforcement in basketball and hockey: and Sobel and Nesbitt (2004) on offsetting behavior in response to safety regulations in National Association for Stock Car Auto Racing.

(4.) In reply Goff, Shughart, and Tollison (1998) responds to the criticisms by referring to the control variables (batting and pitching statistics) in previous and revised regressions that control for this factor.

(5.) This mirrors Levitt's critique on the deterrent effect of capital punishment in Katz, Levitt, and Shustorvich (2003).

(6.) These studies also extend the data into the 1990s, in which the difference in hit batsmen between leagues becomes less stable. However, there are several plausible explanations for the change that we discuss in Section VI.

(7.) Pitches hitting batters have ended several professional baseball careers; seven players (only one in MLB) have died from their injuries. For a sample of serious injuries resulting from players hit by pitches see James (1985).

(8.) Two other unpublished studies of the relationship between the DH and hit batsmen also find support for the deterrence hypothesis. Using differences in hit batsmen between relievers (who rarely bat in the NL) and starting pitchers (who do bat) who switch leagues, Rupert and Stephenson (2002) find some support for the deterrence explanation. Kawaura and La Croix (2002) replicate the GST and TWK estimations on Japanese professional baseball, which also has the DH in only one league. The results are consistent with Goff, Shughart. and Tollison (1998) in support of the deterrence hypothesis.

(9.) The quotation is from Rosenthal (2002).

(10.) A well-documented case of how long teams will wait to retaliate against a pitcher occurred in 2002. Shawn Estes of the New York Mets threw at (and missed) Roger Clemens in retaliation for hitting Mets' star Mike Piazza two seasons prior. Mets' manager Bobby Valentine clearly ordered the retaliation because Estes was not a Met at the time Clemens struck Piazza.

(11.) As a condition of using this data, we must include the following statement. The information used here was obtained free of charge from and is copyrighted by Retrosheet. Interested parties may contact Retrosheet at 20 Sunset Rd., Newark, DE 19711. Turocy (2005) uses Retrosheet play-by-play data to test game-theoretic predictions regarding stolen bases.

(12.) We use baseball designation "plate appearance" as opposed to "at bats," because several results of plate appearances (including being hit by a pitch) do not count as at bats.

(13.) Baseball fans may wonder why we chose this measure as opposed to batting average. Though batting average is positively correlated with offensive output, OPS is a better predictor of run production.

(14.) Trandel (2004) states, "While retaliation is widely believed to exist ... it is now clear that any attempt to debate the method of retaliation should be postponed. Rather, the attention of interested researchers should be focused on finding a way in which retaliation (to the extent that it exists) can be detected."

(15.) Description from ESPN.com (2004).

(16.) A few games in the Retrosheet data are missing for the 1972 and 1973 seasons.

(17.) See Gould (1996) for an interesting theory of how the dispersion of talent causes the best players to excel and the worst players to suffer. Schmidt and Berri (2003) find some support for Gould's hypothesis in regards to the influx of new talent into MLB.

(18.) In the 1993 expansion. AL teams were allowed to protect more players on their rosters than N L teams. See Pappas (1997) for a description of expansion draft rules.

(19.) In December 1993, MLB adopted the On-Field Behavior Policy, which, along with several new policies, created the double-warning rule according to the N.A.P.B.L. Umpire Manual (1996). We would like to thank Jim Porter for identifying the exact timing of this rule change.

JOHN CHARLES BRADBURY and DOUGLAS J. DRINEN *

* An earlier version of this paper circulated under the title "Identifying Moral Hazard: A Natural Experiment in Major League Baseball." Thanks to David Smith and the volunteers of the Retrosheet organization for compiling and sharing the play-by-play data. We also thank seminar participants at Clemson University, session participants at the annual meeting of the American Mathematical Society and the Mathematical Association of America, Kevin Holman, and Jim Porter for helpful comments and suggestions.

Bradbury: Associate Professor, Department of Health, Physical Education, and Sport Science, Kennesaw State University, 1000 Chastain Road #0202, Kennesaw, GA 30144-5591. E-mail jbradbu2@kennesaw.edu

Drinen: Assistant Professor, Department of Mathematics & Computer Science, Sewanee: The University of the South, 735 University Avenue, Sewanee, TN 37383-1000. Phone 1-931-598-3370, E-mail ddrinen@ sewanee.edu

Are you seriously going to throw at somebody when you're facing Randy Johnson?

--Curt Schilling (1)

In 1973, amid waning fan interest in baseball, the American League (AL) of Major League Baseball (MLB) instituted the designated hitter rule as an experiment. The stated goal of this rule change was to boost offensive output by increasing the talent pool of batters in the lineup. Traditionally, the competing teams field nine players who must play defense in the field and bat. Because of the importance of the pitcher, who is responsible for putting every ball in play, teams rely on pitching ability and ignore the hitting ability when choosing the pitcher in the lineup. Therefore, pitchers tend to be very poor hitters. By allowing teams to substitute a player of greater hitting ability known as the designated hitter (DH)--to bat for the pitcher the total offensive output increases. The experiment has since grown into an institution in the AL that differentiates it from the National League (NL), where all players must bat and play in the field.

The DH succeeded in turning the AL into the "power league" as intended, but an unintended consequence of the rule change is that the AL now has more batters hit by pitches than the NL. Traditional baseball lore holds that the lack of retaliatory punishment in the AL for hitting batters is the cause of this phenomenon. Veteran NL manager Dusty Baker describes the deterrent impact from a pitcher's point of view, "You can be bold in (the American) League and get away with [hitting batters]. It's different in our league where you have to hit." (2) Pitchers who do not have to bat (where they might face retaliation) are more willing to risk hitting batters than pitchers who do bat.

A problem with previous studies of the subject is their reliance on yearly aggregate data. Hit batters are rare events--approximately 1% of all plate appearances result in a hit batter--that occur in the course of game where other incentives are quite relevant. This means that identifying small changes in pitcher behavior from specific factors will be difficult to identify with aggregate data. Differentiating between the two competing explanations for the hit batter differential between leagues deterrence and lineup composition--requires micro-level data where we can control for in-game strategic incentives for hitting batters.

Using a unique play-by-play data set that allows us to control for specific factors that affect hit batters, we find that pitchers are still more likely to hit batters with a DH in the lineup than without. And though pitchers are hit more rarely than other players, pitchers do experience retaliation for hitting batters. Retaliation against pitchers who plunk batters, which was not found in earlier studies, is needed to generate an effective deterrent for pitchers in the NL. These findings provide support for the deterrence hypothesis that the threat of retaliation against pitchers who hit batters does impact the probability that a pitcher will hit a batter and demonstrate the importance of informal enforcement mechanisms in deterring criminal behavior. Furthermore, we show that it is likely that the addition of formal administrative punishments for hitting batters by MLB may, in fact, have increased the incidence of hit batters in both leagues. The remainder of the paper is organized as follows. Section II reviews the previous literature, concentrating on the empirical problems with these studies. Section III discusses the strategic incentives for hitting batters. Section IV presents the empirical model with section V discussing the findings. Section VI discusses a peculiar deviation in the pattern of hit batters in the 1990s. Section VII concludes the paper.

II. A REVIEW OF PREVIOUS STUDIES

Goff, Shughart, and Tollison (1997, hereafter, GST) first examined the hypothesis that the cost differential to pitchers for hitting batters induced by the DH is the cause of the interleague difference in hit batsmen. Using annual league time series data from 1901 to 1990, GST analyzes changes in the difference in hit batsmen between leagues before and after the implementation of the DH. The results indicate that, controlling for several factors, the introduction of the DH raised the level of hit batsmen in the AL 15% higher than the NL, consistent with the hypothesis that the DH removes an effective criminal deterrent. However, though the evidence is clear that the DH is associated with an increase in hit batsmen, it is not necessarily a consequence of the lesser deterrent to hitting batters in the AL. Trandel, White, and Klein (1998, hereafter, TWK) and Levitt (1998) are skeptical of the interpretation of GST on two grounds. First, adding the DH raises the marginal benefit of hitting batters due to the existence of an additional good batter in the lineup. Because pitchers are poor hitters, few teams want to risk hitting the other pitcher. Replacing the pitcher with the DH places a batter more worthy of plunking in the lineup. As TWK state, "the difference [in hit batsmen] is largely because the AL batters are (on average) better hitters, and are thus less costly and more beneficial to hit." Therefore, the aggregate increase in hit batsmen in the AL over the NL may be attributed to increased rewards rather than the lowered punishment for hitting batters. (4) Second, the deterrence story seems implausible because pitchers are rarely hit, and therefore, the probabilistic penalty of retaliation for hitting batters is too small to have any real effect in a pitcher's decision calculus. Hitting the pitcher in retaliation may actually reward the instigator by putting the weak-hitting pitcher on base. Given that pitchers know this, it is unlikely that pitchers consider reciprocal retaliation to be a cost. As evidence, Levitt (1998) finds that there is no correlation between pitchers hitting batters and pitchers being hit themselves, and pitchers are hit so rarely that they would have to be hypersensitive to the relative price change induced by the DH. (5) Trandel (2004) finds similar results for the 1974-1977 seasons using the same test and finds little evidence of retaliation against any pitcher. All of these studies conclude that the increase in the frequency of hit batsmen in the AL likely reflects changes in the batting lineup composition and not a deterrent response by pitchers. (6)

The crux of the problem in identifying what factors influence the difference in hit batters across leagues lies with aggregate data, with which researchers can observe only the increase in the frequency of hit batsmen with the DH, a result that both competing hypotheses predict. Aggregate data, though informative, only reveal a quantity hit without regard to different situations where strategic incentives vary. A retaliatory plunking war in one game may be hidden within yearly statistics from a 162-game season, even though such an event may signal a credible threat of retaliation that persists into the future. As a result, the differing hypothesized effects are hard to disentangle from aggregate data. To remedy this we examine the costs and benefits of hitting any particular batter at the time the pitcher chooses a pitch. This allows us to examine pitcher decisions as the strategic incentives for hitting batters vary.

Readers familiar with the previous economics literature on this topic will recall that the earlier studies debate the cause of the DH-hit batter relationship as a moral hazard problem. In the AL, pitchers are actors who are improperly "monitored" by the opposition and therefore are more likely to engage the undesirable behavior of hitting batters than NL pitchers. The manger of the pitcher, in addition to the opposing team, may suffer as a consequence of their shirking agent, creating the potential for moral hazard. In this paper, we focus on the strategic incentives of pitchers as they impact pitcher behavior and shift the debate away from relying on moral hazard as the disputed explanation. The central debate has to do with a behavioral response to the cost of hitting batters. Is requiring pitchers to bat a real deterrent against hitting batters, or is the observed league difference in hit batsman a product of differing lineup compositions in the leagues? That pitchers might shirk in their duty to the manager is a possibility, but it is largely irrelevant to the debate and nearly impossible to identify empirically. Pitchers can still be perfect agents of their managers and act in accordance with their managers' wishes to hit more batters with a DH and fewer batters without a DH. Managers certainly have reason to fear that a pitcher who hits batters may end up injured, suspended, etc. Therefore, even without moral hazard, the cost differential of hitting batters in the leagues is sufficient to explain the league difference in hit batters.

III. THE COSTS AND BENEFITS OF HITTING BATTERS

Getting hit by a pitch is a painful, and sometimes dangerous, event in baseball. (7) To deter the pitcher from accidentally or purposely hitting batters the hit batter is awarded first base. Given this statutory constraint, we know the benefits and costs of hitting a batter. The benefits include preventing a batter from hitting the ball, which might generate more runs than simple base advancement; inflicting injury on a competitor; or decreasing the opposing team's willingness to stand close to the plate to hit outside pitches. The costs include putting a runner on base, which increases the chance of the opposing team scoring; ejections, fines, and suspensions to the pitcher or manager; physical retaliation via assault; retribution against the teammates of the pitcher by plunking them when they bat; and direct reciprocal retribution in the form of the current pitcher being intentionally plunked by the opposing pitcher.

It is this last cost which is the main subject of discussion. AL and NL pitchers bear identical costs and benefits when facing batters except for the reciprocal retribution. AL pitchers do not bat and therefore bear a lower cost for hitting a batter than NL pitchers. GST hypothesizes that this relative price difference induces AL pitchers to hit more batters than NL pitchers hit. (8) This is not to say that any pitcher actively seeks to consume hit batsmen purposely (at least not most of the time). Instead pitchers engage in riskier behavior (e.g., throwing inside, faster) when DHs bat in their place, which results in a greater number of hit batsmen. Cy Young award--winning pitcher Randy Johnson, who has played in both leagues, explains, "If you're the pitcher and you're playing in the American League, then you may have a tendency to throw inside a little bit more knowing when that ninth hole comes up, you won't be hitting. You're protected in that regard." (9) This is a behavioral response similar to automobile drivers in Peltzman (1975), which documents seat belt laws inducing motorists to engage in riskier behavior that leads to car accidents. Lowering the negative consequences of an action leads people to take more risks that result in the negative outcome.

However, the critics of the deterrent hypothesis propose that the expected punishment for plunking is too minor to have much, if any, effect on an NL pitcher's decision when he pitches to batters. On average, hitting the pitcher is not a good idea. If retaliation does provide a deterrent, pitchers would have to be extremely sensitive to that threat. However, circumstances arise where the benefits of hitting a pitcher exceed the costs. This may occur in the course of the same game in which a pitcher hits a batter, or it may occur in some future game. (10) Though pitchers are hit less than other position players, it is possible that pitchers may be hit more than they ought to be, given their hitting abilities. Ultimately, this is an empirical question. Using the data we describe in the following section, we can answer the questions central to the analysis.

IV. EMPIRICAL MODEL

A data-gathering project known as Retrosheet has carefully reconstructed baseball play-by-play data for many seasons. (11) Using a computer program, we extracted data by plate appearance from the Retrosheet data. From this we can observe the exact game situation when a player is hit or not hit; thereby, we can control for the costs and benefits of hitting any particular player during a plate appearance. (12) We use a predictive model to estimate the likelihood that a batter is hit given five categories of influences on hitting batters: deterrence (DH), batter quality (BQ), pitcher quality (PQ), retaliation (R), and game situation (GS). Thus,

[HBP.sub.j] = [[alpha].sub.j] + [beta][DH.sub.j] + [lambda][BQ.sub.j] + [gamma][PQ.sub.j] + [phi][R.sub.j] + [psi][GS.sub.j] + [eta][YR.sub.j] + [[epsilon].sub.j],

where HBP is a dummy variable equal to one when a pitcher hits a batter and zero otherwise. Subscript j represents a plate appearance, [alpha] is the constant, [epsilon] is the error term, and YR is the year. We estimate the equation using both logit and probit estimation techniques.

The variable DH is a dummy equal to one when the DH rule is in effect. According to baseball's conventional wisdom, it is the lower deterrence induced by the DH that explains the hit batsmen differential between leagues. If, after accounting for the many factors we include, the DH coefficient remains positive, it is likely that the deterrent effect of requiring the pitcher to bat explains some of the difference. If it is insignificant when controlling for other factors, it is likely that the deterrence hypothesis is incorrect.

The BQ vector includes two batter quality variables that effect the cost-benefit calculation of hitting batters. Better hitting batters that replace pitchers in the batting order ought to be hit more frequently than pitchers; therefore, we include the season OPS of the batter and a dummy equal to one if a pitcher is hitting. OPS is the sum of the slugging average and on-base percentage. (13) A batter with a higher OPS is more likely to generate offense and is therefore more beneficial to hit than a player with a lower OPS. Even when controlling for OPS, pitchers may be less desirable batters to hit relative to teammates, which is why we also include the pitcher dummy. These variables are important in determining whether composition of the lineup explains the hit batsmen differential.

Though it has not been discussed much in this literature, hit batsmen are largely accidental. Even pitchers who do not mean to brush a player back or pitch on the inside of the plate may accidentally hit batters. The variables in PQ control for the pitcher's propensity to hit batters accidentally. Pitchers who are more likely to hit batters mistakenly are also likely to make mistakes in the form of placing hittable balls in the strike zone when compared to other pitchers. These pitchers will generate more offense for opposing teams relative to more skilled pitchers. We use the pitcher's OPS allowed to hitters as a proxy for pitcher skill. Also, pitchers with less control are more likely to issue walks than pitchers with good control. Therefore, we also include the walk rate of the pitcher to control for pitcher quality. Both of these measures ought to be positively related to hit batsmen.

At the heart of this paper is the issue of using the pitcher to retaliate against the other team for hitting batters. This retaliation may be reciprocal, pitcher hitting pitcher, or pitchers may hit teammates of the pitcher. Trandel (2004) notes that observing retaliation in the data is of fundamental importance to this debate. (14) We have the ability to detect retaliation using the variables in vector R. There are several reasons pitchers retaliate against the other team through plunking, but we will focus on two common reasons that we can capture in the data. If a batter on the current pitcher's team was hit in the previous half-inning, this may provoke a retaliatory plunking of any player on the opposing team. We include a dummy variable equal to one if this occurs. Also, we test specifically for reciprocal retaliation against pitchers. We create a dummy for a pitcher appearing at the plate when a batter was hit in the previous half-inning. If retaliation does occur through pitchers hitting pitchers, this variable should be positive. Also, it is a baseball tradition to hit a batter following a home run, on occasion. We include a dummy equal to one if the previous batter hit a home run.

Finally, we include vector GS of many possible game situations affecting the cost-benefit calculation. The score, inning, number of outs, and base runners threatening to score all affect the costs and benefits of hitting a batter at the particular moment. The following example from an altercation between the Houston Astros and Chicago Cubs in 2004 demonstrates the role of the game situation in the decision to plunk a batter.

Both benches cleared briefly in the second inning of Friday's game between the Houston Astros and the Chicago Cubs when Cubs catcher Michael Barrett began jawing with Astros pitcher Roy Oswalt, 5 days after Oswalt hit him in the back with a pitch. Barrett said after that game that he "thought he [Oswalt] was better than that" and warned: "We're going to see him again." With Wood on the mound Friday in another matchup with Oswalt, Barrett and Oswalt exchanged words when the Houston pitcher came to the plate to lead off the second with the Astros leading 4-2. After the brief altercation, Oswalt grounded out and the two again exchanged a few words after Barrett had run down the first-base line to back up the throw. In the top of the sixth, Oswalt again came up as the leadoff hitter, with the Astros leading 8-4, and he was hit in the thigh by the first pitch from Cubs reliever Kent Mercker. (15)

Score differential and whether a team is winning or losing certainly impact the strategic incentives. A large absolute score difference means there is a low marginal impact of runs. When the score differential becomes large both teams may be more willing to engage in a plunking war than when the game is tight. However, because the costs of retaliation fall as the differential rises, pitchers may be extra careful as the likelihood of retaliation increases. Also, it may be the case that a big loser is more willing to plunk batters. As the score differential grows the losing team may feel the opponent is "rubbing it in," and thus may start plunking the winning team. It is unclear if batters are more likely to be hit early as opposed to later in a game. Plunking early may send a message to players not to "crowd" the inside of the plate throughout the game. However, if retaliation is an ex post enforcement mechanism to prevent plunking, later in the game teams have more incentive to renege on tacit agreements not to hit each others' players. The number of outs affects the probability that any base runner will score. A batter put on base with no outs has a better chance of scoring than a batter who reaches first with two outs. Finally, the number and positioning of base runners can also affect the costs and benefits. The specific predictions between differing runner configurations is a bit complicated and largely irrelevant to the discussion here. Generally, the more runners on base and the further the runners are around the bases, the greater the offensive damage a batter can do by putting a ball in play. Therefore, the seven of eight possible runner configurations included (we exclude empty bases for comparison) should be positively related to hit batsmen. We also include year effects to control for unobservable differences in hit batsmen that may change from year to year. Table 1 lists the summary statistics for the variables in both of our data sets.

V. RESULTS AND DISCUSSION

Retrosheet provides play-by-play data for both leagues from 1972 to 1992 and for the 1969 season. From this data, we extract two four-season data sets--the earliest and most recent available years--to estimate the empirical model. For each plate appearance, we extract data on the specific game situation and the seasonal performance of the players participating in the event. Initially, we focus on the four most recent years of available data, 1989-1992.

Table 2 lists the results of the logit and probit estimations of coefficients and z statistics. For ease of interpretation, we also include the odds ratio (logit) and marginal probability (probit) calculations for each variable. Most of the variables are significant and of the expected sign. Of particular interest, the DH variable is positive and significant when controlling for all other factors, including the lineup composition batter quality variables. When the DH is in effect pitchers are 15%-17% more likely to hit batters than when the DH is not in effect. Consistent with the lineup composition hypothesis, batter quality is positively related with hit batsmen, and pitchers are about 55% less likely to be hit than other batters. Also, pitcher quality is negatively related to hit batsmen. Both proxies for lack of pitch control are positively associated with a greater likelihood of hitting a batter; however, pitcher OPS is not statistically significant.

The retaliation variables tell an interesting story. Preceding events likely to provoke retaliation all tend to increase the likelihood that a pitcher hits a batter. A batter who appears at the plate following a home run is 32% more likely to be hit than when the preceding batter did not hit a home run. In response to hit batters, pitchers are more likely to hit batters the half-inning after the opposing pitcher hits a current pitcher's teammate. Though the results are not significant for players in general, a pitcher is four times more likely to be hit when an opposing player was hit in the previous half-inning. This variable, which is significant, is very important for the deterrence hypothesis. We now observe retaliation against pitchers who hit batters. This is a phenomenon previously unidentified in the aggregate data. Though pitchers are hit less frequently than other players on average, pitchers are more likely to be hit after plunking an opposing player. Therefore, pitchers do bear a very real cost to hitting batters, which is necessary to provide a deterrent to pitchers.

For the game situation variables, outs and innings seem to be relatively unimportant. Though the absolute score differential is unimportant, the relative score differential from the offense to the defense has a small but significant effect on the incidence of hit batsmen. The base runner composition variables appear to be important and consistent with our predictions.

Over this sample, the AL hit batsmen rate is 26% higher than the NL. Controlling for all of the above factors, including batter quality, the DH rule dummy explains approximately 60% of this difference. It is now clear that something other than the lineup composition created by the DH is responsible for the increase in hit batsmen differential between leagues. The most plausible explanation for this difference is the removal of a credible threat of ex post punishment of pitchers in the AL. However, it is possible that some yet unidentified factors unique to the AL may explain the sign and significance of the DH dummy since the DH occurs only in this league. For example, differences in strike zones, stadium configurations, league traditions, etc. between leagues are competing but less satisfying explanations. To be thorough, we examine a different data set that includes the 1969 and 1972-1974 seasons. (16) This is the earliest available data in the Retrosheet archives, plus this period includes some observations of the AL without the DH rule. We are slightly less confident in this data than our more recent data for a few reasons. First, the only two seasons available prior to the introduction of the DH are the noncontiguous years 1969 and 1972. Second, the late 1960s and early 1970s were low offense years for baseball. The DH was the last of several rules instituted to "fix" baseball, which was losing fans. The league responded in 1969 by lowering the pitchers mound, shrinking the strike zone to make it more hitter friendly, and adding two new teams to each league. Luckily, the exogenous events other than the DH apply to both leagues; therefore, we feel the data are useful in isolating the effect of the DH rule on hit batsmen, given that we acknowledge the deficiencies.

We apply the same model for the previous sample to this data, except that the DH variable is now coded zero for some observations in the AL, when the DH was not in effect. Therefore, the DH should now reflect the change in the costs of hitting batters induced by the DH rule itself. The results in Table 3 are quite similar to estimates of the more recent data. For this sample, when the DH is in effect pitchers are 11%-12% more likely to hit batters than without the DH rule, ceteris paribus. These results differ only slightly in magnitude from our previous estimates. We also observe pitchers, as well as position players, being hit in retaliation for hit batters in this sample. Most of the other factors are of similar magnitude and size, though some measures are no longer statistically significant. In the years following the DH, AL pitchers hit batters at a rate 14% higher than NL pitchers; thus, the DH rule dummy explains approximately 80% of the interleague difference in hit batsmen in this sample. Therefore, the results indicate that the removal of the threat of retaliation by the DH rule, not unobserved league-specific factors, is the likely cause.

VI. THE PROBLEM OF THE 1990s

Figure 1 shows the fluctuation in the rate of hit batsmen from 1921 to 2003. It is evident from the figure that the introduction of the DH in 1973 coincides with the AL's continued higher rate of hit batsmen as compared to the NL. From the introduction of the DH until the present, the AL's average yearly hit batsmen rate has exceeded the NL rate by 15%. From the beginning of the modern era of baseball until the 1970s, the hit batsmen rates varied closely, yet randomly, across leagues. For the next 21 years the AL rate exceeded the NL rate every year, which is consistent with both the deterrence and lineup composition hypotheses. But in the mid-1990s, just as economists began to study this issue and the public availability of play-by-play data ceased, a deviation in this pattern emerged. In 1994, the NL hit batsmen rate rose above the AL rate for the first time since the implementation of the DH, and the leagues' hit batter rates began to fluctuate more closely than in the past. Why was there such a dramatic shift and how does this shift impact the perceived relationship between the DH and hit batsmen? There are two important factors that contributed to this change: league expansion and a rule change in the punishment for pitchers hitting batters.

In response to criticism over the recent events of the 1990s, Goff, Shughart, and Tollison (1998) postulates that the NL expansion in 1993 diluted the talent pool of players increasing the number of accidental hit batsmen. We believe the evidence supports this contention. Fringe pitchers and hitters are more apt to make mistakes by hitting batters or getting hit by pitches. (17) In 1993, the NL hit batsmen rate fell to only 10% of the AL rate, while the average difference over the preceding 2 decades was 20%. Additionally, the mid-1990s corresponded with a dramatic increase in hit batsmen in both leagues, doubling over the decade. The expansion hypothesis is consistent with both the change in the difference between leagues and the rise in hit batters overall. In the two rounds of expansion, three of the four expansion teams joined the NL with one existing AL team (Milwaukee) shifting to the NL. But more importantly, the asymmetric expansion draft rules in 1993 favored expansion players coming from NL rosters. Therefore, the expansion diluted the talent pool of both leagues but affected NL rosters to a greater degree than AL rosters. (18) From 1993 to 1997, as the expansion players eased into the league, the NL rate exceeded the AL rate in 3 of the 5 years, with the AL rate being 2% lower than the NL rate, on average. But in 1998, when both leagues added a team, there was no asymmetry between leagues in drafting expansion players. Interestingly, the disruption to the previous pattern was noticeably smaller. Since the last round of expansion in 1998, only once (2000) has the NL rate exceeded the AL rate, with the AL rate exceeding the NL rate by an average of 6.5%.

[FIGURE 1 OMITTED]

The second factor was an important rule change in the punishment for hitting batters that significantly altered the cost of retaliating against pitchers who hit batters. In 1994, MLB adopted the "double-warning" rule, which authorized umpires to warn both teams if an umpire judges that a pitcher intentionally hits or attempts to hit a batter. (19) Once the umpire issues a warning, retaliation results in the immediate expulsion of the current pitcher and manager of the offending team, accompanied by monetary fines. This statutory constraint significantly raised the cost of retaliation, which, in turn, lowered the probability of punishment to pitchers for initially hitting batters. With the double-warning rule, pitchers can take risks that are more likely to result in hit batters than without the rule because pitchers know the opposition will be less likely to retaliate due to the increased penalty after the umpire issues a warning. The relative rise in the NL rate of hit batsmen in the 1990s is consistent with the deterrence hypothesis and therefore is expected. NL pitchers are now more protected from retaliation than before the double-warning rule, thereby inducing NL pitchers to behave more like their counterparts in the AL. A similar phenomenon occurred in Japanese professional baseball. The Nippon Professional Baseball League, which also has the DH in only one league, experienced a similar convergence of hit batsmen after several years of excessive hit batsmen in the DH league. Kawaura and La Croix (2002) find that this convergence correlates with a rule change that increased the penalty for intentionally hitting batters. Thus, the statutory constraint designed to limit hit batters likely had the unintended consequence of doing the opposite.

Finally, using game-level (not play by play) data from 1973 to 2003, Bradbury and Drinen (forthcoming) find that games in which the DH is allowed are associated with more hit batters than non-DH games, controlling for many relevant factors. Over the entire 31-year history of the DH, the 4 years in which the NL rate exceeded the NL rate appear to be exceptions rather than the rule. When relevant historical facts are included in the analysis, it is clear that the 1990s actually lend support to, rather than cast doubt upon, the deterrence explanation for differences in hit batsmen between leagues. The most recent history of hit batsmen rates is indicative of a return to the pre-1990s league differences in hit batsmen, with a tempering of the difference by the double-warning rule.

VII. CONCLUSION

Competing economic theories predict that DH rule is responsible for raising the rate of hit batsmen in the AL above the NE. As Figure 1 illustrates, for the 3 decades following the introduction of the DH the AL hit batsmen rate remained higher than the NL rate for all but 4 years. However, there is less agreement regarding the cause of this difference. While the lower cost of hitting batters for AL pitchers, who do not have to bat and face reciprocal retaliation, is a likely explanation, it is not the only explanation. An alternate hypothesis is that the DH increases the incidence of hit batters by altering the composition of the batting lineup. Replacing a poor-hitting pitcher with a good-hitting DH raises the marginal benefit to hitting an additional batter in the lineup in the AL. Additionally, the retaliation threat to pitchers may be too small to be relevant. If retaliation is not likely, there is no cost differential to pitchers between leagues to explain the hit batsmen difference.

Past empirical analyses of the competing hypotheses to explain the hit batsmen differential have relied on yearly aggregate data. However, the explicit cost-benefit analysis of the pitcher at the time of a pitch is not observable with aggregate data. Using two new unique data sets we have been able to observe the costs and benefits of hitting batters during individual plate appearances. Therefore, we isolate the predicted effects of the competing hypotheses and determine the impact of many relevant factors that influence the pitcher's consumption of risk in hitting a batter every time a hitter steps to the plate. Controlling for variables that proxy batter quality, pitcher quality, retaliation, and game situation, we find that the DH rule increases the likelihood that any batter will be hit during a plate appearance between 11% and 17%. This explains approximately 60%-80% of the hit batsmen rate differential between leagues. Furthermore, we observe retaliation against plunking pitchers, which is necessary for the deterrent explanation. In total, the findings support the fact that requiring pitchers to bat deters them from hitting batters. Though other factors are important in the economics of plunking, the threat of retaliation plays a nontrivial role. Players really do "take care of it" themselves through an informal enforcement mechanism, while official enforcement penalties may have, in fact, exacerbated the problem of hit batters.

ABBREVIATIONS

AL: American League

DH: Designated Hitter

MLB: Major League Baseball

NL: National League

OPS: On-Base Percentage Plus Slugging percentage

doi:10.1093/ei/cb1014

REFERENCES

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Duggan, M., and S. D. Levitt. "Winning Isn't Everything: Corruption in Sumo Wrestling." American Economic Review, 92(5), 2002, 1594-1605.

ESPN.com. 2004. Teams touchy after incident Sunday. August 27, 2004. http://sports.espn.go.com/mlb/ news/story?id=1869441.

Goff, B. L., R. E. McCormick, and R. D. Tollison. "Racial Integration as an Innovation: Empirical Evidence from Sports Leagues." American Economic Review, 92(2), 2002, 16-26.

Goff B. L., W. F. Shughart, and R. D. Tollison. "Batter Up! Moral Hazard and the Effects of the Designated Hitter Rule on Hit Batsmen." Economic Inquiry, 35(3), 1997, 555-61.

--. "Moral Hazard and the Effects of the Designated Hitter Rule Revised." Economic Inquiry, 36(4), 1998, 688-92.

Gould, S. J. Full House: The Spread of Excellence from Plato to Darwin. New York: Harmony Books, 1996.

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James, B. The Bill James Baseball Abstract. New York: Ballantine Books, 1985.

Katz, L., S. D. Levitt, and E. Shustorovich. "Prison Conditions, Capital Punishment, and Deterrence." American Law and Economics Review 5(2), 2003, 31843.

Kawaura, A., and S. J. La Croix. "Baseball in Japan and North America: Same Game, Same Rules, Same Results?" Manuscript, University of Hawaii, 2002.

Levitt, S. D. "The Hazards of Moral Hazard: Comment on Goff, Shughart, and Tollison." Economic Inquiry, 36(4), 1998, 685-7.

McCormick, R. E., and R. D. Tollison. "Crime on the Court." Journal of Political Economy, 92(2), 1984, 223-35.

Morgan, J. 2002. Attention Estes: throw at Clemens. June 14. 2002.

N.A.P.B.L. Umpire Manual. Chicago: Triumph Books. 1996.

Pappas, D. 1997. Terms of major league expansion. Outside the Lines, Fall, 3. http://roadsidephotos. com/baseball/expansion.htm

Peltzman, S. "The Effects of Automobile Safety Regulation." Journal of Political Economy, 83(4), 1975, 677-725.

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Rosenthal, K. 2002. Mets get shot with mighty Clemens at the bat. The Sporting News, June 13.

Rupert, M. E., and E. F. Stephenson, "On Moral Hazard and Hit Batsmen: A Comment on Goff et al. and Trandel et al." Manuscript, Berry College, 2002.

Schmidt, M. B., and D. J. Berri. "On the Evolution of Competitive Balance: The Impact of Increasing Global Search." Economic Inquiry, 41(4), 2003, 692-704.

Sobel, R. S.. and T. M. Nesbitt. "Automobile Safety Regulation and the Incentive to Drive Recklessly: Evidence from NASCAR." Manuscript, West Virginia University, 2004.

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Trandel, G. A. "Hit By Pitches: Moral Hazard, Cost-Benefit. Retaliation. or Lack of Evidence?" Journal of Sports Economics. 5(1). 2004, 87-92.

Trandel, G. A., L. H. White, and P. G. Klein. "The Effect of the Designated Hitter Rule on Hit Batsmen: Pitcher's Moral Hazard or the Team's Cost-Benefit Calculation." Economic Inquiry, 36(4), 1998, 679-84.

Turocy, T. L. "Offensive Performance, Omitted Variables, and the Value of Speed in Baseball." Economics Letters, 89(3), 2005, 283-6.

(1.) The quotation is from Rosenthat (2002).

(2.) The quotation is from Morgan (2002).

(3.) The controlled environment of sports has become a popular arena for testing economic hypotheses. Some examples include Duggan and Levitt (2002) on identifying corruption in sumo wrestling: Goff, McCormick, and Tollison (2002) on racial integration and innovation as measured by winning in baseball and basketball: McCormick and Tollison (1984) and Heckelman and Yates (2003) on crime and enforcement in basketball and hockey: and Sobel and Nesbitt (2004) on offsetting behavior in response to safety regulations in National Association for Stock Car Auto Racing.

(4.) In reply Goff, Shughart, and Tollison (1998) responds to the criticisms by referring to the control variables (batting and pitching statistics) in previous and revised regressions that control for this factor.

(5.) This mirrors Levitt's critique on the deterrent effect of capital punishment in Katz, Levitt, and Shustorvich (2003).

(6.) These studies also extend the data into the 1990s, in which the difference in hit batsmen between leagues becomes less stable. However, there are several plausible explanations for the change that we discuss in Section VI.

(7.) Pitches hitting batters have ended several professional baseball careers; seven players (only one in MLB) have died from their injuries. For a sample of serious injuries resulting from players hit by pitches see James (1985).

(8.) Two other unpublished studies of the relationship between the DH and hit batsmen also find support for the deterrence hypothesis. Using differences in hit batsmen between relievers (who rarely bat in the NL) and starting pitchers (who do bat) who switch leagues, Rupert and Stephenson (2002) find some support for the deterrence explanation. Kawaura and La Croix (2002) replicate the GST and TWK estimations on Japanese professional baseball, which also has the DH in only one league. The results are consistent with Goff, Shughart. and Tollison (1998) in support of the deterrence hypothesis.

(9.) The quotation is from Rosenthal (2002).

(10.) A well-documented case of how long teams will wait to retaliate against a pitcher occurred in 2002. Shawn Estes of the New York Mets threw at (and missed) Roger Clemens in retaliation for hitting Mets' star Mike Piazza two seasons prior. Mets' manager Bobby Valentine clearly ordered the retaliation because Estes was not a Met at the time Clemens struck Piazza.

(11.) As a condition of using this data, we must include the following statement. The information used here was obtained free of charge from and is copyrighted by Retrosheet. Interested parties may contact Retrosheet at 20 Sunset Rd., Newark, DE 19711. Turocy (2005) uses Retrosheet play-by-play data to test game-theoretic predictions regarding stolen bases.

(12.) We use baseball designation "plate appearance" as opposed to "at bats," because several results of plate appearances (including being hit by a pitch) do not count as at bats.

(13.) Baseball fans may wonder why we chose this measure as opposed to batting average. Though batting average is positively correlated with offensive output, OPS is a better predictor of run production.

(14.) Trandel (2004) states, "While retaliation is widely believed to exist ... it is now clear that any attempt to debate the method of retaliation should be postponed. Rather, the attention of interested researchers should be focused on finding a way in which retaliation (to the extent that it exists) can be detected."

(15.) Description from ESPN.com (2004).

(16.) A few games in the Retrosheet data are missing for the 1972 and 1973 seasons.

(17.) See Gould (1996) for an interesting theory of how the dispersion of talent causes the best players to excel and the worst players to suffer. Schmidt and Berri (2003) find some support for Gould's hypothesis in regards to the influx of new talent into MLB.

(18.) In the 1993 expansion. AL teams were allowed to protect more players on their rosters than N L teams. See Pappas (1997) for a description of expansion draft rules.

(19.) In December 1993, MLB adopted the On-Field Behavior Policy, which, along with several new policies, created the double-warning rule according to the N.A.P.B.L. Umpire Manual (1996). We would like to thank Jim Porter for identifying the exact timing of this rule change.

JOHN CHARLES BRADBURY and DOUGLAS J. DRINEN *

* An earlier version of this paper circulated under the title "Identifying Moral Hazard: A Natural Experiment in Major League Baseball." Thanks to David Smith and the volunteers of the Retrosheet organization for compiling and sharing the play-by-play data. We also thank seminar participants at Clemson University, session participants at the annual meeting of the American Mathematical Society and the Mathematical Association of America, Kevin Holman, and Jim Porter for helpful comments and suggestions.

Bradbury: Associate Professor, Department of Health, Physical Education, and Sport Science, Kennesaw State University, 1000 Chastain Road #0202, Kennesaw, GA 30144-5591. E-mail jbradbu2@kennesaw.edu

Drinen: Assistant Professor, Department of Mathematics & Computer Science, Sewanee: The University of the South, 735 University Avenue, Sewanee, TN 37383-1000. Phone 1-931-598-3370, E-mail ddrinen@ sewanee.edu

TABLE 1 Summary Statistics 1989-1992 Standard Variable Mean Deviation Minimum Maximum Hit by pitch 0.005528 0.074145 0 1 DH 0.54037 0.498368 0 1 Batter OPS 0.702851 0.126917 0 3 Pitcher hitting 0.029663 0.169656 0 1 Pitcher OPS 0.705841 0.089025 0 4 Pitcher walk rate 0.087155 0.027912 0 0.5 Batter hit in last 0.022312 0.147695 0 1 half-inning Pitcher hitting after 0.00056 0.023647 0 1 batter hit Previous batter hit 0.019982 0.139937 0 1 home run Relative score (offense -0.04484 2.959261 -20 20 to defense) Absolute relative score 2.017926 2.164995 0 20 Inning 5.035247 2.6935 1 22 Outs 0.977866 0.81623 0 2 Man on first 0.179398 0.383686 0 1 Man on second 0.08862 0.284194 0 1 Man on third 0.031418 0.174445 0 1 Men on first and second 0.068588 0.252753 0 1 Men on first and third 0.032199 0.176528 0 1 Men on second and third 0.022048 0.14684 0 1 Bases loaded 0.022591 0.148594 0 1 1969,1972-1974 Standard Variable Mean Deviation Minimum Maximum Hit by pitch 0.005394 0.073247 0 1 DH 0.253612 0.435078 0 1 Batter OPS 0.687984 0.142806 0 4 Pitcher hitting 0.05493 0.227845 0 1 Pitcher OPS 0.690408 0.079808 0 2.5 Pitcher walk rate 0.08878 0.027886 0 1 Batter hit in last 0.021249 0.144212 0 1 half-inning Pitcher hitting after 0.001058 0.032515 0 1 batter hit Previous batter hit 0.019453 0.138112 0 1 home run Relative score (offense -0.04525 2.924196 -19 19 to defense) Absolute relative score 1.96396 2.166984 0 19 Inning 5.067495 2.735999 1 25 Outs 0.977184 0.816156 0 2 Man on first 0.188647 0.391228 0 1 Man on second 0.083079 0.276003 0 1 Man on third 0.027007 0.162104 0 1 Men on first and second 0.071399 0.25749 0 1 Men on first and third 0.031505 0.174679 0 1 Men on second and third 0.020134 0.140458 0 1 Bases loaded 0.02336 0.151045 0 1 TABLE 2 Estimates of Hit Batsmen, 1989-1992 Category Variable Logit Deterrence DH dummy Coefficient 0.162214 Odds ratio 1.176112 z statistic 4.63 ** Batter Batter OPS Coefficient 0.614065 quality Odds ratio 1.847929 z statistic 3.93 ** Pitcher hitting Coefficient -0.81375 dummy Odds ratio 0.443191 z statistic -4.52 ** Pitcher Pitcher OPS Coefficient 0.235353 quality Odds ratio 1.265355 z statistic 1.24 Pitcher walk Coefficient 2.893827 rate Odds ratio 18.06231 z statistic 4.89 ** Retaliation Hit batter in Coefficient 0.073132 previous Odds ratio 1.075873 half-inning z statistic 0.65 Pitcher hitting Coefficient 1.447219 after hit in Odds ratio 4.251276 previous half- z statistic 2.36 ** inning Previous batter Coefficient 0.283919 hit home run Odds ratio 1.328325 z statistic 2.68 ** Game Relative score Coefficient 0.042483 situation differential of Odds ratio 1.043399 offense to z statistic 7.02 ** defense Absolute relative Coefficient -0.0117 score Odds ratio 0.988371 z statistic -1.38 Inning Coefficient -0.01545 Odds ratio 0.984668 z statistic -2.31 * Outs Coefficient -0.01707 Odds ratio 0.983078 z statistic -0.8 Man on first Coefficient 0.048279 Odds ratio 1.049463 z statistic 1.03 Man on second Coefficient 0.075331 Odds ratio 1.078241 z statistic 1.23 Man on third Coefficient 0.278385 Odds ratio 1.320995 z statistic 3.14 ** Men on first and Coefficient 0.135049 second Odds ratio 1.144593 z statistic 2.02 * Men on first and Coefficient 0.197402 third Odds ratio 1.218234 z statistic 2.19 * Men on second Coefficient 0.251846 and third Odds ratio 1.286398 z statistic 2.4 ** Bases loaded Coefficient 0.132466 Odds ratio 1.14164 z statistic 1.21 Year Year 1990 Coefficient 0.050283 dummies Odds Ratio 1.051568 z statistic 1.02 Year 1991 Coefficient 0.096062 Odds Ratio 1.100827 z statistic 1.97 * Year 1992 Coefficient 0.189908 Odds Ratio 1.209139 z statistic 3.97 ** Goodness Observations 641640 of fit Log-likelihood -21841.72 Likelihood ratio- 264.99 [chi squared] Category Variable Probit Deterrence DH dummy Coefficient 0.056729 Marginal 0.000864 probability z statistic 4.65 ** Batter Batter OPS Coefficient 0.215086 quality Marginal 0.003293 probability z statistic 3.95 ** Pitcher hitting Coefficient -0.26391 dummy Marginal -0.00297 probability z statistic -4.6 ** Pitcher Pitcher OPS Coefficient 0.086929 quality Marginal 0.001331 probability z statistic 1.3 Pitcher walk Coefficient 1.033456 rate Marginal 0.015822 probability z statistic 4.92 ** Retaliation Hit batter in Coefficient 0.023006 previous Marginal 0.000362 half-inning probability z statistic 0.58 Pitcher hitting Coefficient 0.485283 after hit in Marginal 0.013967 previous half- probability inning z statistic 2.18 * Previous batter Coefficient 0.099798 hit home run Marginal 0.001729 probability z statistic 2.62 ** Game Relative score Coefficient 0.01472 situation differential of Marginal 0.000225 offense to probability defense z statistic 7.05 ** Absolute relative Coefficient -0.00368 score Marginal -5.60 x [10.sup.-5] probability z statistic -1.26 Inning Coefficient -0.0055 Marginal -8.40 x [10.sup.-5] probability z statistic -2.36 ** Outs Coefficient -0.00581 Marginal -8.90 x [10.sup.-5] probability z statistic -0.78 Man on first Coefficient 0.016536 Marginal 0.000257 probability z statistic 1.01 Man on second Coefficient 0.026281 Marginal 0.000414 probability z statistic 1.23 Man on third Coefficient 0.098077 Marginal 0.00169 probability z statistic 3.1 ** Men on first and Coefficient 0.047716 second Marginal 0.00077 probability z statistic 2.03 * Men on first and Coefficient 0.068936 third Marginal 0.001147 probability z statistic 2.15 * Men on second Coefficient 0.08975 and third Marginal 0.001534 probability z statistic 2.4 ** Bases loaded Coefficient 0.045277 Marginal 0.000733 probability z statistic 1.17 Year Year 1990 Coefficient 0.017926 dummies Marginal 0.000278 probability z statistic 1.05 Year 1991 Coefficient 0.033389 Marginal 0.000522 probability z statistic 1.97 * Year 1992 Coefficient 0.066534 Marginal 0.001064 probability z statistic 3.99 ** Goodness Observations 641640 of fit Log-likelihood -21841.24 Likelihood ratio- 265.96 [chi squared] Notes: Constants not reported. * Statistical significance at the 5% level. ** Statistical significance at the 1% level. TABLE 3 Estimates of Hit Batsmen, 1969, 1972-1974 Category Variable Logit Deterrence DH Dummy Coefficient 0.116442 Odds ratio 1.123492 z statistic 2.25 * Batter Batter OPS Coefficient 1.415542 quality Odds ratio 4.118717 z statistic 10.07 ** Pitcher hitting Coefficient -0.45747 dummy Odds ratio 0.632884 z statistic -3.68 ** Pitcher Pitcher OPS Coefficient 0.708161 quality Odds ratio 2.030253 z statistic 3.2 * Pitcher walk rate Coefficient 3.173758 Odds ratio 23.89712 z statistic 5.16 ** Retaliation Hit batter in Coefficient 0.334874 previous half-inning Odds ratio 1.397764 z statistic 3.04 ** Pitcher hitting Coefficient 1.630977 after batter hit in Odds ratio 5.108862 previous half-inning z statistic 4.57 ** Previous batter hit Coefficient 0.172655 home run Odds ratio 1.188456 z statistic 1.51 Game Relative score Coefficient 0.046296 situation differential of Odds ratio 1.047385 offense to defense z statistic 7.22 ** Absolute relative Coefficient 0.006134 score Odds ratio 1.006153 z statistic 0.7 Inning Coefficient -0.01564 Odds ratio 0.984482 z statistic -2.26 * Outs Coefficient 0.020185 Odds ratio 1.02039 z statistic 0.89 Man on first Coefficient -0.06239 Odds ratio 0.939517 z statistic -1.25 Man on second Coefficient 0.144596 Odds ratio 1.155572 z statistic 2.29 * Man on third Coefficient 0.09539 Odds ratio 1.100088 z statistic 0.91 Men on first and Coefficient 0.072292 second Odds ratio 1.074969 z statistic 1.04 Men on first and Coefficient -0.09798 third Odds ratio 0.906663 z statistic -0.92 Men on second and Coefficient 0.05883 third Odds ratio 1.060595 z statistic 0.48 Bases loaded Coefficient -0.05206 Odds ratio 0.949273 z statistic -0.43 Year Year 1969 Coefficient 0.191714 dummies Odds ratio 1.211324 z statistic 3.41 ** Year 1972 Coefficient 0.168274 Odds ratio 1.18326 z statistic 2.89 ** Year 1973 Coefficient -0.05735 Odds ratio 0.944265 z statistic -1.11 Goodness Observations 584886 of fit Log-likelihood -19436.32 Likelihood 373.84 ratio-[chi square] Category Variable Probit Deterrence DH Dummy Coefficient 0.039419 Marginal 0.000596 probability z statistic 2.19 * Batter Batter OPS Coefficient 0.509973 quality Marginal 0.007512 probability z statistic 9.87 ** Pitcher hitting Coefficient -0.14043 dummy Marginal -0.00177 probability z statistic -3.41 ** Pitcher Pitcher OPS Coefficient 0.266846 quality Marginal 0.003931 probability z statistic 3.33 * Pitcher walk rate Coefficient 1.150442 Marginal 0.016947 probability z statistic 5.16 ** Retaliation Hit batter in Coefficient 0.120795 previous half-inning Marginal 0.002068 probability z statistic 3.06 ** Pitcher hitting Coefficient 0.581011 after batter hit in Marginal 0.018297 previous half-inning probability z statistic 4.24 ** Previous batter hit Coefficient 0.06083 home run Marginal 0.000966 probability z statistic 1.48 Game Relative score Coefficient 0.016223 situation differential of Marginal 0.000239 offense to defense probability z statistic 7.31 ** Absolute relative Coefficient 0.0027 score Marginal 3.98 x [10.sup.-5] probability z statistic 0.89 Inning Coefficient -0.00546 Marginal -8.10 x [10.sup.-5] probability z statistic -2.27 * Outs Coefficient 0.00656 Marginal 9.66 x [10.sup.-5] probability z statistic 0.83 Man on first Coefficient -0.02133 Marginal -0.00031 probability z statistic -1.24 Man on second Coefficient 0.050283 Marginal 0.000782 probability z statistic 2.25 * Man on third Coefficient 0.034457 Marginal 0.000529 probability z statistic 0.93 Men on first and Coefficient 0.024472 second Marginal 0.00037 probability z statistic 1 Men on first and Coefficient -0.03147 third Marginal -0.00045 probability z statistic -0.85 Men on second and Coefficient 0.021674 third Marginal 0.000328 probability z statistic 0.5 Bases loaded Coefficient -0.0202 Marginal -0.00029 probability z statistic -0.48 Year Year 1969 Coefficient 0.066753 dummies Marginal 0.001027 probability z statistic 3.4 ** Year 1972 Coefficient 0.05901 Marginal 0.000905 probability z statistic 2.91 ** Year 1973 Coefficient -0.02012 Marginal -0.00029 probability z statistic -1.13 Goodness Observations 584886 of fit Log-likelihood -19434.23 Likelihood 378.01 ratio-[chi square] Notes: Constants not reported. * Statistical significance at the 5% level. ** Statistical significance at the 1$ level.

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