Modeling Markets for Sports Memorabilia

 

Robert F. Mulligan

Department of Business Computer Information Systems and Economics

College of Business

office 828-227-3329

fax 828-227-7414

(mulligan@wcu.edu)

 

A.J. Grube

Assistant to the Chancellor for Equal Opportunity Programs

Chancellor's Division

office 828-227-7116

fax 828-227-7047

(agrube@wcu.edu)

 

Western Carolina University

Cullowhee, North Carolina  28723

 

ABSTRACT

 

A simple hedonic pricing model is developed for baseball cards, of the type often used successfully to model prices for artworks.  The model is constructed based on insights contributed by both the sports psychology and finance literatures and is estimated for a dataset of twelve well-known players observed at eight points in time over a span of twenty years.  Dummy variables are used to capture various relevant characteristics of the player or card.  Tobit estimates perform extremely well, explaining most differences among baseball card prices for the cards in the sample.  Among extrinsic variables that represent specific players and card characteristics that differentiate cards issued during the same season, race had a significant positive effect on price for black players.  Batting average and number of World Series appearances had significant positive impacts on price, but surprisingly, rookie cards tended to be worth relatively less than non-rookie cards.  Similarly unexpected findings with respect to players' death and elevation to the Hall of Fame may result from trying to estimate too many characteristics simultaneously on a limited dataset.  Results suggest famous players' cards generally are extremely attractive investment instruments.

 

 

INTRODUCTION

 

The economic literature on appreciation of non-financial investment assets has generally found low rates of return accompanied by high risk.  Assets studied have included real estate, artworks, wines, and sports memorabilia.  Sports memorabilia comprise an especially promising subject for further study. 

 

Although sports memorabilia may be collected solely for its financial aspects, often collectors seek to identify with their heroes by collecting associated memorabilia.  This is one metric in motivating athletes that is seldom examined (White et al 1998), and also motivates non-athletes who seek to emulate athlete behavior in a more general way.  Although athletic performance and ability seem to characterize most of the athletes whose memorabilia is most prized by collectors, demonstrated ability to overcome adversity seems to make athletes especially valued as role models, both to other athletes and to collectors who do not also compete.  Several baseball players in the sample examined here are famous for overcoming injuries or playing with pain over long careers, including Dimaggio and Mantle. 

 

One essential feature rendering sports memorabilia more favorable subjects is the relative homogeneity of collectibles such as baseball cards, a feature clearly not shared by artwork or real estate.  All cards of a certain issue should have their value determined by characteristics intrinsic to the card, such as a card's age, condition, and scarcity, and characteristics extrinsic to the card, such as the particular player's records, fame, and popularity.  Intrinsic characteristics are generally properties of the whole issue and are shared by all cards of a given year printed by a given manufacturer, assuming that equal numbers of each player were printed.  Obscure player's cards will be sought to complete sets of a given issue, and famous or star player's cards will face additional demand to complete sets or enhance partial sets of star player or team cards.

 

This paper develops a simple hedonic pricing model for baseball cards, of the type often used successfully to model auction prices for artworks.  The model is estimated for an illustrative sample of cards for several different years.  The paper is organized as follows: a review of the literature is followed by a development of the hedonic pricing model, a discussion of the data used, a brief introduction to the statistical methodology, presentation of the empirical results, and finally the conclusion.

 

LITERATURE

 

Several categories of scholarly literature were reviewed to develop the background necessary for this study.  First the sport psychology literature on fan identification and behavior is used to develop an explanatory framework for a basic theory of why collectors demand sports memorabilia in the first place.  Next, we discuss possible career characteristics players might possess which might plausibly enhance the desirability of associated memorabilia.  A discussion of issues related to sports injury is provided next.  We argue that athletes are especially prized as role models because they overcome obstacles, and that injury constitutes the most common and archetypal obstacle faced by athletes.  An athlete's memorabilia will be more prized by collectors if the athlete either successfully overcame injury, or even if they failed to do so, but faced the obstacle with superior courage and character.  Finally, after establishing reasons for a base demand for sports memorabilia, we turn to a discussion of purely financial considerations, drawing on the established literature on the investment demand for sports memorabilia and related assets, including artwork. 

 

Identification and Fan Motivation

Fans provide the basic demand for sports memorabilia, at least initially.  This section discusses the sports psychology literature addressing fan motivation in attending sports events and buying memorabilia, and in identifying with particular teams.  In many situations, fans cannot purchase memorabilia unless they attend a sporting event (Jarrell and Mulligan 2002), so attendance, team identification, and base demand for memorabilia are inextricably linked.

 

On the most basic level, divorced from any financial investment considerations, memorabilia seems to be valued for its association with the sport activity, particularly if the memorabilia in question is particularly old and no longer resembles contemporary equipment, such as obsolete golf balls and clubs.  On a higher level, memorabilia is associated with the success of the player or even the team.  Fans value their association with winning teams more highly (End et al 2003a), and this presumably confers more value on associated memorabilia; fans desire the items in order to bask in reflected glory (BIRG) (Cialdini et al 1976; Cialdini and Richardson 1980; Lee 1985; Wann and Branscombe 1990; Wann et al 1993; End et al 1997; 2003b).  An additional source of demand for memorabilia is fans' strong identification with certain teams and players (Tajfel and Turner 1986; Hirt et al 1992; Murrell and Dietz 1992; Wann and Branscombe 1993; Wann, Tucker, and Schraeder 1996; Dietz-Uhler and Murrell 1999).  This effect is enhanced if the team enjoys success, but is present to some extent even for losing teams.

 

Though fan identification with teams and players can be negatively impacted by poor performance or sudden reversal of fortune (Mann 1974; Wann and Dolan 1994), demand for memorabilia such as baseball cards is generally not affected by such reversals.  Much of the value possessed by a baseball card is based on the player's established performance.  A record-holder's card probably does not fall in value when their record is surpassed.  A famous player's later cards are always highly desired, even if their performance falters late in their career. 

 

Motivations among Memorabilia Collectors

We turn next to the sport psychology literature about motivation of athletes, as opposed to fans.  We attempt to draw conclusions from this literature about why memorabilia collectors might identify particularly strongly with certain athletes, and thus why associated memorabilia would be especially prized.

 

The goal perspective approach to explaining motivational processes among athletes (Duda 1989, 1992) emphasizes the differences in how athletes define success and judge their overall performance.  Ames (1984, 1992) identifies two principal goals athletes seek, task orientation and ego orientation.  We suggest here that similar motivating factors vicariously influence memorabilia collectors.  Task-oriented collectors value memorabilia associated with a particular athlete based on the athlete's performance and achievement, but always taking into account extraordinary obstacles the athlete may have overcome.  These collectors seek inspiration from the athletes they admire, and attempt to apply lessons learned from the athletes' live experience to problems faced by the collectors, normally outside the arena of athletic competition.  These collectors particularly value memorabilia associated with athletes who are perceived as having demonstrated superior courage and character, in addition to those who have been particularly successful. 

 

In contrast, ego-oriented collectors seek memorabilia associated with athletes and teams which are most famous or most popular. These collectors seek to bask in the reflected glory and are less likely to seek memorabilia associated with a fine athlete from a team with which they do not identify.  The two goal orientations have supported the discovery of divergent behavioral patterns in athletes (Duda 1992, 1993).  While we suggest that price data for sports memorabilia will not be sufficiently rich to distinguish between the two motivational paradigms for collectors, we believe both motivators exist in addition to purely financial factors to which collectors respond.

 

Athletic Injury: the Archetypal Hardship

The impact of injury on athletes has been extensively studied (Granito 2001).  Although injury is not the only obstacle athletes have to overcome, it is the one most universal experience with which non-athletes can empathize, thus we argue that an athlete's injury response is one of the most important factors determining the value of associated memorabilia.  Several studies found that athletic injuries at all levels of competition contribute to a variety of physical, physiological, and psychological hardship against which athletes struggle (Grossman and Jamieson 1985; Brewer and Petrie 1995; Leddy, Lambert, and Ogles 1994; Smith et al 1993).  A significant literature in sport psychology research focuses on the psychological and emotional impact of athletic injuries (Heil 1993; Taylor and Taylor 1997; Pargman 1999). 

 

The cognitive appraisal approach to explaining how athletes respond to injury emphasizes the athlete's perception of the injury (Brewer 1994; Wiese-Bjornstal et al 1998).  This perception is influenced by interactions among personal factors such as physiological aspects of the injury and  personal characteristics of the athlete, among situational factors including sport related factors, social aspects of competition and training, and among environmental factors (Wiese-Bjornstal and Shaffer 1999; Granito 2001).  Wiese-Bjornstal et al (1998) emphasize that athletes' response to injury is dynamic and can change over time.  Athlete response to injury depends on a large number of hypothesized variables (Wiese-Bjornstal et al 1988; Wiese-Bjornstal, Smith, and LaMott 1995).  Evans and Hardy (1995) suggest that conventional quantitative research methodologies may fail to capture the full complexity of injury recovery.  The cognitive appraisal approach offers an explanation for individual differences in responses to injury (Brewer 1994). 

 

This range in injury recovery success helps explain why different athletes are more admired, and why their memorabilia is more desired by collectors, independently of the athletes' levels of achievement.  Rose and Jevne (1993) and Shelley (1999) document the experience of injury and recovery, finding a four-phase process which is potentially arduous and protracted: 1) injury, 2) acknowledging the injury, 3) dealing with the impact, and 4) achieving a physical and psychosocial outcome, which might consist of recovery, adaptation, or acceptance of the injury.  This process can be considered analogous to a standard archetype for how individuals in all walks of life face and overcome adversity.  Bianco, Malo, and Orlick (1999) document a similar injury recovery process.  Because athletic injury is such a direct metaphor for the hardships we all face, it is small wonder that non-athletes identify with, and strive to emulate, the athletes they admire.  Evidence suggests the most competitive athletes, who identify most strongly with their sport, have an enhanced psychological response to injury (Brewer 1993).

 

Shelley (1999) found athletes' perceptions about injury change over the course of the process and emphasized the importance of the influence of coaches, teammates, and family members on athletes' emotional response.  Social interactions seem to be an important part of a successful emotional response to injury and the frustrations of recovery (Udry et al 1997b; Zimmerman 1999).  Cultural aspects and social influences impact the way an athlete experiences and talks about pain and injury (Young and White 1999).  Since pain influences an individual's emotional state (Udry et al 1997a; Taylor and Taylor 1998; Heil and Fine 1999), it can impact an athlete's ability to overcome injury, and render their recovery that much more admirable.  Certain athletes are particularly admired for their ability to play with pain, in particular Dimaggio and Mantle.  Athletes can perceive benefits from injury, because it provides relief from the stress of competition and the pressure to perform, and often find rehabilitation stressful (Gould et al 1997a).  Rehabilitation may be inherently painful, or an athlete may feel pressured to demonstrate rapid progress in order to return to competition. 

 

Financial Aspects of Collecting Memorabilia

This section discusses some of the relevant economic literature on pricing sports memorabilia and other non-financial investment assets, such as artwork.  Stoller (1984) provides a valuable analysis of the Fleer v. Topps antitrust case as well as a discussion of the underlying economics of the baseball card business.  The loss of Topps' monopoly power in 1980 and the introduction of competition (Stoller 1984, p. 23) may have caused the collapse of a speculative bubble in card prices.  Stoller (1984, p. 19) documents a 31.6 percent annual return on Topps cards. 

 

Nardinelli and Simon (1990) and Andersen and La Croix (1991) both found that a player's race significantly affected the price paid for baseball cards on the secondary market.  These studies focus on the secondary market for sports memorabilia to isolate consumer discrimination from co-worker and employer discrimination.  McGarrity, Palmer, and Poitras (1999) found little evidence of racial discrimination in the market for baseball cards, using a dataset with constant supply and where effects from speculative demand are largely removed by considering only retired players, and using a variety of econometric specifications to allow assessment of robustness of results.  Fort and Gill (2000) study racial discrimination in baseball card markets using continuous, non-binary racial perceptions of market participants, as reported by surveys.  They find evidence of discrimination against black and Hispanic hitters and against black pitchers, but not Hispanic pitchers.  Gill and Brajer (1994) use baseball card prices to demonstrate monopsony exploitation of non-free-agent players.  Comparison of the distribution of salaries among free-agent and non-free-agent players with the competitive secondary market prices of their baseball cards, shows that non-free-agent salaries are systematically depressed. 

 

The literature on pricing artwork has significant implications for sports memorabilia markets.  Ekelund, Ressler, and Watson (2000) examine how an artist's death affects the demand for that artist's work.  They find a clustered rise in artwork's values immediately around the time of the artist's death.  This phenomenon has two implications for the sports memorabilia market.  The supply of baseball cards is effectively frozen for a particular player when the player retires from the game, rather than at death.  Ancillary memorabilia, including autographs, can continue to be supplied until the player's death however, and it seems plausible for death to induce an interest and nostalgia-generated increase in card prices as well. 

 

Rengers and Velthuis (2001) study determinants of artwork prices based on characteristics of the artwork, artist, and gallery.  This approach generalizes fairly readily to baseball cards, which have characteristics attributable to the player, team, year of issue, and card issuer.  Reneboog and Van Houtte (2002) find that artworks significantly underperform compared with financial assets, owing the very high risk of investing in art, the heterogeneity of artworks, high transactions costs, and high costs of insurance, transportation, security, and resale.  It is particularly worth noting that none of these negative features generally applies to sports memorabilia.  Baseball cards of a given player, issue, and condition are always non-unique, homogeneous assets.

 

MODEL

 

This section develops the model tested in the results section in the context of three kinds of data which might be used to estimate a model: time series data, cross-sectional data, and panel data.  Only the cross-sectional model is tested in this paper.

 

Time Series Models

Time series data measure characteristics of an individual member of a population or sample, or of the sample or population as a whole, as they evolve over time.  As more time elapses, more data are observed and more subtle models can be estimated.  An optimal timing model is used to express the value of any asset that appreciates over time.  The value V of an asset at any point in time is an exponential function of the initial value K and the time elapsed t during which the asset appreciates:

 

V = K e√t     [= K exp(t1/2)]

 

Alternatively, the simpler formulation V = K tn can be used.  This class of models is broadly applicable to many different assets, including wine, agricultural crops, renewable natural resources such as lumber forests, and non-renewable natural resources such as petroleum deposits.  The important characteristic of the K tn term is that it can grow at an increasing or decreasing rate, depending on whether n is greater or less than one.  Sports memorabilia should increase in value at a decreasing rate, and formulating the model this way allows for testing whether n < 1.

 

Adapting this model to sports memorabilia, certain differences must be noted.  Unlike wines, baseball cards and other sports memorabilia do not acquire chemical changes as they age which improve their taste, quality, and desirability.  In fact, the chemical changes to which sports memorabilia are subject over time normally detract from their desirability, and collectors attempt to prevent or delay chemical changes. 

 

Nevertheless, cards appreciate in value in a fashion similar to wine, though for different reasons.  The supply of cards of a particular brand, player, and year is initially limited.  Only so many of a particular card were ever printed.  Surviving copies appreciate in value as some of the initially limited supply are lost, destroyed, or decay in condition as time passes.  This gradual diminution of the supply of cards is similar to what happens as vintage wines are consumed, mature forests are harvested for lumber, or petroleum deposits are pumped out of the ground.

 

The prices of sports memorabilia are also affected by changes in demand.  Demand normally increases with an increasing population, and in addition, demand for sports memorabilia increases with interest in the particular sport or athlete, as well as interest in the memorabilia for its own sake and as investment assets.  Demand effects can occasionally be negative, as documented for the collapse of baseball card prices caused by the end of monopoly pricing in 1980 (Stoller 1984, p. 23), but fortunately that has been an exceptional event.

 

Sports memorabilia have unique characteristics which call for generalizing the standard optimal timing model.  Though old baseball cards of comparable significance, condition, and quality are generally more valuable than newer cards, the career performance and general fame of the player make a card more sought after and therefore more valuable.  All cards of a given issue had the same price when new, and appreciate over time.  A rookie card of an average player appreciates much less than that of a more well-known player.  A rookie card of a presumed hot-prospect may appreciate rapidly early on, but plateau or even decline in value as the player's career fails to achieve its initial promise.  Some players' cards are especially desirable due to tragically brief careers. 

 

To capture these kinds of effects, the exponent of the optimal timing model is augmented with a multiplicative vector of exponentially-weighted factors S.  These factors include the player's career longevity, records held, retirement, hall-of-fame induction, and death.  Including the factors which distinguish average from well-known players is accomplished mathematically by inserting the product of each factor variable, each weighted by its own exponent:

 

V = K S tn     [=  K ∏(Sia) tn)  = K ∏(AaBbCc ...) tn]

 

Taking natural logarithms of both sides,

 

ln A(t)  = ln K + a ln A + b ln B + c ln C + ... + n ln t

 

This is the equation of interest for time series estimation.  This can also be considered a generalized hedonic price equation, and the reduced form of supply and demand functions in the same arguments. 

 

Pit = ∑tatXt + ∑tbtZt + et

 

Where X and Z are vectors of observable characteristics, both intrinsic and extrinsic to a specific card.  Extrinsic characteristics are associated with specific players and vary across cards of a specific issue.

 

Cross-sectional Models

Cross-sectional data provides a description of an entire population or sample at a given point in time.  Cross-sectional estimation is appropriate when researchers want to distinguish among factors which influence population behavior or characteristics but do not have observations at many different points in time.  Time series estimates would allow for estimating the price and the return as functions of the explanatory variables.  Cross-sectional estimates only allow for computing the return between two observed cross-sections.  Cross-sectional estimates can also be useful to investors, because they can be used to evaluate the likely change in price whenever one of the explanatory variables changes, for example, if a current player improves his batting average, or appears in the world series, or if a retired player is elected to the Hall of Fame.

 

Building on the significant literature studying race as a determinant of sports memorabilia prices, we include dummy variables for race in the specification.  Batting average is included as the single most important measure of a player's performance.  Note that earned run average would be used for pitchers, who would generally have to be priced with a separate model.  Rookie cards are commonly thought to be more desired by collectors, and generally to be more rare, especially for famous players.  If rookie cards are valued in any way differently from ordinary cards, including a dummy variable for rookie card status should improve the model's forecasting performance. 

 

The player's age serves as a proxy for the age of the card, and generally cards of older players should be more valuable.  Death is measured with a dummy variable, as is hall-of-fame status.  The number of world series appearances improves the desirability of a player's cards, though a player's team is more likely to make it to the world series the better the player's performance, as captured by batting average, for example.  The number of years elapsed from the start, and from the end, of a player's career, like age, proxies for age of the card.  Because the difference of these two variables gives the player's career longevity, if longevity has a positive impact on card price, the expectation is that the more years elapsed from the start, and the fewer elapsed from the end, the higher the price.  This effect can be washed out by the general phenomenon that older cards are more valuable.

 

A hedonic pricing model is often specified in a less restrictive exponential form, and then estimated in its linear logarithmic transformation.  However, because several of the right-hand-side variables are dummies, which can only take on values of zero or one, the model is specified here in levels.

 

P = a + bBLK + cHSP + dBA + eR + fAnn + gDnn + hHOFnn + iWSnn + jSnn + kEnn


This is the model which is estimated in the results section.  Card price is thus asserted to be a function of the player's race, batting average, rookie card status, age, death, hall of fame status, number of world series appearances, and career longevity as captured in years elapsed from start and from end of career.  The variables are described in Table 1.  The race, rookie card, death, and hall-of-fame status are measured with dummy variables which take on values of zero or one depending on whether the relevant condition is satisfied, as described in table 1.

 

{{INSERT TABLE 1 ABOUT HERE}}

 

Panel Data Models

Panel data is the term applied to data which describes the cross-section of the population or sample, but where each characteristic is observed at many points in time.  Thus panel data represent a cross between time-series and cross-sectional data.  These data are also called pooled time-series and cross-sectional data.  Kmenta (1971, pp. 508-517) discusses panel data estimation.  Although panel estimation should provide the best results, it also calls for the most intense computational and data resources.  Mulligan, Jarrell, and Grube (2003) present and interpret panel estimates.



DATA

 

This section documents the data used to estimate the model.  A sample of twelve well-known players, listed in table 2, was chosen to obtain illustrative estimates of the model.  Prices for one card for each player were taken from the Price Guides for eight different years over a twenty-year span from 1982 to 2002. 

 

{{INSERT TABLE 2 ABOUT HERE}}

 

One significant difference between these data and the auction prices used in empirical examinations of artwork prices should be noted.  Artworks are unique and each auction price for a given artwork records a unique transaction at a unique point in time.  In contrast, the Price Guide observations of card price in a given year are taken from dealer surveys.  There is never any specific, single exchange which can be documented at the listed price. 

 

Generally, the Price Guide is used as an authority for dealers to price and update their inventory.  Many transactions occur at the price listed in the Price Guide because it is widely accepted as an authoritative source.  However, the listed card price is logically prior to the prices of actual transactions.  In the art market, in contrast, the auction price is logically prior to any compilation of art values.  A further difference derives from the fact that there are many identical copies of a given card, even in the same grade of condition, but an artwork is always absolutely unique.


METHODOLOGY

 

This section explains the statistical estimation technique used in the results section.  Because the left-hand-side variable, baseball card price, cannot be negative, a censored estimation technique is employed, introduced in econometrics by Tobin (1958) and called the Tobit model.  If left-hand-side variables are limited in some way, ordinary least squares estimates are asymptotically biased (Kennedy 1993, p. 232).  Ordinary least square estimation can provide negative estimates of the left-hand-side variable, which can never be negative in reality, a shortcoming avoided through censored estimation. 

 

McDonald and Moffit 1980 showed that Tobit estimates combine properties of standard linear regression, namely the predicted value of the left-hand-side variable and its changes for observations beyond the relevant limits, with properties of the probit estimator, namely the probabilities and changes in the probabilities of being outside the limits.  Tobit estimates are obtained through maximizing the likelihood function, Greene 1981, p. 508. 

 

Descriptions of the Tobit estimation procedure are provided by Abramovitz and Stegun 1972, p. 299, Amemiya 1981, Greene 1981, Maddala 1983, pp. 151-155.  Davidson and MacKinnon 1993 pp. 537-542, and Judge et al pp. 783-785.  Hall 1984 reviews software available for Tobit estimation.  The Tobit model is an iterative, restricted maximum likelihood estimate. 

 

Estimates are reported below for sample datasets taken from eight different annual Price Guides.  The estimate did not converge for some years, or yielded a near-singular matrix or a negative standard error, probably because the model devours nearly all available degrees of freedom;  eleven coefficients, including the constant, are estimated on twelve observations of each variable.  These problems vanished when one variable was omitted from the specification.  Including more cards in the sample would probably avoid these estimation problems.

 

RESULTS

 

This section presents the results of econometric estimation.  Tables 3 through 10 present estimated Tobit models for cross sectional data samples taken from eight different Price Guides.

 

{{INSERT TABLE 3 ABOUT HERE}}

 

This model was estimated over a sample of twelve well-known players.  Very high R-squares and adjusted R-squares are impressive, but may be due more to small sample properties than any particularly sterling qualities of the specification.  Nevertheless, high R-squares suggest the model should serve investors and collectors as a useful tool. 

 

{{INSERT TABLE 4 ABOUT HERE}}

 

Race coefficients are positive and significant for black players in the sample for 1982, 1983, 1984, 1993, and 2002, but not significant in 1985, 1988, and 1999, suggesting race became less important in determining card price over time, though clearly the positive effect on price remains in the 2002 dataset. In contrast, the race coefficient is negative and significant for the lone Hispanic player, Rod Carew in 1983, 1984, 1993, not significant in 1985, 1988, 1999, and becomes positive and significant in 2002.  This is likely a small sample characteristic which results from the relatively higher prices initially paid for cards of very famous non-Hispanic players in the sample, and the relatively rapid appreciation of the Rod Carew card. 

 

{{INSERT TABLE 5 ABOUT HERE}}

 

Batting average has a very strong positive impact on card price.  The coefficient is always positive and significant and almost always an order of magnitude greater than any other coefficient, except in 1988.  Rookie card status always has a negative impact on price, which is always statistically significant except in 2002.  This is probably a small sample effect.  Player age has a positive and significant impact on price in 1982, 1984, 1985, 1988, 1993, and 1999, but not significant in 1983 or 2002.

 

{{INSERT TABLE 6 ABOUT HERE}}

 

Deceased players' cards generally sell for more than those of still-living players, at least for this limited sample.  This outcome is not surprising in light of the empirical literature on artwork valuation, which shows death of the artist has a positive impact on the value of his or her work. Death is different for card valuation, however, as a player stops generating new card issues when he retires, rather when he dies.  Death is statistically significant and negative only for 1982, significant and positive for 1985, 1988, 1993, and 1999, and not significant for 1983, 1984 and 2002.  The significant positive coefficient on death in 1982 indicates that in that year, for this sample of players, the living players' cards were worth more than dead players'.  The result may have been reversed later on as more star players in the sample passed on.

 

{{INSERT TABLE 7 ABOUT HERE}}

 

Hall of Fame status has a negative but statistically insignificant effect in 1982, 1983, 1984, but its impact becomes positive and significant in 1985, 1993, 1999, and 2002, and is positive but insignificant in 1988.  Insignificant coefficients for many years probably results from multicollinearity;  Hall of Fame status should have a positive impact on card price, but that impact is likely captured better by two other variables included in the model, batting average and the number of world series appearances.

 

{{INSERT TABLE 8 ABOUT HERE}}

 

The number of World Series appearances is always positive and significant as expected.  Two variables are included to capture time elapsed from each player's period of professional activity, and career longevity: years elapsed from the beginning and end of the player's career.  These variables broadly capture the relative age of the card as well.  Years since the start of the player's career is negative and significant in 1982 and 1988, and negative but insignificant in 1983.  Years since the start of the player's career was omitted from the 1984, 1985, 1993, 1999, and 2000 regressions because the Tobit model would not converge without removing one variable from the model.  Statistically significant negative coefficients are surprising, and may be due to multicollinearity with player age and years since the end of the player's career. 

 

{{INSERT TABLE 9 ABOUT HERE}}

 

Years since the end of the player's career is negative and significant in 1982, 1984, 1985, 1993, 1998, and 2002, and statistically insignificant in 1983 and 1988.  This means there is an aura effect which elevates the value of cards for players who have recently retired, and that as more years pass, card price declines, or at least grows less rapidly.  Multicollinearity may also account for this outcome.

 

{{INSERT TABLE 10 ABOUT HERE}}

 

CONCLUSION

 

A conceptual framework to explain the demand for sports memorabilia was developed from the sports psychology and finance literatures, and used to construct a formal hedonic pricing model.  This model was estimated on a sample of twelve baseball cards with prices observed in eight years over a twenty-year period.  This model was estimated separately for each of the eight years and performed extremely well in explaining differences among baseball card prices.  Race had a positive but diminishing effect on card price for black players.  For the only Hispanic player, the effect of race was initially negative but became positive in the last year estimated.  Race effects should not be taken as overturning the results of earlier researchers, as they may be due to small sample properties.

 

Batting average and player age have positive impacts on price, but surprisingly, rookie cards tend to be worth relatively less than non-rookie cards.  A player's death generally increases the value of his cards, but in at least one year, 1982, the reverse was found to be the case.  Hall of Fame status only began to have a significant and positive impact on card value starting in 1985; before that it was not significant.  World series appearances also add to the value of a player's cards.  Career longevity, as measured by years since the start and end of a player's career gave ambiguous results, but results suggest that retirement adds to the value of a player's cards, though years since retirement detracts from card value.  Years since the start of a player's career also detracts from card value, at least where that variable was included in the model.

 

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Table 1

Variables in the Hedonic Pricing Model

P

= card price in current dollars from the Price Guides

BLK

= 1 if player is black, = 0 otherwise

HSP

= 1 if player is Hispanic, = 0 otherwise

R

= 1 if card is a rookie card, = 0 otherwise

BA

= player career batting average

Ann

= player age at year of Price Guide

Dnn

= 1 if player was deceased prior to year of Price Guide, = 0 otherwise

HOFnn

= 1 if player was in Hall of Fame prior to year of Price Guide, = 0 otherwise

WSnn

= number of world series appearances prior to year of Price Guide

Snn

= number of years from start of career to year of Price Guide

Enn

= number of years from end of career to year of Price Guide, = 0 for players who were still playing during year of Price Guide

Price Guides from 1982, 1983, 1984, 1985, 1988, 1993, 1999, and 2002.  nn indicates variables that change from one Price Guide to the next, and serves as a placeholder for the year, e.g., A82, A83, etc.

 

 

Table 2

Sample of Baseball Cards

Player

Years Played

Teams

Card Issuer and Year

Card #

Aaron, Hank

1954-76

MLN ATL MIL

1954 Topps

128

Bench, Johnny

1967-83

CIN

1968 Topps

247

Brett, George

1973-93

KCR

1975 Topps

228

Carew, Rod

1967-85

MIN CAL

1967 Topps

569

Fisk, Carlton

1969-93

BOS CHW

1972 Topps

79

Jackson, Reggie

1967-87

KCR OAK BAL NYY CAL

1969 Topps

260

Mantle, Mickey

1951-68

NYY

1952 Topps

311

Musial, Stan

1941-63

STL

1948 Bowman

36

Robinson, Jackie

1947-56

BRO

1949 Bowman

50

Rose, Pete

1963-86

CIN PHI MON

1963 Topps

537

Williams, Ted

1939-42 & 1946-60

BOS

1950 Bowman

98

Yastrzemski, Carl

1961-83

BOS

1960 Topps

148

 

 

Table 3

Tobit Model of Baseball Card Prices

1982 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-3642.540

360.2244

-10.11186

0.0000

BLK

412.3077

69.14140

5.963253

0.0000

HSP

-245.1361

48.18435

-5.087462

0.0000

BA

9732.350

974.7919

9.984028

0.0000

R

-491.6519

30.37142

-16.18798

0.0000

A82

56.47143

12.96109

4.356997

0.0000

D82

-264.7751

86.78959

-3.050770

0.0023

HOF82

-141.8288

131.5777

-1.077909

0.2811

WS82

156.3141

7.421117

21.06342

0.0000

S82

-64.62543

15.39043

-4.199067

0.0000

E82

-19.12512

3.730426

-5.126792

0.0000

R-squared

0.998321

    Mean dependent var

178.8333

S.D. dependent var

435.0034

    Akaike info criterion

10.93275

Sum squared resid

3494.350

    Schwarz criterion

11.41765

Log likelihood

-53.59649

    Hannan-Quinn criter.

10.75322

 

 

Table 4

Tobit Model of Baseball Card Prices

1983 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-3516.604

910.7125

-3.861377

0.0001

BLK

511.4870

191.1275

2.676156

0.0074

HSP

-266.6450

125.0442

-2.132407

0.0330

BA

10881.65

2679.818

4.060592

0.0000

R

-471.2385

75.38311

-6.251248

0.0000

A83

34.36457

32.34181

1.062543

0.2880

D83

-425.1508

251.7122

-1.689036

0.0912

HOF83

-268.5695

359.6466

-0.746760

0.4552

WS83

158.4635

19.75503

8.021424

0.0000

S83

-43.76523

38.61433

-1.133393

0.2570

E83

-12.61742

9.396278

-1.342810

0.1793

R-squared

0.989107

    Mean dependent var

179.8417

S.D. dependent var

434.6804

    Akaike info criterion

12.72936

Sum squared resid

22640.95

    Schwarz criterion

13.21426

Log likelihood

-64.37613

    Hannan-Quinn criter.

12.54983

 

 

Table 5

Tobit Model of Baseball Card Prices

1984 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-2500.287

367.8762

-6.796544

0.0000

BLK

311.0981

78.05443

3.985656

0.0001

HSP

-173.5807

51.84731

-3.347922

0.0008

BA

8363.918

1064.941

7.853881

0.0000

R

-412.8758

29.58992

-13.95326

0.0000

A84

8.240391

2.254256

3.655481

0.0003

D84

-117.4760

98.94498

-1.187287

0.2351

HOF84

-238.9534

134.0693

-1.782313

0.0747

WS84

142.8158

8.011905

17.82544

0.0000

E84

-31.29713

4.034931

-7.756546

0.0000

R-squared

0.998729

    Mean dependent var

195.6667

Adjusted R-squared

0.986020

    S.D. dependent var

389.8842

S.E. of regression

46.09906

    Akaike info criterion

10.81732

Sum squared resid

2125.123

    Schwarz criterion

11.26182

Log likelihood

-53.90394

    Hannan-Quinn criter.

10.65275

 

 

 

Table 6

Tobit Model of Baseball Card Prices

1985 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-1121.022

189.0700

-5.929136

0.0000

BLK

-5.327959

39.55047

-0.134713

0.8928

HSP

-22.53180

25.55713

-0.881625

0.3780

BA

2895.855

520.7312

5.561133

0.0000

R

-324.4782

16.26105

-19.95432

0.0000

A85

13.28342

1.335820

9.944018

0.0000

D85

350.0922

51.12925

6.847199

0.0000

HOF85

222.6048

68.23334

3.262405

0.0011

WS85

79.80095

4.030060

19.80143

0.0000

E85

-40.01059

2.190951

-18.26175

0.0000

R-squared

0.999029

    Mean dependent var

143.4167

Adjusted R-squared

0.989320

    S.D. dependent var

268.1490

S.E. of regression

27.71181

    Akaike info criterion

9.641930

Sum squared resid

767.9447

    Schwarz criterion

10.08643

Log likelihood

-46.85158

    Hannan-Quinn criter.

9.477361

 

Table 7

Tobit Model of Baseball Card Prices

1988 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-16888.58

6427.465

-2.627565

0.0086

BLK

1672.837

1003.078

1.667703

0.0954

HSP

-658.5319

698.8496

-0.942309

0.3460

BA

29512.94

12756.28

2.313601

0.0207

R

-2089.132

387.4653

-5.391793

0.0000

A88

592.2054

297.6067

1.989893

0.0466

D88

1108.047

1763.161

0.628443

0.5297

HOF88

81.24800

1720.452

0.047225

0.9623

WS88

603.7357

96.71123

6.242664

0.0000

S88

-763.7149

383.8863

-1.989430

0.0467

E88

109.8458

109.2591

1.005369

0.3147

R-squared

0.984770

    Mean dependent var

742.9167

S.D. dependent var

1818.894

    Akaike info criterion

16.13852

Sum squared resid

554255.8

    Schwarz criterion

16.62343

Log likelihood

-84.83114

    Hannan-Quinn criter.

15.95899

 

 

Table 8

Tobit Model of Baseball Card Prices

1993 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-45258.40

5130.998

-8.820584

0.0000

BLK

2246.530

604.2588

3.717827

0.0002

HSP

-2268.757

920.8260

-2.463828

0.0137

BA

120963.4

9041.685

13.37841

0.0000

R

-7969.344

767.4725

-10.38388

0.0000

A93

318.0706

88.38023

3.598889

0.0003

D93

7865.327

2787.320

2.821824

0.0048

HOF93

6467.120

875.5687

7.386193

0.0000

WS93

2352.526

94.24879

24.96081

0.0000

E93

-843.0741

117.7661

-7.158887

0.0000

R-squared

0.996144

    Mean dependent var

2543.333

Adjusted R-squared

0.957583

    S.D. dependent var

6768.140

S.E. of regression

1393.917

    Akaike info criterion

17.44062

Sum squared resid

1943005.

    Schwarz criterion

17.88512

Log likelihood

-93.64374

    Hannan-Quinn criter.

17.27605

 

 

 

Table 9

Tobit Model of Baseball Card Prices

1999 Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-24163.55

3930.125

-6.148290

0.0000

BLK

1112.228

821.5341

1.353842

0.1758

HSP

1004.742

921.0848

1.090824

0.2754

BA

35945.08

14140.94

2.541916

0.0110

R

-3937.315

956.3839

-4.116877

0.0000

A99

368.5283

80.12773

4.599260

0.0000

D99

11184.69

2907.367

3.847017

0.0001

HOF99

2425.958

722.4471

3.357973

0.0008

WS99

1014.841

236.4284

4.292380

0.0000

E99

-511.8389

63.85813

-8.015250

0.0000

R-squared

0.990732

    Mean dependent var

2177.083

Adjusted R-squared

0.898056

    S.D. dependent var

5631.749

S.E. of regression

1798.142

    Akaike info criterion

17.81578

Sum squared resid

3233315.

    Schwarz criterion

18.26028

Log likelihood

-95.89468

    Hannan-Quinn criter.

17.65121

 

 

Table 10

Tobit Model of Baseball Card Prices

2002 Data Cross Section

 

Coefficient

Std. Error

z-Statistic

Prob. 

C

-26918.56

9566.457

-2.813848

0.0049

BLK

1942.812

711.5880

2.730249

0.0063

HSP

1754.909

988.9637

1.774493

0.0760

BA

60821.88

9669.695

6.289948

0.0000

R

-2708.483

3498.638

-0.774153

0.4388

A02

168.1110

112.5893

1.493134

0.1354

D02

2308.123

3814.893

0.605030

0.5452

HOF02

3744.651

1170.177

3.200071

0.0014

WS02

1799.813

161.5804

11.13881

0.0000

E02

-345.4981

121.2549

-2.849354

0.0044

R-squared

0.985945

    Mean dependent var

2004.583

Adjusted R-squared

0.845395

    S.D. dependent var

5060.267

S.E. of regression

1989.689

    Akaike info criterion

17.73438

Sum squared resid

3958864.

    Schwarz criterion

18.17887

Log likelihood

-95.40626

    Hannan-Quinn criter.

17.56981