Seat Value Based Revenue Implications for Baseball
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Seat Value Based Revenue Implications for Baseball
Senthil Veeraraghavan∗• Ramnath Vaidyanathan
OPIM Department, Wharton School, 3730 Walnut St, Philadelphia, PA 19104, USA
senthilv@wharton.upenn.edu • ramnathv@wharton.upenn.edu
March 2008
Abstract
We study how patrons attending a baseball game value different seats in a professional league
stadium based on their location and view. Most surprisingly, we find that customers seated in
symmetric seats on left and right fields might derive very different valuations from the game.
Thus, commonly followed symmetric pricing mechanisms in baseball stadiums might be inef-
ficient from a revenue management perspective. Furthermore, customers perceive higher net
valuations at distantly located upper deck seats and some outfield seats, possibly because of
easy availability and low prices. Consistent with the notion of baseball being an experience
good, more frequent customers to the baseball stadium experience lower variance in their val-
uations of the game compared to first time visitors. Thus, we quantify the impact of repeated
visits on customers’ learning of the valuation a seat might provide. Younger customers have
higher variance in their valuations compared to customers who are older. Our research on cus-
tomer valuations quantifies the significant influence of the seat locations on customers’ ex-post
valuations of the game. The findings provide a novel opportunity for the teams to price tickets
differentially and asymmetrically to accommodate varying seat valuations, using seat value mea-
sures in different sections of the stadium. Keywords: Ordinal Logit Models, Customer Behavior,
Revenue Management, Seat Valuations, Baseball, Experience goods.
∗
Corresponding author. The authors would like to thank the NPB franchise and Yuta Namiki for the survey. We
would like to thank Ken Shropshire, Scott Rosner and the Wharton Sports Business Initiative for their support of
the project. Special thanks to Eric Bradlow, Gerard Cachon, Serguei Netessine, Devin Pope, and the participants at
WSBI seminar for their thoughtful comments.
11 Introduction
There is very limited research on how consumers value different seats in baseball stadiums. Despite
the paucity of research in this area, seat location discussions have been the focus of several articles
in the popular press, especially recently. In 2006, the Oakland Athletics decided to reduce the
capacity of McAfee Coliseum (where their home games are played) by covering several of their
seats with tarpaulin sheets, thus reducing the stadium capacity from 44,000 seats to about 34,077
seats (Urban 2005). The Oakland A’s announced that the decision was made in order to provide an
“intimate” experience to those in attendance, in a smaller field. In fact, when the team moves to a
newer field for the 2008 season, they will play in a stadium that has lesser planned capacity (32,000)
than the currently-used, tarpaulin-covered stadium. Bnet.com quoted “...the fans who are feeling
slighted most are the lower-income brackets who feel the third deck was their last affordable large-
scale refuge for a seat behind home plate, even one so high.”. The team management contended
that people liked the third deck mostly because of availability and price, and perhaps not so much
because of the view (Steward 2006). One article on Slate Magazine criticized the move, stating
“Some of us want to sit far away” (Craggs 2006). Thus, the valuation received by consumers seated
at the upper deck was not only unclear, but also varied among different fans. So is it true that the
consumers seated in the upper deck valued those seats highly? Are the net valuations perceived
by the consumers different across seats? Our research addresses such questions using data from a
professional league stadium in Japan.
Employing statistical analysis of player performance in making drafting and payroll decisions,
and signing free-agents, is a common practice in present-day professional baseball. By re-evaluating
the strategies which produce wins on the field, the Oakland Athletics (A’s) were competitive with
several larger market teams, with approximately less than half the payroll of leading professional
teams. Managing such a high performance team with lower operating expenses has been discussed
in Moneyball (Lewis 2003), and “Sabrematics” popularized in the book is now a part of baseball
folklore. In the wake of the popularization of Sabrematics, several other professional baseball
franchises have employed full time analysts at the front office, thereby diminishing the competitive
edge enjoyed by the A’s. Consequently, several franchises have been examining new levers through
which they can continue to stay competitive while minimizing their expenses and/or increasing
2revenues.
Professional clubs (major leagues) generally have three sources of income: (1) Local income, (2)
Shared Broadcasting and Licensing deals, and (3) Revenue sharing across teams (Source: mlb.com).
Local income includes ticket sales, local broadcasting rights fees, parking fees, and local team
sponsorships; these generally form a significant portion of the revenues for any professional franchise.
According to CFO.com, local revenue is the biggest factor that explains disparity between teams.
However, both theory and application of revenue management in non-travel related industries,
especially in the sports business, has been very limited (See Talluri and van Ryzin (2004) for a
survey of applications of revenue management across various industry profiles). In this work, we
examine new revenue opportunities that would improve customer valuations and stadium revenues.
Through this paper, we seek to provide insights on how customers perceive different seats, which
in turn provides some opportunities for teams to improve customer traction and revenues.
A careful analysis of heterogeneity in customer seat valuations can boost ticketing revenues,
and also increase additional revenues accrued through concession stands and parking fees. In a
league that has no salary cap structure,1 generation of local revenues can assist a team in freeing
up some space for payroll. However, revenue management schemes in the baseball industry are not
employed in full scale, unlike in the airline industry.
This is possibly because there are many challenges in the implementation of revenue management
strategies in baseball stadiums. One primary difference between the baseball and airline industries
is that while airline seats are well defined products, baseball seats are not. Although there are
differences between aisle seats and middle seats, most seats in the same (business or economy) class
provide comparable valuations for consumers. Usually, an airline seat is considered simply as a
conduit for transporting a person from an origin to a destination. Therefore, for the most part, the
price of a ticket in economy class indicates how much a person values the trip, more than how much
he values the seat itself. However, (view from) baseball seats might be thought of as experience
goods. It is unclear how customers’ valuations are distributed across different attributes. The
location of a seat might affect the valuation realized by a consumer from the game. In addition,
there could be a number of factors that might affect how a patron values a seat. For instance, the
nature of the opposing team, the age of the patron, or whether the patron is a regular visitor or
1
The existence of luxury tax might create some ceiling.
3an infrequent visitor, might affect her valuation from the seat. For most stadiums, understanding
heterogeneity in customer valuations is the key to increase local revenues. A clear understanding
of the seat valuations, would lead to the creation of better “fences” that would provide baseball
franchises an opportunity to improve and manage their stadium revenues. Our paper sheds more
light on the key factors influencing customers’ valuations in a baseball stadium.
Based on a study of reported consumer seat valuations in a baseball stadium of a professional
franchise, we provide some insights and measures by which firms (baseball franchises) can improve
customer satisfaction through better handling of ticket pricing, seat rationing, and stadium design
decisions. Our methodology is applicable across a wide variety of products such as performances
in theaters and opera houses, and games in stadiums. We make the following theoretical and
managerial insights:
1. To our knowledge, ours is among the first papers to study the distribution of consumer valu-
ations in a professional baseball stadium, based on various attributes. We set up an empirical
model to study consumer valuations from seats/products to characterize the heterogeneity
in valuations. We find that baseball seats can be thought of as experience goods, whose
valuation uncertainty decreases with every successive visit. Revenue management practice
hinges on the ability to price-discriminate which is possible only if there is heterogeneity in
valuations. We find clear evidence for heterogeneity in valuations in baseball stadiums, and
quantify the heterogeneity based on customer attributes and seat locations of the customers.
2. Repeated visits to the ballpark help consumers “learn” the “true value” of the seats. As
consumers (fans) visit more often, the uncertainty in their valuation of the game, from a
particular seat location, is diminished. This is consistent with the notion that baseball games
are experience goods (with residual uncertainty). Furthermore, we estimate the change in
perceived seat valuation and reduction in uncertainty from repeated visits. In particular,
a customer who has visited the ball park five times has 29% lower variance in his realized
valuation as compared to a first timer.
3. Most surprisingly, we show that the valuations received by consumers in a baseball field are
asymmetric. Consumers seated on the third base side (left field) are less likely to have extreme
valuations, as compared to those seated on the first base side (right field). In particular, the
4consumers on the third base side are 10% less likely to report the net valuations of their seat
location as low. One likely cause was the location of the home-team dugout on the left field
side of the stadium.
4. We find that the consumers seated at the upper deck have higher average ex-post valuations
than those in the lower deck seats. This has two possible implications that are important from
the franchise’s perspective. First, consumers might be responding positively to the availability
of lower priced upper deck seats. Second, even though the upper deck seats are located far
off, they could still be providing comparatively higher ex-post valuations to consumers who
sat there. We find that consumers seated in the upper deck also had higher heterogeneity in
their valuations, with a 52% higher variance in realization, as compared to those in the lower
deck.
5. Age matters. Older customers have less uncertain valuations than younger customers seated
at the same seats controlling for all other factors.
In the following section, we position our paper with respect to the existing literature in §2. In
Section 3, we discuss the research design and description of data. We discuss the main research
issues, and frame our main hypotheses. In §4, we briefly analyze the data, develop our model based
on the data, and summarize the empirical results. Finally, we conclude the paper in section §5 by
discussing the main insights derived in light of our empirical findings.
2 Literature Positioning
To our knowledge, our paper is among the first to analyze seat value perception and its implications
for stadium revenues of sports businesses. Most of the literature in the sports business has been
about secondary markets and ticket pricing in scalping markets (See Courty 2000, for a compre-
hensive survey). There is a stream of literature that focuses on labor market aspects in baseball
(Scully 1974, Rottenberg 1956). In contrast, we focus on seat valuations perceived by consumers,
which can potentially be used to improve traction for the team among the fans, and increase local
revenues for the franchise.
Our paper also contributes to an emerging literature (Shen and Su 2007) on consumer behavior
5in Revenue Management. Su (2007) finds that heterogeneity in consumer valuations, along with
waiting time behavior, influences pricing policies of a monopolist. Dana (1998) shows that advance
purchase discounts can be employed effectively in competitive markets, if consumers’ uncertain
demand for a good is not resolved before the purchase of the good. Shugan and Xie (2000) show that
advanced selling mechanisms can be used effectively to improve firm profits as long as consumers
have to purchase a product ahead of their consumption, and their post-consumption valuation is
uncertain. Due to the inherent heterogeneity in valuations realized by the consumers, we discover
that devising an advance selling mechanism is useful for the team. While this literature is mostly
analytical, ours is one of the first empirical papers analyzing customer valuations and revenue
implications in the sports business.
Methodologically, our paper is related to the literature employing ordinal models to study the
antecedents and the drivers of customer satisfaction in Operations. Kekre et al. (1995) study the
drivers of customer satisfaction for software products by employing an ordinal probit model to
analyze a survey of customer responses. Gray et al. (2008) use an ordinal logit model to empirically
analyze the quality-risk associated with outsourcing to contract manufacturers. They find that
contrary to the conventional wisdom favoring specialization, contract manufacturing plants have
a higher quality-risk than internal plants. We use an ordinal model similar to the aforementioned
papers, but consider extended models that account for heterogeneity in reporting (across customers)
and heterogeneity in the distribution of ex-post valuations (across seats). For a detailed analysis
of customer satisfaction using ordinal Bayesian models, refer to Bradlow (1994).
Anderson and Sullivan (1993) note that relatively few studies investigate the antecedents of
satisfaction, though the issue of post-satisfaction behaviors is treated extensively. They note that
disconfirmation of expected valuation causes lower satisfaction and affects future consumptions.
While previous considerations about a product might affect the experience realized from consuming
it, we mainly focus on how product attributes such as seat location and distance, and personal
attributes such as gender, age and frequency affect customer valuations.
Ittner and Larcker (1998) provide empirical evidence that financial performance measures are
positively associated with customer satisfaction. We use customer valuation measures reported in a
consumer survey to recommend changes that would help the franchise achieve a chosen“service level”
objective on customer satisfaction. We believe this would lead to a better long-run performance of
6the franchise.
3 Data Description and Research Issues
3.1 Research Description
This research is based on a survey conducted by a professional league baseball franchise (equivalent
of Major League Baseball) in Japan. The franchise is located in a mid-small city and hence could
not rely on conventional streams of revenue such as broadcasting, merchandising and advertising.
The franchise management decided to focus on ticket sales as it was one revenue source for which
the league’s intervention was minimal. In order to understand the revenue potential from ticket
sales, the management explored how consumers valued different seats in the stadium. The team
management saw an upside potential in considering improvements in pricing and seating layouts.
As the team was a recently established franchise, the management intended to survey its cus-
tomers to better understand the traction for the team among its fans. The survey discussed in
the paper was conducted by the team based on inputs from various departments and team exec-
utives in the franchise. In the execution phase, the survey was administered to a random sample
of consumers at the stadium on a weeknight game. Only one response was obtained from each
consumer.
3.2 Research Issues
The focus of our research is to understand how the net valuation derived by a consumer from
a baseball game varies based on her seat location and frequency of visits. A large volume of
consumer choice literature focuses on how consumers choose between various options. In our case,
it would mean modeling the consumer’s decision problem of whether to go see a game and how
to choose between the different seats available. However, we only observe the respondents who
purchased a ticket, and hence are not privy to the underlying trade-offs made by the consumer
while arriving at the purchase and seat choice decisions. Consequently, we do not model the
customer’s revealed preferences (with respect to the seats chosen). Instead, we model the ex-post2
net valuations realized, to understand how they differ based on seat location, frequency of visits,
2
The results were collected during the last two innings.
7and other attributes. In the survey, respondents were asked to report their perception of the seats
that they sat in as expensive, fair or cheap. We define Seat Value Index (SVI) as a measure of a
respondent’s ex-post valuation of her experience net of price. SVI takes the values {Low, Medium,
High} corresponding to the responses expensive, fair, and cheap respectively.
To derive sharper insights, we assume that consumers are forward-looking and have rational
expectations, i.e. their beliefs about valuations are consistent with realizations, and that they do
not make systematic forecasting errors. The rational expectations assumption is widely employed
in empirical research in economics (Muth 1961, Lucas and Sargent 1981, Hansen and Sargent 1991)
and marketing literature (for example, Sun et al. 2003).3 Every consumer has some belief on the
distribution of possible valuations that she could realize, conditional on her covariates. For a rational
consumer, the ex-ante distribution of valuations is identical to the distribution of valuations realized
by all consumers with identical covariates. Note that rational expectations does not imply that
consumers are perfectly informed about their true valuations. The rational expectations approach
provides a parsimonious way to exclude ad hoc forecasting rules that some customers in the system
might adopt.
We now focus on three key hypotheses of interest on seat valuations. The first two hypotheses
pertain to the effect of seat location while the third hypothesis considers the effect of repeat pur-
chases or interactions with the service provided. We describe the rest of our conclusions in section
§5.
Unlike airline tickets, where the seat location is secondary to the value realized from being
transported to the destination, the location of the seat may play a critical role in the value a
consumer derives from a baseball game. One of the key factors through which the location of a seat
influences the game experience is the view offered; the better the view, the better the experience,
and the higher the ex-post valuation. However, the seats offering a better experience are also priced
higher. For example, backnet seats are always priced higher than the bleacher seats or upper deck
seats. Hence, it is not clear, in general, whether ex-post net valuations would be higher or lower
for seats with a better view.
Consequently, we focus our attention on seat location attributes. We examine those seat lo-
3
In our paper, the rational expectations assumption does not affect any of the empirical results we derive from
the model. It allows a better interpretation of how the consumers’ seat valuations might influence selling decisions.
8cations where there is an underlying supporting rationale for similar, higher or lower seat value
indices. To preserve focus, we examine two important seat location attributes: namely ‘horizontal’
symmetric locations (left vs right), and ‘vertical’ distance (lower deck vs. upper deck).4
H1: Customers seated on the left (3rd Base) side have identical SVI to those seated on the right
(1st Base) side.
H2: Customers seated at the Upper Deck have higher SVI as compared to those seated at the Lower
Deck.
Our first hypothesis (H1) compares the valuations between two symmetrically located seats on
the 1st Base side (left side of the stadium) and the 3rd Base side (right side of the stadium). They
are priced the same (all professional baseball teams we examined offered identical prices) and offer
similar views. Hence, the expected net valuation of consumers should be identical across the 1st
Base and 3rd Base, once we have controlled for all other variations.
In our second hypothesis (H2), we consider the valuation of consumers located in the far-placed
upper deck seats. Upper deck tickets are usually priced very low (None of the professional major
league teams priced them higher than 30% of the infield seats in the lower deck and for certain teams
uppermost decks were priced as low as 5% of the lower deck seats. A large fraction of the upper deck
seats sell closer to game day to the perceived common availability). While the view is inferior to
the lower deck seats, we believe that the relatively cheap price often more than compensates for the
lower utility a customer may obtain at these seats. Hence, our second hypothesis is that customers
seated in the upper deck derive higher valuations on average as compared to the customers seated
at the lower deck.
H3: More frequent customers are less likely to report extreme SVIs.
Finally, our third hypothesis pertains to the notion that the attendance and viewership at
baseball games are perceived as experience goods. In the traditional literature on experience goods,
customers learn the true valuation of a good by consuming it. We consider baseball games to be
experience goods, whose residual uncertainty of quality decreases with the number of visits made
by a patron. Therefore, a customer who is a regular at all games is expected to have a narrower
distribution of his valuations as compared to a customer who is a first-timer. The consequence of
such learning can push the mean valuation in either direction, causing the more frequent customers
4
See Figure 1.
9to report higher or lower average SVI.
Equipped with the above hypotheses, we examine our data using SVI as a dependent variable
and seat locations and frequency, among other factors, as our covariates.
3.3 Data Definitions
Seat Value Index (SVI): Variable of interest defined earlier.
Age: In order to understand how different age groups valued their ticket, the survey required
respondents to indicate the age group they belonged to, from the buckets 0 − 9, 10 − 19, 20 − 29,
30 − 39, 40 − 49, 50 − 59, and 60 − 69, in which the management was interested. We code Age in
“buckets” as a continuous variable taking the values {1, 2, 3, . . . , 7}, where 1 refers to the age group
0 − 9, 2 represents the age group 10 − 19 and so on.5
Gender: The management felt that male fans valued the experience differently from the female
fans. Understanding the difference was critical to direct selling efforts. In addition, it was essential
to control for gender effects while making inferences about the effect of seat location and frequency of
visits on SVI. Hence, each respondent was asked to indicate their gender in the survey questionnaire.
Hometown: A positive perception from local fans is an important factor for franchise’s long-
term sustenance. Hence, the team was interested in measuring valuations of fans based on their
hometown. In order to characterize a respondent as a local or an outsider, the respondents were
asked to indicate their hometown as {City, Prefecture, Outside} based on whether they live in the
city where the team is based, the prefecture (district) excluding the city where the team is based,
or outside the prefecture, respectively.
Seat: The experience and the resulting valuation are highly dependent on the seat from which
a respondent watched the game. Hence, it was necessary to capture this information by asking
respondents to indicate the location of their seat from twelve possible choices as shown in Figure 1.
We further decomposed each seat into its location attributes Side={1st Base, 3rd Base, Backnet,
Field, Grass}, InOut={Infield, Outfield} and Deck={Upper, Lower}.
Frequency: It was important for the team to understand how their regular customers differed in
their valuations from those who were infrequent customers in order to improve their season tickets
program. Hence, respondents were asked to report how often they attended the team’s baseball
5
We could represent each age-group by its mid-point and the results would still stay the same.
10Figure 1: Stadium Design and Seating Layout
games during the season. The number of visits was reported in frequency buckets: {First Time,
Once, Thrice, Five Times, All Games}.
Visiting Team: Some of the major league teams in the US price individual tickets differently based
on the strength or popularity of the visiting team. The franchise management was interested in
adopting this idea by understanding customer preferences for visiting teams. Hence, the respondents
were asked to indicate the visiting teams they would like to see, from a list of five major teams
{Team1, Team2, Team3, Team4, Team5} in the league that the home team plays against, labeled
in rank order of their finishes in the last two seasons combined.
4 Models and Data Analysis
4.1 Preliminary Data Analysis
From a total of 1397 respondents, 259 responses were dropped due to missing information. A
preliminary analysis revealed that the frequency distribution of seat value index was skewed towards
the right, as shown in Figure 2. This implies that a higher proportion of people reported a low seat
11valuation index.6
Figure 2 also reveals several interesting insights. The seat value index reported by older respon-
dents seem to be more homogeneous. Customers seated in Grass seats report higher SVI, while
respondents seated at Backnet seem to have a lower SVI. Infield and Lower Deck seats seem to
have a higher proportion of respondents reporting a Low SVI as compared to Outfield and Upper
Deck seats. Finally, the season regulars attending all games seem to have more homogeneous SVIs
as compared to the first-timers.
843 71 119 137 324 235 150 102
800
0.0 0.4 0.8
400
216
79
0
Low Medium High 0−9 20−29 40−49 >60
Seat Value Index Age
479 659 124 313 701
0.0 0.4 0.8
0.0 0.4 0.8
Female Male Outside City
Gender Hometown
257 483 198 92 108 653 485
0.0 0.4 0.8
0.0 0.4 0.8
1st 3rd Backnet Grass Infield Outfield
Side Field
623 515 221 326 306 166 119
0.0 0.4 0.8
0.0 0.4 0.8
Lower Upper First Time Once Three Times All Games
Deck Frequency
Figure 2: Distribution of Respondents by the covariates and their Seat Value Index. The width of
the histogram signifies frequency. The number of respondents is indicated on the top for each value
of the covariate.
We tested for the usual symptoms of multi-collinearity (Greene 2003): (1) high standard errors,
(2) incorrect sign or implausible magnitude of parameter estimates, and (3) sensitivity of estimates
to marginal changes in data. We found no evidence of these symptoms. We computed the Vari-
ance Inflation Factors (VIF) for every covariate and found all of them to be less than two, i.e.
6
The result of the game might have influenced the skewness observed.
12(max(V IF ) < 2), which again suggests that multi-collinearity is not an issue. We also introduced
random perturbations in the data and found that our estimates were robust to such changes, thus
allaying the possible multi-collinearity related issues.
4.2 Model Analysis
The dependent variable of interest is the Seat Value Index (SVI). It is reported on a three point scale
and is a rank-ordered ordinal variable, as there is a well-understood ordering in the respondent’s
mind.
We could treat SVI as continuous and estimate a multiple regression model using ordinary
least squares (OLS). However, this approach is flawed as (1) the OLS estimates are inefficient and
the predictions cannot be restricted to the interval [1, 3] (Kmenta 1986), and (2) the regression
estimates will roughly correspond to the correct ordered model only if differences in the valuations
between the two consecutive indices are identical. For additional discussion on the limitations of
OLS regressions, see Judge et al. (1980).
Alternately, we could treat the dependent variable as categorical and employ a multinomial
logit or probit model. This overcomes the limitations of OLS regression, but is still inefficient as
it throws away valuable information by ignoring the ordinal nature of SVI. Hence, the appropriate
model for our purpose is an ordinal regression model that takes into account the categorical nature
of the data as well as the ordering information contained.
According to the ordinal regression model, the values of the observed ordinal dependent variable,
Vi ∈ {1, 2, . . . , J}, are determined by collapsing a continuous unobserved (latent) variable, Vi∗ ,
into categories defined by the boundaries (thresholds) {τi0 , τi1 , ..., τiJ }, where it is understood that
τi0 = −∞ and τi0 = +∞ for purposes of identification. Mathematically, Vi = j, if and only if
τij−1 < Vi∗ ≤ τij , for j = 1, 2, . . . , J. The model specification is completed by assuming that
Vi∗ = xTi β + i , where xi is a vector of covariates (excluding a constant), β is the associated vector
of parameters, and i is the stochastic error term. Based on the distribution of i being normal or
logistic, we get the ordinal probit model (McKelvey and Zavoina 1975), or the ordinal logit model
(McCullagh 1980). Also see Zaslavsky and Bradlow (1999) for a detailed discussion for ordinal
models.
We found very little difference between the results obtained from the probit and logit models.
13McCullagh (1980) shows that the ordinal probit and logit models are qualitatively similar and that
the fits are indistinguishable for any given data set; hence the selection of an appropriate distribution
should be primarily based on ease of interpretation. We chose the logistic error distribution as it
allowed us to interpret the regression coefficients in terms of log-odds.
4.2.1 Ordinal Logit Model
The standard ordinal logit model assumes that the thresholds are identical for all customers, i.e.
τij = τ j , ∀i, and that the error terms are independent identically distributed logistic random vari-
ables with mean zero. As Vi∗ is not observed, the variance of the error terms are not identified and
π2
hence set to 3 , the variance of the standard logistic distribution. Given these assumptions, the
cumulative probability distribution of Vi can be written as
Pr(Vi ≤ j | xi ) = Pr(Vi∗ ≤ τ j | xi ) = Pr(xTi β + i ≤ τ j | xi )
j −xT β)
e(τ i
= Pr(i ≤ τ j − xTi β) = j T
1 + e(τ −xi β)
j T
= Λ(τ − xi β), j = 1, 2, . . . , J − 1 (1)
Note that when J = 2, this reduces to a logistic regression model. The conditional probability of
Vi = j is given by:
Pr(Vi = j | Xi ) = Λ(τ j − xTi β) − Λ(τ j−1 − xTi β) (2)
For the probabilities to be well-defined, the threshold parameters need to satisfy the condition
τ 1 < τ 2 < . . . < τ J−1 (3)
We can now express the log-likelihood function for the standard ordinal logit model as
i=N
X
I(Vi = j) Λ(τ j − xTi β) − Λ(τ j−1 − xTi β) ,
LL(β, τ | V, X) = (4)
i=1
where I is the indicator function. The parameters of the model are estimated by maximizing the
log-likelihood in equation (4) subject to the constraints in condition (3).
The standard ordinal logit model is equivalent to estimating J − 1 logistic regressions of the
14form Pr(Vi ≤ j | Xi ) = Λ(τ j − xTi β j ), with the assumption that the slope coefficients, β j , are
identical across all equations, i.e. β j = β, j = 1, 2, . . . J − 1. Then, we can rewrite Equation (1) in
terms of the log-odds of {Vi ≤ j} to obtain log (Odds(Vi ≤ j | Xi )) = τ j − xTi β. This implies that
the ratio of odds for two different levels j1 and j2 ,
Odds(Vi ≤ j1 | x1 )
= exp(τ j1 − τ j2 ), (5)
Odds(Vi ≤ j2 | x1 )
is independent of the covariate x1 . For this reason, the standard ordinal logit model is also referred
to as the Proportional Odds Model (POM).
In our data set, Vi∗ represents respondent i’s underlying valuation of his seat, net of the price
paid, while Vi is the reported SVI that can take the rank-ordered values j = 1, 2, 3 corresponding
to Low, Medium and High, respectively. The mapping between a respondent’s underlying valuation
and reported SVI is illustrated in Figure 3.
Distribution of V*
Probability Density f( V*i )
SVIi =1 SVIi =2 SVIi =3
τ0 = − ∞ τ1 τ2 τ3 = + ∞
Net Valuation (V*i )
Figure 3: Mapping between Net Valuation Vi∗ (latent) and Seat Value Index SV Ii (observed)
The vector of covariates xi consists of Age, Gender, Hometown, Side, Deck, InOut, Frequency
15and Team 1 7 . We can express the linear predictor as:
xTi β = β1 Agei + β2 Malei + β3 Cityi + β4 Prefecturei + β5 3rdBasei + β6 Backneti +
β7 Fieldi + β8 Grassi + β9 UpperDecki + β10 Outfieldi + β11 Frequencyi + β12 T eam1i
We use the OLOGIT routine in STATA 10.0 to estimate the parameters of the proportional odds
model. The results are summarized in Table 1. We can reject the null model consisting only of the
intercepts τ j in favor of the proportional odds model, as the model χ2 = 149.02 is significant at the
p < 0.0001 significance level. A standard measure of fit for logit models is the McFadden pseudo-R2
LLP OM
which is defined as 1 − LLN U LL . It indicates the improvement in likelihood due to the explanatory
variables over the naive (null) model. For the proportional odds model we find the pseudo-R2 to
be 9.06%. This value needs to be interpreted with caution as it is not directly comparable to the
R2 obtained in OLS. Moreover, it is not uncommon to obtain low values for the pseudo-R2 even
when the explanatory power of the model is good (Gray et al. 2008).
Before we make any inferences based on this model, we must validate the proportional odds
property stated in (5), which implies that all respondents have the same ratio of odds of reporting a
low SVI to odds of not reporting a high SVI. While it might be reasonable to assume that customers
sitting in different seats might inherently have the same propensity to find higher (or lower) value,
one would expect that customers ‘learn’ their valuation through repeated visits to the stadium, and
hence would have different odds ratios based on the number of visits. Hence, we need to investigate
the validity of the implicit proportional odds assumption made by the standard ordinal logit model.
The standard approach to test the proportional odds assumption is to fit two models: a reduced
and a full model, where the former is the proportional odds model and the latter is an expanded
non-proportional odds model (NPOM) that allows the βs to depend on j, and perform a likeli-
hood ratio test. The null hypothesis being tested is H0 : β j = β, j = 1, 2, . . . , J − 1. The test
statistic −2 {ln(P OM ) − ln(N P OM )} has a χ2(k) distribution, where k is the number of additional
parameters in the full model. Applying the likelihood ratio test to our data set, we find that the
proportional odds model is strongly rejected in favor of the full model (χ2(12) = 46.74, p < 0.0001).
7
We considered all visiting teams in our specification, but retained only the statistically significant ones in order
to keep the model parsimonious. Our results and conclusions do not change significantly even if we include all visiting
teams as predictors.
16Variable Standard Generalized Heteroskedastic
Ordinal Logit Ordinal Logit Ordinal Logit
j = 1, 2 j=1 j=2 j = 1, 2
Threshold: Low-Medium τ1 -1.215*** -0.761** -0.748***
Threshold: Medium-High τ2 3.387*** 2.071*** 2.067***
Age β1j 0.048 0.127** -0.172** 0.034
Male β2j -0.019 -0.026 -0.026 -0.034
City (vs. Outside) β3j 0.083 0.029 0.029 0.011
Prefecture (vs. Outside) β4j 0.192 0.166 0.166 0.102
3rd Base (vs. 1st Base) β5j 0.428** 0.873*** -0.727** 0.145
Backnet (vs. 1st Base) β6j -0.730*** -0.678*** -0.678*** -0.440***
Field (vs. 1st Base) β7j -0.893*** -0.824*** -0.824*** -0.509***
Grass (vs. 1st Base) β8j 1.816*** 1.206*** 1.206*** 0.919***
Outfield β9j 0.215 0.211 0.211 0.171
j
Upper Deck β10 0.246 0.066 0.947*** 0.263**
j
Frequency β11 -0.126** -0.093 -0.234** -0.081**
j
Team 1 β12 0.249* 0.250* 0.250* 0.185**
Age γ1 -NA- -0.075***
3rd Base (vs. 1st Base) γ5 -NA- -0.324***
Upper Deck γ10 -NA- 0.208***
Frequency γ11 -NA- -0.057*
Log Likelihood LL -748.12 -727.18 -726.27
Likelihood Ratio χ2 LR 149.02 190. 90 192.72
No. of Parameters 12 16 16
McFadden Pseudo R2 9.06% 11.60% 11.71%
*** p < 0.01, ** p < 0.05, * p < 0.1
Table 1: Parameter Estimates for All Models
The likelihood ratio test is an omnibus test that the βs across threshold levels are equal for all the
covariates simultaneously. However, it is possible that the proportional odds assumption is violated
only for a subset of the covariates.
A Wald test developed by Brant (Brant 1990) allows us to test the proportional odds assumption
for each covariate individually. The key idea of the Brant test is to fit separate logistic regressions
Pr(Vi ≤ j | xi ) = Λ(τ j − xTi β j ) for each of the J − 1 threshold levels, and test the equality of the βs
by constructing a test statistic based on the estimated coefficients and the asymptotic covariance
matrix. Conducting the Brant test on our data-set, we find that the proportional odds assumption
is violated8 for the coefficients β1 , β5 and β9 . A likelihood ratio test confirms that a partially
8
We applied the Brant test to ordinal regression models with different link functions such as probit, log-log and
complementary log-log to ensure that the violations are not on account of a misspecified link. We find that the
proportional odds assumption is violated for the same coefficients β1 , β5 and β9 in all cases.
17constrained model that allows only for β1 , β5 and β9 to depend on j cannot be rejected in favor of
an unconstrained model that allows all the β’s to depend on j (χ2(9) = 6.33, p = 0.71).
In order to validate our hypotheses, we now consider two different modifications to the standard
ordinal logit model that do not suffer from the proportional odds restriction. The first model is
a generalized threshold model that relaxes the assumption that the category boundaries, τij , are
identical for all respondents, while the second model is a heteroskedastic model that allows for the
variance of the error term, i , to systematically vary across respondents.
The generalized thresholds model discussed in §4.2.2 addresses the different thresholds that
customers might use in reporting their responses, and usually pertain to the individual attributes
of customers that affect their response thresholds. The heteroskedastic model, discussed in §4.2.3
usually addresses the inherent differences in the distribution of product valuations as perceived by
the customers based on a particular covariate.
4.2.2 Generalized Threshold Ordinal Logit Models
The standard ordinal logit model assumes that all respondents use the same category boundaries in
reporting their underlying valuations. However, it is not uncommon for respondents of a survey to
use different thresholds in reporting their responses. For example, older respondents tend to assess
their health as better, as compared to younger respondents with similar characteristics (Groot 2000;
van Doorslaer and Gerdtham 2003). This has been referred to as response category threshold shift,
reporting heterogeneity, state-dependent reporting bias (Lindeboom and van Doorslaer (2004)), or
scale-usage heterogeneity (Rossi et al. 2001). This is the central motivation for the generalized
threshold model.
The generalized threshold ordinal logit model retains the idea that respondents have underlying
2
valuations drawn from a common distribution, Vi∗ ∼ Λ(xTi β, π3 ), but allows respondents to use
different thresholds, τij , while collapsing them into categories. A common approach to model
generalized thresholds is to make the threshold parameters linear functions (Maddala 1983, Peterson
and Harrell 1990), or polynomial functions (Rossi et al. 2001) of the covariates. We follow Maddala
(1983) and let τij = τ̃ j + xTi δ j , where xi is the set of covariates and δ j , j = 1, 2, . . . J − 1 are vectors
of the associated parameters that capture the effect of the covariates in shifting the thresholds. For
2
example, δ3rdBase captures the shift in the threshold τ̃ 2 for respondents seated along the 3rd Base
18side relative to those seated along the 1st Base side. Substituting the expression for τij in place of
τij in Equation (1), we can write the defining equation of the generalized ordinal logit model as
Pr(Vi ≤ j | xi ) = Λ(τ̃ j − xTi β j ), (6)
where β j = β − δ j and it is understood that τ̃ 0 = −∞ and τ̃ J = ∞ as before, for the purposes of
identification. The parameters of the model are estimated by maximizing the likelihood subject to
the constraints τ̃ j−1 − xTi δ j−1 ≤ τ̃ j − xTi δ j , j = 1, 2, . . . , J, which are required for the probabilities
to be well-defined.
The parameter β captures the real effect of the covariates on the valuation. According to the
generalized threshold model, the βs manifest as significantly different β j s in the expression for
log-odds, on account of the different thresholds used by the respondents. The β j ’s can hence be
interpreted as the net of the real effect (β) and the threshold-shifting effect (δ j ) of the covariates
on the log-odds. However, the two effects cannot be separately identified in this model.
The generalized threshold model is very flexible. When the β j ’s are allowed to differ across
threshold levels for all covariates, it is referred to as the Unconstrained Ordinal Logit Model. The
generalized threshold model nests the standard ordinal logit model under the restriction that β j =
β, j = 1, 2, 3, . . . , J − 1. It is also possible to constrain the coefficients for a subset of covariates
zi ⊂ xi , by restricting βkj = βk , j = 1, 2, 3, . . . , J − 1, k ∈ zi , to obtain a Partially Constrained
Ordinal Logit Model.
Given that the covariates Age, 3rd Base and Upper Deck violated the Brant test, we include
them in zi . In addition, we include the covariate Frequency in zi , as we believe that repeated visits
help respondents learn the true value of the game experience and would induce them to use different
thresholds. Hence, we let zi = {Age, 3rd Base, Upper Deck, Frequency}. We use the GOLOGIT2
routine (Williams 2006a) in STATA 10.0 to estimate the parameters of the generalized threshold
model. The results are summarized in Table 1.
A likelihood ratio test comparing the generalized threshold model with the standard ordinal
logit model confirms that we can reject the latter in favor of the former (χ2(4) = 41.89, p < 0.0001).
The McFadden pseudo-R2 has also improved from 9.06% to 11.60%, which indicates a better fit.
We observe that the predictors Side and Frequency continue to be significant. In addition, Age
191st Base, 3rd Base
Probability Density f( V*i )
τ13rd τ11st τ21st τ23rd
Net Valuation (V*i )
Figure 4: Comparison of Thresholds for 3rd Base vs. 1st Base, when β3rdBase = β1stBase
is also a significant predictor. As noted before, the net effect, βkj , cannot be decomposed into the
real effect (βk ) and the threshold-shifting effect (δkj ) for any covariate k. If we believe that the real
effect βk = 0, then the δkj ’s can be identified as δkj = −βkj , j = 1, 2. This case is illustrated for the
covariate 3rd Base in Figure 4.
Though we cannot separately identify δ 1 and δ 2 , their difference δ 2 − δ 1 = β 1 − β 2 is identified.
We can use this to compute the effect of the threshold-shifting on the size of the threshold interval
for being in the category SVI=Medium. For example, the size of the threshold interval for a
respondent on the 1st Base side is given by τ̃ 2 − τ̃ 1 = 2.832, while the same for a respondent on
the 3rd Base side is given by τ̃ 2 + δ 2 − τ̃ 1 − δ 1 = τ̃ 2 − τ̃ 1 + β 1 − β 2 = 4.432. This implies that the
respondents on the 3rd Base side are a lot less likely to report extreme values of SVI as compared
to respondents on the 1st Base side. This is also evident from Figure 4.
The generalized threshold model implies that two groups of customers might have identical
valuation distributions, but their distribution of seat value indices might differ because of different
reporting thresholds. This explanation might be apt for a covariate such as age, where older
customers might inherently have lower thresholds for higher valuations (high SVI) as compared to
younger patrons, although they have identical distribution of valuations. However, we find such an
explanation less plausible for the difference in SVI distributions between customers who only differ
20in the location of their seat (left field vs. right field).
In the next section §4.2.3, we consider the possibility that consumers might report different SVI
despite using the same thresholds, because the valuations that they experienced are distributed
differently across different seat locations.
4.2.3 Heteroskedastic Ordinal Logit Model
In ordinal regression models, the error variances are not determined and need to be explicitly
specified for the model to be identified. The standard ordinal logit and probit models assume that
π2
the error terms have a constant variance and set it to 3 and 1, respectively. As the choice of this
constant is arbitrary, the β parameters of the model are identified only up to a scale factor. In
β
other words, we essentially estimate σ, and identify β by fixing σ arbitrarily.
When the errors are homoskedastic, the relative effect of any two covariates on the outcome
β̂1 /σ
variable, given by , is correctly identified, independent of the choice of σ. However, these com-
β̂2 /σ
parisons become invalid in the presence of heteroskedasticity and can lead to erroneous conclusions.
For example, if β1st Base = β3rd Base , but σ1st Base = 2σ3rd Base , then assuming equal variance would
lead us to the erroneous conclusion that β̂1st Base = 0.5β̂3rd Base .
Yatchew and Griliches (1985) show that heteroskedasticity in binary regression models leads to
parameter estimates that are biased, inconsistent, and inefficient. This extends to ordinal regression
models and is in contrast to OLS estimates for ordinary linear regression, which remain unbiased
and consistent even in the presence of heteroskedasticity. Hence accounting for heteroskedasticity
is critical and this is the central motivation for the Heteroskedastic Ordinal Logit Model.
The heteroskedastic ordinal logit model assumes that the error variance differs systematically
across respondent groups. For example, frequent customers might have less heterogeneous valu-
ations as compared to first-timers, as they are more likely to have learned the true valuation of
their experience through repeated visits. This dependence of the error variance on the covariates
is captured by a skedastic function h(.) that scales the iid error terms in the standard ordinal logit
model. Mathematically, we write Vi∗ = xTi β + h(zi )i , where zi is the vector of covariates upon
which the residual error variance is believed to depend. Following Harvey (1976), we parametrize
21h(.) as an exponential skedastic function given by
h(zi ) = exp(zTi γ)
Given these assumptions, we can rewrite Equation (1) to obtain the defining equation of the
heteroskedastic ordinal logit model:
τj − xTi β
Pr(Vi ≤ j | xi ) = Λ (7)
exp(xTi γ)
The heteroskedastic ordinal logit model belongs to a larger class of models known as location-scale
models, and the reader is directed to McCullagh and Nelder (1989) for more details. The parameters
of the model are estimated using the maximum likelihood approach as before.
The heteroskedastic ordinal logit model does not display proportional odds for the covariates
in zi . This can be seen by writing out the expression for log-odds of Vi ≤ j given xi ,
τj − xTi β
log(Odds(Vi ≤ j | xi )) = ,
exp(zTi γ)
and observing that the effect of the covariates zi on the log-odds is no longer independent of the
threshold level j.
The heteroskedastic ordinal logit model nests the standard ordinal logit model under the re-
striction γ = 0. Hence it is possible to compare the two models using a likelihood ratio test, and
identify the sources of heterogeneity that are statistically significant.
Note that the different β j s manifesting in the ordinal logit model could be explained by differ-
ences in residual error variance. Hence we include in zi the covariates Age, 3rd Base and Upper
Deck, all of which violated the Brant test. In addition, we include the covariate Frequency in zi ,
as we believe that repeated visits should help respondents learn the “true value” of the game expe-
rience, and as a consequence reduce the residual variation in their valuations. Hence, we let zi =
{Age, 3rd Base, Upper Deck, Frequency} as in the generalized threshold model. We use the OGLM
routine (Williams 2006b) in STATA 10.0 to estimate the parameters of the heteroskedastic ordinal
logit model. The results are summarized in Table 1.
A likelihood ratio test comparing the heteroskedastic ordinal logit model with the standard
22ordinal logit model confirms that we can reject the standard ordinal logit model in favor of the
former (χ2(4) = 43.7, p < 0.0001). The McFadden pseudo-R2 has also improved from 9.06% to
11.71%, which indicates a better fit.
We observe that the significant β coefficients correspond to the covariates Frequency, Side
(except 3rd Base) and Upper Deck. All the γ coefficients included in the variance equation are
significant. We can draw several interesting inferences from these results.
3rd Base
Probability Density f( V*i )
1st Base
τ1 τ2
Net Valuation ( V*i )
Figure 5: Net Valuation of Respondents on the 1st Base side vs. the 3rd Base side
Controlling for heteroskedasticity, respondents at the 3rd Base seem to have the same mean
valuations as respondents at the 1st Base, as β̂5 is no longer significant. However, that(β̂10 =
0.263, p = 0.04) implies that respondents seated on the 3rd Base side have significantly less hetero-
geneous valuations (standard deviation is 1-exp(γ̂5 ) = 28% lower) as compared to those seated on
the 1st Base side. This could be due to the location of the home team dugout and/or the relative
incidence of foul balls/home runs on the left field. Figure 5 illustrates the valuations implied by
the heteroskedastic ordinal logit model for the customers in first base side vs. third base side.
ˆ = 0.263, p = 0.04)
Respondents seated in the upper deck have a higher mean valuation (β 10
as well as higher heterogeneity (γ̂10 = 0.208, p = 0.0408) as compared to lower deck tickets. More
ˆ
frequent customers have a lower mean valuation (β 11 = −0.081, p = 0.028) and lower variance
(γ̂11 = −0.058, p = 0.074) in their valuations. This finding is consistent with the experience goods
23literature where customers learn the value of a product through repeated interactions. Age of a
respondent does not affect the mean valuation, but older respondents tend to have less variance in
their valuations.
4.3 Robustness Checks
We made several implicit assumptions in our model specification. These assumptions might impact
the results and insights. Hence, we carried out a sequence of tests to confirm the robustness of our
models to relaxation of those assumptions. We find that our results are robust to these assumptions.
• We treat the ordinal explanatory variables Age and Frequency as continuous in our model
specification. While this treatment is justified for Frequency, it is possible that the effect of
Age on SVI is non-monotonic. In such cases, specifying Age as a categorical variable might
be apt. Hence, we estimated all models presented here with separate parameter estimates for
each level of Age. A likelihood ratio test rejects the categorical specification in favor of the
continuous specification used before (χ2(10) = 13.38, p = 0.22).
• Field seats are located on both sides of the home base, and hence can be classified into 1st
base and 3rd base seats. However, we did not have the data to perform this classification.
Consequently, the co-variate Field, in our current model specification, captures the aggregate
effect of being on the left field or right field, and could be the cause for the asymmetry
observed. Hence, we dropped all Field seats and ran the models discussed above. Our
qualitative insights did not change, confirming that the inability to classify field seats into
left field and right field is not driving our insights.
• We treat Hometown as a categorical variable in our model specification. However, it is also
possible to use the distance of the hometown from the stadium as a covariate, and treat it as
a continuous variable taking the values {0,1,2} which refer to City, Prefecture and Outside
respectively. We estimated a model treating Hometown as a continuous variable and find
that the results are almost identical to the earlier specification of Hometown as a categorical
variable.
• We found the valuation of the customers independent of the opposing team, except for one
24team (Team1) which they preferred to see. The franchise team under consideration played
the opposing team (Team1) frequently, and Team1 had also been the strongest in the division
over the past two years. The higher valuation the customers experienced when they listed
Team1 is possibly a reflection of their informed-ness about the game and the team’s season
performance. None of the other teams were significant in their influence of the valuation
perceived by the customers.
4.4 Marginal Probabilities from the Models
Using our models, we can measure how a change in a covariate impacts the distribution of the
response variable using marginal probabilities. If we let xil denote the value of the lth covariate for
∂ Pr(Vi =j|Xi )
respondent i, then the marginal probability effect for a continuous covariate is given by ∂xil .
In the case of a categorical covariate like Side, the marginal probability effect is given by the change
in probability ∆ Pr(Vi = j | xi ) when compared to a reference category. While measuring the
marginal probability effects of any covariate, we fix the rest of the covariates at their mean value
(or median value for categorical covariates). We use the MFX2 routine in STATA 10.0 to estimate
the marginal probability effects based on the three models, and the results are summarized in Table
2. Table 2 indicates insignificant differences between the two non-proportional models, which allows
us to use threshold interpretation for consumer attributes, and heterogeneity interpretation for seat
attributes.
5 Conclusions and Insights
We use a latent variable approach to measure post-consumption valuations of baseball seats. Con-
sumers with different descriptive attributes (age, gender, etc) might have different thresholds for
reporting their valuation as high or low. Based on the intrinsically different realizations of the value
of the experience due to product attributes (such as seat location), consumers might end up with
varying seat value indices. Hence, it is critical to account for differences in reporting thresholds
and heterogeneity in the distribution of valuations. Otherwise, the estimate of the real effects of a
covariate could be biased. For instance, the standard ordinal logit model (that ignores both of these
effects) leads us to an incorrect conclusion that the age of the fan does not affect his valuations.
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