The Demand for
Programs at a College Football Game:
OLS and LMS
Estimates of Optimal Prices
JOURNAL OF SPORT
MANAGEMENT (2002) 16(3): 209229
Stephen
Jarrell and Robert F. Mulligan
Corresponding author:
Stephen Jarrell
Department of Management &
International Business
College athletic directors face the
difficulty of setting a price for goods and services they provide to the
public. One complementary good provided
as a part of major college sports events is gameday programs. This paper estimates a demand function for
football programs using 11 years of data for an NCAA Division IAA college.
Least median of squares (LMS), a new outlierresistant estimation technique, is
used to refine the model and provide a more useful estimate of the demand
function. The revenue and profitmaximizing
program price is found and compared with prices actually charged throughout the
sample period.
Along with other complementary goods, such as concessions
and paraphernalia, gameday programs, rosters, and/or magazines, provide
athletic managers an opportunity to capitalize on the captive crowd inside a
stadium. Considering the growth in
stadium capacities and interest, athletic directors and promotion managers need
to understand the increased potential market and factors that influence sales
levels to make the most of these changes.
College athletic programs, like commercial firms, pursue
goals of profit and revenue maximization.
Also like commercial firms, colleges seek to enhance their reputation
and consumer awareness. Despite their
monopolist position as sellers of concessions and ancillary paraphernalia, they
cannot ignore demand, setting an arbitrary price, especially a high one, and
expect to maximize revenues or profits.
This is especially true in the case of programs, which, unlike food, can
be consumed by more than one spectator.
A practical demand model and cost estimates would provide information
for managers to select a price that achieves their goals.
This paper estimates such a demand function based on 11
years of home game data for a division IAA collegiate football team[1]. A new outlierresistant estimation technique,
Least Median of Squares (LMS), supplements the traditional Ordinary Least
Squares (OLS) to produce a better estimate of the demand model. It unveils the influence of single outlying
games or groups of them, allowing practitioners to adapt the model or price for
such conditions.
Researchers have studied the demand for athletic events
for many years but not the subsequent demand for complementary goods inside the
athletic arena (Cairns 1990; Schofield 1983).
In the case of football, most of the demand studies investigated
professional soccer in Great Britain (Peel & Thomas 1988; Cairns 1987), though
there have been a few studies that involved the NFL in the United States (Cebula, Austin, Wildener, &
Belton 1997; Welki & Zlatoper
1994; Cairns, Jennet, & Sloan 1986; Siegfried & Hinshaw
1979).
2. Income
3. Uncertainty
of outcome
4.
Importance of outcome for team standing
5. Quality
of teams
6. Size
of potential audience
7. Weather
8. Broadcast
influences
LMS is especially suited to pinpointing such games so
researchers and managers can study them for additional information about
spectator behavior. Unlike many other
outlierresistant techniques, LMS discovers clusters of outliers that together
overly influence estimates, while most other techniques, such as Studentized residuals and DFFITS, only check the influence
of one point at a time. Generally, these
other techniques fail to recognize groups of outliers, and thus ignore
potential information and causal variables that segregate observations into a
cluster (Rousseeuw & Leroy 1987; Rousseeuw 1984, 1997).
From an economic standpoint and given the funding goals
that pervade most programs, especially at the Division IAA level, profit
maximization or revenue maximization would be useful and not unexpected
goals. Secondary goals might include
maximizing unit sales to promote longterm allegiance to the team and school,
or providing information that broadens or heightens interest in the current and
future games along atypical dimensions, such as a specific player’s performance
(on and/or off the field), a chance for the league or national title, a new
coach with a new style or emphasis, and upcoming events in other sports. The
athletic administration decides the price and quantity it expects to sell to
achieve these goals.
Simply put, demand for programs (Q) depends on their
price (P) and the potential market measured by game attendance (A). Program consumption behavior is different at
certain games than at others, so dummy variables are added. F = 1 for the first game of a season. The first game of the new season represents a
special sales opportunity when uncertainty about the team and individual
players is high and interest in football is renewed rather than routine, as it
would be later in the season. H = 1 for
homecoming, another special interest game.
Our assumptions imply demand adheres to the following functional form:
(1) Q = f(P, A, F, H)
Complexity derives from the demand for the game, A, which
is influenced by the variables mentioned by
(2) A = f(RSP, I, U, IMP, QUAL, S, W)
where RSP = reserved
seat price[3]
I
= income
U = uncertainty of outcome
IMP = importance of
outcome to teams
QUAL = quality
of teams
S =
size of potential audience
W = weather condition variables
Consequently, substituting
equation (2) into equation (1), the reduced form function for program demand
is:
(3) Q = f(P, F, H, RSP, I, U, IMP, QUAL, S, W).
Here, some independent variables
are expressed as general categories to be further delineated in the data
section that follows, namely U, IMP, QUAL, S, and W.
The program sales total revenue function is the product
of equation (3) and P. From the total
revenue function, optimization techniques should produce the price and quantity
that would maximize revenue from program sales.
Using a marginal cost value provided by the administration, the
approximate price and quantity to maximize profit results. Traditional statistical techniques are used
to evaluate the model. As mentioned
earlier, the LMS technique is employed to determine the influence of outliers
and discover additional factors that influenced demand. In addition, we use a sample of more recent
games to evaluate the precision of the model.
Two
kinds of capacity constraints are present in this context. One is imposed by the print run negotiated
before the season starts. Demand and
attendance forecasts inform this decision.
The second constraint is imposed by the venue, which physically limits
attendance, limiting in turn the maximum possible program sales. However, neither constraint is relevant in
this particular case because ticketsellout and programsellout games are
omitted from the data set.
Data
To operationalize the general
expression (3) so we can estimate the model for program sales, we must measure
the general factors, such as importance of the game, mentioned earlier (U, IMP,
QUAL, S, and W). To arrive at a
practical formulation for all the variables, summarized below as equation (4),
we explain our breakdown of these variables and the data.
Three variables, P, RSP, and T, measure the cost of
attending the game and purchasing the program.
Both prices are expressed in real terms (19821984 is the base year). A nominal price that changes each year but
remains constant over each season might well pick up all seasontoseason
variation, not just price effects.
However, the nominal price seldom changed during the time span of this
study. Furthermore, because we deflated
the price variable by the monthly CPI, the inflationadjusted price variable
changes monthly rather than by season.
More formal seasonal adjustment would be problematic due to the
relatively short college football season.
Mapquest supplies T, travel time between the
opponent’s and hometeam’s locations, an element of trip cost (Siegfried &
Eisenberg 1980). Another basic factor, income
(I), also influences demand.
Measuring uncertainty is itself an uncertain process
(Siegfried & Hinshaw 1979). A ratio of win records reflects the
uncertainty (U) of the outcome in this model.
The winning percentage of the team with the poorest record is always in
the numerator and the better win percentage is the denominator. Consequently, U ranges between 0 and 1. Game outcome is generally more uncertain the
larger the U value because teams with identical records yield a value of one.[4]
Following earlier work, the impact of team quality and
game importance on attendance is considered (Siegfried & Hinshaw 1979; Rottenberg
1956). The individual quality of the
teams draws spectators excited by the chance to see a good team. Individual win percentages for the home team
(W) and for opponents (WO) measure these aspects of attendance that flow into
the program purchase decision. Because
the home team seldom figured in championships at any level, though some
opponents did, this variable reflects the importance of the outcome. A team usually struggles to attain the best
record possible, and a better win record motivates continued spectator
interest. This vector of home and
visitor win percentages is cumulative for the current season, except when first
home game is the season opener, when the previous season’s win percentage is
used.
A vector of current enrollments at the home team’s
university (E) and at the opponent’s (EO) represents potential market
size. These also proxy alumni base. Siegfried & Eisenberg (1980) discuss
alternative measures of market size. The vector (T for gametime temperature, R
for rain) reflects the weather.
Siegfried & Hinshaw (1979) discuss the
impact of rain on attendance. Dummy
variables (F) and (H) represent the first home game of the season and the annual
homecoming game.
Together, these refinements imply the following detailed
reformulation of equation (3):
(4) Q = f(P, RSP, T, I, U, W, WO, E, EO, TMP, R, F, H)
The empirical section presents
estimates of this reduced form equation.
Table 1 lists the variable definitions and descriptive statistics. The data is on a per game basis for home
games played during the 1989 through 1999 seasons for a total of 56 games. Games where programs sold out are
omitted. Table 2 provides a correlation
matrix.
Estimation Techniques
First, we estimate
the model with ordinary least squares (OLS).
OLS models minimize the sum of squared residuals. However, squaring values inflates or weights
each value by itself. Extreme values or
outliers receive especially heavy influence in the process. Accordingly, OLS shifts the line to accommodate
these outliers and reduce their residuals.
This process distorts estimated coefficients. Consequently, OLS often masks outliers with
smaller residuals than they deserve. In
the process OLS hides information from analysts, most notably the undistorted
estimate revealed by typical data as well as unacknowledged factors, such as
games against particular teams, that influence program sales.
Next, to
counteract this OLS shortcoming and improve on the OLS estimates, least median
of squares (LMS), an outlierresistant technique, is employed. This technique identifies outliers by
producing a set of coefficient estimates that minimizes the median of the
squared errors produced by these coefficients.
To determine if the outliers provide further information about program
sales, we search for their common traits or causes. This step suggests additional factors to
include in the model estimated with all available data, which subsequently
reduces the OLS objective function.
The structural
program demand function (Equation 1) is estimated by OLS, twostage
leastsquares (TSLS), and blockrecursive estimation in Table 3. Twostage estimates display low Rsquares,
and more troubling, positive coefficients on both ticket and program
price. Because the estimates go against
economic intuition, they cannot support managerial decision making. Indeed, positive coefficients on program
price suggest managers who want to increase program sales should increase
program price. Such counterintuitive
results could be due to problems such as multicollinearity,
errors in data measurement, omission of one or more important independent
variables, or some other form of specification error. Since White specification tests on the
reduced form equation indicate no statistically significant specification error
in the reduced form (see Table 4), it should be recalled that in general, a
reduced form may be mathematically derivable from more than one structural
equation. Therefore, it is possible for
the structural equations to be incorrectly specified while the reduced form
derived from them is correctly specified.
Consequently, it is useful for explaining and predicting behavior of the
dependent variable. In fact, it is a
plausible first approach that adequately illustrates the model and optimization
process. The model is parsimonious,
passes specification tests, produces good statistical results, and produces
results that agree well with expectations and economic theory. Because the reduced form is simpler to
estimate, practitioners will probably find it more useful.
We proceed with
estimates of the reduced form program demand equation which are used to refine
the model and illustrate optimal pricing.
Empirical Findings
OLS estimates of
the proposed linear model appear in column 1 of Table 4.[5] All coefficients possess the expected sign,
except income (I) and enrollment for the hometeam institution (E). Income is a problem because national trends
do not necessarily reflect local and regional patterns. In addition, other researchers find mixed and
generally little support for income as a factor in attendance (Cairns 1986, p.
15).
Not unexpectedly,
the insignificant enrollment coefficient suggests current students are not big
program customers. In fact, the coefficient
sign indicates their presence reduces sales, but the coefficient is not
significantly different form 0 for a onetail test. Please note that we adjusted the twotail
pvalues in Table 4 for onetail tests.
The negative sign on homeinstitution enrollment is unexpected, and so
far inexplicable, but also insignificant in the onetail test. While the correlation coefficient between P
and E, –0.76, is higher than we would like, the standard errors of both variables
are sufficiently small to produce T statistics both above 0.05 critical values
(though in the wrong direction for E).
It seems reasonable to search elsewhere for an explanation for the
negative sign. Flooding the student
section with program sellers would probably not be the best use of limited
sales personnel, because nonstudent spectators are more likely to have funds
to buy. Heightened interest and loyalty,
evidenced by their decision to spend funds and travel to the game, in turn,
reflects increased likelihood of a program purchase. It is entirely plausible that higher
enrollment results in lower program sales as student attendees crowd out these nonstudents
who are more likely to purchase programs.
Onetailed tests
suggest that of the three customer cost variables (P, RSP, and T) only program
price (P) significantly affects program sales.
This coefficient is significant at the 0.05 level, onetail. Opponents with large win records
significantly impact sales (at the 0.01 level, onetail), while hot days (at
the 0.05 level, onetail) and rain (0.01 level, onetail) counteract the
positive factor. Special games, such as
the first home game of the season (0.01 level, onetail) and homecoming (0.05
level, onetail), increase sales and offer opportunities for increased revenue
and profits.[6]
The estimates
provide some promise for understanding program sales but disappoint as
well. Although the F is significant (0.01
level), the R^{2} of 57.8% suggests problems in capturing major factors
in the sales process. Also one game in
the data set produces sales of only 90 programs—a very unusual value compared
to all the other sales values (with a mean of 360 and a standard deviation of
21). OLS standardized residuals conceal
this problem (count of outliers from standardized residuals shown in Table
4). The shaded cells in Table 5 show the
outliers that Studentized residuals and DFITS
techniques identify from the OLS model. This OLS model potentially masks
aberrant sales values, because they unduly influence OLS to minimize the sum of
squared residuals. We employed an
alternative estimation technique, LMS, to investigate these possibilities and
improve on the mixed evaluation results.
To improve on the
original model and incorporate other important factors, we employ least median
of squares (LMS), an outlierresistant technique mentioned earlier. This technique identifies outliers by
producing a set of coefficient estimates that minimizes the median of the squared
errors produced by these coefficients.
Each LMS iteration requires 14 observations from the data set (based on
13 coefficients for explanatory variables and one for the intercept). These observations form a set of equations
with 14 unknowns or estimates. LMS sorts
the squared residuals that result from each iteration and then locates the
median of these values. We present the
solution produced from the subset of 14 observations with the smallest median
squared error among the sample of subsets attempted. Because of the unusually large number of
iterations (subsets) and computer resources required to find the exact
solution, the LMS program employs random subsets rather than all possibilities
in an attempt to locate this solution (Rousseeuw
& Leroy, 1987). We enhance this
process by rearranging our data set and starting new searches. The eventual solution subset in this paper
identifies 14 statistical outliers (see Table 5).[7]
For
comparison purposes (with the eventual rival model in the next section), we
report a reweighted least squares estimate (RWLS) in
Table 4. We remove the outliers,
including the 90program one mentioned in the last section. Model statistics improve.[8] Three additional coefficients are significant
at the 0.01 level (travel time T, home win percentage W, and opponent
enrollment EO). Opponent win percentage
WO becomes insignificant. Programbuying
spectators appear more often when the home team wins.
To determine if
the outliers provide further information about program sales, we search for
their common traits or causes. This step
suggests additional factors to include in the model. While this RWLS model may appear a suitable
stopping point in model estimation, it is actually a plateau from which we
launch an improved model that incorporates any newly discovered factors.
LMS identified 14
outliers. With those observations
dropped from the data set, RWLS estimates provide new and hopefully better
insight into the impact of the independent variables on program sales. At this stage of the research process,
analysts might consider the RWLS estimate a working model of practical value. To do so may alarm other analysts concerned
that information is lost or not incorporated in the result—especially in this
case, where 25% of the entire data set drops from the scene. On the other hand, we view this situation as
ripe with more information about program sales than either the original OLS or
the RWLS model incorporates. Perhaps the
fourteen observations stand out from the other 42 for some common
reason—perhaps the OLS and RWLS models are misspecified
because important factors are excluded from the model.
Studentized residuals and DFITS find more outliers than
standardized residuals, because they are based on single elimination of data
points. These techniques attempt to
avoid the OLS masking problem mentioned earlier. However, because Studentized
residuals and DFITS are singleobservation techniques, and because OLS
potentially masks any remaining extreme points, these techniques identify fewer
outliers (shaded cells of Table 5) than LMS.
These limits do not affect LMS, so there is greater leverage for identifying
variables missing from a model.
No characteristics
appear outstanding and uniformly applicable to all or even a majority of these
fourteen games. However, games with a
nearby conference rival appear five times—more than onethird of the LMS outliers. The rival, Furman, does not stand out in the
singleobservation techniques as it does with LMS. This team is not the traditional or
archrival, but is nearby (about 2.5 hours away) and both it and the home team
are perennial conference rivals and were major rivals on the national scene earlier. During the 19891999 period, the rival played
seven games in the home team’s stadium.[9]
To account for
this team’s special effect, we estimate a new OLS model. It includes a dummy variable for games
against this particular rival.[10] This model does locate the 90program sales
value as an outlier. The rival effect is
statistically and practically significant.
Expected program sales rise by 135 programs for games with this rival
compared to other games where the other variable values are identical. Mean sales without the rival are 340
programs, so 135 programs represents a 40% increase over expectations for games
against other opponents.
Ramsey's (1969) RESET
test fails to reject the null hypothesis of no specification error, suggesting
all variables should be retained in the model.
Most coefficient test results remain the same as for the first OLS
model, but the hometeam’s win percentage, W, is now close to significance at
the 0.05 level (actually at the 0.065), onetail. The coefficient on the dummy variable
indicating the rival is significant at the 0.05 level, onetail.
From the model, we
can generally conclude that one realdollar increase in program price (about
$1.73 currently) will diminish sales by 354 programs. Uncertainty draws little extra attention from
spectators that eventually results in programs sales. Rather sales appear to provide information to
fans at their first opportunity to view the team (first home game of the season
and homecoming—often the first and only game attended by fans who travel
farther). On average, 241 more programs
reach patrons at the first home game each season, while an average of 90 extras
sell at each homecoming. Bad weather
reduces sales about 157 programs; while sweaty fans lose interest on hot days—about
4 programs for each degree Fahrenheit.
Overall, the
improved results appear satisfactory for further analysis of management
decisions about prices and the associated quantities they expect to sell.
Fixed Prices
The optimal program
price depends on goals of the sellers and characteristics of the game listed in
the independent variables. Because
managers often employ a single program price for every game, which they
determine before the season begins, we first compute the optimal fixed price using a typical game
situation. That typical game for this
study is one with the year 2000 hometeam enrollment, 6459 students
(headcount), current win percentage equal to the result of previous season
(37.5%), real disposable income (6508.8 billion 1996 dollars), and
reservedseat price ($12 deflated to $6.95).
Opponent variables (T, U, WO, and EO) assume mean values (see Table
1). Mean temperature without rain sets
the stage for this game. A typical game
is against a nonrival and is neither the first game of the season nor
homecoming.
All variables
included in the rival model figure in the optimal pricing exercise that
follows, though some are effectively deleted when they take on typical values
of zero, e.g., rain (R), RIVAL, and homecoming (H). The model provides the general expression Q =
A + bP for the demand function. In practice, A incorporates particular (often
average) effects of all the variables (except program price) and 0 or 1 for
dummy variables appropriate to the situation being optimized:
_{}
Solving for P yields _{}, which slopes downward because b < 0. Total revenue (TR) and marginal revenue (MR)
as functions of Q are: _{} and _{}.
The optimal Q (Q*)
that will maximize revenue occurs when MR = 0.
This will be Q* = A/2. Attempts
to sell a quantity larger than Q* will diminish total revenue. The P function provides the price that clears
the market of the Q* programs.
For the RivalOLS
model and the typical game described in the first paragraph of this section
(note that RIVAL = 0), these equations are:
(5) Q = 923.24 – 353.86 P
(6) P = 2.61 – 0.0028 Q
(7) TR = 2.61 Q – 0.0028 Q^{2}
(8) MR = 2.61 – 0.0056 Q
Consequently, for
a typical game, the optimal quantity to prepare for sale is 462 programs and
the optimal nominal (real) price is about $2.25 ($1.30 in 19921994 dollars). See the prices and quantities in Table 6
under Fixed Price. If managers seek to maximize circulation, or
advertising revenue, where that is proportional to circulation, they should
charge the revenuemaximizing price.
To maximize
profit, MR must be the same as MC. The
optimal quantity that does this must satisfy:
_{}
and the respective P^{**}
derives from the above P equation (6) again.
The average nominal (real) cost of a program during the 90s is about
$1.95 ($1.30).[11] If all extra programs can be ordered at this
cost, the AC curve will be flat and MC = AC.
Assuming this to be the case for demonstration purposes, the optimal nominal
(real) prices and quantity from the RivalOLS model would be $3.37 ($1.95) and
232 programs. (See the lower half of the
Fixed price column in Table 6).
The optimal P is similar to the current $3 price.
Overall profits
may not be maximized, if advertising revenue fluctuates with readership. Diminished readership may subvert other goals
of informing and increasing interest as well.
Regardless of the optimal advertising situation (a major issue apart
from program sales because many advertisers buy space from loyalty and
community service considerations[12]),
optimal prices and quantities remain at issue for managers. The closeness of these optimal prices to the
current price must please the current management. The results even suggest room for price
increases.
Flexible Prices
Although it is not
standard practice among college football programs, one strategy a
profitmaximizing monopolist should adopt whenever possible is market
segmentation by demand elasticity. One
way to accomplish this segmentation would be to charge different prices for
programs for “special issues,” such as for the first game of the season and
homecoming.
If the
administration allows flexible pricing from game to game, the model provides
easy solutions. The A value in the
formula would change, because dummy variables assume a value of one for the
unique characteristic of the game (e.g. F = 1 for the first home game, so Q =
1164.13 – 353.86P). Table 6 displays the
different possibilities.
The RivalOLS
model suggests a nominal price of $2.84 (say $2.75 or $3 for ease of making
change) to maximize revenue and $3.96 to maximize profits at the home
opener. If the rival is the opponent,
this price further increases to $3.16 to maximize revenue and $4.28 to maximize
profit.
For homecoming,
$2.47 maximizes revenue, while $3.59 maximizes profits. If the rival is the homecoming opponent, the
price rises to $2.79 to maximize revenue and $3.91 to maximize profit. Changing prices during the special atmosphere
of homecoming might not surprise spectators, so managers could find this a
suitable game for flexing their "monopoly power." Otherwise, flexiblepricers
risk loss of goodwill, a particularly severe cost to college athletic programs.
If a typical game
(not homecoming or the first game) is played against the rival, the revenue and
profit maximizing prices are $2.57 and $3.69.
All of the special game situations, mentioned so far, positively
influence optimal prices.
If flexibility
allows managers to set the price close to game time, they can include the
weather forecast as well. Of course,
rain lowers sales expectations, which requires lower prices to realize those sales
as Table 6 shows.
This price
flexibility will not help if the quantity available for sale is not also
flexible. If the goal is to sell all
available programs, the model provides a price that achieves this as well. The manager simply substitutes the quantity
available for sale in the appropriate P function.
Forecasts
The
home team’s overall quantity decisions occur in the spring when the manager
contracts with the printer. This is the
first opportunity for a forecast to benefit decisionmaking. The model could inform pricing decisions
relying on quantity forecasts, especially in a fixedprice regime. Managers could use the sum of individual game
forecasts, from the appropriate model for each game, which would sum to an
overall contract quantity. However,
managers can change individual game run totals as the season progresses. Later orders allow for more accurate
individualgame forecasts with more uptodate information about the game
situation and specific independent variable values, such as win record and
weather.
For instance,
given the current nominal (real) price of $3 ($1.74) and the specific values
mentioned earlier in the optimization computations, the RivalOLS model
forecasts 527 programs sold for the 2000 season opener. Programs sold out with about 335
programs. Table 7 shows forecasts for
the other games that were not sellouts (recall sellout games are omitted from
the sample used to estimate the model).
Analysis of the
forecast errors for the year 2000 outofsample games yields mixed results
(Table 7). The mean absolute error (MAD)
of these forecasts is about 109 programs and the rootmeansquareerror is
about 121 programs. Not surprisingly,
both values are larger than the same measures for the withinsample
observations used to estimate the Rival model (Table 4). But they are both surprisingly close to the
standard error of the estimate for the withinsample observations, 111
programs. The mean percentage absolute
error (MAPE) is about 41%, which indicates the forecasts stray compared to the
withinsample value of 27%. While these
values are not what we prefer, remember they represent forecasts for games that
lie outside the sample we use to estimate the model. On a more positive note, when we form 95%
prediction intervals using the standard error of the estimate from the Rival
model, all five predictions fall in their respective intervals. In fact, all five are within 1.5 standard
errors of the actual.
These results may
also reflect structural changes that accompany a new athletic management. Often new managements (and new coaches)
represent the end of one regime and the start of a new one. In such cases, spectators may withhold
program purchases or even stay at home, because they expect the first season in
the new regime to be a poor (or rebuilding) one, and hence uninteresting. Others may be upset by personnel changes and
the loss of favored coaches or individual players. Specifically, in addition to lowering ticket
prices, the new management improved recruiting, and made a variety of minor
changes. Lower ticket prices go against
the secular trend present in the 19891999 data we employ, when ticket prices
only increase, never fall. This also
indicates that the year 2000 season is different from the preceding eleven.
Other chances for
errors increase when forecasting sales for games outside the estimating
sample. November 2000 was the coldest
November on record, and two of the games were especially affected by extremes. In one case, the temperature was 64 at the
start of the game but dropped drastically during the game while the wind
surged.
Other information
might improve the error evaluations. For
instance, the first game sold out, so its forecast error is unobservable. Otherwise, the overall forecast error
measures might decrease. In fact, use of
this model would have initiated a larger printing for the first game and
reduced the chance of a sellout. The error
for the traditional rival (not the rival identified in the LMS stage) decreases
if we adjust the forecast with the rival coefficient. Finally, data for other forecasting methods,
especially prior forecasts by athletic managers, are not available for further
comparison.
Conclusions
Using OLS and LMS estimation techniques, we derive and
refine a model for the sales of programs at home football games. Previous research on the modeling of
attendance at such games appears in the reduced form model we estimate.
The outlierresistant estimation technique, LMS, provides
significant information, which improves results for program sales. In particular, LMS distinguishes outliers,
which we examine for common characteristics not included in the model. These findings suggest a modified model that
accounts for the effect of playing a particular annual rival. Surprisingly, the prevalent preconception
that excess demand occurs at games with the traditional rival proves
unfounded. The estimated model already incorporates
any such special effect.
Mindful of other program sales objectives, such as
enhancement of the institution’s image, prestige, and community presence,
athletic directors can use the improved model to assist their efforts to set
prices in order to maximize revenues and/or profits. Optimal fixed program prices exceed the
actual price over almost all of the sample years. However, revenuemaximizing prices fell below
the 2000 season price, while profitmaximizing prices exceeded it. Allowing flexible prices from game to game,
managers should charge higher prices for programs on special event days, such
as the first game of the season, homecoming game, and games against special
rivals. This paper illustrates a
technique that quantifies and controls for these factors, then takes advantage
of them to illustrate the attainment of typical management goals.
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TABLE 1
VARIABLE INFORMATION
Symbol 
Variable 
Measure 
Mean 
St. Dev. 
Q 
Program
Sales per game 
Number
of programs 
359.77

156.95

P 
Program
Price 
Real
19821984 dollars 
1.47 
0.2 
RSP 
Reserved
Seat Price 
Real
19821984 dollars 
8.14 
0.67 
T 
Travel
Costs 
Hours
between opponent's & hometeam's sites 
4.35 
1.74 
I 
Income 
Average
Real Disposable Income19821984 dollars 
5545.86 
489.47 
U 
Uncertainty 
Ratio
of Win Percentages* 
0.52 
0.31 
W 
Home
Win Percentage 
Percentage
of Games won** 
40 
22 
WO 
Opponent
Win Percentage 
Percentage
of Games won** 
55 
31 
E 
Home
Enrollment 
Fall
Headcount 
6622 
209 
EO 
Opponent
Enrollment 
Fall
Headcount 
6191 
4544 
TMP 
Temperature
during game 
Degrees
F 
65.9 
9.49 
R 
Rain 
1
signifies rain during game; 0 for no rain 
0.14 
0.35 
F 
First
Home Game of Season 
1
signifies first home game of season; 0 otherwise 
0.2 
0.4 
H 
Homecoming 
1
signifies a homecoming game; 0 otherwise 
0.2 
0.4 
RIVAL 
Opponent
with special impact 
1
signifies special rival (not traditional rival) 
0.04 
0.19 
Notes: * Lowest win
percentage/highest win percentage
** Cumulative for season or last
season record if season opener is at home
Table 2
Correllation
Matrix

Q 
P 
F 
H 
A 
RSP 
T 
I 
U 
W 
WO 
E 
EO 
Tmp 
R 
Rival 
Q 
1.0000 















P 
0.0075 
1.0000 














F 
0.3530 
0.0017 
1.0000 













H 
0.1201 
0.0096 
0.2444 
1.0000 












A 
0.7367 
0.0079 
0.0549 
0.2647 
1.0000 











RSP 
0.2745 
0.3868 
0.0115 
0.0301 
0.1096 
1.0000 










T 
0.1777 
0.1095 
0.1393 
0.1027 
0.3103 
0.0232 
1.0000 









I 
0.2507 
0.3758 
0.0588 
0.0262 
0.0955 
0.0523 
0.0106 
1.0000 








U 
0.0608 
0.1876 
0.1294 
0.0822 
0.0921 
0.2694 
0.1382 
0.0174 
1.0000 







W 
0.0273 
0.2490 
0.1321 
0.0386 
0.1921 
0.2577 
0.2853 
0.1296 
0.3988 
1.0000 






WO 
0.3292 
0.0098 
0.0413 
0.0966 
0.3465 
0.2114 
0.0013 
0.1199 
0.0224 
0.0240 
1.0000 





E 
0.1948 
0.7645 
0.0006 
0.0006 
0.0223 
0.3706 
0.1706 
0.0398 
0.3199 
0.3024 
0.0003 
1.0000 




EO 
0.0138 
0.0138 
0.2926 
0.2003 
0.1292 
0.0886 
0.0203 
0.0758 
0.1951 
0.1631 
0.2016 
0.0812 
1.0000 



Tmp 
0.1206 
0.1128 
0.4253 
0.0095 
0.0340 
0.0282 
0.0887 
0.0696 
0.2779 
0.1984 
0.1505 
0.1448 
0.1733 
1.0000 


R 
0.3203 
0.2248 
0.1835 
0.2018 
0.3491 
0.3282 
0.1501 
0.1204 
0.2297 
0.1731 
0.0540 
0.2235 
0.2311 
0.1698 
1.0000 

Rival 
0.2119 
0.0288 
0.3892 
0.0951 
0.1247 
0.1766 
0.2161 
0.1700 
0.0930 
0.0130 
0.1250 
0.0523 
0.1243 
0.0391 
0.1964 
1.0000 
Table 3 

STRUCTURAL ESTIMATES OF PROGRAM DEMAND EQUATIONS 


OLS 

TSLS 

Block Recursive 

Variable 
Coef 
Twotail PValue 
Coef 
Twotail PValue 
Coef 
Twotail PValue 

P 
10.1471 
0.1518 

10.4298 
0.8992 

19.4225 
0.8155 

A or Ahat^{1} 
0.04490 
0.0000 

0.04908 
0.0000 

0.05007 
0.0000 

F 
123.46 
0.0007 

120.1294 
0.0063 

181.722 
0.0000 

H 
2.8800 
0.9354 

4.8617 
0.9143 

73.5060 
0.0862 

constant 
97.4487 
0.3831 

134.594 
0.3657 

184.512 
0.2417 



Standard
Error 
97.6 


119.6 


120.9 


RSQUARE 
64.1 


46.1 


44.9 


F 
227604 
0.0000 

10.905 
0.0000 

10.416 
0.0000 

DOF 
51 


51 


51 


N 
56 


56 


56 


^{1} Actual attendance (A) is used in the OLS
regression. Fitted values (Ahat) are
used in the TSLS and block recursive regressions. TSLS instruments are RSP, T, I, U, W, WO,
E, EO, Tmp, R, P, F, and H. In the block recursive estimate, Ahat is
the fitted value from a regression of
RSP, T, I, U, W, WO, E, EO, Tmp, and R on A. 

Table 4 

REDUCEDFORM PROGRAM DEMAND ESTIMATES 


OLS 

RWLS 

RIVALOLS 

Variable 
Coef 
Twotail PValue 
Coef 
Twotail PValue 
Coef 
Twotail PValue 

P 
321.8655 
0.0735 

450.907 
0.0000 

353.8582 
0.0404 

RSP 
25.61065 
0.3851 

30.07882 
0.0697 

26.45297 
0.3460 

T 
11.18703 
0.2862 

20.99088 
0.0012 

1.39254 
0.9019 

I 
0.00205 
0.9649 

0.00581 
0.8056 

0.01250 
0.7801 

U 
23.07945 
0.7110 

36.11487 
0.2219 

4.23241 
0.9435 

W 
1.06479 
0.2364 

1.88172 
0.0002 

1.30929 
0.1310 

WO 
1.58098 
0.0064 

0.07125 
0.8189 

1.25182 
0.0263 

E 
0.33615 
0.0425 

0.57693 
0.0000 

0.39180 
0.0151 

EO 
0.00124 
0.7760 

0.01063 
0.0002 

0.00515 
0.2541 

TMP 
3.82874 
0.0672 

6.77738 
0.0000 

3.97454 
0.0468 

R 
180.8259 
0.0021 

157.7383 
0.0000 

156.7327 
0.0055 

F 
229.88 
0.0000 

409.4503 
0.0000 

240.8869 
0.0000 

H 
89.45138 
0.0556 

90.4235 
0.0010 

90.01424 
0.0435 

Rival 






130.9947 
0.0247 

constant 
3391.803 
0.00549 

5181.333 
0.0000 

3659.983 
0.0020 



Standard
Error 
116.6 


87.1 ^{1} 


110.9 


RSQUARE 
57.8 


91.6 


62.8 


F 
4.433 
0.0001 

23.572 
0 

4.938 
0 

DOF 
42 


28 


41 


N 
56 


42 


56 




#
OF OUTLIERS 









Included Data 
0^{2} 


0 


1^{2} 


Excluded Data 



14 







White
Test 
26.08 
0.3 

22.63 
0.48 

31.82 
0.13 

Ramsey
RESET F 









Second Order 
0.075 
0.78 

0.004 
0.95 

0.119 
0.73 

Third Order 
0.707 
0.49 

0.065 
0.94 

1.560 
0.22 

Fourth Order 
2.113 
0.11 

1.866 
0.16 

1.320 
0.28 

Fifth Order 
1.565 
0.20 

1.476 
0.24 

1.404 
0.25 



^{1}
This is a scale estimate produced by LMS and similar to the typical standard
error of the estimate. 

^{2}
These are standardized residuals. For
other outlier techniques see Table 5. 

Table 5
Identified Outliers by
Technique
Table 7 

Sales Forecast Results 



Forecast 
Forecast Interval 
Actual 
Error 
Standardized Error 
344 
(126.6, 561.4) 
240 
104 
0.9 
376 
(158.6, 593.4) 
345 
31 
0.3 
391 
(173.6, 608.4) 
307 
84 
0.8 
329 
(111.6, 546.4) 
499 
170 
1.5 
334 
(116.6, 551.4) 
176 
158 
1.4 



Error Analysis 



Within Sample 
Outside Sample 

MAD 
76.3 
109.4 

MAPE 
27.2 
40.7 

MSE 
9006.7 
14539.4 

RMSE 
94.9 
120.6 

Standard
Error 
110.9 

Endnotes
We would like to thank various
people for their assistance: Brenda
Moore in the Interlibrary Loan section of Hunter Library at Western Carolina
University; William Crocker, Derik Briggs, and Muniza Haq, graduate assistants
in the MBA program; Kevin Ayers, Director of the WCU Sport Management Program;
Jeff Compher, Athletic Director; Jody Jones, Sports
Information Director; and especially two people: Hunter Yurachek,
Senior Associate Athletic Director and Steve White, retired Sports Information
Director and current encyclopedia of sport lore and detailed facts at WCU. All assisted us generously and without them
we could never have produced this paper.
Normally, authors accept responsibility for all errors, but we have much
greater confidence in our work thanks to Hunter and Steve. We also wish to thank Nick Rupp, Gerald Granderson, and three anonymous referees of the Journal
of Sport Management for much helpful comment and advice. Their involvement greatly improved the
quality of the paper.
[1]
For generalization, we refer to Western Carolina University (WCU) as the home
team and
[2] We assume
that broadcasts do not influence attendance.
[3] For
almost all years, general admission seat tickets were reserved seat tickets, so
we used reserve seat price for all years.
[4] The
computational problem posed when both teams had a zero win record does not
occur in the sample; however, this uncertainty measure may present another
problem. It does not distinguish the
uncertainty of a team with a 100% win record playing a nowin team from a game
where the comparison is small, say a 1% winning record against a team with no
wins.
[5] White's
(1980) heteroskedasticity test fails to reject the
null hypothesis of homoskedastic errors. Examination of crosscorrelation coefficients
suggests an absence of multicollinearity. Although other mathematical specifications
seem relevant, Ramsey (1969) RESET tests fail to reject the null hypothesis of
no specification error at the 0.10 significance level. Results from the RESET tests suggest (a) all
relevant variables have been included in the model, (b) the estimated
specification is acceptable, and (c) there is no correlation between the RHS
variables and the error term.
Correlation among the RHS variables and the errors, if present, could
have been caused by (a) measurement error, (b) simultaneity bias, (c) serial
correlation, or (d) any combination.
Absence of significant correlation among RHS variables and errors suggests
none of the four are present. In light
of these results and in the interests of parsimonious application by
practitioners, we continue with the standard linear functional form.
[6] Enrollment
is not significant at conventional levels in a onetail test because it has the
wrong sign. This could be a result of multicollinearity, because the correlation between P and E
is –0.76. (The next largest correlation
coefficient among the variables is 0.43).
[7] With 56
observations and 13 independent variables (the limit on the LMS program), there
are nearly six trillion possible subsets or iterations required to get the
estimates that actually minimize the median of the squared residuals. This would require immense, if not
impossible, current computer resources.
Consequently, the program employs a default of 3000 random samples of 14
observations. In an attempt to overcome
this limitation, we shuffled the data (top observation sequentially moved to
end of data set) and reran the program.
After six such reruns, the solutions repeated. This may only be a local minimum, but
randomization should improve the chances of approaching the global minimum.
[8] LMS
identifies outliers as those observations with standardized residuals, based on
a LMS scale estimate (similar to the standard error of the estimate), of 2.5 or
more in absolute value. Rousseeuw and Leroy suggest that users subsequently rerun
OLS with a diminished data set—with identified outliers deleted. Without the outliers, the reweighted
least squares (RWLS) results approximate OLS distributions and inferences are
more “trustworthy” than applying OLS to the original data set (Rousseeuw & Leroy 1987, pp. 4546).
[9]
Enrollments at the two schools are very similar, but the rival is a private
school, so the outliers may represent a unique income effect.
[10] The
limit of 13 independent variables in LMS software precluded comparisons between
LMS solutions of this model with the first, so only OLS results appear here as
the RivalOLS model in column 3 of Table 4.
[11] Conversation with Steve White
[12]
Comment from Jeff Compher, Athlethic
Director of Western