Choice Behavior under Dynamic Quality Changes:
State Dependence Versus 'Play-It-By-Ear' in Selecting Ski Resorts
Klaus Moeltner
Jeffrey Englin
Paper Submission for the
Second World Congress of Environmental and Resource Economists
Monterey, CA
June 24-27, 2001
(Long Paper Presentation)
Corresponding Author:
Klaus Moeltner
Assistant Professor
Department of Applied Economics and Statistics
University of Nevada, Reno
MS 204
Reno, Nevada 89557
Phone: (775) 784-4803
Fax: (775) 784-1342
e-mail: moeltner@unr.edu
web page: www.ag.unr.edu/moeltner
JEL Codes: C15, C35, D12, Q26
ABSTRACT AND KEY WORDS
The literature on brand loyalty has focused on products that exhibit constant quality over time.
In this study we consider ski resorts, for which quality attributes change frequently. This requires
a model that includes dynamic quality features and indicators for state dependence, while
controlling for individual heterogeneity. We show that purchase history and dynamic choice
characteristics have a significant and offsetting effect on repurchase decisions. This suggests a
third category of consumer in repeated choice settings next to habit formers and variety seekers:
the play-it-by-ear type who, unaffected by purchase history, moves across brands in pursuit of
consistent quality.
KEY WORDS: Repeated brand choice; Dynamic quality features; Random parameters;
Simulated choice probabilities.
1
1. INTRODUCTION
When consumers repeatedly choose over several products, past choices can affect the
probability of selecting a given product again at a later occasion. This phenomenon is commonly
referred to as ‘state dependence’ (Heckman 1981a). Generally, state dependence can increase a
consumer’s propensity to repurchase a specific good (habit formation), or decrease the
probability of repurchase (variety seeking). A key element of the research on the effects of state
dependence to date has been the stability of characteristics for goods under consideration. This
study examines the role of purchase history for products with both fixed and time-variant
attributes. Our empirical application is based on a set of ski areas in the Sierra Nevada. The
dynamic quality attributes are snow and temperature.
In the marketing literature, state dependence is often labeled as purchase carryover or
purchase-event feedback (Allenby and Lenk 1995; Keane 1997). Understanding these forces
that guide consumer choice is important to managers when making marketing and pricing
decisions. As pointed out in Keane (1997), temporary promotional efforts may affect consumer
behavior well into the future if people are susceptible to habit formation. On the other hand, if
consumer choice is relative insensitive to past purchases, or if variety seeking is the dominant
element of state dependence, the promotional impact on sales may be short-lived.
Interest in the effect of state dependence on choice behavior has also found entry into the
recreation literature in recent years. Some examples are McConnell, Strand, and Bockstael
(1990), Adamowicz (1994), and Smith (1997). In those studies, the ‘products’ people are
choosing from are not food or household items, as commonly investigated in marketing research,
but rather recreation sites such as fishing spots (Adamowicz 1994; Swait, Adamovicz, and van
Bueren 2000), or beach destinations (McConnell, Strand et al. 1990).
In this context,
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understanding demand effects attributable to state dependence aid public land managers in
making policy decisions on site access, pricing, and quality.
Regardless of the application, researchers must take care to disentangle ‘true’ state
dependence (Heckman 1981a) from the effect of time-variant exogenous variables and consumer
heterogeneity. Specifically, the effect of true state dependence may be inflated if consumer
preferences are erroneously assumed to be homogeneous, or dynamic exogenous variables are
omitted (Heckman 1981a; Keane 1997; Erdem and Sun 2001). In recent marketing contributions
heterogeneity has been explicitly captured in repeated choice models by introducing random
coefficients into random utility models (RUMs) (Allenby and Lenk 1995; Erdem 1996; Keane
1997). Allenby and Lenk (1995), Keane (1997), and Erdem and Sun (2001) also include time
varying price and marketing variables (display and advertising) to disentangle their effect on
purchase decisions from true state dependence.
For the household products generally considered in marketing applications (ketchup,
peanut butter, detergents, etc.), price and marketing variables may well be the main candidates
among time varying exogenous factors that could lead to erroneous inferences about state
dependence if omitted from a multi-choice model. Other product attributes, such as quality
features, usually do not change over the research period for a specific brand item. Thus, they can
plausibly be held constant over time in a repeated choice specification. However, there are
consumer products that are inherently susceptible to quality changes over time, within the same
brand label. One such example would be wine, where the same winery and appellation (the
‘brand’ or ‘label’) can sell products of varying quality, depending on vintage. If one were to
ignore the dynamics in these attributes when examining consumer loyalty or variety seeking for
wine labels over time, the above mentioned problem of ‘spurious state dependence’ (Heckman
3
1981a) may transpire, even after controlling for heterogeneity and marketing effects. This issue
becomes even more important if the goods under consideration are recreation sites. As these
sites are not the end product of a highly controlled manufacturing process, they are by nature
susceptible to a multitude of quality changes over time. The importance of these changes for
repeated site choices will depend on the type of recreational activity, and individual preferences.
However, to our knowledge there does not yet exist a multi-site recreation study with focus on
state dependence that includes either dynamic quality variables or individual heterogeneity.
This research extends existing marketing and recreation studies by analyzing the separate
effects of quality changes and state dependence on consumer choices, while allowing for
individual heterogeneity. In addition, we examine if consumers who place a relatively large
weight on a specific time and site-varying attribute are less likely to form habits. Conversely, we
investigate if what appears to be behavior driven by variety seeking is in fact a manifestation of a
‘play-it-by-ear’ attitude fueled by strong preferences for the changing attribute in question. This
requires a product that is purchased relatively frequently, and exhibits both time variant and time
invariant features. Ski resorts are well suited for this purpose. Their terrain and level of
difficulty remain unchanged over time, while on a given day the quality of a visit to the resort
will be heavily affected by day-specific attributes such as snow conditions and weather. To our
knowledge, this is the first such application for a multi-choice model with state dependence.
The remainder of this text is structured as follows: In the next section we develop an
econometric model of state dependence, dynamic quality effects, and heterogeneity.
The
empirical part of this study then discusses data, estimation results, and operational implications
for ski resort managers. Concluding remarks and a summary of key findings are given in the last
section.
4
2. MODEL FORMULATION
Following the majority of brand choice studies, we embed our model in a random utility
(RUM) framework. Specifically, we assume that the utility individual i derives from a visit to
resort j at time t is given by



U ijt  A j   i  Pijt   i  Q jt   i  Sijt   i   ijt  Vijt   ijt
i  1 N ,
j  0  J , t  1T ,
(1)
where Aj is a vector of time-invariant resort attributes, Pijt is the price to i for visiting resort j at
time t, Qjt is a vector of quality characteristics that change over resorts and time, and Sijt is a
vector of variables associated with state dependence. The symbols i, i, i, and i denote
individual-specific coefficient vectors, and ijt is an i.i.d. random error term.
Time periods are generally defined as purchase occasions in brand-choice studies (e.g.
Allenby and Lenk 1995; Erdem 1996; Keane 1997). This preempts an investigation of interpurchase time effects on choice decisions. As shown in Papatla and Krishnamurthi (1992),
Chintagunta (1998), and Chintagunta and Prasad (1998), the length of time between purchases
can affect the nature and intensity of state dependence. To capture such effects in our model,
and in synchronicity with the nature of our data (day trips) we choose days as the relevant time
unit. It should be noted that for household goods it is often assumed that a given product is being
consumed continuously throughout the inter-purchase period (e.g. Papatla and Krishnamurthi
1992). This is different for recreation sites, where actual consumption ends with the visit. In
5
fact, non-consumption at time t becomes a separate choice. This is usually modeled as ‘stayinghome’ option or ‘nonparticipation’ in a RUM specification (e.g. Morey, Shaw, and Rowe 1991;
Morey, Rowe, and Watson 1993). We follow Adamowicz (1994) by modeling nonparticipation
as an additional alternative to actual sites with associated utility

U i 0t   i 0  S i 0t   i   i 0t ,
(2)
where the “0” subscript indicates the stay-home option. We reduce all quality indicators to a
constant, and set price to zero for this choice. We do, however, retain variables measuring state
dependence as described below.
While some studies on repeated choice let state dependence work through attributes of
brands purchased in the past (e.g. Trivedi, Bass, and Rao 1994; Erdem 1996), we follow recent
contributions in marketing (Keane 1997; Chintagunta and Prasad 1998; Erdem and Sun 2001)
and recreation (Adamowicz 1994; Swait, Adamovicz et al. 2000) by defining variables for state
dependence based on brands (sites) chosen at previous purchase occasions. The question then
arises as to how far back into a consumer’s purchase history the model should reach. At one
extreme, one could include only the choice decision made in the preceding period (‘1st order’
process, Erdem and Sun 2001). At the other extreme, one could explicitly model the individual
effect of all past choices made by the consumer (Heckman 1981b). Some authors have proposed
a middle ground by using a weighted average of past purchases to model choice history
(Guadagni and Little 1983), or by including the number of uninterrupted times (‘run’) a given
brand was chosen prior to the current time period (Heckman 1981b; Bawa 1990). For our
application the number of time periods during the skiing season of interest (151 days) is far too
6
large to allow the separate inclusion of all previous choices. However, we a priori concur with
McConnell, Strand et al. (1990) that recent visits ought to weigh more heavily for current site
decisions than visits further in the past. We therefore include two indicators for past site choices
in the model: the total number of times a given resort was chosen prior to t (Nijt), as in
Adamowicz (1994), and the consecutive number of times a given resort was chosen up to t
uninterrupted by any visits to other destinations (Rijt). This variable conveys the notion of ‘run’
mentioned above. Since our data do not include many day-to-day runs, we allow for interruption
by the stay-home option for this indicator. Our hypothesis is that whatever manifestation of state
dependence, if any, drives individual i ought to manifest itself more strongly through Rijt than
Nijt. We also specify two analogous indicators for the stay-home option: the number of times
during the season prior to t an individual chose not to participate (Hit) (Adamowicz 1994), and
the number of consecutive days of non-skiing immediately preceding t (Dit) (Provencher and
Bishop 1997; Swait, Adamovicz et al. 2000). Thus, the elements of Sijt, j=0...J, materialize as
t


d ijt



t 1
 t 1

 N ijt     s


 R      d ij ,t l  1  hij ,t l   hij ,t l   1  hij ,t l 

 ,
S ijt   ijt    s 1   l 1
 H it  
 t


  d ijt   hijt
  

 t 1

 Dit  

t 1 s


hij ,t l



s 1 l 1
(3)
where dijt , j = 0...J, is a zero / one dummy taking the value of one if resort j was chosen by i at
time t, while hij,t is a dummy equal to one if j is the stay-home option, and equal to zero if j is an
actual resort. This implies that Rijt and Dit equal zero for j = 0, while Hit only applies to
nonparticipation. While somewhat counterfactual, setting Dit, the number of days since the last
7
ski trip, to zero for the stay-home option is necessary to preserve this indicator in a RUM
specification.
In essence, Dit can be interpreted as the relative effect of prolonged
nonparticipation on choice probabilities for resorts versus the probability to stay home in the
current period as well.
Theoretically, both nonparticipation indicators could measure a building up of
‘eagerness’, i.e. an increased probability of skiing as they grow large, or ‘rustiness’, i.e. the
opposite effect. In essence, ‘eagerness’ is the equivalent to ‘variety seeking’ for actual sites,
while ‘rustiness’ corresponds to ‘habit formation’, or ‘inertia’. Adamowicz (1994) for example,
finds that eagerness dominates behavior in his ‘Rational Model’ (increased probability of visits
as Hit increases), while Provencher and Bishop (1997) find that as more time elapses since the
last visit the probability of participation decreases. Swait, Adamovicz et al. (2000), in turn,
report initial rustiness following a preceding visit that turns into eagerness after about ten weeks
of nonparticipation. As we will show, the stay-home counter reaching through the entire season
(Hit) is a much more robust indicator of state dependence for nonparticipation than the recent
‘run’ of days at home. Specifically, the latter measure masks a skier’s preferences for timevariant attributes if they are omitted from the model. In other words, what appears to be shortterm rustiness turns out to be a wait for the right conditions, or a ‘play-it-by-ear’ effect in our
terminology. This is one of the key findings in this study.
As mentioned previously, to correctly estimate and interpret indicators for state
dependence and the effect of time-variant exogenous factors, we need to control for individual
heterogeneity in our model. The rationale behind this requirement is that probabilities associated
with repeated choices made by the same person will be correlated due to unobserved individual
characteristics and preferences. If ignored, such inherent ‘tastes’ may be incorrectly interpreted
8
as an indication for state dependence. In recent contributions to the brand choice literature,
heterogeneous reactions to pricing and marketing variables were found to be a significant factor
in the process that drives repeated purchase decisions (Allenby and Lenk 1995; Erdem 1996;
Keane 1997; Erdem and Sun 2001). Erdem (1996) and Erdem and Sun (2001) explicitly allow
for and find significant heterogeneity in the way past brand choices and attributes affect
individuals’ repurchase decisions.
In a RUM framework, it is convenient to introduce
heterogeneity through random coefficients (Revelt and Train 1998; McFadden and Train 2000).
In contrast to Allenby and Lenk (1995), and Keane (1997), who allow for time-varying taste
parameters, and following Erdem (1996), and Erdem and Sun (2001), we assume individual
preferences remain constant throughout our research period (151 days). Collecting i, i, i, and
i into a single coefficient vector  i, we stipulate that parameters are distributed multivariate
normal with
E  i    ,

E   i   i   


0
i j,
(4)
i j
Thus, we estimate a vector of parameter means,  , and the elements of the variance-covariance
matrix . In contrast to previous studies, the covariance terms in  are of major interest and
importance in this paper. Specifically, we a priori expect a negative sign for covariances
between (presumed positive) coefficients associated with dynamic quality attributes and
(positive) coefficients for state dependence, if habit formation dominates. Conversely, if a
coefficient for state dependence is negative (indicating variety seeking tendencies), its
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covariance with coefficients for dynamic quality should be positive.
In other words, we
hypothesize that the stronger the effect of quality seeking for a given individual, the smaller will
be the absolute value of the coefficient for state dependence drawn for this individual. The
intuition behind this premise is that quality-sensitive skiers, i.e. the play-it-by-ear types, should
be less likely to be influenced by past resort choices. As we will show below, our results
generally confirm this stipulation. This constitutes the second key finding flowing from this
research.
We assume that remaining serial correlation in our model is accounted for by observed
time-varying site attributes and indicators for state dependence. This implies that ijt in (1) is a
truly random error term uncorrelated with the elements of i. The stipulated density of this error
will dictate the specification of choice probabilities. Two frequently used distributions in the
choice literature are normal, resulting in a multivariate probit specification (Hausman and Wise
1978; Keane 1997), and type I extreme value. The latter distribution, in combination with the
random coefficients in i, yields a random parameter logit, or ‘mixed logit’ model (Revelt and
Train 1998; Brownstone and Train 1999). In either case, the estimation of choice probabilities
requires solving a high-dimensional integral. As discussed in Layton (2000), the dimension of
integration proliferates with choice occasions in the multivariate probit model, and with the
number of random parameters in a mixed logit specification. In our application, each individual
faces 1359 choice occasions in a model with a limited number of random coefficients. For the
sake of computational tractability we therefore choose the mixed logit approach for our
application. Thus, the probability of skier i choosing option j at time t, conditional on φi, is given
by (McFadden 1974)
10
Pijt  i , ,   
exp Vijt  i , ,  
J
 exp V  , , 
ikt
k 0
,
(5)
i
where Vijt is defined as in (1). The conditional probability of observing an individual’s entire
sequence of trip decisions is therefore (Erdem 1996)
T
J
dijt
Pi  i ,  ,    Pijt  i ,  ,  ,
(6)
t 1 j 0
where dijt is defined as in (3). Relaxing conditionality on coefficients yields
Pi 
 P 
i

i
,  ,   f  i  d i ,
(7)
i
where f(i) is the multivariate normal distribution. The dimension of the integral in (7) is
commensurate to the number of elements in i. Since the evaluation of high-dimensional
intergrals is computationally impractical beyond an order of three or four given existing software
capabilities, researchers have proposed simulation methods to estimate such probabilities and
associated likelihood function (Börsch-Supan and Hajivassiliou 1993; Keane 1994; McFadden
and Train 2000). We follow the procedure outlined in Brownstone and Train (1999) by drawing
a set of i from f(i), with some arbitrary starting values for  and . This allows computation
of (6) for all individuals. The process is repeated R times, yielding the simulated choice
probability (Erdem 1996; Layton 2000)
11
~ 1 R
Pi    Pir  i , ,  
R r 1
(8)
and simulated log-likelihood function
 
~ N
~
l   ln Pi ,
(9)
i 1
where N denotes the number of individuals in the sample. The elements of  and  are updated
throughout the optimization process, and are part of the estimation output together with fixed
coefficients, if present.
Aside from allowing for the examination and interpretation of potentially revealing
covariance terms in , our random parameter specification provides two additional advantages:
First, as discussed in Train (1998), and noted in Allenby and Lenk (1995), by introducing
correlation across choice probabilities we eliminate the problem of independence of irrelevant
alternatives (IIA) that plagues standard conditional logit models. Second, as shown in Heckman
(1981b), maximum likelihood procedures with random parameters will yield consistent estimates
even under arbitrarily set initial conditions for state dependence, if N and T tend to infinity. This
is important in our application, since we do not have information on visits prior to the sampling
period. Instead, we assume all skiers start out with a ‘blank memory’, and set the elements of Sijt
to zero for the first day of the season. Keane (1997) takes a similar approach, and shows
robustness for his maximum likelihood estimators under fixed initial conditions for a sample
with large N (1,150 individuals), but rather small T (about five purchase occasions, on average).
12
In our application, both N and T are reasonably large (131 and 151, respectively), allowing us to
invoke Heckman's (1981b) finding as well.
3. EMPIRICAL ANALYSIS
1. Data
Our data stem from a spring 1998 survey of 131 randomly selected skiers and
snowboarders (referred to as snow riders in the remainder of this text) at the University of
Nevada, Reno. Each individual was asked to complete a chronology of trips to nine resorts in
the Lake Tahoe area during the preceding 1997 / 98 ski season. The exact time period spans 151
days from the end of November 1997 to the end of April 1998. All nine resorts were open
during this period. Resort-specific information was collected from brochures and web sites
associated with these ski areas. The source for our meteorological data is the SNOTEL web site
established by the US Department of Agriculture’s Natural Resources Conservation Service
(www.wcc.nrcs.usda.gov/snotel). All seven SNOTEL sites considered for this research are
located in close vicinity of a given resort or pair of resorts. Table one summarizes the data and
variables used in our final model specification: Overall, our data set comprises 19781 choice
occasions, including the stay-home option. Of these trips, 1195 (6%) result in actual ski resort
visits. As can be seen from the table, the lion’s share of real trips (56%) were made to Mt. Rose,
followed by Alpine Meadows (11%) and Squaw Valley (8%). Actual trips can be further divided
into 210 visits (17.5%) by season pass (SP) holders, and 718 visits (60.1%) made on holidays
(HD).
The holiday dummy takes the value of one for designated university holidays and
weekends, and is set to zero otherwise. It is also held at zero for the stay-home option. The next
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four columns in table one show the number of acres of beginner (green), intermediate (blue) and
expert (black) terrain for a given resort, as well as the total size of the area. The variable PRICE
includes round-trip travel costs from an individual’s ZIP code area to a given resort, plus ticket
price for a specific resort, day, and individual. While travel costs to and from a specific
destination are assumed constant for a given rider during the research period, ticket prices vary
by day-of-the-week, time-of-season, gender, and age for some destinations. In addition, ticket
prices are set to zero for season pass holders. Our price variable reflects these dynamic changes.
Time-varying quality variables are given in the next two columns. We hypothesize that an
individual’s riding experience will be strongly affected by weather and snow conditions on a
given day. We use the average daily temperature in Fahrenheit (TEMP) to capture weather
effects, and the cumulative water content of snowfall in inches (‘pillow’) during the seven days
preceding a given date to describe snow conditions. The resulting variable is labeled SNOW7 in
the table. Data on actual snow volume were not available on a daily basis for our time period
and sites.
Since volume is a complex function of water content, barometric pressure,
temperature, and other unobserved meteorological factors, we settle for pillow as a proxy for
actual volume.
In addition, some aspects of snow quality will also be captured by the
temperature variable. Specifically, if snowfall has been abundant, lower temperatures allow for
better and longer lasting ‘powder’ conditions. On the contrary, if little or no snow has recently
been added to the overall pack, low temperatures may result in hard and icy surfaces. The sevenday accumulation period in SNOW7 was chosen to allow for reasonable delays in skiers’
reaction to fresh snowfall while preserving a modicum of the ‘freshness’ quality. As shown in
the first row of the table, TEMP and SNOW7 are arbitrarily set to zero and 32 degrees,
respectively, for the nonparticipation option.
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The remaining four columns in table one show mean and maximum values across sites
for state dependence variables Nijt (PASTVIS) and Rijt (AREARUN) as defined in equation (3).
The first row of PASTVIS corresponds to variable Hit, the number of days an individual had
chosen not to ski throughout the season prior to time t (labeled PASTHOME in the remainder of
this text). As mentioned above, AREARUN is set to zero for the stay-home option. For actual
sites, PASTVIS is largest for Mt. Rose for an average visitor-day in our sample, with close to
three prior visits. Mt. Rose also shows the highest maximum for previous visits by a given snow
rider (100), followed by Kirkwood (49) and Diamond Peak (44). In different order, these three
resorts also lead in maxima for uninterrupted previous visits (last column of table one). As
indicated by the first row of PASTVIS, the sample mean and maximum for PASTHOME are
70.12 and 150, respectively. The interpretation is as follows: On an average visitor-day based on
our sample, an individual has not visited any of the nine resorts for 70 days since the start of the
research period. The maximum of 150 indicates that there is at least one individual who skied /
boarded only once during the 151-day long season. The remaining variable measuring state
dependence is Dit, the consecutive number of nonparticipation days prior to a given choice
occasion. This variable is labeled DAYSHOME in our output. Since this measure is invariant
over actual resorts, it is not reported in table one. Its sample mean and median are 20.4 and 11
days, respectively. Therefore, the average lag preceding an actual ski trip is roughly three weeks
for our sample of riders, while over 50 % ski / board at least once in a two week period.
The structure of our data results in 197810 observations for the RUM framework
outlined in section two. Our full specification has 13 explanatory variables: A nonparticipation
dummy (D), season pass (SP), holiday (HD), PRICE, skill-adjusted terrain shares in natural log
form (LNGREEN, LNBLUE, LNBLACK), the climate variables TEMP and SNOW7, and the
15
indicators for state dependence PASTVIS, PASTHOME, AREARUN, and DAYSHOME. Skill
adjusted terrain (SAT) is computed as total acres times the percent of terrain assigned to a
specific skill category (green = beginner, blue = intermediate, black = expert) times the percent
of time during a season an individual uses any of the three categories. This concept is similar to
Morey's (1981) ‘effective physical characteristics’.
However, in contrast to Morey who
implicitly assumes that a skier will use terrain appropriate for his skill level 100% of the time we
have the benefit of actually knowing seasonal usage shares through direct elicitation.
Each of these regressors is associated with a specific mean coefficient.
In a fully
unrestricted model, the variance-covariance matrix  would hold 91 additional parameters. The
estimation of such a large number of parameters is infeasible, especially under the burden of high
dimensional integration at each function evaluation (Keane 1997). For the sake of computational
efficiency, and to conserve on parameters we therefore impose fixed coefficients for D, SP, HD,
PRICE, the SAT variables, and PASTHOME. This leaves random coefficients for the main
variables of interest: Time-variant quality attributes and the three remaining indicators for state
dependence, PASTVIS, AREARUN, and DAYSHOME.
Adding the 15 unique variance-
covariance terms in  to the 13 elements of coefficient vector  yields a total number of 28
model parameters.
2. Estimation Results
We estimate three different models: Model one includes all 13 explanatory variables
mentioned above. In Model two the dynamic quality variables TEMP and SNOW7 are dropped.
16
Model three, in turn, includes these quality attributes, but omits DAYSHOME for reasons
discussed below. Estimation results for all three specifications are summarized in Table two.
The first two columns in the table show parameter estimates and standard errors for model one.
Ten out of 13 coefficients are significant at the 1% level. DAYSHOME is the only variable that
is not significant at or below the 10% level. The signs and relative magnitude of coefficients are
as expected: The stay-home dummy D has a strong positive effect on choice probabilities. This
as expected, given that 94% of individual decisions result in this outcome. Season pass (SP) and
holiday (HD) also have a positive effect on trip decisions. To the extent that season pass
purchases are based on experience with a given resort in previous years, SP can be interpreted as
initial condition on state dependence, and may indicate trans-seasonal loyalty.
The estimated effect of skill-adjusted terrain variables conveys a preference for more
difficult slopes for our sample of snow riders: SATs for green and blue slopes have a negative
effect on choice probabilities, while the coefficient for black runs is positive and significantly
larger in magnitude. This indicates that the typical rider in our sample may use green and blue
slopes over the season, but does not necessarily consider them desirable assets of a given resort.
For example, a skier / boarder may use such runs predominately to ‘shuttle’ between different
areas of expert terrain. In essence, these results reflect the relatively large share of advanced
riders in our sample (about 50% of individuals spend more than 40% of their time on black
diamond runs).
Mean coefficients for dynamic quality attributes TEMP and SNOW7 are comparable in
magnitude, but of opposite sign. On average, riders prefer lower temperatures and more snow.
Again, this may reflect the relatively high skill level of participants: An abundance of fresh
snow is especially desirable on more difficult and out-of-bounds runs. As discussed above, the
17
quality of recent additions to the snow pack depends, in turn, on daily temperatures with lower
preferred to higher. The remaining regressors in model one are indicators for state dependence.
As expected, more recent experience with a given destination (AREARUN) has a stronger effect
on repurchases than long-run visitation history (PASTVIS). Both parameter means are positive,
indicating an overall dominance of habit-forming over variety seeking.
Long-run habit
formation is much reduced for the nonparticipation option, as reflected by the negative sign for
PASTHOME: Although significant, the overall long-run effect of nonparticipation on current
destination choices is close to zero. In contrast, short run eagerness or rustiness, as measured by
DAYSHOME, was not found to be a significant driving force for participation in our full model.
Turning to variance-covariance terms for the random segment of parameters in model
one, we note that most elements of Ω are arbitrarily close to zero. There are, however, some
interesting exceptions: The variance terms for TEMP, SNOW7, and AREARUN are highly
significant and of non-trivial magnitude. This becomes apparent when we translate variances
into standard deviations, which are .034, .08, and .115, respectively, for these three variables.
This clearly indicates heterogeneity across individuals with respect to both dynamic quality and
state dependence, and is consistent with recent findings by Allenby and Lenk (1995), Erdem
(1996), Keane (1997), and Erdem and Sun 2001. Based on our assumption of normality for all
random parameters, we can follow Erdem (1996) and approximate the share of habit-followers
and variety seekers using the mean and variance for the AREARUN coefficient. This reveals
that the segment of variety seekers, as measured by AREARUN, is only about 0.1 %. Clearly, an
uninterrupted series of past visits to a given resort has a strong habit-forming effect for the vast
majority of riders. A similar argument holds for long term experience, considering the positive
mean (0.56) and trivially small standard deviation (0.007) for PASTVIS.
By analogous
18
computation, the share of riders that prefer higher temperatures equals 18%. Furthermore, for
about 29% of snow riders recent snowfall reduces the probability of visiting a given resort. To
some extent, this may capture adverse impact on highway conditions associated with abundant
snowfall.
In addition to these variance terms, two statistically significant covariance elements of Ω
emerge in model one: The first one is the covariance between parameters for TEMP and
SNOW7. It is significant at 1% and has a negative sign. Expressed as correlation coefficient,
the magnitude of this interaction is -0.74. This supports again our hypothesis on riders’ quality
preferences: Skiers and boarders to whom fresh snow constitutes a positive (negative) site
attribute prefer lower (higher) temperatures.
The second covariance term of interest and
significance is between SNOW7 and AREARUN. As for the previous case, the covariance is
highly significant and negative.
The corresponding correlation coefficient is –0.54.
The
interpretation is as follows: Riders who have relatively high (low) preferences for fresh snow are
less (more) likely to be affected by visitation history as measured by uninterrupted recent visits
to a specific destination. This key result supports our initial hypothesis that dynamic quality
attributes and state dependence have a joint and potentially offsetting impact on repurchase
decisions.
Parameter estimates for model two are given in the two center columns of table two. The
key finding flowing from this model is that the omission of dynamic quality attributes results in
augmented absolute values for all coefficients corresponding to state dependence. This is a
strong indication that these variables absorb the effect of these omitted characteristics. In
consequence, their impact on repurchase decisions becomes inflated, or ‘spurious’ (Heckman
1981a). Perhaps the most obvious manifestation of this phenomenon is the increased magnitude
19
and significance of DAYSHOME compared to model one. In essence, model two suggests that
prolonged nonparticipation induces rustiness, i.e. a decreased probability of visiting any resort at
a specific choice occasion. However, as indicated by model one, what appears to be increasing
rustiness is in fact a wait for the right quality conditions, i.e. a manifestation of our hypothesized
play-it-by-ear effect. Once TEMP and SNOW7 are introduced to the model, DAYSHOME
becomes insignificant.
The inferiority of model two compared to model one is also highlighted by its much
lower log-likelihood value. A corresponding likelihood-ratio (LR) test at 11 degrees of freedom
(since some elements of Ω are also constraint to zero) clearly rejects the null hypothesis that
dynamic quality attributes have no effect on repeated site choices. It should be noted that some
parameters are restricted under the alternative hypothesis (since variances cannot be negative).
As discussed in Barlow, Bremner, and Brunk (1972) and Chen and Cosslett (1998), the resulting
LR statistic follows a mixed  2 distribution, and standard LR test results may be biased towards
not rejecting the null hypothesis. However, the LR-value in this case (245.8) is well above the
upper bound for the critical  2 0.05 value for such a mixed distribution. Thus, the adjustment
procedure proposed by Chen and Cosslett (1998) would not affect our test result in this case.
Based on these findings we drop DAYSHOME from the original specification, which
leads to model three. As can be seen from the last two columns of table two, the magnitude and
significance of parameter estimates for this model are very similar to the ones in the unrestricted
specification. A comparison of log-likelihood values yields a LR statistic of 9.4. The lower
bound for the critical  2 0.05 value for the corresponding mixed test distribution is 11.07. We
therefore conclude that the constraint of setting the effect of DAYSHOME to zero holds for our
application.
20
4. SHARE PREDICTIONS AND MARKETING SIMULATIONS
Due to the predominance of stay-home choices in our data, the comparison of model
predictions based on the ‘highest probability principle’ (Greene 1997, p. 917) and hit rates is
problematic (Ben-Akiva and Lerman 1985, p. 92). Instead, we follow Allenby and Lenk's
(1995) approach of estimating average choice probabilities over all individuals and choice
occasions for all destinations. These aggregate choice shares for models two and three are
presented in table three together with actual sample shares. Since our models allow for random
parameters, estimates are based on simulations using 1000 different vectors of ̂ , drawn from
the multivariate normal with mean ̂ and variance-covariance matrix ̂ . Approximations of
trip figures corresponding to estimated shares are derived by multiplying shares by the total
number of observed trips (19781). Both models predict shares for Mt. Rose and Squaw Valley
fairly well, while generally over-predicting trips to other resorts, and under-predicting
nonparticipation choices. In the spirit of Keane (1997) the table also includes  2 -statistics based
on squared deviations between actual and predicted shares. As indicated by these values, the
predictive power of model three is superior to that of model two.
A resort manager may be interested in estimating the impact of promotional efforts and
associated state dependence effects on daily and seasonal market shares. In a model with wellbalanced choice distribution this could be accomplished by simulating the promotion, reestimating the model using ̂ and the altered data, and empirically predicting new total visits
based on the ‘highest probability’ principle. In our context, the preponderance of the stay-home
option makes it very unlikely that the predicted probability for an actual resort will surpass the
21
probability of nonparticipation, even under substantial marketing effort. This leads to an underprediction of seasonal visits for a given resort, as well as the marketing impact.
This is
especially problematic in a model with state dependence, since any choices made in the past
affect the values of state dependence indicators for all subsequent periods and destinations. For
example, failure to correctly predict a trip at time t to resort j will bias all subsequent trip
probabilities for j downward if habit formation dominates. This, in turn, compounds the problem
of underestimating seasonal visits based on ‘highest probabilities’.
To avoid this pitfall we propose a combined empirical - analytical approach to elicit the
impacts of marketing and state dependence on daily and seasonal shares for a given destination.
We demonstrate this process using as an example a one-time price reduction at destination j at
time t and focusing on state dependence indicator Nijt (PASTVIS): First, choice probabilities are
re-computed in period t using estimated coefficients and the new price set, while indicators for
state dependence remain unchanged. In the following period, the value for PASTVIS needs to
be updated based on the outcome of period t. Instead of adding ‘1’ to Nj,t+1 for the destination
with the highest choice probability at time t (and ‘0’ to the value of PASTVIS for all other sites)
we compute expected values for Nj,t+1 for all sites as follows (dropping rider-specific subscripts
for convenience):


N *j ,t 1  p*j ,t  N j ,t  1  1  p*j ,t  N j ,t  N j ,t  p*j ,t
j  0J ,
(10)
where N*j,t+1 is the updated value for PASTVIS, and p*j,t is the predicted probability of visiting j
at time t after the price change. This adjusted value of PASTVIS, in turn, changes the original
22
choice shares at time t+1 for all resorts, yielding adjusted shares p*j,t+1 for all j. Extending this
recursive process to some time period t+s leads to the following simple updating rule for Nj:
N *j ,t s  N *j ,t s1  p*j ,t s1
j  0 J
.
(11)
Analogous updating rules for the remaining state dependence indicators Hit (PASTHOME), Dit
(DAYSHOME) and Rijt (AREARUN) are given in the Appendix. To allow for a comparison of
choice shares between scenarios with and without marketing efforts, and between marketing
efforts implemented over different time periods during the season, this updating process is
implemented starting at t=2 for all scenarios and state dependence indicators .
Given that most actual trips were made to Mt. Rose ski area, and considering the good fit
of our estimated models for this destination (see table three), we will focus on this resort for our
marketing scenarios. Specifically, we simulate six pricing scenarios: Scenarios one and two
correspond to a price reduction of $5 and $10, respectively, to all visitors for days 12 through 18
(Monday, December 8 through Sunday, December 14). This time slot was chosen to leave ample
time for state dependence to take effect throughout the remainder of the season. Also, actual
visitation shares for Mt. Rose had reached standard levels by that week. Scenarios three and four
simulate the same price reductions with extension through day 25. During the 1997/98 ski
season, Mt. Rose offered two weekday specials that continued through the entire ski season:
‘Ladies Thursdays’ ($15 day pass for female visitors), and ‘Student Wednesdays’ ($10 day pass
for students). In scenarios five and six we investigate the impacts of ‘undoing’ these promotions.
Specifically, scenario five imposes a price increase of $23 to reach the standard day pass price of
$38 for all skiers and boarders that had received a ladies’ discount. Similarly, scenario six
23
increases day pass prices to student day beneficiaries by $28. The six scenarios are summarized
in table four. We implement these scenarios using both models two and three. For each model,
we also estimate a baseline scenario without marketing changes, but with analytically derived
state dependence indicators as described above. For each scenario, in turn, we predict average
daily shares as
 N *
pijt,r

1 R 
*
i 1
p jt   
R r 1  N








j  0 J ,
t  1T ,
(12)
where p*ijt are predicted choice shares, and r is an index for a specific vector of coefficients
drawn from ( ̂ , ̂ ). As before, the total number of repetitions over coefficient vectors (R)
equals 1000. We further define the relative percentage change in daily shares compared to the
baseline scenario as
 p *jt ,s  p *jt ,b 
~
  100
p jt ,s  
*


p
jt ,b


j  0 J ,
t  1T ,
(13)
where s and b indicate a specific marketing scenario and the baseline scenario, respectively. To
estimate the average relative change in daily shares over the entire season, we further aggregate
(13) over time periods, i.e.
24
T
~
Pj ,s 
 ~p
t 1
jt , s
j  0 J
T
s  1 6
(14)
~
In essence, Pj ,s can be interpreted as the percentage change in market shares relative to the
baseline scenario on a typical day of the season. For simplicity, we assume that any change in
daily shares observed on days of actual price changes are pure price effects, and changes in
shares on other days are pure state dependence effects. This allows for decomposition of (14)
into an average price effect and an average state dependence effect for a typical season day, i.e.
~
~
~
Pj ,s  Pjm,s  Pjsd,s 
 ~p
t m
T
jt , s

 ~p
t m
T
jt , s
j  0 J
s  1 6 .
(15)
Superscript m denotes marketing or price effects, and superscript sd indicates effects related to
state dependence. The indices under the first and second summation sign refer to all days with
actual price changes, and all other days, respectively.
These aggregate effects for Mt. Rose and scenarios one through six are summarized in
table five. The upper half of the table is based on parameter estimates generated by model three,
our preferred specification. The lower half presents results associated with model two. The first
column in the table following the scenario labels shows the season-averaged daily share (i.e. the
mean of (12) over all season days) for the baseline scenario. For model three, the value of this
statistic (3.07%) is close to the estimate for the average daily share based on original data (3.3%
- see table three). This indicates that the replacement of actual state dependence measures with
our analytical updates essentially preserved the predictive power of the model. Thus, on an
25
average day in the season, Mt. Rose captures approximately 3% of the market, including the
stay-home option. If only shares for actual resorts are considered, Mt. Rose attracts close to
30% of riders on a typical day. This ‘real market’ base share is shown in the third column of
table five. The next two columns list price and state dependence effects as defined in (15). The
interpretation is as follows: The pure price effect associated with a $5 discount from day 12
through 18 (scenario one) increases the daily choice share for Mt. Rose by 0.59% on an average
day, relative to the baseline scenario. Pure state dependence effects add another 0.06% to this
increase. This yields a total increase in shares relative to baseline of 0.65% on a typical day
(column seven). The ratio of state dependence over total effect is shown in column six. Finally,
total changes in average daily market shares relative to baseline and actual ski resorts only (real
market) are depicted in the last column of the table. Standard errors stemming from R repetitions
of these simulations are given in parentheses for each effect.
As expected, gains in daily shares increase with magnitude and duration of the discount.
Generally, price effects far outweigh state dependence effects, which constitute between seven
and 9% of total impact for the first four scenarios under model three. Also, raising the discount
from $5 to $10 over one week has a slightly larger effect than extending the $5 promotion over a
second week, as can be seen by comparing results for scenarios two and three. While the
average gain in daily shares based on scenarios one through four are relatively modest when all
choices are considered (between 0.7% and 2.8% relative to baseline), they reach substantial
magnitude when measured relative to actual destinations only. For example, reducing day pass
prices by $10 for a week (scenario two) increases daily choice shares for Mt. Rose by 23%
relative to its original ‘real market’ share of close to 30 percent. Eliminating “Ladies’ Day”
(scenario five) leads to a total reduction in shares of 2.3% on an average day, of which about
26
0.1% are attributable to state dependence. The reduction in shares is more pronounced if
‘Student Day’ discounts are dropped (scenario six): on an average day in the season, Mt. Rose
would lose close to 8% of its original share. State dependence effects account for about 0.3% of
this loss. Interestingly, ‘real market’ effects relative to actual resorts are slightly smaller than
total effects for the last two scenarios. This suggests that most of the loss is absorbed by the
nonparticipation option. Results for model two generally mirror the findings for model three.
As expected, the relative impact of state dependence (column six) is slightly larger, given the
inflated coefficients for AREARUN and PASTVIS (see table two). The predicted daily base
share for model two (2.55%) is substantially lower than the one given in table three, suggesting
that our analytical updating of state dependence indicators diminished the predictive ability of
the model to some extent.
It should be noted that the share gains presented in table five are ‘smeared’ over the entire
season as indicated in (15) regardless of the actual days during which they take effect. Share
gains for a specific day can be much more pronounced than the seasonal average. This can be
seen in figures one and two, which show the magnitude of ~p jt , s for each day of the season for
scenarios one through four based on model three. The peaks of the time series indicate share
gains through price effects, which amount to over 35% relative to the same time segment in the
baseline scenario. The tails in the figures correspond to pure state dependence effects, since
these gains occur after the end of the simulated promotion. As can be seen in figure one, the
effect of state dependence is relatively short-lived for scenarios one and two. It starts at about
0.5 to 1% share gain immediately following the discount period, and diminishes essentially to
zero within the following two to three weeks. Figure two, which displays daily percentage gains
for the two-week promotions simulated in scenarios three and four, shows a slightly higher initial
27
magnitude and longer impact of state dependence compared to shorter promotions. As indicated
by the double-peaked price effect in figure two relative share gains for Mt. Rose are most
pronounced on student Wednesdays (days 14 and 21).
Figures three and four show the impact of ladies’ day and students’ day discounts on
daily share gains relative to scenarios five and six, which eliminate these repeated weekly
promotions. The solid lines are based on model three, while the dotted lines show share gains
stemming from model two. These graphs are illustrative in three ways: First, they show that oneday promotions appear to be too short to induce any significant gains attributable to state
dependence effects, as indicated by the flat segments of the graphs between peaks generated by
price effects. Second, as state dependence builds up throughout the season for both the baseline
models and the elimination scenarios, the relative impact of these discounts diminishes almost
monotonically over the season. Third, this reduction in promotional impact is more pronounced
for model two, which assigns more weight to state dependence effects as discussed above. For
ladies Thursdays (figure three), relative share gains start at about 30% for both models, and end
at 10% and 5% for models three and two, respectively. Daily share gains attributable to student
Wednesdays (figure four) amount to as much as 95% for week one and both models, and
decrease to 30 % and 20% for models three and two, respectively, by the end of the season.
The question arises as to what extent quality variables may affect the short and long run
impact of these promotions. Specifically, one may hypothesize that during a promotion week
with good snow conditions a relatively larger proportion of play-it-by-ear types are attracted to
the resort. This should lead to smaller state dependence effects relative to price effects compared
to a promotion week with average or poor snow conditions. We found no empirical evidence
that moving the discount periods for scenarios one through four to different weeks of the ski
28
season with different snow and temperature levels had any significant effects on the results
presented above. In essence, this reflects the inability of our model to actually distinguish
between different types of riders. To differentiate between promotional effects on play-it-by-ear
types versus, say, habit formers, one would need a data set and model that clearly identifies the
two groups. Such a model could then be used to generate separate seasonal time series of share
gains for each visitor type, and to investigate how the two groups differ in their reaction to price
discounts under varying quality scenarios. This might constitute a promising avenue for future
research.
5. CONCLUSION
Repeated consumer choices over substitute products can induce habit formation or
variety seeking. These tendencies, when recognized, aid managers in marketing and pricing
decisions. To date, the literature on state dependence has focused on goods that exhibit constant
quality over a given research period. For such products, the main analytical and econometric
challenge when investigating the forces of state dependence is to distinguish between true effects
of past purchases and individual heterogeneity. In this paper, we consider consumer goods for
which quality attributes change frequently over time. Ski areas are a good example for such
commodities, as daily snow and weather conditions strongly affect the level of resort benefits
flowing to visitors.
To disentangle the impact of quality changes, state dependence, and
consumer preferences on destination choice, we propose a model that explicitly includes
dynamic quality features as well as indicators for state dependence, while allowing for individual
heterogeneity.
29
Several key results flow from our analysis: We show that dynamic quality attributes have
a strong and significant impact on consumer choice. Omitting these characteristics leads to an
overestimation of state dependence. Such a misspecified model also inflates the importance of
prolonged non-participation on future purchase decisions. A cross-model comparison reveals that
what may falsely be interpreted as consumption lethargy or ‘rustiness’ may well be a wait for
appropriate quality conditions. Generally, state dependence remains a strong force in a correctly
specified model. In our application, habit formation clearly dominates variety seeking. The
lion’s share of this effect is linked to recent purchase history. Most importantly, our results
indicate that dynamic quality and habit formation have an offsetting effect on consumer choice.
Specifically, visitors to whom quality matters more are less likely to develop habits. This
suggests that aside from habit formers and variety seekers there exists a third category of
consumer in repeated choice settings: the play-it-by-ear type who, relatively unaffected by
purchase history, will move across brands in pursuit of consistent quality. Identifying these
quality seekers and analyzing how their reactions to pricing and marketing strategies differ from
those exhibited by the traditional state dependence types will be subject to future research.
ACKNOWLEDGEMENTS
We thank J. Scott Shonkwiler for helpful suggestions and comments. Support from the Nevada
Agricultural Experiment Station is acknowledged.
30
APPENDIX: ANALYTICAL UPDATING OF STATE DEPENDENCE INDICATORS
FOLLOWING PRICE SHOCKS
In contrast to PASTVIS, changes in PASTHOME (Hit) only apply to the stay-home
option. Its updated value at time t+s is therefore
N0*,t s  h jt  N *j ,t s ,
(A1)
where hjt is the dummy for nonparticipation as in (3), and N*j,t+s is defined in (11).
For
DAYSHOME (Dit), which is constrained to zero for nonparticipation, equal in value across all
remaining choices, and set back to one the day after an actual trip was made, the updated values
in periods t+1 and t+2 following a price reduction at site j are


D *j ,t 1  p0*,t  D j ,t  1  1  p0*,t  p0*,t  D j ,t  1 and

 

 

D *j ,t  2  p0*,t  p0*,t 1  D j ,t  2  p0*,t  1  p0*,t 1  1  p0*,t  1  p0*,t 1  1  p0*,t  p0*,t 1  2 


 p0*,t 1  p0*,t  D *j ,t  1  1
(A2)
j  0
where p* indicates adjusted probabilities based on preceding updates and price changes. This
leads to the general updating rule for Dit :


D*j ,t s  p0*,t s1  D*j ,t s1  1
j  0 .
(A3)
31
The remaining state dependence indicator is AREARUN (Rijt). This variable is always zero for
j=0. For actual destinations, it increases by one following a trip to a given site, remains constant
throughout nonparticipation, and is set back to zero if a rival destination was visited the previous
day. Accordingly, the updated values following a price reduction at some site j for the first two
periods are


R *j ,t 1  p *j ,t  R j ,t  1  p0*,t  R j ,t  p *j ,t  p0*,t  R j ,t  p *j ,t
R *j ,t  2  p *j ,t  p *j ,t 1 R j ,t  2  p *j ,t  p0*,t 1  R j ,t  1  p0*,t  p *j ,t 1  R j ,t  1 




 1  p *j ,t  p0*,t  p *j ,t 1  p *j ,t 1  p0*,t 1  R *j ,t 1  p *j ,t 1
and
(A4)
j  0 ,
leading to the general updating rule for AREARUN of


R*j ,t s  p*j ,t s1  p0*,t s1  R*j ,t s1  p*j ,t s1
j  0 .
(A5)
32
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35
Table 1. Description of Data
Trips
SP
HD
Resort
Green
None
Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
18586
673
27
96
34
69
67
43
134
52
0
117
0
14
0
49
0
7
23
0
0
342
22
74
27
59
49
36
65
44
Total
19781
210
718
0
220
255
1000
860
345
136
618
340
90
Terrain (acres)
Price
Temp
Snow7
Blue Black Total mean ($) mean (F) mean (in.)
0
480
645
1800
1935
1150
347
1235
800
165
0
300
600
1200
1505
805
272
618
860
45
0
1000
1500
4000
4300
2300
755
2470
2000
300
0.0
60.30
59.60
78.00
80.54
73.15
87.62
58.00
66.91
50.27
32.0
24.2
28.6
27.2
25.0
29.9
27.7
31.2
27.2
28.6
0.0
2.1
2.2
3.8
1.7
2.3
0.2
1.0
3.8
2.2
Pastvis
Arearun
mean
max mean max
70.12
2.75
0.11
0.42
0.15
0.30
0.28
0.17
0.51
0.20
150
100
6
14
5
49
44
7
9
5
0.00
1.25
0.03
0.12
0.06
0.04
0.19
0.09
0.18
0.04
0
25
2
6
5
24
44
7
9
3
36
Table 2. Estimation Results
Parameters
Model 1
Coeff.
s.e.
D
SP
HD
PRICE
LN_GREEN
LN_BLUE
LN_BLACK
PASTHOME
TEMP
SNOW7
PASTVIS
AREARUN
DAYSHOME
3.165
0.582
0.720
-0.041
-0.058
-0.042
0.133
-0.047
-0.037
0.043
0.056
0.349
-0.003
-(.200)
(.185)
(.070)
(.003)
(.029)
(.024)
(.025)
(.005)
(.006)
(.016)
(.005)
(.031)
(.004)
***
Variance / Covariance Terms
TEMP
TEMP / SNOW7
SNOW7
TEMP / PASTVIS
SNOW7 / PASTVIS
PASTVIS
TEMP / AREARUN
SNOW7 / AREARUN
PASTVIS / AREARUN
AREARUN
TEMP / DAYSHOME
SNOW7 / DAYSHOME
PASTVIS / DAYSHOME
AREARUN / DAYSHOME
DAYSHOME
0.001
-0.002
0.007
0.000
0.000
0.000
0.000
-0.005
0.000
0.040
0.000
0.000
0.000
0.001
0.000
(.000)
(.001)
(.002)
(.000)
(.000)
(.000)
(.001)
(.002)
(.000)
(.012)
(.000)
(.001)
(.000)
(.001)
(.000)
***
Log-likelihood
5759.570
Model 2
Coeff.
s.e.
***
***
***
**
*
***
***
***
***
***
***
***
***
**
***
***
Model 3
Coeff.
s.e.
2.858
0.569
0.573
-0.037
-0.045
-0.049
0.148
-0.057
0.067
0.386
-0.007
(.197)
(.169)
(.067)
(.003)
(.026)
(.020)
(.023)
(.005)
(.005)
(.034)
(.004)
***
0.000
0.000
0.038
0.000
0.002
0.000
(.000)
(.000)
(.010)
(.000)
(.001)
(.000)
-
5882.490
***
***
***
*
**
***
***
***
***
*
**
*
***
*
*
3.211
0.614
0.712
-0.041
-0.048
-0.042
0.127
-0.048
-0.037
0.042
0.057
0.351
-
(.206)
(.178)
(.069)
(.003)
(.029)
(.024)
(.025)
(.005)
(.006)
(.016)
(.005)
(.033)
-
***
0.001
-0.002
0.007
0.000
0.000
0.000
0.000
-0.005
0.000
0.041
-
(.000)
(.001)
(.002)
(.000)
(.000)
(.000)
(.001)
(.002)
(.000)
(.013)
-
***
5764.300
NOTE: *significant at 10% level; ** significant at 5% level; ***significant at 1% level
***
***
***
*
*
***
***
***
***
***
***
-
***
***
***
***
***
-
37
Table 3. Seasonal Shares
Resort
None
Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
Total
chi-square:
Sample
Trips
Shares
Model 3
Trips
Shares
Model 2
Trips
Shares
18586
673
27
96
34
69
67
43
134
52
19781
18088
643
117
95
63
119
127
129
259
140
17991
677
125
91
65
142
144
160
245
138
0.940
0.034
0.001
0.005
0.002
0.003
0.003
0.002
0.007
0.003
0.914
0.033
0.006
0.005
0.003
0.006
0.006
0.007
0.013
0.007
319.60
0.910
0.034
0.006
0.005
0.003
0.007
0.007
0.008
0.012
0.007
381.03
38
Table 4. Discount Scenarios
Scenario
Time period
Price change
1
2
3
4
5
6
Days 12-18
Days 12-18
Days 12-25
Days 12-25
Every Thursday
Every Wednesday
-$5
-$10
-$5
-$10
+$23
+$28
39
Table 5. Average daily changes in choice shares
Model 3
Scenario
Base share
1
3.07%
29.50%
2
3.07%
29.50%
3
3.07%
29.50%
4
3.07%
29.50%
5
3.07%
29.50%
6
3.07%
29.50%
Model 2
Scenario
Base share
1
2.55%
26.56%
2
2.55%
26.56%
3
2.55%
26.56%
4
2.55%
26.56%
5
2.55%
26.56%
6
2.55%
26.56%
Real market share Price effect SD effect SD/Total Total effect
0.59%
(.04%)
1.31%
(.09%)
1.16%
(.10%)
2.60%
(.21%)
-2.23%
(.97%)
-7.35%
(3.49%)
0.06%
(.03%)
0.13%
(.07%)
0.09%
(.04%)
0.21%
(.09%)
-0.10%
(.05%)
-0.33%
(.14%)
9.23%
9.03%
7.20%
7.47%
4.29%
4.30%
0.65%
(.04%)
1.44%
(.09%)
1.25%
(.11%)
2.81%
(.23%)
-2.33%
(.97%)
-7.68%
(3.52%)
Real market share Price effect SD effect SD/Total Total effect
0.56%
(.03%)
1.23%
(.05%)
1.11%
(.06%)
2.45%
(.14%)
-1.83%
(.65%)
-6.21%
(2.46%)
0.06%
(.02%)
0.13%
(.04%)
0.11%
(.04%)
0.24%
(.09%)
-0.12%
(.04%)
-0.40%
(.18%)
9.68%
9.56%
9.02%
8.92%
6.15%
6.05%
0.62%
(.03%)
1.36%
(.08%)
1.22%
(.09%)
2.69%
(.20%)
-1.95%
(.67%)
-6.61%
(2.59%)
Real market effect
10.58%
(7.43%)
23.00%
(16.16%)
19.87%
(14.10%)
43.29%
(30.85%)
-1.94%
(.93%)
-6.30%
(3.20%)
Real market effect
8.16%
(2.34%)
17.64%
(5.06%)
15.19%
(5.39%)
32.93%
(11.67%)
-1.63%
(.61%)
-5.44%
(2.18%)
40
Figures 1 and 2: Daily market share gains due to price discounts. Price effects far outweigh state
dependence effects. State dependence for longer periods of discount have a slightly higher initial
magnitude and duration than such effects for shorter promotion periods.
41
Figures 3 and 4: The build-up of state dependence diminishes the impact of repeated weekly
discounts over the season. This effect is stronger for model two due to its inflated coefficients for
state dependence.