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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Choice Behavior Under Time-Variant Quality
Klaus Moeltner & Jeffrey Englin
To cite this article: Klaus Moeltner & Jeffrey Englin (2004) Choice Behavior Under
Time-Variant Quality, Journal of Business & Economic Statistics, 22:2, 214-224, DOI:
10.1198/073500104000000109
To link to this article: http://dx.doi.org/10.1198/073500104000000109

Published online: 01 Jan 2012.

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Choice Behavior Under Time-Variant Quality:
State Dependence Versus “Play-It-by-Ear” in
Selecting Ski Resorts
Klaus M OELTNER and Jeffrey E NGLIN

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Department of Resource Economics, University of Nevada, Reno, NV 89557 (moeltner@cabnr.unr.edu)
In past studies of consumer loyalty changes in brand attributes over time were generally unobservable
and treated as additional model parameters. In this study we consider ski resorts, for which observable
quality attributes change frequently. Using a repeated-purchase model with observed time-variant brand
attributes, indicators for state dependence, and individual heterogeneity, we show that purchase history
and time-variant site characteristics have a signiŽcant and offsetting effect on repurchase decisions. This
suggests a third category of consumer along with habit formers and variety seekers, the “play-it-by-ear”

type, who, unaffected by purchase history, moves across brands in pursuit of high quality.
KEY WORDS: Random parameters; Repeated brand choice; Simulated choice probabilities; Timevariant quality features.

1.

INTRODUCTION

When consumers repeatedly choose among several products,
past choices can affect the probability of selecting a given product again at a later occasion. This phenomenon is commonly
called “state dependence” (Heckman 1981a). Generally, state
dependence can increase a consumer’s propensity to repurchase a speciŽc 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 observed characteristics for goods under consideration. The present study examines the role of purchase
history for products with both Žxed and time-variant observed
attributes. This allows us to disentangle effects attributable to
state dependence from those associated with quality variation.
Our empirical application is based on a set of ski areas in the
Sierra Nevada; the time-variant quality attributes are snow and
temperature.
In the marketing literature, state dependence is often called

“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. Keane (1997) pointed out that
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, then the promotional impact on sales may be
short-lived.
The effect of state dependence on choice behavior has also
become a topic of interest in the recreation literature (see, e.g.,
McConnell, Strand, and Bockstael 1990; Adamowicz 1994;
Smith 1997). In those studies, the “products” from which people are choosing are not food or household items, as commonly
investigated in marketing research, but rather recreation sites,
such as Žshing spots (Adamowicz 1994; Swait, Adamovicz,
and van Bueren 2000) or beach destinations (McConnell et al.
1990). In this context, 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. The estimated effect of true state dependence
may be inated if consumer preferences are erroneously assumed to be homogeneous, or if time-variant exogenous variables are omitted (Heckman 1981a; Keane 1997; Erdem and
Sun 2001). In recent marketing contributions,heterogeneityhas
been explicitly captured in repeated-choice models by introducing random coefŽcients into random utility models (RUMs)
(Allenby and Lenk 1995; Erdem 1996; Keane 1997). Allenby
and Lenk (1995), Erdem (1996), Keane (1997), and Erdem
and Sun (2001) also included time-varying exogenousvariables
(price and marketing effort) to disentangle their effect on purchase decisions from true state dependence.
Some marketing studies have also given consideration to
the possibility that quality attributes of a given brand may
change over time and/or may not be fully revealed to the consumer. Meyer (1982), Roberts and Urban (1988), and Erdem
and Keane (1996) proposed brand choice models that capture
perceived variability in product attributes over time. In the approaches of Roberts and Urban (1988) and Erdem and Keane
(1996), this uncertainty about quality from the consumer’s
perspective is rooted in both incomplete information about a
given product and in inherent product variability (i.e., unintentional changes in attributes due to aberrations in the production
process). As agents gain more information through advertising,

word of mouth, or direct consumption,they update their beliefs
about the quality levels of each brand. This updating process,
in turn, affects expected utility levels and choice probabilities.
Meyer (1982) also allowed for incomplete information on product features, but added the possibility of weight shifts across
attributes as consumers’ choice sets change during a sequential
elimination process. In each of these studies, attribute variability is captured in form of stochastic model components, which

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© 2004 American Statistical Association
Journal of Business & Economic Statistics
April 2004, Vol. 22, No. 2
DOI 10.1198/073500104000000109

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Moeltner and Englin: Choice Behavior Under Time-Variant Quality

215

are estimated together with the effects of exogenous choice
factors.
The common element in these marketing contributionsis that
from the researcher’s perspective, actual changes in attributes
for a given brand over time are unobserved. For the household
products generally considered in marketing applications, such
as ketchup, peanut butter, and detergents, quality uctuations
over time for a given brand are likely to be subtle and costly
to measure. Recreation sites, on the other hand, can have pronounced changes in quality over time. Because these “brands”
are not the end product of a highly controlled manufacturing
process, they are by nature susceptible to a multitude of attribute changes over time. The importance of these changes for
repeated site choices will depend on the type of recreational
activity and on individual preferences. Ignoring the dynamics in site characteristics when examining consumer loyalty
or variety-seeking associated with a given recreational activity
bears the risk of erroneous inferences about state dependence
even when controlling for heterogeneity and marketing effects.
This research extends existing marketing and recreation
studies by analyzing the separate effects of observed quality
changes and state dependence on consumer choice, while allowing for individual heterogeneity. In addition, we examine
whether consumers who place a relatively large weight on a

speciŽc 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 that exhibits both timevariant and time-invariant features. Ski resorts are well suited
for this purpose. Their terrain and level of difŽculty remain unchanged over time, and on a given day the quality of a visit
to the resort will be heavily affected by day-speciŽc attributes,
such as snow conditions and weather. Exploiting this natural
experiment allows us to estimate a novel multichoice model
with state dependence.
The remainder of this manuscript is structured as follows.
In Section 2 we develop an econometric model of state dependence, observed time-variant quality effects, and heterogeneity.
We then discuss the data and estimation results in Section 3, and
operational implications for ski resort managers in Section 4.
We provide concluding remarks and a summary of key Žndings
in Section 5.
2.

MODEL FORMULATION

Following most brand choice studies, we embed our model in
a RUM framework. SpeciŽcally, we assume that the utility that
individual i derives from a visit to resort j at time t is given by
Uijt D D0j ¢ ´i C A0j ¢ ® i C Pijt ¢ ¯ i C Q0jt ¢ ° i C S0ijt ¢ ± i C "ijt
D Vijt C "ijt ;

i D 1; : : : ; N; j D 0; : : : ; J; t D 1; : : : ; T;
(1)

where Dj is a vector of site-speciŽc intercepts, 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 , ° i ,
and ± i denote individual-speciŽc coefŽcient vectors, and "ijt is
an iid random-error term.
Time periods are generally deŽned as purchase occasions
in brand-choice studies (see, e.g., Allenby and Lenk 1995;
Erdem 1996; Keane 1997). This preempts an investigation of

inter-purchase time effects on choice decisions. As shown by
Papatla and Krishnamurthi (1992), Chintagunta (1998), and
Chintagunta and Prasad (1998), the 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 interpurchase period (see, e.g., Papatla
and Krishnamurthi 1992). This is different for recreation sites,
where actual consumption ends with the visit. In fact, nonconsumption at time t becomes a separate choice. This is
usually modeled as the “staying-home” option or “nonparticipation” in a RUM speciŽcation (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
Ui0t D ´i0 C S0i0t ¢ ± i C "i0t ;

(2)

where the “0” subscript indicates the stay-home option. We reduce all quality indicators to a constant, and set the price to 0
for this choice. We do, however, retain variables measuring state
dependence, as described later.
Although 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 et al. 2000) by deŽning 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 [i.e., the “Žrst-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 (i.e., “run”) a given brand
was chosen before 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 large
to allow the separate inclusion of all previous choices.
However, we a priori concur with McConnell et al. (1990)
that recent visits should 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 that a given resort was chosen before t.Nijt /, as in
Adamowicz (1994), and the consecutive number of times that
a given resort was chosen up to t uninterrupted by any visits
to other destinations (Rijt /. This variable conveys the notion of

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216

Journal of Business & Economic Statistics, April 2004

“run” mentioned earlier. Because 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 form of state dependence,
if any, drives individual i should manifest itself more strongly
through uninterruptedtrips Rijt than total trip count Nijt . We also
specify two analogous indicators for the stay-home option: the
number of times during the season before t that an individual chose not to participate (Hit / (Adamowicz 1994), and the
number of consecutive days of nonskiing immediately preceding t.Dit / (Provencher and Bishop 1997; Swait et al. 2000).
Thus the elements of Sijt , j D 0; : : : ; J, materialize as
2
3
Nijt
6R 7
Sijt D 4 ijt 5
Hit
Dit
2
3
t
X
dijt
6
7
6
7
tD1
6
ÁÁ s
!7
6X
t¡1
Y¡
¢ 7
6
7
6
d ij;t¡l ¢ .1 ¡ hij;t¡l / C hij;t¡l 7
6
7
6 sD1
7
lD1
!
6
7
6
7
6
7 ; (3)
D6
¢.1 ¡ hij;t¡l /
7
6
7
Á t
!
6
7
6
7
X
6
7
dijt ¢ hijt
6
7
6
7
tD1
6
7
6
7
s
t¡1
XY
4
5
hij;t¡l
sD1 lD1

where dijt , j D 0; : : : ; J, is a 0/1 dummy taking the value of 1
if resort j was chosen by i at time t and hij;t is a dummy equal
to 1 if j is the stay-home option and equal to 0 if j is an actual
resort. This implies that Rijt and Dit equal 0 for j D 0, whereas
Hit applies only to nonparticipation. Setting Dit , the number of
days since the last ski trip, to 0 for the stay-home option preserves this indicator in a RUM speciŽcation. 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,” an increased probability of
skiing as they grow large, or “rustiness,” the opposite effect.
In essence, “eagerness” is the equivalent to “variety seeking”
for actual sites, whereas “rustiness” corresponds to “habit formation” or “inertia.” Adamowicz (1994), for example, found
that eagerness dominated behavior in his “rational model”
(increased probability of visits as Hit increases), whereas
Provencher and Bishop (1997) found that as more time elapses
since the last visit, the probability of participation decreases.
Swait et al. (2000) reported initial rustiness following a preceding visit that turns into eagerness after about 10 weeks of nonparticipation. As we show, both eagerness and rustiness play a
role in our model, with about 50% of riders adhering to each
tendency.
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 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 signiŽcant 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 allowed for and found signiŽcant heterogeneity in how past brand choices and attributes
affect individuals’ repurchase decisions.
In a RUM framework, it is convenient to introduce heterogeneity through random coefŽcients (Revelt and Train 1998;
McFadden and Train 2000). In contrast to Allenby and Lenk
(1995) and Keane (1997), who allowed for time-varying taste
parameters, and following Erdem (1996) and Erdem and Sun
(2001), we assume that individual preferences remain constant
throughout our research period. Collecting ´i , ®i , ¯ i , ° i , and ±i
into a single coefŽcient vector ’ i , we stipulate that parameters
are distributed multivariate normal with
E.’ i / D ’;
N
»
Ä;
E.’ i ¢ ’ 0i / D
0;

iDj

(4)

i 6D j.

N and the
Thus we estimate a vector of parameter means, ’,
elements of the variance-covariance matrix Ä. In contrast to
previous studies, the covariance terms in Ä are of major interest and importance in this article. SpeciŽcally, we a priori
expect a negative sign for covariances between (presumed positive) coefŽcients associated with time-variant quality attributes
and (positive) coefŽcients for state dependence, if habit formation dominates. Conversely, if a coefŽcient for state dependence is negative (indicating variety-seeking tendencies), its
covariance with coefŽcients for time-variant quality should be
positive. In other words, we hypothesize that the stronger the
effect of quality seeking for a given individual, the smaller the
absolute value of the coefŽcient for state dependence drawn for
this individual.The intuition behind this premise is that qualitysensitive skiers (i.e., the play-it-by-ear types) should be less
likely to be inuenced by past resort choices. As we show later,
our results generally conŽrm this stipulation. This constitutes
the key Žnding owing 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
speciŽcation of choice probabilities. Two frequently used distributions in the choice literature are normal, resulting in a multivariate probit speciŽcation (Hausman and Wise 1978; Keane
1997), and type I extreme value. The latter distribution, in
combination with the random coefŽcients in ’ i , yields a random parameter logit, or “mixed logit” model (Revelt and Train
1998; Brownstone and Train 1999). In either case, estimation
of choice probabilities requires solving a high-dimensional integral. As discussed by Layton (2000), the dimension of integration proliferates with choice occasions in the multivariate

Moeltner and Englin: Choice Behavior Under Time-Variant Quality

217

probit model, and with the number of random parameters in
a mixed logit speciŽcation. In our application, each individual
faces 1,359 choice occasions in a model with a limited number
of random coefŽcients. 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 ’ i , is given by McFadden (1974)
N Ä//
exp.Vijt .’i ; ’;
;
Pijt .’ i ; ’;
N Ä/ D PJ
N Ä//
kD0 exp.Vikt .’ i ; ’;

(5)

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where Vijt is deŽned as in (1). The conditional probability of
observing an individual’s entire sequence of trip decisions is
therefore (Erdem 1996)
Pi .’ i ; ’;
N Ä/ D

J
T Y
Y

Pijt .’ i ; ’;
N Ä/dijt ;

(6)

tD1 jD0

where dijt is deŽned as in (3). Relaxing conditionality on coefŽcients yields
Z
Pi D
Pi .’ i ; ’;
N Ä/ ¢ f .’ i / d’ i ;
(7)
Ái

where f .’ i / is the multivariate normal distribution. The dimension of the integral in (7) is commensurate with the number
of elements in ’ i . Because the evaluation of high-dimensional
integrals 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 functions (Börsch-Supan and
Hajivassiliou 1993; Keane 1994; McFadden and Train 2000).
We follow the procedure outlined by Brownstone and Train
(1999) by drawing a set of ’ i from f .’ i /, with some arbitrary
starting values for ’N and Ä. This allows us to compute (6) for
all individuals. The process is repeated R times, yielding the
simulated choice probability (Erdem 1996; Layton 2000)
R

PQ i D

1X
Pir .’ i ; ’;
N Ä/
R

(8)

rD1

and simulated log-likelihood function
Ql D

N
X

ln.PQ i /;

(9)

iD1

where N denotes the number of individuals in the sample.
The elements of ’N and Ä are updated throughout the optimization process and constitute the estimation output.
Aside from allowing for the examination and interpretation
of potentially revealing covariance terms in Ä, our random parameter speciŽcation provides two additional advantages. First,
as discussed by Train (1998) and noted by Allenby and Lenk
(1995), by introducing correlation across choice probabilities,
we eliminate the problem of independenceof irrelevant alternatives that plagues standard conditional logit models. Second, as
shown by 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 inŽnity. This is important in our application,
because we do not have information on visits before the sampling period. Instead, we assume that all skiers start out with
a “blank memory” and set the elements of Sijt to 0 for the
Žrst day of the season. Keane (1997) took a similar approach
and showed robustness for his maximum likelihood estimators under Žxed initial conditions for a sample with large N
(1,150 individuals) but rather small T (about Žve purchase occasions, on average). In our application, both N and T are reasonably large (131 and 151), allowing us to invoke Heckman’s
(1981b) Žnding as well.
3.
3.1

EMPIRICAL ANALYSIS

Data

Our data stem from a spring 1998 survey of 131 randomly
selected skiers and snowboarders (called “snow riders” hereinafter) 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 1997–1998
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-speciŽc information was collected from brochures and websites associated with these ski areas. The source for our meteorologic
data is the SNOTEL website established by the U.S. 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 1 summarizes the data and variables used in our Žnal model speciŽcation. Overall, our dataset comprises 19,781

Table 1. Description of Data

Resort

Trips

SP

HD

Green

None
18,586
Mt. Rose
673
Sugar Bowl
27
Squaw Valley
96
Heavenly
34
Kirkwood
69
Diamond Peak
67
Northstar
43
Alpine Meadows
134
Boreal
52

0
117
0
14
0
49
0
7
23
0

0
342
22
74
27
59
49
36
65
44

0
220
255
1,000
860
345
136
618
340
90

Total

210

718

19,781

Terrain (acres)
Blue
Black
0
480
645
1,800
1,935
1,150
347
1,235
800
165

0
300
600
1,200
1,505
805
272
618
860
45

Total

PRICE
mean ($)

TEMP
mean ( ±F)

0
1,000
1,500
4,000
4,300
2,300
755
2,470
2,000
300

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

SNOW7
PASTVIS
mean (in.) Mean Max
0
2.1
2.2
3.8
1.7
2.3
.2
1.0
3.8
2.2

70.12
2.75
.11
.42
.15
.30
.28
.17
.51
.20

150
100
6
14
5
49
44
7
9
5

AREARUN
Mean Max
0
1.25
.03
.12
.06
.04
.19
.09
.18
.04

0
25
2
6
5
24
44
7
9
3

Downloaded by [Universitas Maritim Raja Ali Haji] at 00:27 13 January 2016

218

choice occasions, including the stay-home option. Of these
trips, 1,195 (6%) resulted in actual ski resort visits. As can
be seen from the table, the lion’s share of actual 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 1 for designated university holidays and
weekends and is set to 0 otherwise. It is also held at 0 for the
stay-home option. The next four columns in Table 1 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 speciŽc resort, day, and individual. Although travel
costs to and from a speciŽc 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 0 for season
pass holders. Our price variable reects these dynamic changes.
The next two columns in Table 1 give time-varying quality
variables. We hypothesizethat 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 7 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. Because volume is a complex function of water content, barometric pressure, temperature, and other unobserved
meteorologic factors, we settle for pillow as a proxy for actual
volume. Some additional aspects of snow quality will also be
captured by the temperature variable. SpeciŽcally, if snowfall
has been abundant, then lower temperatures allow for better and
longer lasting “powder” conditions. On the other hand, if little
or no snow has recently been added to the overall pack, then low
temperatures may result in hard and icy surfaces. The 7-day accumulation period in SNOW7 provides for reasonable delays in
skiers’ reaction to fresh snowfall while preserving a modicum
of the “freshness” quality. As shown in the Žrst row of the table,
TEMP and SNOW7 are arbitrarily set to 0 and 32 degrees for
the nonparticipation option.
The remaining four columns in Table 1 show mean and maximum values across sites for state-dependence variables Nijt
(PASTVIS) and Rijt (AREARUN) as deŽned in (3). The Žrst
row of PASTVIS corresponds to variable Hit , the number of
days an individual had chosen not to ski throughout the season before time t (termed PASTHOME hereinafter). As mentioned earlier, AREARUN is set to 0 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 earlier visits. Mt. Rose also had the highest maximum 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 1).
The structure of our data results in 197,810 observations for
the RUM framework outlined in Section 2. Our full speciŽcation has 21 explanatory variables, including 9 intercept terms

Journal of Business & Economic Statistics, April 2004

for actual sites, SP, HD, PRICE, skill-adjusted terrain shares
in natural log form (LNGREEN, LNBLUE, and LNBLACK),
the climate variables TEMP and SNOW7, and the indicators
for state dependencePASTVIS, PASTHOME, AREARUN, and
DAYSHOME. Skill-adjusted terrain (SAT) is computed as total
acres times the percent of terrain assigned to a speciŽc skill category (green D beginner, blue D intermediate, black D expert)
times the percentage of time during a season that 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 assumed that a skier will
use terrain appropriate for his skill level 100% of the time,
we have the beneŽt of actually knowing seasonal usage shares
through direct elicitation.
All regressors are treated as random and are associated
with a speciŽc mean coefŽcient. To conserve on parameters,
we estimate standard deviations for all regressors and a full
variance-covariance matrix for the main variables of interest:
time-variant quality attributes and state-dependence indicators
PASTVIS, AREARUN, and DAYSHOME. This yields a total of 52 model parameters for our unrestricted speciŽcation
(model 1 in the next section).
3.2

Estimation Results

Estimation results for coefŽcient means and the estimated elements of Ä for three different speciŽcations are summarized
in Tables 2 and 3. Model 1, our most general speciŽcation, includes all regressors mentioned earlier plus resort intercepts.
As discussed by Keane (1997), such intercepts may capture the
effect of unobserved brand-speciŽc features. In addition, it is
likely that consumers’ preferences are heterogeneous with respect to these unobserved attributes as well. If this is the case,
or if unobserved brand attributes are correlated with included
regressors, then omission of brand intercepts will bias estimation results. Just as is the case if heterogeneityis ignored for observed regressors, the effect of state-dependenceindicators may
be inated in the absence of such intercept terms. In our application, unobserved resort features could include the availability
and quality of off-slope amenities (e.g., restaurants, lodges,
shopping opportunities, day care for children), speed and comfort of chair lifts, and so forth.
Model 2 omits site-speciŽc terms and instead uses a nonparticipation intercept (D). Model 3 is identical to model 1
except for the omission of the time-variant quality indicators
TEMP and SNOW7. For ease of interpretation,the variability of
slope coefŽcients in Table 3 is given in terms of standard deviations, whereas their interdependence is shown in form of correlations. To be precise, maximum likelihood estimation produces results for Ä in form of its Cholesky factorization 0 (to
impose the necessary constraints for Ä during optimization).
For coefŽcients with covariances, the standard deviations and
correlation terms reported in Table 3 are nonlinear functions of
elements of 0. Their reported sampling standard errors (values
in parentheses in Table 3) are derived using the Delta method
(e.g., Greene 1997).
The Žrst two columns in both tables show parameter estimates and standard errors for model 1. All coefŽcient means
for site intercepts are highly signiŽcant. The negative signs for
all intercepts are expected, because these terms implicitly com-

Moeltner and Englin: Choice Behavior Under Time-Variant Quality

219

Table 2. Estimation Results (parameter means)

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Parameters
d
Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
SP
HD
PRICE
LNGREEN
LNBLUE
LNBLACK
PASTHOME
TEMP
SNOW7
PASTVIS
AREARUN
DAYSHOME
Log-likehood
NOTE:

CoefŽcient

Model 1
Standard error

CoefŽcient

Model 2
Standard error

3.244
(.333) ***
(.458) ***
(.423) ***
(.542) ***
(.538) ***
(.954) ***
(.664) ***
(.407) ***
(.316) ***
(.205)
(.095) ***
(.004) ***
(.040) ***
(.035)
(.042) ***
(.005)
(.005)
(.014) ***
(.005)
(.014) **
(.003)

¡3.263
¡7.343
¡5.169
¡6.823
¡7.148
¡8.141
¡8.997
¡5.661
¡5.755
.153
.750
¡.055
.173
¡.007
.403
.004
¡.003
.083
¡.003
.032
.003
5,107.101

1.288
.986
¡.047
¡.056
¡.237
.225
¡.038
¡.043
.072
.046
.271
.012

(.236) ***

(.238) ***
(.109) ***
(.004) ***
(.051)
(.046) ***
(.038) ***
(.005) ***
(.005) ***
(.013) ***
(.005) ***
(.025) ***
(.004) ***

5,650.518

CoefŽcient

Model 3
Standard error

¡3.209
¡7.310
¡5.026
¡7.065
¡6.898
¡8.204
¡8.232
¡5.547
¡5.570
.240
.697
¡.052
.204
¡.046
.403
.005

(.341) ***
(.525) ***
(.416) ***
(.538) ***
(.558) ***
(.732) ***
(.589) ***
(.433) ***
(.325) ***
(.229)
(.100) ***
(.004) ***
(.046) ***
(.032)
(.044) ***
(.004)

¡.004
.042
¡.001

(.004)
(.012) ***
(.004)

5,140.579

* signiŽcant at the 10% level; ** signiŽcant at the 5% level; *** signiŽcant at the 1% level.

pare the probability of resort visitation with the probability of
nonparticipation, the dominant choice in our dataset. In addition, all estimated standard deviations for site dummies are
signiŽcant as well (Table 3), supporting the hypothesis of pro-

nounced consumer heterogeneity in preferences for unobserved
resort features.
A similar result holds for HD. Although the effect of a holiday on the decision to visit a given resort is positive for the

Table 3. Estimation Results (parameter standard deviations and correlations)
Standard deviations/
correlations
d
Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
SP
HD
PRICE
LNGREEN
LNBLUE
LNBLACK
PASTHOME
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
Log-likelihood
NOTE:

CoefŽcient

Model 1
Standard error

CoefŽcient
.440

1.072
1.108
.356
1.992
1.458
4.607
2.496
1.142
1.010
.246
1.122
.011
.010
.056
.189
.000
.004
.781
.049
.208
¡.189
.004
¡.907
¡.730
.126
.035
.414
.029
.372
¡.331
.021
5,107.101

(.082) ***
(.213) ***
(.214) *
(.259) ***
(.266) ***
(.615) ***
(.291) ***
(.147) ***
(.176) ***
(.106) **
(.083) ***
(.001) ***
(.029)
(.018) ***
(.013) ***
(.001)
(.006)
(.203) ***
(.015) ***
(.271)
(.330)
(.001) ***
(.108) ***
(.298) **
(.397)
(.012) ***
(.201) **
(.338)
(.294)
(.274)
(.004) ***

.760
1.041
.016
.060
.048
.212
.004
.028
¡.997
.048
.349
¡.274
.012
.076
¡.022
.403
.290
.239
¡.219
.354
.202
.030
5,650.518

* signiŽcant at the 10% level; ** signiŽcant at the 5% level; *** signiŽcant at the 1% level.

Model 2
Standard error
(.113) ***

(.137) ***
(.116) ***
(.001) ***
(.043)
(.015) ***
(.020) ***
(.003) *
(.007) ***
(.030) ***
(.016) ***
(.094) ***
(.354)
(.002) ***
(.031) **
(.238)
(.153) ***
(.036) ***
(.159)
(.189)
(.209)
(.245)
(.006) ***

CoefŽcient
1.106
1.171
.485
2.265
1.418
4.676
2.258
1.118
.928
.250
1.210
.010
.063
.120
.186
.001

.002

Model 3
Standard error
(.084) ***
(.233) ***
(.233) **
(.257) ***
(.255) ***
(.382) ***
(.367) ***
(.180) ***
(.165) ***
(.135) *
(.102) ***
(.001) ***
(.036) *
(.015) ***
(.012) ***
(.001)

(.001)

¡.762
.035

(.132) ***
(.012) ***

.581
¡.641
.023

(.209) ***
(.174) ***
(.003) ***

5,140.579

Downloaded by [Universitas Maritim Raja Ali Haji] at 00:27 13 January 2016

220

average participant (as indicated by the positive sign for the
coefŽcient mean of HD in Table 2), there are individuals for
whom this effect may well switch to a negative sign given the
relatively high standard deviation for HD relative to its mean
(Table 3). This is intuitively sound, because the occurrence of
a holiday signiŽes the relaxation of time constraints, but also
implies increased crowdedness at most resorts.
The estimated effect of skill-adjusted terrain variables conveys a preference for beginner runs (LNGREEN) and more
difŽcult slopes (LNBLACK) for our sample of snow riders.
This may indicate an acceleration effect in the acquisition of
riding skills. Green areas are the only terrain accessible to
novice skiers and boarders; thus, resorts with extensive acreage
of beginner terrain will be preferred by this group for an extended period. After this initiation period, riders advance to
intermediate slopes. However, at this point the learning curve
becomes much steeper, especially for the age group captured
in our sample. This shortens the total riding time required to
progress to advanced terrain. Once riders can access these areas,
black terrain becomes a positive resort feature and remains desirable for an indeŽnite time horizon. In other words, intermediate slopes are sought out only during a relatively short transition
period for this particular population of riders.
Of the time-varying quality attributes, only the mean coefŽcient for snow is signiŽcantly different from 0. Clearly, riders
generally prefer more snow, but preferences are again heterogeneous for this attribute, as indicated by the relatively large and
signiŽcant standard deviation for SNOW7 in Table 3. SpeciŽcally, for about 10%–20% of riders, recent snowfall has a negative effect on visitation, applying our assumption of normality
for random parameters (Erdem 1996). This may be related
to difŽcult driving conditions during and immediately after
winter storms.
The state-dependence indicators PASTVIS, PASTHOME,
and DAYSHOME do not have a signiŽcant mean effect on
site choice in model 1. In contrast, the mean coefŽcient for
AREARUN is highly signiŽcant. Its positive sign indicates that
the average rider exhibits habit-forming tendencies in the short
run. However, the correspondingnontrivial and signiŽcant standard deviation for this indicator (Table 3) suggests the possibility of variety-seekingfor about 20% of our population.It should
be noted that although PASTVIS and DAYSHOME do not affect destination choices for the average rider, the signiŽcant
estimated standard deviations for these parameters (Table 3)
indicate that they may affect site decisions for some individuals. This is especially true for the coefŽcient of DAYSHOME,
which exhibits a standard deviation that is substantially larger
than its estimated mean of 0. This in turn implies that about
equal shares of riders are affected by eagerness and rustiness
due to prolonged nonparticipation.
As can be seen in Table 3, model 1 also yields some signiŽcant correlation terms. In general, a signiŽcant correlation
between two random coefŽcients associated with some explanatory variables would be indicative of an interactive effect
of the two regressors on site choices. Given that both moments
of the marginal distribution for the slope coefŽcient associated
with temperature are estimated to be arbitrarily close to 0, the
correlation terms for TEMP/SNOW7, TEMP/AREARUN, and

Journal of Business & Economic Statistics, April 2004

TEMP/DAYSHOME probably should not be given much importance. In contrast, the relatively large, negative, and significant correlation term for SNOW7/AREARUN is as expected;
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 speciŽc destination. This key result supports our initial hypothesis that timevarying quality attributes and state dependencemay have a joint
and potentially offsetting impact on repurchase decisions.
The mean parameter estimates associated with the second
speciŽcation (model 2) are provided in the center columns of
Table 2. The omission of site-speciŽc intercept terms in this
model has a strong effect on estimated parameters. Most notably, coefŽcients for all state-dependence indicators increase
substantially and exhibit high statistical signiŽcance. In light
of the signiŽcant means and standard deviations for site intercepts in model 1, the parameters generated by model 2 manifest the omitted variable problems discussed at the beginning
of this section. Clearly, what is interpreted as “long-run habit
forming” (PASTVIS, PASTHOME) and “short-run build-up of
eagerness” (DAYSHOME) in model 2 appears to be primarily
a reection of preferences for some unobserved resort-speciŽc
attributes for the representative rider. Thus model 2 is plagued
by “spurious state dependence” (Heckman 1981a).
The omission of site intercepts also affects the coefŽcient
means of SP and TEMP. Both are highly inated and estimated
to be signiŽcantly different from 0. For SP, this can be explained by the fact that the vast majority of season passes are
held for Mt. Rose and Alpine Meadows. In model 2, the effect of having a season pass on choice probability is smeared
across all destinations. Because Mt. Rose and Alpine Meadows
are by far the most-visited sites for SP holders and nonholders
alike, the coefŽcient for SP in model 1 likely captures the effect of omitted quality attributes for these areas. With respect
to TEMP, a possible explanation for this outcome is the fact
that is the variability of average daily temperature is much less
pronounced within and across resorts than is the variability in
recent snowfall. Accordingly, the effect of TEMP is largely absorbed by the site intercepts in model 1. Given the bias in the
mean coefŽcients produced by model 2, the elements of Ä for
this model are equally awed. A discussion of these terms is
thus omitted.
Our Žnal speciŽcation, model 3, reinstates site intercepts
but omits the time-variant regressors TEMP and SNOW7. The
resulting parameter estimates are shown in the last set of
columns of Tables 2 and 3. They are similar to those generated by model 1, with one noteworthy exception: the estimated
mean coefŽcient for short-run state dependence (AREARUN)
gains in both magnitude and signiŽcance. This is yet another
manifestation of Heckman’s (1981a) spurious state-dependence
problem. In this case, the coefŽcient for AREARUN absorbs
some of the effect of the omitted indicators for time-variant
weather conditions.
The inferiority of models 2 and 3 compared with model 1
is also highlighted by their much lower log-likelihood values.
Corresponding likelihood ratio (LR) tests clearly reject the null
hypotheses that brand intercepts and time-variant quality attributes have no effect on repeated site choices. It should be

Moeltner and Englin: Choice Behavior Under Time-Variant Quality

221

Table 4. Seasonal Shares (including nonparticipation)
Sample
Resort
None
Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
Total

Model 1

Trips

Shares

18,586
673
27
96
34
69
67
43
134
52
19,781

.940
.034
.001
.005
.002
.003
.003
.002
.007
.003

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Chi-squared

Trips

Model 2
Shares

17,585
1,177
34
138
101
40
356
59
226
65

.889
.060
.002
.007
.005
.002
.018
.003
.011
.003

1,914.98

Model 3
Shares

17,130
833
243
186
166
188
158
239
404
229

.866
.042
.012
.009
.008
.010
.008
.012
.020
.012

4,856.40

noted that some parameters are restricted under the alternative hypotheses (because variances cannot be negative). As discussed by Barlow, Bremner, and Brunk (1972) and Chen and
Cosslett (1998), the resulting LR statistics follow a mixed chisquared distribution, and standard LR test results may be biased
toward not rejecting the null hypothesis. In this application, the
resulting LR values are well above the upper bound for the
critical chi-squared(.05) value for such a mixed distribution.
Thus, the adjustment procedure proposed by Chen and Cosslett
(1998) would not affect our test results in this case.
4.

Trips

SHARE PREDICTIONS AND
MARKETING SIMULATIONS

To compare the predictive power of our models, 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
1, 2, and 3, together with actual sample shares, are presented in
Tables 4 and 5. Table 4 captures all choices, including the nonparticipation option. Table 5 focuses on actual trips to ski areas.
Because our models allow for random parameters, estimates
O
are based on simulations using 1,000 different vectors of ’,
drawn from the multivariate normal distribution with mean ’ON
O Approximations of trip Žgand variance-covariance matrix Ä.
ures corresponding to estimated shares are derived by multiplying shares by the total number of observed choice occasions
(19,781 for Table 4 and 1,195 for Table 5). When nonparticipation is included, all models generally overpredict trips to actual

Trips

Shares

17,609
1,137
32
135
123
44
380
71
188
63

.890
.058
.002
.007
.006
.002
.019
.004
.010
.003

2,131.55

resorts by varying degrees and slightly underpredict nonparticipation choices. When only actual trips are considered, models 1 and 3 predict trips to most resorts fairly well, whereas
predictions generated by model 2 deviate more signiŽcantly
from sample results for most destinations. In the spirit of Keane
(1997), the tables also include chi-squared statistics based on
squared deviations between actual and predicted shares. As indicated by these values, the overall predictive power of model 1,
our most general speciŽcation, is superior to that of models
2 and 3 with and without nonparticipation.
A resort manager may be interested in estimating the impact
of promotional efforts and associated state-dependence effects
on daily and seasonal market shares. Given that most actual
trips were made to Mt. Rose ski area, and considering the good
Žt of our estimated models for this destination (see Table 5),
we focus on this resort for our marketing scenarios. SpeciŽcally,
we simulate six pricing scenarios. Scenarios 1 and 2 correspond
to price reductions of $5 and $10, to all visitors for days 12–18
(Monday, December 8–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 3 and 4 simulate the same price reductions
with extension through day 25. During the 1997–1998 ski season, Mt. Rose offered two weekday specials that continued
through the entire ski season: “Ladies Thursdays” (a $15 day
pass for female visitors) and “Student Wednesdays” (a $10 day
pass for students). Scenarios 5 and 6 investigate the impacts of
“undoing” these promotions. SpeciŽcally, scenario 5 imposes a

Table 5. Seasonal Shares (actual trips only)
Sample

Model 1

Resort

Trips

Shares

Mt. Rose
Sugar Bowl
Squaw Valley
Heavenly
Kirkwood
Diamond Peak
Northstar
Alpine Meadows
Boreal
Total

673
27
96
34
69
67
43
134
52
1,195

.563
.023
.080
.028
.058
.056
.036
.112
.044

Chi-squared

Trips
641
18
75
55
22
194
32
123
36
303.06

Model 2

Shares
.536
.015
.063
.046
.018
.162
.027
.103
.030

Trips
376
110
84
75
85
71
108
182
104
607.26

Model 3

Shares
.315
.092
.070
.063
.071
.060
.090
.152
.087

Trips
626
17
74
67
24
209
39
103
35
387.93

Shares
.524
.015
.062
.056
.020
.175
.033
.087
.029

222

Journal of Business & Economic Statistics, April 2004
Table 6. Discount Scenarios

Scenario

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1
2
3
4
5
6

Time period

Price change

Days 12–18
Days 12–18
Days 12–25
Days 12–25
Every Thursday
Every Wednesday

¡$5
¡$10
¡$5
¡$10
C$23
C$28

price increase of $23 to reach the standard day pass price of $38
for all skiers and boarders who had received a ladies’ discount.
Similarly, scenario 6 increases day pass prices to student day
beneŽciaries by $28. The pricing scenarios are summarized in
Table 6. We implement these scenarios using model 1, our preferred speciŽcation.
For each person, the simulations are conducted as follows.
First, all exogenous price changes are incorporated into the
data. Second, visitation probabilities Pjt , j D 0; : : : ; J, are updated for the Žrst day of the scenario implementation (t D a).
A choice or “hit” for day a is then simulated by dividing the
unit interval into J C 1 segments corresponding in width to Pja ,
j D 0; : : : ; J, drawing a random term from the uniform .0; 1/
distribution, noting the segment into which it falls, and matching the segment with the associated site. Conditionalon the outcome of this step, all state-dependencecounters for all sites are
then updated for the following day (t D a C 1) according to the
deŽnitions in (3). Next, all choice probabilities for that day are
adjusted to incorporate the updated state-dependence counters.
A new site is drawn, state-dependence counters are updated
for t D a C 2, and so on, until the end of the research period
(t D T). For each rider, this process is repeated R D 1,000 times.
For each day a through T , simulated daily visitation shares are
then determined by averaging updated daily probabilities over
the R repetitions and over all respondents. Finally, average seasonal shares for all sites are derived by averaging these aggregate daily shares over time periods. The resulting simulated
shares are then compared with daily and seasonal