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UTILITY IN WILLINGNESS TO PAY SPACE:ATOOL
TO ADDRESS CONFOUNDING RANDOM SCALE
EFFECTS IN DESTINATION CHOICE TO THE ALPS
RICCARDO SCARPA,MARA THIENE,AND KENNETH TRAIN
We compare two approaches for estimating the distribution of consumers’ willingness to pay (WTP)
in discrete choice models. The usual procedure is to estimate the distribution of the utility coefficients
and then derive the distribution of WTP, which is the ratio of coefficients. The alternative is to estimate
the distribution of WTP directly. We apply both approaches to data on site choice in the Alps. We find
that the alternative approach fits the data better, reduces the incidence of exceedingly large estimated
WTP values, and provides the analyst with greater control in specifying and testing the distribution of
WTP.
Key words: destination site choice, mixed logit, nonmarket valuation, outdoor recreation, random
parameters, random willingness to pay, travel cost.
Nonmarket values of qualitative changes in
sites for outdoor recreation are often inves-
tigated by estimating random utility mod-
els (RUMs) of site selection (Bockstael,
Hanemann, and Kling 1987; Morey, Rowe,
and Watson 1993). Most recent applications
address the issue of unobserved taste het-
erogeneity by using continuous (Train 1998,
1999) or finite mixing (Provencher, Baeren-
klau, and Bishop 2002; Scarpa and Thiene
2005) of individual taste distributions by means
of panel mixed logit models. Such approaches
are shown to produce more informative and
realistic estimates of nonmarket values than
models without taste heterogeneity and are
now part of the state of practice in the pro-
fession. However, models with conveniently
tractable distributions for taste coefficients,
such as the normal and the log-normal, often
obtain estimates that imply counter-intuitive
distributions of marginal willingness to pay
(WTP). This is due to the fact that the an-
alytical expression for WTP involves a ratio
Riccardo Scarpa is professor in the Department of Economics,
Waikato Management School, University of Waikato, Hamilton,
New Zealand. Mara Thiene is assistant professor in the Dipar-
timento Territorio e Sistemi Agroforestali, University of Padua,
Italy. Kenneth Train is adjunct professor in the Department of
Economics, University of California-Berkeley, U.S.A.
Senior authorship is equally shared. The authors would like
to acknowledge useful comments from three anonymous review-
ers, the associated editor Stephen Swallow and Timothy Gilbride.
They are also grateful to Kelvin Balcombe for making avail-
able Gauss routines to graphically assess convergence of the
Gibbs sampler. The usual disclaimer on the remaining errors
applies.
where the denominator is the cost coefficient.
Values of the denominator that are close to
zero (which are possible under most standard
distributions such as the lognormal) cause the
ratio to be exceedingly large, such that the de-
rived distribution of WTP obtains an unten-
ably long upper tail. The mean and variance of
the skewed distribution are both raised artifi-
cially by these implausibly large values.
One solution is to assume that the cost coef-
ficient is constant and not random (e.g., Revelt
and Train 1998; Goett, Train, and Hudson
2000; Layton and Brown 2000; Morey and
Rossmann 2003). This restriction allows the
distributions of WTP to be calculated eas-
ily from the distributions of the nonprice
coefficients, since the two distributions take
the same form. For example, if the coeffi-
cient of an attribute is distributed normally,
then WTP for that attribute, which is the at-
tribute’s coefficient divided by the price co-
efficient, is also normally distributed. The
mean and standard deviation of WTP is sim-
ply the mean and standard deviation of the
attribute coefficient scaled by the inverse of
the (fixed) price coefficient. The fixed cost
coefficient restriction also facilitates estima-
tion. For example, Ruud (1996) suggests that
a model specification with all random coeffi-
cients can be empirically unidentified, espe-
cially in data sets with few observed choices for
each decision-maker (short panels). However,
this restriction is counter-intuitive as there
are very good theoretical and common-sense
Amer. J. Agr. Econ. 90(4) (November 2008): 994–1010
Copyright 2008 Agricultural and Applied Economics Association
DOI: 10.1111/j.1467-8276.2008.01155.x
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 995
reasons as to why response to costs should
vary across respondents according to factors
that can be independent of observed socioeco-
nomic covariates.
Train and Weeks (2005) note on the topic
that assuming a fixed price coefficient implies
that the standard deviation of unobserved util-
ity (i.e., the scale parameter) is the same for
all observations. On the other hand, it is im-
portant to recognize that the scale parame-
ter can, and in many situations clearly does,
vary randomly over observations. Estimation
practices that ignore such source of variation
may lead to erroneous interpretation and pol-
icy conclusions. For example, in the context of
destination choice modeling, if the travel cost
coefficient is constrained to be fixed when in
fact scale varies over observations, then the
variation in scale will be erroneously attributed
to variation in WTP for site attributes.
Another solution is to re-parameterize the
model such that the parameters are the
(marginal) WTP for each attribute rather than
the utility coefficient of each attribute. That
is, instead of the usual approach of parame-
terizing the model in “preference space”(i.e.,
coefficients in the utility), the model is pa-
rameterized in “WTP space.”This alterna-
tive procedure has recently been utilized to
represent taste heterogeneity by Train and
Weeks (2005) and Sonnier, Ainslie, and Otter
(2007). However, the idea of specifying util-
ity in the WTP space is not new. For example,
readers familiar with the analysis of discrete-
choice contingent valuation data may recall
the so-called variation function or expenditure
function approach suggested by Cameron and
James (1987) and Cameron (1988), which—as
discussed in some more detail by McConnell
(1995)—in some cases boils down to a simple
re-parameterization of the RUM model pro-
posed by Hanemann (1984, 1989).
Train and Weeks (2005) and Sonnier,
Ainslie, and Otter (2007) extended the ap-
proach by Cameron and James to multinomial
choice models with random tastes, where dis-
tributional assumptions and restrictions can
be placed on the WTP’s. They point out
that the two approaches are formally equiv-
alent because any distribution of coefficients
translates into some derivable distribution
of WTP’s, and vice versa. However, the ap-
peal of the approach is that it allows the
analyst to specify and estimate the distribu-
tions of WTP directly, rather than deriving
them indirectly from distributions of coeffi-
cients in the utility function. To researchers
in nonmarket valuation this is an important
advantage.
Comparisons of estimates obtained from the
two parameterizations on an identical data set
have already been investigated using hierar-
chical Bayes (HB) estimation on stated pref-
erence (SP) data. Train and Sonnier (2005)
compared the estimates of the two approaches
and the implied WTP for attributes related to
cars with different fuel (fossil, hybrid, and elec-
tric). Sonnier, Ainslie, and Otter (2007) inves-
tigate the same issues in the context of stated
preference data for car brand choice and pho-
tographic cameras. Both studies found that the
specifications in the preference space fit their
in-sample data better but produce distribu-
tions of WTP with fatter tails than specifica-
tions in the WTP space. The studies differed
for out-of-sample fit, as Train and Weeks found
that their model in preference space fit out-of-
sample better than their model in WTP space,
while Sonnier, Ainslie, and Otter found the
reverse.1
We apply these concepts to revealed-
preference (RP) data, the first such applica-
tion to our knowledge. In order to ensure that
results are not dependent on the estimation
method, we estimate our models by both HB
and maximum simulated likelihood (MSL). To
our knowledge, this is the first application of
MSL to random coefficient models in WTP
space. We find, like in both previous stud-
ies, that models in WTP space imply WTP
distributions with lower incidence of extreme
values than models in preference space, a fea-
ture that some analysts may associate with en-
hanced model plausibility. Further, we show
with an example that assuming bounded distri-
butions of coefficients in the preference space
does not avoid the “fat tail”problem of WTP
distributions when the cost coefficient is as-
sumed to be random over a support approach-
ing zero. Unlike previous studies, though, we
find that the models in WTP space also fit the
in-sample data better than the models in pref-
erence space. This improved fit arises with both
HB and MSL estimation. Our findings indicate
1Importantly for the practice of RUM estimation, Train and
Weeks emphasize how assuming independence across utility coef-
ficients in the presence of a scale parameter which varies across
visitors implies dependence (correlation) across implied WTP dis-
tributions, and vice versa. This issue may escape the attention
of analysts, and it is worth bearing in mind for its consequences
in interpretation of results, because in general neither marginal
WTPs for attributes nor their taste intensities are independently
distributed, and hence correlation matrices should be estimated
whenever the data allow it, regardless of the choice of utility
specification.
996 November 2008 Amer. J. Agr. Econ.
that in our data there is no trade-off between
goodness of fit and density over extreme val-
ues: the model in WTP space outperforms on
both criteria.
Specification
In this section, we start with the conventional
specification of utility in the preference space,
and describe the implications for correlation
of utility coefficients and implied WTPs. We
then re-parameterize the model in WTP space
and discuss the implications. Throughout, the
notation and language is adapted for our ap-
plication to Alpine site choice.
Day trippers are indexed by n, destination
sites by j, and choice situations by t. To ease
the illustration, we specify utility as separable
in price, p, and a vector of nonprice attributes,
x:
Unjt =−npnjt +
nxnjt +njt
(1)
where scalar nand vector nvary randomly
over day visitors and njt is Gumbel dis-
tributed. The variance of njt is visitor-specific:
Var(njt )=2
n(2/6), where nis the scale pa-
rameter for day visitor n. Since utility is ordi-
nal one can divide equation (1) by the scale
parameter to obtain its scale-free equivalent.
This division does not affect behavior and yet
it results in a new error term that has the same
variance for all decision makers:
Unjt =−(n/n)pnjt +(n/n)xnjt +εnjt
(2)
where εnjt is i.i.d. type-one extreme value, with
constant variance 2/6. The utility coefficients
are defined as n=(n/n) and cn=(n/n),
such that utility may be written:
Unjt =−npnjt +c
nxnjt +εnjt.(3)
Note that if nvaries randomly, then the util-
ity coefficients are correlated, since nenters
the denominator of each coefficient. Specify-
ing the utility coefficients to be independent
implicitly constrains nto be constant. If the
scale parameter varies and nand nare fixed,
then the utility coefficients vary with perfect
correlation. If the utility coefficients have cor-
relation less than unity, then nand nare nec-
essarily varying in addition to, or instead of, the
scale parameter. Finally, even if ndoes not
vary over visitors (e.g., the standard deviation
in unobserved factors over sites and trips is the
same for all visitors), utility coefficients can be
correlated simply due to correlations among
tastes for various attributes.
The specification in equation (3) parameter-
izes utility in “preference space.”The implied
WTP for a site attribute is the ratio of the
attribute’s coefficient to the price coefficient:
wn=cn/n=n/n. Using this definition, util-
ity can be rewritten as:
Unjt =−npnjt +(nwn)xnjt +εnjt
(4)
which we name “utility in WTP space,”while
Sonnier, Ainslie, and Otter (2007) called it the
“surplus model.”In a context in which scale
can vary over people—such as in our alpine
destination choice—this specification is very
useful for distinguishing WTP variation (i.e.,
the distributional features of wn) from varia-
tion in scale. To what extent this distinction af-
fects the derived welfare estimates remains an
empirical question, and one of the objectives
of our investigation. We note that, although
any coefficient can be used as the base that
incorporates scale, the reason to focus on the
travel cost coefficient in this case is that the
scale-free terms can be directly interpreted as
WTPs, which are easy to rationalize. This util-
ity specification is distinctive for another rea-
son as it gives a nonlinear-in-the-parameter
utility function, which poses some computa-
tional challenges in the context of MSL esti-
mation (and is probably the reason MSL has
not been previously used for models in WTP
space). In contrast, nonlinearity is readily ac-
commodated in HB estimation.
The utility expressions are behaviorally
equivalent and any distribution of nand cn
in (3) implies a distribution of nand wnin (4),
and vice versa. The general practice in nonmar-
ket valuation and elsewhere has been to spec-
ify distributions in preference space, estimate
the parameters of those distributions, and de-
rive the distributions of WTP from these esti-
mated distributions in preference space (Train
1998). While fully general in theory, this prac-
tice is usually limited in implementation by the
use of computationally convenient distribu-
tions for utility coefficients. However, empiri-
cally tractable distributions for coefficients do
not necessarily imply convenient distributions
for WTP, and vice versa. For example, if the
travel cost coefficient is distributed log-normal
and the coefficients of site attributes are nor-
mal, then WTP is the ratio of a normal term
to a log-normal term. Similarly, in (4), normal
distributions for WTP and a log-normal for the
(negative of) travel cost coefficient imply that
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 997
the utility coefficients are the product of a log-
normal variate and a normal one: n×wn.
A similar asymmetry exists for the place-
ment of restrictions on patterns of correlations
(such as independence). In the travel cost site
selection literature it is fairly common for re-
searchers to specify uncorrelated utility coef-
ficients. However, this restriction implies that
scale is constant, as stated above, and more-
over that WTP is correlated in quite a particu-
lar way via the common variation in the price
coefficient. Researchers might not be aware of
such implications of their choice of specifica-
tion, as few papers discuss its consequences.
Symmetrically, specifications assuming uncor-
related WTP imply a pattern of correlation in
utility coefficients that is difficult to implement
in preference space. We know of only one other
application of travel cost RUMs that assumes
a random scale parameter, but in that case
the authors do not explicitly address correla-
tion across WTP estimates (Breffle and Morey
2000).
The issue becomes: does the use of con-
venient distributions and restrictions in pref-
erence space or WTP space result in more
accurate models? The answer is necessarily
situationally dependent, since the true distri-
butions differ across applications. However,
some insight into this issue can be obtained by
comparing alternative specifications on a given
data set under alternative estimators. Descrip-
tion of our data is the topic of the next section.
Table 1. Site-Specific Data
Site Attributes
Descriptive Statistics of Trips Degree of Easy Hard
Destination Sites Mean Std. Dev. Visits Percentage Difficulty Ferratas Trails Shelters Trails
1. Vette Feltrine 0.7 1.5 642 7.0 3 3 0.61 25 0.07
2. P. Dolomiti-Pasubio 2.1 4 1,808 19.6 1 4 0.54 13 0.17
3. Alpago-Cansiglio 0.5 1.7 414 4.5 3 4 0.86 10 0.08
4. Asiago 1.5 2.8 1,318 14.3 1 0 1 13 0
5. Grappa 0.9 2.1 757 8.2 1 1 0.99 5 0.01
6. Baldo-Lessini 1.2 3.6 1,045 11.3 1 2 0.76 18 0.02
7. Antelao 0.3 0.7 244 2.6 3 0 0.68 6 0.08
8. Pelmo 0.3 0.6 243 2.6 3 0 0.66 9 0.04
9. Cortina 0.3 0.8 220 2.4 2 22 0.53 32 0.11
10. Duranno-Cima Preti 0.1 0.3 44 0.5 3 0 0.33 4 0.09
11. Sorapis 0.1 0.5 128 1.4 3 4 0.36 9 0.23
12. Agner-Pale S.Lucano 0.1 0.5 112 1.2 3 2 0.51 7 0.14
13. Tamer-Bosconero 0.2 0.6 188 2.0 3 0 0.3 6 0.06
14. Marmarole 0.2 0.7 161 1.7 2 1 0.51 9 0.07
15. Tre Cime-Cadini 0.6 1.2 547 5.9 2 4 0.6 9 0.08
16. Civetta-Moiazza 0.7 1.3 561 6.1 2 4 0.34 16 0.11
17. Pale S.Martino 0.7 1.3 564 6.1 2 11 0.46 14 0.14
18. Marmolada 0.3 0.7 225 2.4 3 2 0.21 13 0.25
Survey and Site Attribute Data
The data for our analysis were collected
through a survey of 858 members of the local
(Veneto Region) chapter of the CAI (Italian
Alpine Club). Respondents provided infor-
mation on the mountain visits that they took
during the year 1999. Importantly for this ap-
plication, respondents were asked the total
number of trips they took to each of 18 sites
in the last twelve months. A total of 9,221 trips
were reported, for which table 1 gives summary
statistics. The most visited sites are Piccole
Dolomiti, Asiago, Lessini-Baldo, which are
located in the pre-Alps, and Civetta, Pale
S.Martino and Tre Cime, all of which are in
the Dolomites. Unsurprisingly the most fre-
quently attended sites are those closest to the
urban centers located in the plains.
The interviewers contacted the CAI mem-
bers at club meetings taking place in the mu-
nicipalities of the Veneto region. The various
parts of the questionnaire were explained to
the group, and then the members filled out the
questionnaire on their own. Respondents were
asked questions about their mountaineering
abilities and experience (i.e., when they started
mountain recreation, whether they attended
mountaineering training courses, and the kind
of activities they usually undertook at the sites,
etc.). Respondents also provided socioeco-
nomic information about themselves and their
households.
998 November 2008 Amer. J. Agr. Econ.
Round-trip distance from the respondent’s
residence to each of the destinations in the
choice set was calculated using the software
package “Strade d’Italia e d’Europa.”These
data were used to estimate the “travel cost”
to each destination. Distance was converted
into monetary values using a rate of €0.35 per
km, which was the car running cost at the time.
Each reported trip was a “day out”(i.e., a trip
lasting one day and not involving overnight
stay), as is customary for this generic form of
local outdoor recreation. The eighteen moun-
tain destinations differ substantially from both
a morphological and mountaineering point of
view, but they can provide both specialist and
nonspecialist outdoor recreation, and so are all
destinations for local visitors planning “a day
out.”
Two broad geographically determined
groups of destinations can be distinguished.
Destinations 1–6 (table 1) belong to the
Prealps, which are mountains with gentler
slopes and lower peaks separating the plane
from the proper Alps. Because of their distinct
nature, the Prealps are often chosen for dif-
ferent recreational objectives than the Alps.
Destinations 7–18 are in the Northeastern
Alps, in the mountain chain of the Dolomites,
which is an extended rocky area mostly made
of dolomite rocks. This rare and distinguished
rock type is geologically well-defined as it
originates from coral reefs. Mountains made
of this rock are scenically quite attractive as
they tend to show orange–pink reflections at
sunset.
Some of the recreational attributes describe
the land-use of the sites, while others provide
specific information about hiking conditions
at each destination. “Degree of difficulty”is a
score from 1 to 3 describing the degree of tech-
nical difficulty of the trailing itineraries that
are available at each destination. This score
takes into account not only the total length
of the trails network but also the average de-
gree of adversity of the mountain environment
at each destination. “Ferrata”is the number
of trails equipped with safety ropes, to which
visitors can secure themselves in the ascent
toward hard-to-reach vantage points. “Alpine
shelters”is the number of equipped alpine
shelters accessible in the destination area.
The recreational attractiveness of a desti-
nation is also measured as percent of total
trail length that is “easily”walkable (“%of
easy trails”). These trails require lower-than-
average physical effort and are selected on the
basis of a composite set of measurements, such
as width, incline and accessibility. At the other
extreme of the spectrum, we use the percent
of total trail length that is “hard”walkable
(“% of hard trails”), requiring higher-than-
average physical effort and fitness. Finally,
because the “Prealps”offer an experience dis-
tinctively different from the Dolomites, an
alternative-specific constant is included for the
“Prealps”to capture this difference.
Method
Revelt and Train (1998) derived the mixed
logit specification in the context of repeated
choices by individuals with continuous taste
distributions, the so-called panel mixed logit.
In our alpine destination choice context, visitor
nfaces a choice among Jdestination alterna-
tives in each of Tntrips taken over an outdoor
season. Jin our case is 18 while we have a maxi-
mum of Tn=40 which represents a reasonable
maximum number of days out over a year. We
have an unbalanced panel since the number of
trips varies across individuals, hence the sub-
script n.
To ensure that the range of variation of the
travel cost coefficient is negative we define
n=−exp(vn), where vncan be considered
the latent random factor underlying such co-
efficient.2Let ndenote the random terms
entering utility, which are vnand cnfor the
model in preference space (equation (3)) and
vnand wnfor the model in WTP space (equa-
tion (4)). Similarly, let utility be written Unjt =
Vnjt(n)+εnjt , with Vnjt(n) being defined by
either equation (3) or (4), depending on the
parameterization.
Visitor nchooses destination iin period tif
Unit >Unjt ∀j= i. Denote the visitor’s cho-
sen destination in choice occasion tas ynt,
the visitor’s sequence of choices over the Tn
choice occasions as yn=yn1,...,ynTn. Con-
ditional on n, the probability of visitor n’s se-
quence of choices is the product of standard
logit formulas:
L(yn|n)=
t=Tn
t=1
eVnyntt(n)
jeVnjt(n).(5)
The unconditional probability is the integral of
L(yn|n) over all values of nweighted by its
density:
2We note that in our estimations without random “travel cost”
coefficient the point estimate is negative, as expected.
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 999
Pn(yn)=L(yn|n)g(n)dn
(6)
where g(·) is the density of nwhich depends
on parameters to be estimated. This uncon-
ditional probability is called the mixed logit
choice probability, since it is a product of logits
mixed over a density of random factors reflect-
ing tastes.
Mixed Logit Estimation via Hierarchical
Bayes
Because MSL estimation of mixed logit mod-
els is well-documented (e.g., Train 2003), in
this section we mostly focus on HB estima-
tion. For the MSL estimation we just mention
that to deal with nonlinearity of Vnit we used
BIOGEME (Bierlaire 2002, 2003) and the al-
gorithm CFSQP Lawrence, Zhou, and Tits
(1997) so as to avoid the problem of local op-
tima. All MSL estimates were obtained using
100 quasi-random draws via Latin-hypercube
sampling (Hess, Train, and Polak 2006).
The Bayesian procedure for estimating the
model with normally distributed coefficients
was developed by Allenby (1997) and imple-
mented by Sawtooth Software (1999). This es-
timation method was also applied by Rigby
and Burton (2006) to derive transforms that
address mass distribution at zero (indifference
to attributes) of utility coefficients (not WTP
coefficients) for choice over GM food prod-
ucts in the United Kingdom. Related methods
for probit models were developed by Albert
and Chib (1993), McCulloch and Rossi (1994),
Allenby and Rossi (1999). Layton and Levine
(2005) made a contribution in the context of se-
quential learning from previous applications.
A review of applications to marketing meth-
ods is found in Rossi, Allenby, and McCulloch
(2005).
We specify the density of nto be normal
with mean band variance Ω, denoted g(n|b,
Ω). Although terminology differs over authors
and fields,3we call band Ω“population pa-
rameters”since they describe the distribution
of visitor-level n’s in the population. With
this usage, the distribution g(n|b,Ω) is inter-
preted as the actual distribution of tastes for
the recreational attributes of destination sites
in the population of the regional branch of the
3In Bayesian applications band Ωtend to be called hyperpa-
rameters,with the n’s themselves being the parameters of interest.
Sometimes, however, the n’s are called nuisance parameters, to
reflect the concept that they are incorporated into the analysis to
facilitate estimation of band Ω.
Italian Alpine Club, from which we drew the
sample. Note that, given the expression above
for the price coefficient, the specification of
normal nimplies that the price coefficient is
log-normally distributed.
In Bayesian analysis, a prior distribution is
specified for the parameters. We lack previous
information on the type of visitors in our sam-
ple4and therefore specify the prior on bto be
a diffuse normal, denoted N(b|0, Θ), which
has zero mean and a sufficiently large variance
Θsuch that the density is essentially flat from
a computational perspective.5A normal prior
on bhas a computational advantage since it
provides a conditional posterior on b(i.e., con-
ditional on n∀nand Ω) that is also normal
and hence easy to draw from, while the large
variance ensures that the prior has minimal
(effectively no) influence on the posterior, re-
flecting the absence of a-priori knowledge, es-
pecially in the presence of large samples, such
as in our case. The standard diffuse prior on
Ωis inverted Wishart with low degrees of free-
dom. This specification is also computationally
advantageous as it provides a conditional pos-
terior on Ωthat is also Inverted Wishart and
hence easy to draw from. The conditional pos-
terior on n∀n, given band Ω,is
(n|,b,Ω)∝
n
L(yn|n)·g(n|b,Ω).
(7)
Information about the posterior is obtained
by taking draws from the posterior and calcu-
lating relevant statistics, such as moments, over
these draws. Draws from the joint posterior
are obtained by Gibbs sampling (Casella and
George 1992). In particular, a draw is taken
from the conditional posterior of each param-
eter, given the previous draw of the other
parameters. The sequence of draws from the
conditional posteriors converges, after a suffi-
cient number of iterations (called “burn-in”),
to draws from the joint posterior. Technical in-
formation about the algorithm can be found in
Train and Sonnier (2005) and Train (2003).6
4The only other study we know of on the region is Scarpa and
Thiene (2005) and it focused on rock-climbers and not generic
day-out visitors.
5The final results are not sensitive to the choice of prior as when
we started the search algorithm from different prior values we
obtained very similar results. This is unsurprising given the large
sample size available.
6For the HB models in preference space, we used the
GAUSS code that is available on K. Train’s website at
http://elsa.berkeley.edu/train/software.html. We adapted this code
appropriately for the HB models in WTP space.
1000 November 2008 Amer. J. Agr. Econ.
It is worth reminding the reader not fa-
miliar with Bayesian estimation that the
Bernstein–von Mises theorem states that, un-
der quite unrestrictive conditions, the mean of
the Bayesian posterior of a parameter is a clas-
sical estimator that is asymptotically equiva-
lent to the maximum likelihood estimator of
the parameter. Similarly, the variance of the
posterior distribution is the asymptotic vari-
ance of this estimator. See Train (2003) for
an extended explanation with citations. Hence,
the results obtained by Bayesian procedures
can be interpreted from a purely classical per-
spective. In the tables below, results are pre-
sented in the way that is standard for classi-
cal estimation, giving the estimate and stan-
dard error for each parameter. These statistics
are the mean and standard deviation, respec-
tively, of the draws from the posterior for each
parameter.
Estimation Results
We discuss the preference space estimation re-
sults first and those in WTP space afterwards.
Preference Space
The estimates for models in the preference
space (i.e., equation (3)) are reported in ta-
ble 2. The left column reports the HB esti-
mates, while the right column the MSL ones.
Estimates with uncorrelated coefficients are
Table 2. Hierarchical Bayes (HB) and Maximum Simulated Likelihood (MSL) Estimates for
Preference Space Models
Statistics of HB post. Distribution Statistics of MSL Estimates
Prefer. Parameters
ln( ˆ
) and ˆ
cMean Std. err. Std Dev. Std. err. Mean Std. err. Std. Dev. Std. err.
ln( ˆ
)−1.29 0.04 0.73 0.25 −1.41 0.06 0.71 0.06
Degree of difficulty −0.76 0.04 0.72 0.24 −0.51 0.04 0.48 0.06
Ferrata −0.12 0.01 0.09 0.03 −0.07 0.01 0.02 0.01
% of easy trails 0.02 0.002 0.06 0.001 0.01 0.001 0.01 0.002
Alpine shelters 0.11 0.005 0.08 0.001 0.07 0.01 0.03 0.01
% of hard trails 0.09 0.01 0.10 0.03 0.05 0.005 0.07 0.005
Prealps ASC −1.54 0.10 1.28 0.46 −0.98 0.11 0.98 0.09
Uncorrelated: ln L∗at means of post. dist. −20,774 ln L∗at convergence −20,470
ln( ˆ
)−1.22 0.05 0.88 0.28 −1.43 0.07 0.92 (∗)
Degree of difficulty −1.16 0.07 1.17 0.39 −0.67 0.12 0.73
Ferrata −0.19 0.01 0.23 0.06 −0.10 0.01 0.11
% of easy trails 0.04 0.004 0.11 0.03 0.01 0.002 0.01
Alpine shelters 0.15 0.01 0.18 0.05 0.09 0.01 0.08
% of hard trails 0.14 0.01 0.19 0.06 0.07 0.01 1.95
Prealps ASC −2.74 0.16 2.84 0.94 −1.62 0.25 0.07
With correlation: ln L∗at means of post. dist. −20,384 ln L∗at convergence −20,148
Note: (∗) Std. err. for the elements of the Cholesky matrix are reported in Scarpa, Thiene, and Train (2008).
reported in the top part and estimates with cor-
related coefficients are reported in the bottom
part, while the estimates of the Cholesky ma-
trix associated with the MSL correlated model
are available in Scarpa, Thiene, and Train
(2008). Allowing for full correlation amongst
coefficients increases the log-likelihood simu-
lated at the posterior means from −20,773.59
to −20,383.65 in the HB case. Similarly in
the MSL case, the value of the simulated
log-likelihood improves from −20,469.86 to
−20,147.91.
In interpreting the figures in tables 2, re-
call that the coefficient for “travel cost”is
log-normally distributed, such that the esti-
mated mean and standard deviation are the
mean of the latent normally distributed ran-
dom factor underlying the travel cost coeffi-
cient. The other coefficients were all normally
distributed, such that their means and stan-
dard deviations are estimated directly. The es-
timated mean and standard deviation together
determine the proportion of the population
implied to have coefficients of each sign.
The estimated means have the same signs
and orders of magnitude across models
(with and without correlation) and estima-
tors (HB and MSL). The signs are plausible
considering that the population of reference
are the members of the Italian Alpine Club
selecting days out in the Alps. A negative
mean is observed for the “Degree of technical
difficulty.”To tackle technically difficult sites
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 1001
requires rigorous training and experience, and
it is expected that in general visitors are not
attracted by technically challenging destina-
tions. The negative mean for the number of
“Ferrata”seems reasonable when one bears in
mind that the number of “Ferrata”is mostly
a consequence of strategic access for the mili-
tary, established during the World War I period
against invading Austrians, and not necessar-
ily designed to facilitate tourist access to such
vantage points.
Destinations with many “Alpine shelters”
tend to be liked more than those with few.
“Alpine shelters”are often themselves the des-
tinations of days out in the Alps and offer op-
portunities to encounter other visitors and eat
local specialities, as well as providing shelter
for unexpected bad weather. Everything else
equal, one would be more inclined to plan a
day out to a destination with shelters.
Sites with higher percent of easily walka-
ble trails (“% easy Trails”) and hard walka-
ble trails (“% hard trails”) are, on the average,
both liked by visitors from the Alpine Club,
but with large estimated taste variation. Trail-
walking is still one of the most popular activi-
ties in the Alps because it is cheap and attracts
visitors of all ages and abilities. These results
indicate that visitors like destinations with easy
as well as more challenging trails and that there
is considerable heterogeneity in visitors’re-
sponse to trails’features. For example, we note
that the MSL estimate imply that nearly 50%
of the population do not like hard trails. Per-
haps the nature of trails helps in sorting the
composition of the visiting party or the pur-
pose of the recreational visit.
Table 3. Statistics of Simulated WTPs from Models in Preference Space in €
From HB Estimates From MSL Estimates
Site Attributes Median Mean Std. Dev. Median Mean Std. Dev.
Uncorrelated models
Degree of difficulty −2.35 −3.62 5.44 −1.65 −2.99 6.49
Ferrata −0.39 −0.58 0.77 −0.21 −0.40 1.22
% of easy trails 0.06 0.11 0.35 0.03 0.06 0.79
Alpine shelters 0.34 0.51 0.65 0.22 0.42 1.42
% of hard trails 0.28 0.44 0.73 0.16 0.32 2.18
Prealps ASC −4.83 −7.34 10.07 −3.31 −5.75 10.10
Correlated models
Degree of difficulty −3.04 −4.52 8.77 −2.08 −3.12 7.10
Ferrata −0.48 −0.67 1.65 −0.31 −0.29 1.08
% of easy trails 0.09 0.18 0.84 0.05 0.09 0.16
Alpine shelters 0.36 0.40 1.29 0.26 0.21 0.83
% of hard trails 0.35 0.53 1.44 0.21 0.34 0.70
Prealps ASC −6.93 −7.87 19.52 −4.72 −2.67 20.77
Using the estimates for the means of the
latent normal variables and their variance-
covariance matrices, one can simulate the im-
plied distribution of WTP in the population
of visitors. The means, medians and standard
deviations are given in table 3. The implied
distribution of WTP is highly skewed, as ev-
idenced by the absolute values of the mean
WTP being considerably larger than those of
the median for all attributes. Importantly, the
estimates imply a fairly large proportion of
visitors have WTP values that might appear
to some as excessively large when compared
to the sample distribution of round-trip travel
costs ( ¯x=€9.41,s2=€4.29, 95th quantile =
€16.73). Among these are the “Degree of dif-
ficulty”of excursions, the number of “Ferrata”
and the “% of hard trails.”For example, the
MSL model in preference space implies that
10% of visitors are, on the margin, WTP over
€20 to avoid 1 extra level of difficulty per
choice occasion, 5% are WTP more than €3
to avoid an extra “Ferrata”at destination, and
10% are willing to pay over €30 to have the
network of trails classified as difficult increased
by 10%. Similar results are typically found in
specification searches for applications in which
the price coefficient is allowed to vary across
agents, and indeed it often motivates the as-
sumptions of a fixed “travel cost”coefficient,
which, despite the low conceptual plausibility,
are currently prevailing in the published liter-
ature.
The correlation matrices across WTPs ob-
tained by simulating the population distribu-
tion of the utility parameters according to
these estimates are reported in the lower
1002 November 2008 Amer. J. Agr. Econ.
Table 4. WTP Correlations
Site Attributes HB Estimates
Degree of difficulty 1 0.60 −0.35 −0.40 −0.59 0.73
Ferrata 0.43 1 −0.30 −0.80 −0.42 0.61
% of easy trail −0.13 −0.12 1 0.04 0.68 −0.51
Alpine shelters −0.20 −0.48 0.04 1 0.27 −0.40
% of hard trail −0.32 −0.27 0.34 0.14 1 −0.46
Prealps ASC 0.63 0.48 −0.14 −0.38 −0.21 1
MSL Estimates
Degree of difficulty 1 0.80 −0.80 −0.66 −0.73 0.71
Ferrata 0.57 1 −0.52 −0.93 −0.46 0.83
% of easy trail 0.16 −0.07 1 0.41 0.68 −0.64
Alpine shelters −0.38 −0.97 0.11 1 0.33 −0.75
% of hard trail −0.21 −0.02 −0.04 0.02 1 −0.32
Prealps ASC 0.63 0.70 −0.01 −0.67 0.31 1
Note: Upper triangular from WTP space and lower triangular from preference space.
triangular part of table 4, with the top part
of the table showing the HB estimates, and
the bottom part showing the MSL ones. These
estimates mostly concord in signs across esti-
mators with only 2 out 15 correlations being
different. A large positive correlation is found
between WTP for the number of “Ferrata”
at destination and the “Degree of difficulty,”
which is behaviorally very plausible, and sim-
ilarly plausible is the strong negative correla-
tion between WTP for “Alpine shelters”and
the “Degree of difficulty.”“Alpine shelters”
are particularly valuable to hikers embarking
on routes leading to peaks and vantage points,
Table 5. Hierarchical Bayes (HB) and Maximum Simulated Likelihood (MSL) Estimates for
WTP Space Models in €
Statistics of HB post. Distribution Statistics of MSL Estimates
WTP Parameters
ln( ˆ
) and ˆ
wMean Std. err. Std. Dev. Std. err. Mean Std. err. Std. Dev. St. err.
ln( ˆ
)−1.41 0.04 0.74 0.24 −1.22 0.06 0.67 0.05
Degree of difficulty −2.80 0.16 2.24 0.83 −1.99 0.20 2.19 0.33
Ferrata −0.37 0.02 0.21 0.08 −0.31 0.03 0.06 0.04
% of easy trails 0.07 0.01 0.09 0.03 0.07 0.01 0.03 0.01
Alpine shelters 0.35 0.01 0.17 0.06 0.32 0.02 0.12 0.02
% of hard trails 0.30 0.02 0.23 0.08 0.28 0.03 0.16 0.01
Prealps ASC −4.54 0.32 4.60 1.72 −4.39 0.46 3.97 0.39
Uncorrelated: ln L∗at means of post. dist. −20,471 ln L∗at convergence −20,420
ln( ˆ
)−1.81 0.05 0.74 0.25 −1.16 0.04 0.04 (∗)
Degree of difficulty −5.59 0.34 5.87 2.25 −2.85 0.16 2.98
Ferrata −0.60 0.05 0.74 0.28 −0.37 0.02 0.37
% of easy trails 0.16 0.02 0.27 0.09 0.10 0.01 0.08
Alpine shelters 0.53 0.04 0.51 0.19 0.36 0.02 0.23
% of hard trails 0.56 0.05 0.70 0.26 0.37 0.02 0.38
Prealps ASC −7.37 0.78 13.78 5.13 −5.76 0.36 6.57
With correlation: ln L∗at means of post. dist. −20,326 ln L∗at convergence −20,068
Note: (∗) Std. err. for the elements of the Cholesky matrix are reported in Scarpa, Thiene, and Train (2008).
many of which are accessible via “Ferrata.”On
the other hand, those who enjoy sites with a
high “Degree of difficulty”are likely to be less
dependent on the proximity of shelters for pro-
tection from sudden change of weather as they
would probably be better equipped.
WTP Space
A salient feature of the WTP space model
is that estimated parameters are also the pa-
rameters of the implied WTP distributions.
In table 5 we report HB and MSL estimates
of models parameterized in WTP space, i.e.,
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 1003
according to equation (4). Estimates for the
model without correlation are reported in the
top part of the table, while the estimates with
full correlation are in the bottom part of the
table. The simulated log-likelihood is higher
for the models in WTP space than in pref-
erence space: −20,470.89 versus −20,773.59
for uncorrelated terms, and −20,325.55 com-
pared to −20,383.65 for models with corre-
lated terms using HB. This result, which differs
from the findings of Train and Weeks (2005)
and Sonnier, Ainslie, and Otter (2007) on SP
data, indicates that it is possible for models in
WTP space to statistically outperform models
in preference space. A similar improvement is
found for the MSL estimates reported in the
right column of the table. The associated esti-
mates for the Cholesky matrix are available in
Scarpa, Thiene, and Train (2008).
The MSL estimates imply smaller WTP vari-
ation than the HB ones for all attributes, but
means have identical signs and very similar
magnitudes. Models with correlation also uni-
formly imply smaller WTP variation in the
population, with exclusion of the “Prealps
ASC”in the MSL model. Examining the up-
per triangular sections of table 4 we note that
estimated correlation match perfectly in sign
between the HB (upper part of the table) and
MSL estimates.
The estimated standard deviations of WTP
are uniformly lower for the models in WTP
space than the models in preference space.
For example, in the HB models with corre-
lated terms, the standard deviation of WTP
for “Alpine shelters”is 1.29 for the model in
preference space and 0.51 for the model in
WTP space. However, the estimated means
are not consistently higher or lower under ei-
ther parameterization: with correlated terms,
the HB model in WTP space gives a higher
mean than the model in preference space for
three attributes and a lower mean for the
other three. The share of extreme values for
WTP is much smaller with the models in WTP
space than with the models in preference space.
For example, the correlated preference space
HB model implies that 5% of the popula-
tion is willing to pay at least €1.41 for 1% in-
crease in easy trails.7In contrast, the correlated
model in WTP space implies a more plausible
€0.60.
7The distribution of
WTP was simulated with 100,000 draws
from the distributions evaluated at the estimated location and scale
parameters.
Figure 1. Distributions of WTP for one addi-
tional “Alpine shelter”
This point is visually described in figure 1
obtained with the SM package in R (Bowman
and Azzalini 1997). Here we plot the kernel
smoothing with cross-validated bandwidth of a
simulation of 100,000 draws from each model’s
WTP for an extra alpine shelter at destination.
The densities implied by the models in WTP
space are much “tighter”than those implied
by the models in preference space. As a result,
some analysts could find the WTP distributions
implied by WTP space models more plausible.
Comparison
The two estimation methods produced very
similar results, but the estimation via MSL was
1004 November 2008 Amer. J. Agr. Econ.
more difficult and took much longer than HB.
When the MSL estimation was started at the
convergence values of the HB procedures, the
WTP space model with correlation took four
times as long to run as the equivalent model
using HB. With other starting values, it took
even longer. For these reasons, HB estimation
has an advantage. The potential disadvantage
of HB estimation is in determining whether
the sampler has actually converged. We eval-
uated convergence of the HB sampler both
visually and formally using the test suggested
by Geweke (1992) and Koop (2003). The tests
were passed for all coefficients at conventional
significance levels of 10%.
The correlated models always show signif-
icantly better fit than the uncorrelated ones.
These models are also more reasonable a pri-
ori, since one would expect correlations among
both WTP and utility coefficients. In particular,
the scale parameter induces correlation among
utility coefficients, such that allowing correla-
tion in models in preference space is recom-
mended even if the analyst believes that WTP
is not correlated, as discussed more extensively
in Train and Weeks (2005).
The “long tails”of the WTP distributions
implied by the models in preference-space
might be thought to be a direct consequence
of the distributional assumptions invoked for
the utility coefficients. In particular, they might
be thought to be linked to the long tail of the
lognormal distribution used to model the vari-
ation of the travel cost coefficient and to the
normal distribution, which is known to span
the entire real line. If this were the case such
effect would disappear if the assumed distribu-
tions were bounded over a plausible range of
values for each utility coefficient. In order to
empirically investigate this issue we used the Sb
distribution, which was introduced in this lit-
erature by Train and Sonnier (2005) and used
previously by Rigby and Burton (2006), which
is defined over an upper and lower bound.8
We estimate two models with fully correlated
taste intensities using the HB estimator—one
in preference space and the other in WTP
space where the Sbdistribution was assumed
for the price coefficient and the coefficients
for Alpine shelters. All other taste intensities
were assumed normally distributed.9In both
8If s∼Sbthen s=a+(b−a)(1 +e−x)−1and x∼N(,2)
while aand bare the lower and upper bound of the range.
9The simulated log-likelihood at the posterior distribution was
−20,706 for the preference space model and −20,177 for the WTP
space model, hence showing that this bounded distributions imply
a slight worsening of statistical fit. The entire set of model estimates
are reported in Scarpa, Thiene, and Train (2008).
specifications the ∼Sb[0, 2], while the distri-
bution for Alpine shelters was cshelter
n∼Sb[0, 2]
in preference space and wshelter
n∼Sb[0, 1.5] in
WTP space. The different upper bound reflects
differences in the intervals associated with the
highest density observed in estimation results
with unbounded distributions illustrated in
figure 1.
The results are best represented graphically.
Figure 2 reports the histograms describing the
implied WTP distributions for the two speci-
fications. The distribution from the model in
WTP space (in the top panel) is contained in
the interval €[0,1.5], which are the bounds of
the Sbdistribution that was specified for it.
However, the use of bounded distributions in
preference space does not have the same ef-
fect: the implied distribution of WTP (in the
bottom two panels) shows a very long tail.
In the top two panels, the units on the x-axis
are the same and cover the interval €[0,1.5].
However, the preference space WTP distribu-
tion extends far beyond 1.5; the bottom panel
shows the extension of the histogram for this
distribution over the interval €[1.5,16]. These
results illustrate that assuming bounded distri-
butions for utility coefficients does not elim-
inate the long tails of the WTP distributions.
More importantly, it does not imply bounded-
ness of such distributions. This is because the
skewness arises from the WTP formula being
a ratio of two random coefficients. When the
denominator is close to zero, the ratio becomes
arbitrarily large, even when the numerator is
bounded. The ability to invoke and specify dis-
tributional assumptions directly in WTP-space
is an important practical advantage afforded to
analysts. It avoids the analyst achieving such
control by assuming a conceptually undesir-
able fixed “travel cost”coefficient or bound-
ing the price coefficient to be some arbitrary
distance from zero.
Policy Implications and Conclusions
This study investigated destination choices of
an inherently diverse population of visitors to
alpine destinations in the northeast of Italy:
those members of the local (Veneto) chap-
ter of the alpine club visiting the Alps for
day-out trips. Using a panel data set of 858
respondents who took a total of 9,221 trips,
we estimate WTP distributions for key site
attributes using models parameterized in pref-
erence space and in WTP space. Because pa-
rameters enter nonlinearly in the model in
WTP space and the number of parameters
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 1005
Figure 2. Histogram of WTP for one addi-
tional “Alpine shelter”
is large when correlations are allowed, previ-
ous studies used hierarchical Bayes estimation
procedures, which are computationally faster
than maximum simulated likelihood for mod-
els of this form. We contrasted HB and MSL
estimates and found them to produce similar
results, with the latter implying smaller varia-
tion of taste and hence of values. However, we
note that MSL estimates are significantly more
time consuming to derive.
Our results confirm previous findings ob-
tained by Train and Weeks (2005) and Sonnier,
Ainslie, and Otter (2007) that the models in
WTP space provide estimates of WTP distri-
butions that have lower densities associated
with extreme WTP values than the models in
preference space. As a result some analysts
might judge WTP space models more reason-
able. However, unlike these previous studies,
we find that the specification in WTP space sta-
tistically outperforms that in preference space.
This means that practitioners need not face
a trade-off between plausibility of WTP esti-
mates and model fit to the data, as was pre-
viously suggested. We also illustrate how the
undesirable skewness of WTP distributions
derived from preference space specifications
based on a random travel cost coefficient mod-
els is not eliminated by assuming bounded dis-
tributions. WTP space specifications therefore
emerge as a natural choice when the analyst
wants to directly control the distributions of
marginal WTP.
Although the main objective of the paper
is methodological, the estimation results from
the MSL model in WTP space with correlated
terms—which fits the data best—provide some
interesting implications. About 83% of day vis-
itors are estimated to dislike sites with high
“Degree of difficulty”of hiking activities. Only
about 17% show a positive WTP value for this
attribute. Similarly, a large number of “Fer-
rata”at the site is attractive to only about
16% of the population of day-out visitors. The
presence of “Alpine shelters”is preferred by
the vast majority of visitors: only 5% of visi-
tors prefer sites without the shelters. For most
members of the Italian Alpine Club, the site
becomes more attractive as the percent of trails
that are classified as easily walkable and hard
walkable (as opposed to those with mixed clas-
sification) rises. Finally, visitors are found to be
willing to pay more to visit the Dolomites than
the “Prealps,”which—given the popularity of
these sites—is a perhaps foregone conclusion,
but it is nevertheless confirmed by the nega-
tive sign of the alternative specific constant for
“Prealps.”
Further research and applications of mod-
els specified in the WTP space should ad-
dress their performance in models that include
nested recreation choices (e.g., participation
decisions) and in the derivation of consumer
surplus measures for site attribute changes that
vary across sites.10 This will lead to a more com-
plete understanding of this category of models
10 We are grateful to an anonymous referee for this suggestion.
1006 November 2008 Amer. J. Agr. Econ.
which we believe enrich further the tool kit of
analysts interested in nonmarket valuation.
[Received August 2007;
accepted February 2008.]
Appendix
Utility in WTP Space: A Tool to Address
Confounding Random Scale Effects in
Destination Choice to the Alps
The following tables collect auxiliary estimates for
the above mentioned study.
Table A.1 reports the summary statistics for the
ML estimates of the basic MNL model. As can be
seen by comparing the log-likelihood value at the
maximum with those reported in the paper and ob-
tained by MSL (or simulated at the posterior in the
case of HB), the RPL models produce a large im-
provement in fit.
Table A.1. Summary of MNL Model
Model : Multinomial Logit
Number of estimated parameters : 7
Number of observations : 9,221
Number of individuals : 9,221
Null log-likelihood : −26,652.12
Init log-likelihood : −50,407.59
Final log-likelihood : −21,754.39
Likelihood ratio test : 9,795.45
Rho-square : 0.1838
Adjusted rho-square : 0.1835
Final gradient norm : +1.739e−003
Variance-covariance : From analytical hessian
Table A.2. Estimates of MNL Model
Robust
Variable Coeff. Asympt.
Number Description Estimate Std. Error t-Stat p-Value WTP
1 Travel cost −0.2835 0.0057 −49.3 0.00
2 Degree of difficulty −0.5600 0.0208 −26.8 0.00 −1.975
3 Ferrata −0.0793 0.0046 −17.3 0.00 −0.280
4 % of easy trails 0.0157 0.0013 12.3 0.00 0.055
5 Alpine shelters 0.0885 0.0032 27.2 0.00 0.312
6 % of hard trails 0.0797 0.0033 24.1 0.00 0.281
7 Prealps ASC −0.8917 0.0619 −14.4 0.00 −3.145
Table A.2 reports the estimated ML parame-
ters of the MNL model. The WTP estimates show
similar magnitudes to the means of their RPl
counterparts.
Table A.3 reports the estimated Cholesky ma-
trix for the MSL estimate in WTP space. From this
one can derive the variance-covariance matrix of
the multivariate distribution of WTPs, and the as-
sociated correlation matrix.
Table A.4 reports the estimated Cholesky ma-
trix for the MSL estimate in preference space. From
one can derive the variance-covariance matrix of the
multivariate distribution of taste intensities for site
attributes. These, along with the mean estimates can
this be used to simulate draws which in turn can be
sued to compute WTP distributions.
Table A.5 reports the estimates of the WTP space
model with bounded distributions for ln() and
number of Alpine shelters.
Table A.6 reports the estimates of the preference
space model with bounded distributions for ln()
and number of Alpine shelters.
Table A.7 reports the correlations of the latent
variables for both the bounded specifications.
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 1007
Table A.3. Cholesky Matrix from MSL Estimates in WTP Space
Degree % Easy Alpine % Hard Prealps
Parameters ln of Diff. Ferrata Trails Shelters Trails ASC
ln −0.043
(21.5)
Degree of difficulty 0.193 −2.977
(1.7) (19.4)
Ferrata 0.067 −0.291 0.220
(2.9) (11.1) (9.3)
% of easy trail −0.007 0.060 0.015 −0.043
(1.1) (7.5) (1.3) (12.5)
Alpine shelters −0.037 0.148 −0.149 −0.003 −0.081
(2.2) (8.8) (8.5) (0.3) (7.0)
% of hard trail 0.011 0.279 0.070 −0.038 0.024 −0.244
(0.5) (11.2) (2.2) (4.3) (2.1) (10.5)
Prealps ASC 2.520 −4.517 2.449 1.605 −0.014 −1.312 2.490
(7.9) (11.4) (7.2) (5.3) (1.6) (4.2) (14.2)
Note: |z-values|in brackets.
Table A.4. Cholesky Matrix from MSL Estimates in Preference Space
Degree % Easy Alpine % Hard Prealps
Parameters ln of Diff. Ferrata Trails Shelters Trails ASC
ln 0.92
(20.4)
Degree of difficulty −0.19 0.70
(3.9) (19.7)
Ferrata −0.06 0.05 −0.08
(5.5) (6.5) (7.3)
% of easy trail 0.00 0.00 0.00 0.01
(0.4) (1.3) (1.0) (0.7)
Alpine shelters 0.06 −0.02 0.06 −0.00 −0.00
(8.1) (3.5) (9.6) (0.1) (0.7)
% of hard trail 0.01 −0.01 −0.02 0.00 −0.03 0.06
(2.8) (2.2) (2.0) (0.9) (6.1) (10.8)
Prealps ASC −1.29 0.92 −0.34 −0.07 −0.02 1.08 −0.01
(7.3) (8.2) (2.4) (0.5) (4.0) (14.8) (0.04)
Note: |z-values|in brackets.
Table A.5. Estimates of WTP Space Model with Sb.lnL−20,177.50
HB Estimates
Site Attributes
DISTR. PARAM. Mean St.dev. Var. St.dev. Var.
Sb[0, 2] ×c0.292 0.188 0.604 0.076
Normal Degree of difficulty −3.341 3.359 10.957 1.624
Normal Ferrata −0.450 0.419 0.176 0.024
Normal % of easy trails 0.119 0.156 0.023 0.003
Sb[0, 1.5] Alpine shelters 0.417 0.240 0.632 0.116
Normal % of hard trails 0.429 0.402 0.162 0.022
Normal Prealps ASC −5.952 8.132 62.996 8.949
1008 November 2008 Amer. J. Agr. Econ.
Table A.6. Estimates of Preference Space Model with Sb.lnL−20,706.25
HB Estimates
Site Attributes
DISTR. PARAM. Mean St.dev. Var. St.dev. Var.
Sb[0, 2] 0.383 0.291 1.076 0.128
Normal Degree of difficulty −0.920 0.932 0.877 0.105
Normal Ferrata −0.151 0.153 0.023 0.002
Normal % of easy trails 0.031 0.076 0.006 0.000
Sb[0, 2] Alpine shelters 0.133 0.097 0.572 0.082
Normal % of hard trails 0.119 0.142 0.021 0.002
Normal Prealps ASC −2.196 2.284 4.937 0.613
Table A.7. Correlations from HB Estimates of Models with Bounded Distributions
Correlation Matrix for Random WTP for WTP Space Model with Sb.
Site Attributes
PARAM. ln ˆ
Deg. of Diff. Ferrata % Easy Trails Alp. Shelters % Hard Trails Prealps
ln ˆ
1−0.2737 −0.1885 0.042 0.079 0.046 −0.4014
Degree of diff. −0.2737 1 0.6441 −0.3248 −0.4825 −0.5496 0.7706
Ferrata −0.1885 0.6441 1 −0.2434 −0.7887 −0.3309 0.6533
% of easy trails 0.042 −0.3248 −0.2434 1 0.1551 0.6308 −0.4181
Alpine shelters 0.079 −0.4825 −0.7887 0.1551 1 0.1682 −0.5409
% of hard trails 0.046 −0.5496 −0.3309 0.6308 0.1682 1 −0.3489
Prealps ASC −0.4014 0.7706 0.6533 −0.4181 −0.5409 −0.3489 1
Correlation Matrix for Utility Coefficients of Preference Space Model with Sb
Site Attributes
PARAM. ln ˆ
Deg. of Diff. Ferrata % Easy Trails Alp. Shelters % Hard Trails Prealps
ln ˆ
1−0.1956 −0.3122 0.0237 0.6219 0.1775 −0.4204
Degree of diff. −0.1956 1 0.4955 −0.0801 −0.4014 −0.3038 0.6971
Ferrata −0.3122 0.4955 1 −0.1066 −0.5936 −0.2441 0.6077
% of easy trails 0.0237 −0.0801 −0.1066 1 0.1282 0.3609 −0.2097
Alpine shelters 0.6219 −0.4014 −0.5936 0.1282 1 0.2207 −0.6711
% of hard trails 0.1775 −0.3038 −0.2441 0.3609 0.2207 1 −0.2447
Prealps ASC −0.4204 0.6971 0.6077 −0.2097 −0.6711 −0.2447 1
References
Albert, J., and S. Chib. 1993. “Bayesian Analysis of
Binary and Polychotomous Response Data.”
Journal of the American Statistical Association
88:669–79.
Allenby, G. 1997. “An Introduction to Hierarchical
Bayesian Modeling.”Tutorial notes, Advanced
Research Techniques Forum, American Mar-
keting Association.
Allenby, G., and P. Rossi. 1999. “Marketing Mod-
els of Consumer Heterogeneity.”Journal of
Econometrics 89:57–78.
Bierlaire, M. 2002. “An Introductory Tutorial
to BIOGEME. Federal Polytechnic School
of Lausanne, Switzerland.”http://rosowww.
epfl.ch/mbi/biogeme.
——. 2003. “BIOGEME: A Free Package for
the Estimation of Discrete Choice Models.”
Paper presented at the 3rd Swiss Transport
Research Conference Monte Verita, Ascona,
March 2003.
Bockstael, N.E., W.M. Haneman, and C.L. Kling.
1987. “Estimating the Value of Water Qual-
ity Improvements in a Recreational De-
mand Framework.”Water Resources Research
23:951–60.
Bowman, A.W., and A. Azzalini. 1997. Applied
Smoothing Techniques for Data Analysis: The
Kernel Approach with S-Plus Illustrations. Ox-
ford, UK: Oxford University Press.
Breffle, W.S., and E.R. Morey. 2000. “Investigat-
ing Preference Heterogeneity in a Repeated
Discrete-choice Recreation Demand model of
Atlantic Salmon Fishing.”Marine Resource
Economics 15:1–20.
Cameron, T. 1988. “A New Paradigm for Valuing
Non-market Goods using Referendum Data:
Scarpa, Thiene, and Train Modeling Destination Choice in the Alps in WTP Space 1009
Maximum Likelihood Estimation by Censored
Logistic Regression.”Journal of Environmen-
tal Economics and Management 15:355–79.
Cameron, T., and M. James. 1987. “Efficient Esti-
mation Methods for Closed-ended Contingent
Valuation Survey Data.”Review of Economics
and Statistics 69:269–76.
Casella, G., and E. George. 1992. “Explaining
the Gibbs Sampler.”The American Statistician
46:167–74.
Geweke, J. 1992. “Evaluating the Accuracy of
Sampling-Based Approaches to the Calcula-
tion of Posterior Moments.”In J. Bernardo, J.
Berger, A. Dawid, and F. Smith, eds. Bayesian
Statistics. New York: Oxford University Press,
pp. 169–93.
Goett, A., K. Train, and K. Hudson. 2000. “Cus-
tomer Choice Among Retail Energy Sup-
pliers: The Willingness-to-Pay for Service
Attributes.”Energy Journal 21:1–28.
Hanemann, W.M. 1984. “Welfare Evaluations in
Contingent Valuation Experiments with Dis-
crete Response Data.”American Journal of
Agricultural Economics 66:332–41.
——. 1989. “WelfareEvaluations in Contingent Val-
uations Experiments with Discrete Response
Data: A Reply.”American Journal of Agricul-
tural Economics 71:1057–61.
Hess, S., K.E. Train, and J.W. Polak. 2006. “On the
Use of a Modified Latin Hypercube Sampling
(MLHS) Method in the Estimation of a Mixed
Logit Model for Vehicle Choice.”Transporta-
tion Research B 40:147–67.
Koop, G. 2003. Bayesian Econometrics. New York:
Wiley.
Lawrence, C., J. Zhou, and A. Tits. 1997.
“User’s Guide for CFSQP Version 2.5: A C
Code for Solving (large scale) Constrained
Nonlinear (minimax) Optimization Problems,
Generating Iterates Satisfying all Inequality
Constraints.”Technical Report TR-94-16r1,
Institute for Systems Research, University of
Maryland, College Park, MD 20742, 1997.
Layton, D.F., and G. Brown. 2000. “Heteroge-
neous Preferences Regarding Global Climate
Change.”The Review of Economics and Statis-
tics 82(4):616–24.
Layton, D., and R. Levine. 2005. “Bayesian
Approaches to Modeling Stated Preference
Data.”In R. Scarpa and A. Alberini, eds. Ap-
plications of Simulation Methods in Environ-
mental and Resource Economics. Dordrecht,
The Netherlands: Springer Publisher, chapter
10, pp. 187–208.
McConnell, K. 1995. “Consumer Surplus from Dis-
crete Choice Models.”Journal of Environmen-
tal Economics and Management 29:263–70.
McCulloch, R., and P. Rossi. 1994. “An Exact
Likelihood Analysis of the Multinomial Pro-
bit Model.”Journal of Econometrics 64:207–
40.
Morey, E., and K. Rossmann. 2003. “Using Stated-
Preference Questions to Investigate Variations
in Willingness to Pay for Preserving Marble
Monuments: Classic Heterogeneity, Random
Parameters, and Mixture Models.”Journal of
Cultural Economics 27(3/4):215–29.
Morey, E., R. Rowe, and M. Watson. 1993. “A Re-
peated Nested-Logit Model of Atlantic Salmon
Fishing.”American Journal of Agricultural
Economics 75:578–92.
Provencher, B., K. Baerenklau, and R.C. Bishop.
2002. “A Finite Mixture Model of Recreational
Angling with Serially Correlated Random
Utility.”American Journal of Agricultural Eco-
nomics 84(4):1066–75.
Revelt, D., and K. Train. 1998. “Mixed Logit with
Repeated Choices.”Review of Economics and
Statistics 80(4):647–57.
Rigby, D., and M. Burton. 2006. “Modeling Disin-
terest and Dislike: A Bounded Bayesian Mixed
Logit Model of the UK Market for GM Food.”
Environment and Resource Economics 33:485–
509.
Rossi, P.E., G. Allenby, and R. McCulloch. 2005.
Bayesian Statistics and Marketing. New York:
Wiley.
Ruud, P. 1996. “Simulation of the Multinomial Pro-
bit Model: An Analysis of Covariance Matrix
Estimation.”Working paper, Department of
Economics, University of California.
Sawtooth Software. 1999. “The CBC/HB Mod-
ule for Hierarchical Bayes.”Technical pa-
per available on the web site of Sawtooth
Software.
Scarpa, R., and M. Thiene. 2005. “Destination
Choice Models for Rock Climbing in the
Northeastern Alps: A Latent-Class Approach
Based on Intensity of Preference.”Land Eco-
nomics 85(3):426–44.
Scarpa, R., M. Thiene, and K. Train. 2008. AJAE
Appendix: “Utility in WTP Space: A Tool to
Address Confounding Random Scale Effects
in Destination Choice to the Alps.”Available
at: http://agecon.lib.umn.edu/.
Sonnier, G., A. Ainslie, and T. Otter. 2007. “Hetero-
geneity Distributions of Willingness-to-Pay in
Choice Models.”Quantitative Marketing Eco-
nomics 5(3):313–31.
Train, K. 1998. “Recreation Demand Models with
Taste Differences over People.”Land Eco-
nomics 74:185–94.
——. 1999. “Mixed Logit Models for Recreation
Demand.”In J. Herriges and C. Kling, eds.
1010 November 2008 Amer. J. Agr. Econ.
Valuing Recreation and the Environment,pp.
121–40. Northampton, MA: Edward Elgar.
——. 2003. Discrete Choice Methods with Simula-
tion. New York: Cambridge University Press.
Train, K., and G. Sonnier. 2005. “Mixed Logit with
Bounded Distributions of Correlated Part-
worths.”In R. Scarpa and A. Alberini, eds.
Applications of Simulation Methods in En-
vironmental and Resource Economics. Dor-
drecht, The Netherlands: Springer Publisher,
chapter 7, pp. 117–34.
Train, K., and M. Weeks. 2005. “Discrete Choice
Models in Preference Space and Willing-to-
Pay space.”In R. Scarpa and A. Alberini,
eds. Applications of Simulation Methods in
Environmental and Resource Economics. Dor-
drecht, The Netherlands: Springer Publisher,
chapter 1, pp. 1–16.
