Supplement
Discrete choices, such as tourism destination choices, have been of interest to
researchers from a variety of disciplines (psychology, marketing, outdoor
recreation, etc.) for many years. For this reason, probabilistic models based on
the principle of utility maximization have been developed to model choices made
by individuals from a set of mutually exclusive alternatives. However, since the
true utilities derived from the different alternatives are not observable, they are
considered to be random variables and hence the probability of an alternative
being chosen is defined as the probability of it having the greatest utility of all the
available alternatives in the choice set (Ben-Akiva and Lerman, 1985).
Different approaches to the well-known random utility model, such as the
conditional or the nested logit model, have been used to model destination
choice decisions in different contexts (Train, 2003). A generalization of the
standard logit model, known as the random parameter logit (RPL), has become
very common in this kind of analysis, mainly because it enables researchers to
account for the heterogeneous preferences of tourists within a discrete choice
framework, while also allowing for the representation of different correlation
patterns among non-independent alternatives (McFadden and Train, 2000;
Nicolau and Mas, 2005).
Following McFadden (1974), the utility Uni that tourist n derives from choosing to
visit destination i on a given occasion, when a choice set i =1,...,I exists, is
assumed to take the form of the conditional indirect utility function which,
following a linear specification, can be expressed as:
U ni   n xni   ni
[1]
where βn’xni is the non-stochastic portion of the indirect utility received during the
choice occasion if destination i is visited. Therefore, xni are observed attributes
characterizing the alternatives available to tourists and βn is the vector of
estimated coefficients for tourist n representing his tastes. Finally, the error term
εni captures the variation in preferences among tourists in the population. As the
individual is assumed to visit the destination yielding the greatest utility, the
probability πni of him choosing the i-th alternative is:
1
(
)
p ni = Pr bn¢ x ni + eni > bn¢ x nj + enj "j ¹ i
[2]
In addition, as it is usually assumed that the error term εni follows an independent
and identically distributed extreme value type-I distribution, the probability of
destination i being chosen in equation (2) can be expressed as (McFadden,
1978; Train, 2003):
¢
p ni =
e bn x ni
I
åe b
¢
n x nj
[3]
j =1
As βn is unknown to the researcher, a probability function for the coefficient vector
has to be specified and the parameters of such distribution have to be estimated.
In this way, by assuming that the coefficients of the explanatory variables vary
randomly among tourists, the model accounts for preference heterogeneity
regarding the variables included with random coefficients. At this point, although
more complex distributions can be used, it is quite common to specify it as a
normal β~N(b,W) with parameters b and W (Revelt and Train, 1998; McFadden
and Train, 2000). Whichever distribution is used, the choice probability of tourist n
visiting province i becomes the integral of expression (3). Consequently:
æ
ç b¢ x
ç e n ni
p ni = ò ç I
b¢ x
çç å e n nj
è j =1
ö
÷
÷
÷ f (b )db
÷÷
ø
[4]
Finally, the log-likelihood function for a given value of the parameter vector β
takes the form:
N
I
LL (b ) = åå y ni log p ni (b )
n=1 i=1
2
[5]
where N represents the number of tourists in the sample, πni(β) are the choice
probabilities from equation (4) and yni equals one when the n-th tourist chooses
province i and 0 otherwise. As the solution to expression (5) involves the
evaluation of a multiple-dimensional integral that does not have a closed-form,
the estimation of such a model requires the use of simulation methods like a
simulated maximum likelihood estimation (Bhat, 1998; Revelt and Train, 1998).
Once the model has been estimated, the recovered parameter vector β can be
used to simulate the destination choice probabilities in equation (3). In this way,
this kind of discrete choice analysis is not only useful in explaining tourists’
observed decisions, but also in evaluating the effects in the future of hypothetical
changes in the explanatory variables. Then, using data for future temperature
scenarios, the impact of climate change on tourism allocations can be
investigated.
References
Ben-Akiva, M., and Lerman, S.R. (1985). Discrete choice analysis: theory and
application to travel demand. Cambridge, Massachusetts: The MIT Press.
Bhat, C.R. (1998). Accommodating variations in responsiveness to level-ofservice measures in travel mode choice modeling. Transportation Research
Part A, 32(7):495-507.
Nicolau, J.L., and Más, F.J. (2005). Stochastic modeling. A three-stage tourist
choice process. Annals of Tourism Research 32(1):49-69.
McFadden, D. (1974). The Measurement of Urban Travel Demand. Journal of
Public Economics, 3:303-328.
McFadden, D. (1978). Modelling the choice of residential location. In McFadden,
D., et al. (Eds.) Spatial Interaction Theory and Planning Models. NorthHolland Publishing, Amsterdam, pp. 75-96.
McFadden, D., and Train, K.E. (2000). Mixed MNL Models for Discrete
Response. Journal of Applied Econometrics, 15(5): 447-470.
Revelt, D., and Train, K. (1998). Mixed Logit with Repeated Choices:
Households' Choices of Appliance Efficiency Level. The Review of
Economics and Statistics, 80(4):647-657.
Train, K.E. (2003). Discrete choice methods with simulation. Cambridge
University Press, Cambridge.
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