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Romain Crastes (University of Leeds)
Using shifted lognormal distributions in order to avoid “exploding” willingness-to-pay distributions in mixed logit models
Discrete choice experiments (also known as stated preference surveys or conjoint analysis) are a popular method for analysing preferences and measuring Willingness-To-Pay for different attributes of a policy or an environmental good. The literature on model specification mainly proposes two approaches for accounting for unobserved taste heterogeneity: (1) the preference space approach, where the distribution of WTP is derived from the distribution of coefficients specified by the analyst in the utility function and (2) the WTP space approach, where the analyst directly specifies the distribution of WTP, from which the distributions of coefficients can be derived. It is generally found that models in preference space fit the data better but models in WTP space provide more reasonable WTP distributions. Train and Weeks (2005) in a seminal paper suggest that further work is needed to identify distributions that either yield better fit in WTP space or deliver more reasonable WTP distributions in preference space.
WTP distributions in preference space are derived by dividing the coefficient of a given non-monetary attribute by the price coefficient. Unreasonable WTP distributions are often found when the price attribute is assumed to be (negative) lognormal or loguniform. This is because the lognormal and loguniform distributions have a point-mass near zero, leading to the issue known as the “exploding implicit price” problem (Giergiczny et al., 2002). At the same time, specifying the price attribute as being lognormal or loguniform in preference space often leads to specifications which outperform their counterparts in WTP space in terms of goodness-of-fit. This is especially true when most of the non-monetary attributes are assumed to be normally distributed.
In this paper, we propose to use a new distribution for the cost attribute, which we call the mu-shifted lognormal distribution, in order to provide more reasonable WTP distributions in preference space. The mu-shifted distribution is inspired from the three-parameter lognormal distribution originally suggested by Cohen and Whitten (1980). The three-parameter distribution features an additional shift parameter which can contribute to move the point mass of the (negative) lognormal distribution away from zero. The mu-shifted distribution simply consists in replacing the shift parameter by the logarithm of the mean of the negative lognormal distribution, which provides two desirable features:
i. It prevents the shift parameter from being positive, which is both behaviourally implausible and leads to WTP distributions with no existing moments given that the distribution of the price spans on both sides of zero
ii. It leads to a model which is more parsimonious in parameters
We test the proposed distribution on 10 datasets and compare it to seven other specifications including WTP space, a lognormal cost, a loguniform cost and a multinomial logit model, leading to a total of 80 models and 414 WTP distributions. The mu-shifted lognormal distribution is found to yield similar results than the lognormal distribution for the cost in terms of goodness-of-fit but provides much more reasonable WTP distributions. More precisely, we find that the WTP estimates derived from lognormal models are between 4.6 and 23 times higher than the WTP estimates derived from WTP space models, while the mu-shifted lognormal models yield WTP estimates which are only between 0.08 and 1.32 times higher than those derived from WTP space models. At the same time, we find no significant differences in terms of goodness-of-fit between the mu-shifted lognormal models and the lognormal models. Both distributional assumptions lead to reductions of the Bayesian Information Criterion comprised between 12% and 0.5% compared to WTP space models.
Read the paper (PDF)