Confound it! That pesky little scale constant messes up our convenient assumptions
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Since the early 1990s there has been much progress in understanding and taking into account preference heterogeneity in probabilistic discrete choice models (e.g., Wedel and Kamakura '1999;McFadden and Train 2000). The vast majority of models applied in marketing and applied economics try to represent heterogeneity as some type of discrete or continuous distribution of preferences. These relatively new types of statistical models have done well in comparisons against simpler model forms like conditional multinomiallogit in terms of inand out-of-sample !fits,with fit performance often assessed against so-called hold-out sets. It is fair to say that these models are long on statistical theory, but short on behavioral theory; the latter aspect is the focus of this paper.Keywords:
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Multinomial logit (MNL) models are widely used in marketing research to analyze choice data, but it is not generally recognized that the unit of the utility scale in a MNL model is inversely related to the error variance. This means that, for instance, parameters of two identical utility specifications estimated from different data sources with unequal variances will necessarily differ in magnitude, even if the true model parameters that generated the utilities are identical in both sets. Despite a growing number of papers that compare MNL coefficients, no examples of appropriate tests of the joint and separate hypotheses of scale and parameter equality in MNL models exist in the marketing literature. The purpose of this paper is to address the proper procedure for MNL parameter comparisons between different data sets and to propose a simple relative scaling test that can be implemented with standard MNL estimation software. Several examples are given to illustrate the approach.
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We show how to combine statistically efficient ways to design discrete choice experiments based on random utility theory with new ways of collecting additional information that can be used to expand the amount of available choice information for modeling the choices of individual decision makers. Here we limit ourselves to problems involving generic choice options and linear and additive indirect utility functions, but the approach potentially can be extended to include choice problems with non-additive utility functions and non-generic/labeled options/attributes. The paper provides several simulated examples, a small empirical example to demonstrate proof of concept, and a larger empirical example based on many experimental conditions and large samples that demonstrates that the individual models capture virtually all the variance in aggregate first choices traditionally modeled in discrete choice experiments.
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The mixed or heterogeneous multinomial logit (MIXL) model has become popular in a number of fields, especially marketing, health economics, and industrial organization. In most applications of the model, the vector of consumer utility weights on product attributes is assumed to have a multivariate normal (MVN) distribution in the population. Thus, some consumers care more about some attributes than others, and the IIA property of multinomial logit (MNL) is avoided (i.e., segments of consumers will tend to switch among the subset of brands that possess their most valued attributes). The MIXL model is also appealing because it is relatively easy to estimate. Recently, however, some researchers have argued that the MVN is a poor choice for modelling taste heterogeneity. They argue that much of the heterogeneity in attribute weights is accounted for by a pure scale effect (i.e., across consumers, all attribute weights are scaled up or down in tandem). This implies that choice behaviour is simply more random for some consumers than others (i.e., holding attribute coefficients fixed, the scale of their error term is greater). This leads to a “scale heterogeneity” MNL model (S-MNL). Here, we develop a generalized multinomial logit model (G-MNL) that nests S-MNL and MIXL. By estimating the S-MNL, MIXL, and G-MNL models on 10 data sets, we provide evidence on their relative performance. We find that models that account for scale heterogeneity (i.e., G-MNL or S-MNL) are preferred to MIXL by the Bayes and consistent Akaike information criteria in all 10 data sets. Accounting for scale heterogeneity enables one to account for “extreme” consumers who exhibit nearly lexicographic preferences, as well as consumers who exhibit very “random” behaviour (in a sense we formalize below).
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This paper considers mixed, or random coefficients, multinomial logit (MMNL) models for discrete response, and establishes the following results. Under mild regularity conditions, any discrete choice model derived from random utility maximization has choice probabilities that can be approximated as closely as one pleases by a MMNL model. Practical estimation of a parametric mixing family can be carried out by Maximum Simulated Likelihood Estimation or Method of Simulated Moments, and easily computed instruments are provided that make the latter procedure fairly efficient. The adequacy of a mixing specification can be tested simply as an omitted variable test with appropriately defined artificial variables. An application to a problem of demand for alternative vehicles shows that MMNL provides a flexible and computationally practical approach to discrete response analysis. Copyright © 2000 John Wiley & Sons, Ltd.
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Abstract 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 .
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