Non-Linear Mixed Logit

We develop an extension of the familiar linear mixed logit model to allow for the direct estimation of parametric non-linear functions defined over structural parameters. Classic applications include the estimation of coefficients of utility functions to characterize risk attitudes and discounting functions to characterize impatience. There are several unexpected benefits of this extension, apart from the ability to directly estimate structural parameters of theoretical interest. † Department of Economics, Copenhagen Business School, Copenhagen, Denmark (Andersen); Department of Risk Management & Insurance and CEAR, Robinson College of Business, Georgia State University, USA (Harrison); Department of Economics, University of Sheffield, UK (Hole); Durham Business School, Durham University (Lau); and Robinson College of Business and Department of Economics, Andrew Young School of Policy Studies, Georgia State University, USA (Rutström). E-mail contacts: [email protected], [email protected], [email protected], [email protected] and [email protected]. We thank the U.S. National Science Foundation for research support under grants NSF/HSD 0527675 and NSF/SES 0616746, the Danish Social Science Research Council for research support under project 275-08-0289, and the Carlsberg Foundation under grant 2008_01_0410. 1 Of course all logit models are non-linear in the sense that the probabilities are non-linear functions of the latent index, which is a linear function of the explanatory variables and the estimated coefficients. We use the term “non-linear mixed logit” to refer to mixed logit models with a latent index that is a non-linear function of these variables and coefficients. -1The linear mixed logit model provides a valuable inferential tool, allowing unobserved individual heterogeneity in the population to be characterized by random coefficients following some parametric distribution. There are several advantages, however, from considering the extension to the parametric non-linear mixed logit model.1 First, and foremost, one can directly identify and estimate parameters from non-linear structural models, such as coefficients that reflect risk aversion and discounting behavior in individual choice settings. Second, one can easily include nonlinear transformations of familiar parametric families, such as the Normal, that allow theoretically constrained and flexible characterizations of population distributions. Third, one can directly estimate coefficients on demographic explanatory variables without having to “interact” them with task characteristics. We state the formal non-linear mixed logit model in section 1, consider the immediate extension to allow flexible population distributions for the random parameters in section 2, and offer illustrative applications in sections 3 and 4. In section 5 we explain the differences between a parametric non-linear specification and semi-parametric and flexible-functional-form alternatives that have been proposed long ago in the econometric literature. 1. Linear and Non-Linear Mixed Logit For pedagogic purposes, we consider the formulation of parametric, non-linear mixed logit models to estimate structural parameters of a decision-making model of risk attitudes. One natural reason for this focus is that mainstream theory defines risk attitudes in terms of the non-linearity of the utility function. Even non-mainstream utility theories typically allow for non-linearity in utility and/or probability weighting functions, so this application is arguably canonical for the estimation of structural parameters applied in non-linear functional forms. The methodology could just as easily be applied to recover structural parameters in production functions. The overall objective is to be