A discrete choice model for partially ordered alternatives

In this paper, we analyze a discrete choice model for partially ordered alternatives. The alternatives are differentiated along two dimensions: the first an unordered 'horizontal' dimension, and the second an ordered 'vertical' dimension. The model can be used in circumstances in which individuals choose among products of different brands, wherein each brand offers an ordered choice menu, for example, by offering products of varying quality. The unordered-ordered nature of the discrete choice problem is used to characterize the identified set of model parameters. Following an initial nonparametric analysis that relies on shape restrictions inherent in the ordered dimension of the problem, we then provide a specialized analysis for parametric specifications that generalize common ordered choice models. We characterize conditional choice probabilities as a function of model primitives with particular analysis focusing on cases in which unobservable taste for quality of each brand offering is multivariate normally distributed. We provide explicit formulae used for estimation and inference via maximum likelihood, and we consider inference based on Wald and quasi-likelihood ratio statistics, the latter of which can be robust to a possible lack of point identification. An empirical illustration is conducted using data on razor blade purchases in which each brand has product offerings vertically differentiated by quality.