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 amongst 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 a parametric generalization of the ordered probit model. Conditions for point identification are established when the distribution of unobservable heterogeneity is known, but remain elusive when the distribution is instead restricted to the multivariate normal family with parameterized variance. Rather than invoke the restriction that the distribution is known, or simply assume that model parameters are point identified, we consider the use of inference methods that allow for the possibility of set identification, and which are therefore robust to the possible lack of point identification. A Monte Carlo analysis is provided in which inference is carried out using a method proposed by Chen, Christensen, and Tamer (2018), which is insensitive to the possible lack of point identification and is found to perform adequately. An empirical illustration is then conducted using consumer purchase data in the UK to study consumers' choice of razor blades in which each brand has product offerings vertically differentiated by quality.