  We propose a framework for nonparametric identification and estimation of
discrete choice models with unobserved choice sets. We recover the joint
distribution of choice sets and preferences from a panel dataset on choices. We
assume that either the latent choice sets are sparse or that the panel is
sufficiently long. Sparsity requires the number of possible choice sets to be
relatively small. It is satisfied, for instance, when the choice sets are
nested, or when they form a partition. Our estimation procedure is
computationally fast and uses mixed-integer optimization to recover the sparse
support of choice sets. Analyzing the ready-to-eat cereal industry using a
household scanner dataset, we find that ignoring the unobservability of choice
sets can lead to biased estimates of preferences due to significant latent
heterogeneity in choice sets.
