Crossnested

Crossnested is a term used in discrete choice modeling, referring to the cross-nested logit (CNL) model. The CNL generalizes the standard nested logit by allowing each alternative to belong to multiple nests with partial membership weights. This structure provides a flexible way to capture correlation among alternatives that share attributes or travel modes, without forcing a rigid, mutually exclusive nesting.

In a crossnested specification, the choice space is organized into a set of nests, but unlike traditional nested logit, an alternative i can contribute to several nests simultaneously. Each association is governed by a membership weight, typically between zero and one, which reflects how strongly the alternative is tied to a given nest. The resulting choice probabilities are obtained from a generalized logit that aggregates across nests using these weights and nest-specific parameters. The model can reproduce various correlation patterns among alternatives, from the non-overlapping case of standard nested logit to highly overlapping structures.

Special cases and estimation: If each alternative belongs to exactly one nest (with binary weights), the crossnested logit collapses to the standard nested logit. If membership weights are set to produce mostly overlap, the model approaches the flexibility of a multinomial logit with complex correlation. Estimation is typically conducted via maximum likelihood or Bayesian methods, often requiring numerical optimization and, in some specifications, simulation due to the involved integrals. Practitioners rely on CNL when simpler models fail to capture observed substitution patterns among closely related choices.

Applications and history: The crossnested logit has been applied in transportation planning, retail and consumer choice, housing markets, and other areas where alternatives share attributes. It extends the nested logit framework introduced in the late 20th century, offering a middle ground between fixed nests and fully flexible random-coefficients models.