  The multinomial probit model is a popular tool for analyzing choice behaviour
as it allows for correlation between choice alternatives. Because current model
specifications employ a full covariance matrix of the latent utilities for the
choice alternatives, they are not scalable to a large number of choice
alternatives. This paper proposes a factor structure on the covariance matrix,
which makes the model scalable to large choice sets. The main challenge in
estimating this structure is that the model parameters require identifying
restrictions. We identify the parameters by a trace-restriction on the
covariance matrix, which is imposed through a reparametrization of the factor
structure. We specify interpretable prior distributions on the model parameters
and develop an MCMC sampler for parameter estimation. The proposed approach
significantly improves performance in large choice sets relative to existing
multinomial probit specifications. Applications to purchase data show the
economic importance of including a large number of choice alternatives in
consumer choice analysis.
