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Random Projection Estimation of Discrete-Choice Models with Large Choice Sets

We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a lower-dimensional Euclidean space using random projections. Subsequently, estimation proceeds using cyclic monotonicity moment inequalities implied by the multinomial choice model; the estimation procedure is semi-parametric and does not require explicit distributional assumptions to be made regarding the random utility errors. The random projection procedure is justified via the Johnson-Lindenstrauss Lemma -- the pairwise distances between data points are preserved during data compression, which we exploit to show convergence of our estimator. The estimator works well in simulations and in an application to a supermarket scanner dataset.