Optimal linear discriminators for the discrete choice model in growing dimensions

Abstract

Manski’s celebrated maximum score estimator for the discrete choice model, which is an optimal linear discriminator, has been the focus of much investigation in both the econometrics and statistics literatures, but its behavior under growing dimension scenarios largely remains unknown. This paper addresses that gap. Two different cases are considered: p grows with n but at a slow rate, that is, $\mathit{p}/\mathit{n}\to 0$; and $\mathit{p}\gg \mathit{n}$ (fast growth). In the binary response model, we recast Manski’s score estimation as empirical risk minimization for a classification problem, and derive the ${\ell }_2$ rate of convergence of the score estimator under a new transition condition in terms of a margin parameter that calibrates the level of difficulty of the estimation problem. We also establish upper and lower bounds for the minimax ${\ell }_2$ error in the binary choice model that differ by a logarithmic factor, and construct a minimax-optimal estimator in the slow growth regime. Some extensions to the multinomial choice model are also considered.