The estimation problem in this paper is motivated by the maximum score estimation of preference parameters in the binary choice model under uncertainty in which the decision rule is affected by conditional expectations. The preference parameters are estimated in two stages. We estimate conditional expectations nonparametrically in the first stage. Then, in the second stage, we estimate the preference parameters based on the maximum score estimator of Manski, using the choice data and first-stage estimates. This setting can be extended to maximum score estimation with nonparametrically generated regressors. In this paper, we establish consistency and derive the rate of convergence of the two-stage maximum score estimator. Moreover, we also provide sufficient conditions under which the two-stage estimator is asymptotically equivalent in distribution to the corresponding single-stage estimator that assumes the first-stage input is known. We also present some Monte Carlo simulation results for the finite-sample behaviour of the two-stage estimator.
Centre for microdata methods and practice