Distribution theory for the analysis of binary choice under uncertainty with nonparametric estimation of expectations

In analyzing discrete choice under uncertainty, the practice has been to specify expectations and preferences up to a finite-dimensional parameter. Recently, Manski proved the consistency of a two-stage, semiparametric estimator applicable if expectations are fulfilled and are conditioned only on variables observed by the researcher. The first stage estimates expectations nonparametrically, and the second stage uses choice data and the expectations estimate to make parametric, quasi-maximum-likelihood inference on preferences. This paper proves that the estimate of preference parameters converges at rate √N to a limiting normal distribution if the expectations estimate is chosen appropriately. The estimate is √N-asymptotically unbiased. Its asymptotic variance exceeds the inverted Fisher information for the preference parameter.