ECONOMETRICS: Finite Sample Inference for the Maximum Score Estimand ; Professor Adam M. ROSEN (Duke University)

Abstract

We provide a finite sample inference method for the structural parameters of the semiparametric binary response model under a conditional median restriction originally studied by Manski (1975, 1985). This is achieved by exploiting distributional properties of observable outcomes conditional on the observed sequence of exogenous variables. The model is one of many semiparametric models in which point identification is predicated on the existence of exogenous variables with continuous and possibly large support. These support conditions cannot hold in the empirical distribution obtained in any finite sample. We show that this carries an important implication for finite sample inference, namely that conditional on any size n sequence of the exogenous covariates, there is a set of observationally equivalent parameter values from which the true population parameter cannot be distinguished, and against which only trivial power can be guaranteed. Nonetheless, our finite sample inference method is valid for any sample size and irrespective of whether the structural parameters are point identified or partially identified. Moment inequalities conditional on the size n sequence of exogenous covariates are constructed, and the test statistic is a monotone function of violations of the corresponding sample moment inequalities. The critical value used for inference is provided by the appropriate quantile of a known function of n independent Rademacher random variables, and does not require the use of a cube root asymptotic approximation to an appropriately centered point estimator of the target parameter. Simulation studies compare the performance of the test to two alternative tests using an infeasible likelihood ratio statistic and Horowitz's (1992) smoothed maximum score estimator.