This paper extends results regarding smoothed median binary regression to general smoothed binary quantile regression, discusses the interpretation of the resulting estimators under alternative assumptions, and shows how they may be used to obtain semiparametric estimates of counterfactual probabilities. The estimators are applied to a model of labour force participation of married women in the USA. We find that the elasticity with respect to non-labour income is significantly negative only for women that belong to the middle of the conditional willingness-to-participate (WTP) distribution. In comparing the quantile models with parametric logit and semiparametric single-index specifications, we find that the models agree closely for women around the centre of the WTP distribution, but there are considerable disagreements as we move towards the tails of the distribution.
