We consider nonparametric estimation of a regression function that is identified byrequiring a specified quantile of the regression “error” conditional on an instrumentalvariable to be zero. The resulting estimating equation is a nonlinear integral equation ofthe first kind, which generates an ill-posed-inverse problem. The integral operator anddistribution of the instrumental variable are unknown and must be estimatednonparametrically. We show that the estimator is mean-square consistent, derive its rateof convergence in probability, and give conditions under which this rate is optimal in aminimax sense. The results of Monte Carlo experiments show that the estimator behaveswell in finite samples.
Centre for microdata methods and practice