Nonparametric Inference for a Triangular System of Equations for Quantile Regression

In this study, we consider nonparametric estimation and inference for quantile regression (QR) with endogenous regressors. We extend the semiparametric triangular model for QR in Lee (2007) to a nonparametric one, and the identification of the structural parameters is achieved via a control function approach. Based on the identification result, we propose the use of the penalized sieve minimum distance procedure of Chen and Pouzo (2015) and develop an asymptotic theory. The inferential theory is valid regardless of whether or not the functional of the structural parameter is n-estimable, where n denotes the number of observations. We also establish the asymptotic theory for sieve quasi-likelihood ratio test statistics, enabling us to avoid estimating the asymptotic variance. A Monte Carlo simulation study shows that the proposed estimator performs well in finite samples.