## Abstract

We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression "error" conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill-posed inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean-square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples.

Original language | English (US) |
---|---|

Pages (from-to) | 1191-1208 |

Number of pages | 18 |

Journal | Econometrica |

Volume | 75 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2007 |

## Keywords

- Endogenous variable
- Instrumental variable
- Nonlinear integral equation
- Nonparametric regression
- Optimal rate
- Statistical inverse

## ASJC Scopus subject areas

- Economics and Econometrics