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Last edited on
Oct 18, 2021 by JJ
.
Thesis:
Master in Mathematics

Author:
Rouven Behnisch

Title:
Nonparametric instrumental variables estimation of a quantile regression model

Supervisor:
Jan JOHANNES

Abstract:
In this thesis we consider the problem of estimating the relationship g in a quantile regression model Y = g(X)+U. It is a well-known fact, that solving a quantile regression model with exogeneuous variables is computationally complex and numerically challenging. Horowitz and Lee (2007) have made a first attempt to deal with a quantile regression model with endogeneuous variables. In this paper they showed that this model leads to an ill-posed integral operator equation T g = f. Since the integral operator and the underlying distribution are unknown, they introduced a nonparametric kernel density estimator and based on that they constructed an estimator of g by using Tikhonov regularization, a standard method in the theory of ill-posed inverse problems. We will present this approach, discuss the published results and introduce another kernel estimator of T. Based on the methods used in Horowitz and Lee (2007) we will show, that under relatively strong assumptions, our estimator for g is consistent and derive its rate of convergence.

Reference:
J.L. Horowitz and S. Lee. Nonparametric instrumental variables estimation of a quantile regression model. Econometrica, 75(4):1191–1208, 2007.