We introduce a class of instrumental quantile regression methods for heterogeneous treatment effect models and simultaneous equations models with nonadditive errors and offer computable methods for estimation and inference. These methods can be used to evaluate the impact of endogenous variables or treatments on the entire distribution of outcomes. We describe an estimator of the instrumental v...

Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically , it obtains the large sample properties of extremal (ext...

Quantile regression (QR) has become a popular method of data analysis, especially when the error term is heteroscedastic, due to its relevance in many scientific studies. The ubiquity of high dimensional data has led to a number of variable selection methods for linear/nonlinear QR models and, recently, for the single index quantile regression (SIQR) model. We propose a new algorithm for simult...

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one partic...

We consider both ℓ0-penalized and ℓ0-constrained quantile regression estimators. For the estimator, we derive an exponential inequality on tail probability of excess prediction risk apply it to obtain non-asymptotic upper bounds mean-square parameter function estimation errors. also analogous results for estimator. The resulting rates convergence are nearly minimax-optimal same as those ℓ1-pena...