ABSTRACT
This chapter focuses on quantile regression models within the latter regime. High-dimensional models arise from the need for practitioners to improve the accuracy and validity of current models and to handle the increasing availability of data. Large models can arise from using a very flexible specification with many parameters when the functional form is unknown or from having a data-rich environment with many variables that need to be incorporated into the model. To better capture finite-sample behavior of high-dimensional models, it has been proposed to conduct an asymptotic analysis where the model can change with the sample size. The main issue in the high-dimensional setting is the lack of full rank of the design matrix, which creates a lack of identification. In high-dimensional settings, Belloni et al. were the first to use an orthogonality condition in an instrumental variables model with many instruments.
