Uniform confidence sets in high dimensional linear models with heteroscedasticity and endogeneity using linear programming and instrumental variables

Abstract

In this talk we consider estimation of linear models with many regressors, possibly much more than the sample size and endogenous regressors. We propose a method based on instrumental variables. It is pivotal in the sense that it does not require the knowledge of the variance of the errors. It does not rely on model selection. We obtain nested confidence sets for given prior upper bounds on the number of nonzero coefficients, without imposing identification. They are easy to obtain numerically because they only rely on linear programming. This allows to handle numerically very large models. The confidence sets have finite sample validity and are uniform in a wide class of sparse models and distributions of the error terms. If time will permit we will present extensions to the detection of endogenous instruments and confidence sets based on 2-stage methods.