High-dimensional instrumental variables regression and confidence sets

This article considers inference in linear models with K regressors, some or many could be endogenous, and L instruments. L can range from less than K to any order smaller than an exponential in the sample size and K is arbitrary. For moderate K, identification robust confidence sets are obtained by solving a hierarchy of semidefinite programs. For larger K, we propose the STIV estimator. The analysis of its error uses sensitivity characteristics which are sharper than those in the literature on sparsity. Data-driven bounds on them and robust confidence sets are obtained by solving K linear programs. Results on rates of convergence, variable selection, and confidence sets which “adapt” to the sparsity are given. We generalize our approach to models with approximation errors, systems, endogenous instruments, and two-stage for confidence bands for vectors of linear functionals and functions. The application is to a demand system with many endogenous regressors.