High-dimensional instrumental variables regression and confidence sets

We propose an instrumental variables method for inference in high-dimensional structural equations with endogenous regressors. The number of regressors K can be much larger than the sample size. A key ingredient is sparsity, i.e., the vector of coefficients has many zeros, or approximate sparsity, i.e., it is well approximated by a vector with many zeros. We can have less instruments than regressors and allow for partial identification. Our procedure, called STIV (Self Tuning Instrumental Variables) estimator, is realized as a solution of a conic program. The joint confidence sets can be obtained by solving K convex programs. We provide rates of convergence, model selection results and propose three types of joint confidence sets relying each on different assumptions on the parameter space. Under the stronger assumption they are adaptive. The results are uniform over a wide classes of distributions of the data and can have finite sample validity. When the number of instruments is too large or when one only has instruments for an endogenous regressor which are too weak, the confidence sets can have infinite volume with positive probability. This provides a simple one-stage procedure for inference robust to weak instruments which could also be used for low dimensional models. In our IV regression setting, the standard tools from the literature on sparsity, such as the restricted eigenvalue assumption are inapplicable. Therefore we develop new sharper sensitivity characteristics, as well as easy to compute data-driven bounds. All results apply to the particular case of the usual high-dimensional regression. We also present extensions to the high-dimensional framework of the two-stage least squares method and method to detect endogenous instruments given a set of exogenous instruments.