Sparse Linear Models and l1−Regularized 2SLS with High-Dimensional Endogenous Regressors and Instruments

We explore the validity of the 2-stage least squares estimator with l1−regularization in both stages, for linear models where the numbers of endogenous regressors in the main equation and instruments in the first-stage equations can exceed the sample size, and the regression coefficients belong to lq−“balls” for q in [0, 1], covering both exact and approximate sparsity cases. Standard high-level assumptions on the Gram matrix for l2−consistency require careful verifications in the two-stage procedure, for which we provide detailed analysis. We establish finite-sample bounds and conditions for our estimator to achieve l2−consistency and variable selection consistency. Practical guidance for choosing the regularization parameters is provided