Sparse linear models andl1-regularized 2SLS with high-dimensional endogenous regressors and instruments

Abstract

We explore the validity of the 2-stage least squares estimator with l 1 −regularization in both stages, for linear regression 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 are sufficiently sparse. For this l 1 −regularized 2-stage least squares estimator, finite-sample performance bounds are established. We then provide a simple practical method (with asymptotic guarantees) for choosing the regularization parameter. We show that this practical method can produce an l 2 −consistent 2SLS estimator whose rate of convergence can be made as arbitrarily close as the scaling of our finite-sample performance bounds under quite standard conditions. JEL Classification: C14, C31, C36

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