Ill-posed estimation in high-dimensional models with instrumental variables

This paper is concerned with inference about low-dimensional components of a high-dimensional parameter vector β0 which is identified through instrumental variables. We allow for eigenvalues of the expected outer product of included and excluded covariates, denoted by M, to shrink to zero as the sample size increases. We propose a novel estimator based on desparsification of an instrumental variable Lasso estimator, which is a regularized version of 2SLS with an additional correction term. This estimator converges to β0 at a rate depending on the mapping properties of M. Linear combinations of our estimator of β0 are shown to be asymptotically normally distributed. Based on consistent covariance estimation, our method allows for constructing confidence intervals and statistical tests for single or low-dimensional components of β0. In Monte-Carlo simulations we analyze the finite sample behavior of our estimator. We apply our method to estimate a logit model of demand for automobiles using real market share data.