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This paper is concerned with inference about low-dimensional components of ahigh-dimensional parameter vector $\beta^0$ which is identified throughinstrumental variables. We allow for eigenvalues of the expected outer productof included and excluded covariates, denoted by $M$, to shrink to zero as thesample size increases. We propose a novel estimator based on desparsificationof an instrumental variable Lasso estimator, which is a regularized version of2SLS with an additional correction term. This estimator converges to $\beta^0$at a rate depending on the mapping properties of $M$ captured by a sparse linkcondition. Linear combinations of our estimator of $\beta^0$ are shown to beasymptotically normally distributed. Based on consistent covariance estimation,our method allows for constructing confidence intervals and statistical testsfor single or low-dimensional components of $\beta^0$. In Monte-Carlosimulations we analyze the finite sample behavior of our estimator.

Ill-posed Estimation in High-Dimensional Models with Instrumental Variables