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

Christoph Breunig, Enno Mammen and Anna Simoni

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Abstract: This paper is concerned with inference about low-dimensional components of a high-dimensional parameter vector $\beta^0$ 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 $\beta^0$ at a rate depending on the mapping properties of $M$ captured by a sparse link condition. Linear combinations of our estimator of $\beta^0$ 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 $\beta^0$. In Monte-Carlo simulations we analyze the finite sample behavior of our estimator.

Date: 2018-06, Revised 2020-08
New Economics Papers: this item is included in nep-ecm and nep-knm
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