Ill-posed Estimation in High-Dimensional Models with Instrumental Variables
Enno Mammen and
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This paper is concerned with inference about low-dimensional components of a high-dimensional parameter vector beta_0 which is identified through in- strumental 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 desparsi- fication 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 sta- tistical tests for single or low-dimensional components of beta_0. In Monte-Carlo simulations we analyze the finite sample behavior of our estimator.
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