In this paper we propose an $\ell_1$-regularized GLS estimator for high-dimensional regressions with potentially autocorrelated errors. We establish non-asymptotic oracle inequalities for estimation accuracy in a framework that allows for highly persistent autoregressive errors. In practice, the Whitening matrix required to implement the GLS is unkown, we present a feasible estimator for this matrix, derive consistency results and ultimately show how our proposed feasible GLS can recover closely the optimal performance (as if the errors were a white noise) of the LASSO. A simulation study verifies the performance of the proposed method, demonstrating that the penalized (feasible) GLS-LASSO estimator performs on par with the LASSO in the case of white noise errors, whilst outperforming it in terms of sign-recovery and estimation error when the errors exhibit significant correlation.

翻译：本文提出了一种适用于高维回归且误差项可能存在自相关的$\ell_1$-正则化广义最小二乘（GLS）估计量。在允许高度持久性自回归误差的框架下，我们建立了估计精度的非渐近Oracle不等式。由于实践中实现GLS所需的白化矩阵未知，我们提出了一种可行的矩阵估计方法，推导了其一致性结果，并最终展示了所提出的可行GLS如何紧密逼近LASSO的最优性能（即误差为白噪声时的表现）。模拟研究验证了所提出方法的性能，结果表明：当误差为白噪声时，惩罚（可行）GLS-LASSO估计量与LASSO表现相当；而当误差存在显著相关性时，其在符号恢复和估计误差方面优于LASSO。