We study an $\ell_{1}$-regularized generalized least-squares (GLS) estimator for high-dimensional regressions with autocorrelated errors. Specifically, we consider the case where errors are assumed to follow an autoregressive process, alongside a feasible variant of GLS that estimates the structure of this process in a data-driven manner. The estimation procedure consists of three steps: performing a LASSO regression, fitting an autoregressive model to the realized residuals, and then running a second-stage LASSO regression on the rotated (whitened) data. We examine the theoretical performance of the method in a sub-Gaussian random-design setting, in particular assessing the impact of the rotation on the design matrix and how this impacts the estimation error of the procedure. We show that our proposed estimators maintain smaller estimation error than an unadjusted LASSO regression when the errors are driven by an autoregressive process. 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 when the errors exhibit significant autocorrelation.

翻译：我们研究了一种用于具有自相关误差的高维回归的$\ell_{1}$正则化广义最小二乘估计器。具体而言，我们考虑误差被假定服从自回归过程的情况，同时研究了一种可行的广义最小二乘变体，它以数据驱动的方式估计该过程的结构。该估计过程包含三个步骤：执行LASSO回归，对实现的残差拟合自回归模型，然后在旋转后的数据上运行第二阶段LASSO回归。我们在亚高斯随机设计设置下检验了该方法的理论性能，特别评估了旋转对设计矩阵的影响以及这如何影响该过程的估计误差。我们证明，当误差由自回归过程驱动时，我们提出的估计器比未经调整的LASSO回归保持了更小的估计误差。一项模拟研究验证了所提方法的性能，表明惩罚性（可行）广义最小二乘-LASSO估计器在白噪声误差情况下与LASSO表现相当，而在误差表现出显著自相关时则优于后者。