  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.
