Lasso with long memory regression errors

•We investigate properties of Lasso in a linear model with long memory dependent errors.•Theory in both non-asymptotic and asymptotic settings is extended to the present setup.•Finite sample error bounds are obtained for the Lasso in the high dimensional setup.•Sign consistency of Lasso is established in the high dimensional setup.•Oracle property of adaptive Lasso is established in the p

Lasso is a computationally efficient approach to model selection and estimation, and its properties are well studied when the regression errors are independent and identically distributed. We study the case, where the regression errors form a long memory moving average process. We establish a finite sample oracle inequality for the Lasso solution. We then show the asymptotic sign consistency in this setup. These results are established in the high dimensional setup (p>np>n) where p can be increasing exponentially with n  . Finally, we show the consistency, n1/2−d-consistencyn1/2−d-consistency of Lasso, along with the oracle property of adaptive Lasso, in the case where p is fixed. Here d is the memory parameter of the stationary error sequence. The performance of Lasso is also analysed in the present setup with a simulation study.