LAUSR

In recent years, extensive research has focused on the `1 penalized least squares (Lasso) estimators of high-dimensional linear regression when the number of covariates p is considerably larger than the sample size n. However, there is limited attention paid to the properties of the estimators when the errors and/or the covariates are serially dependent. In this paper, we investigate the theoretical properties of the Lasso estimators for linear regression with random design and weak sparsity under serially dependent and/or non-sub-Gaussian errors and covariates. In contrast to the traditional case in which the errors are i.i.d and have finite exponential moments, we show that p can be at most a power of n if the errors have only finite polynomial moments. In addition, the rate of convergence becomes slower due to the serial dependence in errors and the covariates. We also consider sign consistency for model selection via Lasso when there are serial correlations in the errors or the covariates or both. Adopting the framework of functional dependence measure, we provide a detailed description on 1 Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing)