Penalized regression models with autoregressive error terms

Penalized regression methods have recently gained enormous attention in statistics and the field of machine learning due to their ability of reducing the prediction error and identifying important variables at the same time. Numerous studies have been conducted for penalized regression, but most of them are limited to the case when the data are independently observed. In this paper, we study a variable selection problem in penalized regression models with autoregressive (AR) error terms. We consider three estimators, adaptive least absolute shrinkage and selection operator, bridge, and smoothly clipped absolute deviation, and propose a computational algorithm that enables us to select a relevant set of variables and also the order of AR error terms simultaneously. In addition, we provide their asymptotic properties such as consistency, selection consistency, and asymptotic normality. The performances of the three estimators are compared with one another using simulated and real examples.