  Variable selection is a widely studied problem in high dimensional
statistics, primarily since estimating the precise relationship between the
covariates and the response is of great importance in many scientific
disciplines. However, most of theory and methods developed towards this goal
for the linear model invoke the assumption of iid sub-Gaussian covariates and
errors. This paper analyzes the theoretical properties of Sure Independence
Screening (SIS) (Fan and Lv [J. R. Stat. Soc. Ser. B Stat. Methodol. 70 (2008)
849-911]) for high dimensional linear models with dependent and/or heavy tailed
covariates and errors. We also introduce a generalized least squares screening
(GLSS) procedure which utilizes the serial correlation present in the data. By
utilizing this serial correlation when estimating our marginal effects, GLSS is
shown to outperform SIS in many cases. For both procedures we prove sure
screening properties, which depend on the moment conditions, and the strength
of dependence in the error and covariate processes, amongst other factors.
Additionally, combining these screening procedures with the adaptive Lasso is
analyzed. Dependence is quantified by functional dependence measures (Wu [Proc.
Natl. Acad. Sci. USA 102 (2005) 14150-14154]), and the results rely on the use
of Nagaev-type and exponential inequalities for dependent random variables. We
also conduct simulations to demonstrate the finite sample performance of these
procedures, and include a real data application of forecasting the US inflation
rate.
