# Variable Screening for High Dimensional Time Series

## TL;DR

This paper extends variable screening methods to high-dimensional linear models with dependent and heavy-tailed data, introducing a new GLSS procedure that leverages serial correlation for improved performance.

## Contribution

It analyzes the theoretical properties of SIS under dependence and heavy tails, and proposes GLSS, a novel screening method utilizing serial correlation, with proven sure screening properties.

## Key findings

- GLSS outperforms SIS in many dependent data scenarios.
- Both procedures achieve sure screening under certain dependence and moment conditions.
- Simulations and real data application demonstrate effectiveness of the proposed methods.

## Abstract

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.