High-dimensional Inference For Linear Model With Correlated Errors

With the rapidly development of science and technology,high-dimensional statis-tical methods have been fully applied in different scientific areas including information technology,bioinformatics,astronomy and so on.Due to its simplicity and ease of in-terpretation,linear model has become the prior choice of practitioners in the statistical analysis of high-dimensional data.Although the current literature on high-dimensional statistical inference is developing rapidly,most of them assume that the data are ob-served independently.However,temporally correlated error process is commonly en-countered in practice.Direct application of the method based on i.i.d.data to time-series data will lead to unreasonable estimation and inference results.Thus,both high dimen-sionality and temporal correlation pose significant challenges in statistical inference for the regression parameters.This paper conducts low dimensional inference for high dimensional linear mod-els with stationary errors.We adopt the framework of functional dependence measure for adequate accommodation of the error correlation.A new desparsifying Lasso based testing procedure is developed by incorporating a banded estimator of the error auto-covariance matrix.Asymptotic normality of the proposed estimator is established by demonstrating the consistency of the banded autocovariance matrix estimator.The re-sult indicates how the range of p is substantially narrower if the moment condition of error weakens or the dependence becomes stronger.We further develop a data driven choice of the banding parameter.The simulation studies illustrate the satisfactory finite-sample performance of our proposed procedure.Finally,a real data example is also pre-sented for illustration.We apply the methodology to the macroeconomic data.The goal is to discover which macroeconomic variables are associated with the unemployment rate.