Inference of high-dimensional linear models with time-varying coefficients

TL;DR

This paper introduces a novel pointwise inference method for high-dimensional linear models with time-varying coefficients, combining kernel smoothing and bias-corrected ridge regression to handle non-stationarity and dependence in errors.

Contribution

It develops a new inference algorithm that accounts for non-stationarity and dependence, providing asymptotic null distributions and FWER control in high-dimensional, time-varying settings.

Findings

Effective bias correction for time-varying coefficients

Asymptotic null distribution characterization for various error types

Successful application to brain connectivity analysis in Parkinson's disease

Abstract

We propose a pointwise inference algorithm for high-dimensional linear models with time-varying coefficients. The method is based on a novel combination of the nonparametric kernel smoothing technique and a Lasso bias-corrected ridge regression estimator. Due to the non-stationarity feature of the model, dynamic bias-variance decomposition of the estimator is obtained. With a bias-correction procedure, the local null distribution of the estimator of the time-varying coefficient vector is characterized for iid Gaussian and heavy-tailed errors. The limiting null distribution is also established for Gaussian process errors, and we show that the asymptotic properties differ between short-range and long-range dependent errors. Here, p-values are adjusted by a Bonferroni-type correction procedure to control the familywise error rate (FWER) in the asymptotic sense at each time point. The finite sample size performance of the proposed inference algorithm is illustrated with synthetic data and an application to learn brain connectivity by using the resting-state fMRI data for Parkinson's disease.