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

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. First, due to the non-stationarity feature of the model, dynamic bias-variance decomposition of the estimator is obtained. Second, by a bias-correction procedure according to our fundamental representation, the local null distribution of the proposed estimator of the time-varying coefficient vector is characterized to compute the p-values for the iid Gaussian and heavy-tailed errors. In addition, the limiting null distribution is also established for Gaussian process errors and we show that the asymptotic properties have dichotomy features between short-range and long-range dependent errors. Third, the p-values are further adjusted by a Bonferroni-type correction procedure, which is provably to control the familywise error rate (FWER) in the asymptotic sense at each time point. Finally, finite sample size performance of the proposed inference algorithm on a synthetic data and a real application to learn brain connectivity by using the resting-state fMRI data for Parkinson's disease are illustrated.