High dimensional generalized linear models for temporal dependent data

High dimensional generalized linear models are widely applicable in many scientific fields with data having heavy tails. However, little is known about statistical guarantees on the estimates of such models in a time series setting. In this article, we establish statistical error bounds and support recovery guarantees of the classical ℓ1 regularized procedure for generalized linear model with temporal dependent data. We also propose a new robust M-estimator for high dimensional time series. Properties of the proposed robust procedure are investigated both theoretically and numerically. As an extension, we introduce a robust estimator for linear regression and show that the proposed robust estimator achieves nearly the optimal rate as that for i.i.d sub-Gaussian data. Simulation results show that the proposed method performs well numerically in the presence of heavy-tailed and serially dependent covariates and/or errors, and it significantly outperforms the classical Lasso method. For applications, we demonstrate, in the supplementary material, the regularized robust procedure via analyzing high-frequency trading data in finance.