Model selection for unit-root time series with many predictors

This paper studies model selection for general unit-root time series, including the case with many exogenous predictors. We propose a new model selection algorithm, FHTD, that leverages forward stepwise regression (FSR), a high-dimensional information criterion (HDIC), a backward elimination method based on HDIC, and a data-driven thresholding (DDT) approach. Under some mild assumptions that allow for unknown locations and multiplicities of the characteristic roots on the unit circle of the time series and conditional heteroscedasticity in the predictors and errors, we establish the sure screening property of FSR and the selection consistency of FHTD. Our theoretical analysis relies on two novel technical contributions, namely a functional central limit theorem for multivariate linear processes and a uniform lower bound for the minimum eigenvalue of the sample covariance matrices, both of which are of independent interest. Simulation results corroborate the theoretical properties and show the superior performance of FHTD in model selection. We apply the proposed FHTD to model U.S. monthly housing starts and unemployment data, showcasing its practical utility.