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Model Selection for Unit-root Time Series with Many Predictors

Shuo-Chieh Huang, Ching-Kang Ing, Ruey S. Tsay

TL;DR

This work addresses model selection for unit-root time series with many predictors by developing the FHTD algorithm, which fuses forward stepwise regression, high-dimensional information criteria, backward elimination, and data-driven thresholding. It proves that FSR achieves sure screening and that FHTD attains selection consistency under strong sparsity and mild moment conditions, even with complex unit roots and conditional heteroscedasticity. Theoretical breakthroughs include a functional central limit theorem for multivariate linear processes and a uniform eigenvalue bound for sample covariances in high dimensions. Simulation and two macroeconomic applications demonstrate superior variable selection and forecasting performance relative to competing high-dimensional methods. These results provide a practical, principled framework for selecting relevant lagged and exogenous predictors in nonstationary, high-dimensional time series with economic applications.

Abstract

This paper studies model selection for general unit-root time series, including the case with many exogenous predictors. We propose FHTD, a new model selection algorithm 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. Central to our analysis are two key technical contributions, a new 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 showcase the application of the proposed FHTD by modeling U.S. monthly housing starts and unemployment data.

Model Selection for Unit-root Time Series with Many Predictors

TL;DR

This work addresses model selection for unit-root time series with many predictors by developing the FHTD algorithm, which fuses forward stepwise regression, high-dimensional information criteria, backward elimination, and data-driven thresholding. It proves that FSR achieves sure screening and that FHTD attains selection consistency under strong sparsity and mild moment conditions, even with complex unit roots and conditional heteroscedasticity. Theoretical breakthroughs include a functional central limit theorem for multivariate linear processes and a uniform eigenvalue bound for sample covariances in high dimensions. Simulation and two macroeconomic applications demonstrate superior variable selection and forecasting performance relative to competing high-dimensional methods. These results provide a practical, principled framework for selecting relevant lagged and exogenous predictors in nonstationary, high-dimensional time series with economic applications.

Abstract

This paper studies model selection for general unit-root time series, including the case with many exogenous predictors. We propose FHTD, a new model selection algorithm 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. Central to our analysis are two key technical contributions, a new 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 showcase the application of the proposed FHTD by modeling U.S. monthly housing starts and unemployment data.
Paper Structure (22 sections, 8 theorems, 250 equations, 2 figures, 5 tables)

This paper contains 22 sections, 8 theorems, 250 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The FHTD algorithm
  3. Screening and selection consistency
  4. Model assumptions
  5. The sure screening property of FSR
  6. Selection Consistency
  7. Simulation studies
  8. Applications
  9. Housing starts in the U.S.
  10. U.S. unemployment rate
  11. Concluding remarks
  12. Key technical tools
  13. Comments on Assumptions (A1)--(A6), (SS$_{{\rm X}}$), and (SS)
  14. Main proofs
  15. Proofs of Theorems A.1--A.3 and discussions
  16. ...and 7 more sections

Key Result

Theorem 3.1

Assume that (A1)--(A6) and (SS$_{{\rm X}}$) hold. Then, for where $1/3<\varsigma<1/2$,

Figures (2)

  • Figure 1: Time plots of U.S. monthly housing starts and unemployment series
  • Figure 2: Time plots of logarithim of monthly U.S. Housting Starts, $h_{t}$, of selected windows

Theorems & Definitions (26)

  • Theorem 3.1
  • Example 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Example 3.2
  • Example 3.3
  • Example 4.1
  • Example 4.2
  • Theorem A.1
  • Theorem A.2
  • ...and 16 more