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Difference-Based High-Dimensional Long-Run Covariance Matrix Estimation for Mean-shift Time Series

Yanhong Liu, Fengyi Song, Long Feng

Abstract

We consider estimation of high-dimensional long-run covariance matrices for time series with nonconstant means, a setting in which conventional estimators can be severely biased. To address this difficulty, we propose a difference-based initial estimator that is robust to a broad class of mean variations, and combine it with hard thresholding, soft thresholding, and tapering to obtain sparse long-run covariance estimators for high-dimensional data. We derive convergence rates for the resulting estimators under general temporal dependence and time-varying mean structures, showing explicitly how the rates depend on covariance sparsity, mean variation, dimension, and sample size. Numerical experiments show that the proposed methods perform favorably in high dimensions, especially when the mean evolves over time.

Difference-Based High-Dimensional Long-Run Covariance Matrix Estimation for Mean-shift Time Series

Abstract

We consider estimation of high-dimensional long-run covariance matrices for time series with nonconstant means, a setting in which conventional estimators can be severely biased. To address this difficulty, we propose a difference-based initial estimator that is robust to a broad class of mean variations, and combine it with hard thresholding, soft thresholding, and tapering to obtain sparse long-run covariance estimators for high-dimensional data. We derive convergence rates for the resulting estimators under general temporal dependence and time-varying mean structures, showing explicitly how the rates depend on covariance sparsity, mean variation, dimension, and sample size. Numerical experiments show that the proposed methods perform favorably in high dimensions, especially when the mean evolves over time.
Paper Structure (28 sections, 14 theorems, 291 equations, 2 figures, 7 tables)

This paper contains 28 sections, 14 theorems, 291 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Difference-based long-run covariance matrix estimation
  4. Sparse Estimators
  5. Selection of tuning parameters
  6. Discussion of Means
  7. Piecewise-Hölder mean class (with possible jumps)
  8. Bounded-variation mean class (no changepoint parametrization)
  9. Simulation
  10. Model I (tridiagonal cross-sectional dependence).
  11. Model II (Toeplitz cross-sectional dependence).
  12. Model III (Permuted Block Diagonal Structure).
  13. Data Application
  14. Conclusion
  15. Appendix: Proofs of Theorems
  16. ...and 13 more sections

Key Result

Theorem 2.1

Suppose Assumptions ass:kernel_diff--ass:mean_remainder hold and $\log p_n = o(n)$. Then where $a_n$ is defined in eq:an_def.

Figures (2)

  • Figure 1: Heatmaps of sparse long run covariance matrix estimators (all entries are multiplied by 10000).
  • Figure 2: Normalized trajectories of the modified $\bm{S}_{n,p}$ statistic based on hard-thresholding, soft-thresholding, and tapering estimators of the sparse long-run covariance matrix. The dashed vertical line marks the common maximizer of the three curves.

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Corollary 2.1: Bandwidth-balanced rate under negligible mean remainder
  • Theorem 2.2: Thresholded estimation under weak row-sparsity
  • Theorem 2.3: Soft-thresholding under weak row-sparsity
  • Definition 2.1: Tapering weights and tapering estimator
  • Theorem 2.4: Tapering under bandability
  • Proposition 2.1: Generic deterministic bound for $\mathbf B_{\mu,n}$
  • Proposition 2.2: Generic stochastic bound for $\mathbf R_{\mu,n}$
  • ...and 12 more