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Sparse High-Dimensional Vector Autoregressive Bootstrap

Robert Adamek, Stephan Smeekes, Ines Wilms

TL;DR

This work develops a sparse high-dimensional VAR multiplier bootstrap to approximate the distribution of the max-mean statistic $Q=\max_{1\le j\le N}|\frac{1}{\sqrt{T}}\sum_{t=1}^T x_{j,t}|$ when $N$ may exceed $T$. It combines lasso-based VAR estimation with a multiplier bootstrap on residuals to generate bootstrap samples that preserve dependence, and proves bootstrap consistency under sub-Gaussian and finite-moment error assumptions via a high-dimensional CLT for linear processes. A key theoretical contribution is a Gaussian approximation for the maximum mean of linear processes, enabling asymptotically exact inference for high-dimensional means and related statistics. The simulations show generally good size and power, with performance varying across DGPs, particularly under high persistence; the framework is positioned as a foundation for more general high-dimensional time-series inference and potential bias-correction extensions.

Abstract

We introduce a high-dimensional multiplier bootstrap for time series data based on capturing dependence through a sparsely estimated vector autoregressive model. We prove its consistency for inference on high-dimensional means under two different moment assumptions on the errors, namely sub-gaussian moments and a finite number of absolute moments. In establishing these results, we derive a Gaussian approximation for the maximum mean of a linear process, which may be of independent interest.

Sparse High-Dimensional Vector Autoregressive Bootstrap

TL;DR

This work develops a sparse high-dimensional VAR multiplier bootstrap to approximate the distribution of the max-mean statistic when may exceed . It combines lasso-based VAR estimation with a multiplier bootstrap on residuals to generate bootstrap samples that preserve dependence, and proves bootstrap consistency under sub-Gaussian and finite-moment error assumptions via a high-dimensional CLT for linear processes. A key theoretical contribution is a Gaussian approximation for the maximum mean of linear processes, enabling asymptotically exact inference for high-dimensional means and related statistics. The simulations show generally good size and power, with performance varying across DGPs, particularly under high persistence; the framework is positioned as a foundation for more general high-dimensional time-series inference and potential bias-correction extensions.

Abstract

We introduce a high-dimensional multiplier bootstrap for time series data based on capturing dependence through a sparsely estimated vector autoregressive model. We prove its consistency for inference on high-dimensional means under two different moment assumptions on the errors, namely sub-gaussian moments and a finite number of absolute moments. In establishing these results, we derive a Gaussian approximation for the maximum mean of a linear process, which may be of independent interest.
Paper Structure (19 sections, 16 theorems, 181 equations, 5 figures, 2 algorithms)

This paper contains 19 sections, 16 theorems, 181 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Vector Autoregressive Bootstrap
  3. Bootstrap for High-Dimensional VARs
  4. Bootstrap Inference on (Approximate) Means
  5. HDCLT for Linear Processes
  6. Application to VAR Models
  7. Bootstrap Consistency
  8. Bootstrap Consistency for VAR Estimation by the Lasso
  9. Simulations
  10. Setup
  11. DGP1: Diagonal VAR(1)
  12. DGP2: Block-diagonal VAR(1)
  13. DGP3: Weakly sparse VAR(1)
  14. DGP4: Factor model with sparse idiosyncratic component
  15. Conclusion
  16. ...and 4 more sections

Key Result

Theorem 1

Consider a linear process $\boldsymbol{x}_t$ as in eq:DGPVMA, let ass:covariance hold, and define $\tilde{S}:=\sum\limits_{j=0}^{\infty} \left\lVert\boldsymbol{B}_{j}\right\rVert_\infty$, $S_q:=\sum\limits_{j=0}^{\infty}\left(\sum\limits_{k=j+1}^\infty \left\lVert\boldsymbol{B}_{k}\right\rVert_\inft where $\boldsymbol{z}\sim N(\boldsymbol{0},\boldsymbol{\Sigma})$.

Figures (5)

  • Figure 1: DGP1: Diagonal VAR(1), size and power.
  • Figure 2: Block-diagonal VAR(1), size and power.
  • Figure 3: Weakly sparse VAR(1), size and power.
  • Figure 4: Factor model, size and power.
  • Figure 5: Pattern within the blocks of $\boldsymbol{A}$ and $\boldsymbol{\Sigma}_{\epsilon}$

Theorems & Definitions (37)

  • Remark 1
  • Remark 2
  • Theorem 1: Gaussian approximation for linear processes
  • Remark 3
  • Theorem 2
  • Theorem 3: Gaussian approximation for the bootstrap process
  • Theorem 4
  • Corollary 1: Finite absolute moments
  • Corollary 2: Sub-gaussian moments
  • Lemma A.1
  • ...and 27 more