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Sparse VARs Do Not Imply Sparse Local Projections: Robust Inference for High-Dimensional Granger Causality

Eugene Dettaa, Endong Wang

Abstract

This paper studies multi-horizon Granger causality using high-dimensional local projections in sparse Vector Autoregressive (VAR) systems. Since local projection coefficients are nonlinear transformations of the underlying VAR parameters, existing approaches, such as de-biased least absolute shrinkage and selection operator (LASSO) and post-double-selection methods applied directly to local projections, lack a general justification, as sparsity of the VAR does not always propagate to higher horizons. We propose a two-step framework that avoids imposing sparsity at each horizon and delivers valid inference without relying on heteroskedasticityand autocorrelation-consistent (HAC) corrections. We establish large sample theory for the proposed estimators and develop feasible Wald tests. Monte Carlo experiments demonstrate improved size control across horizons relative to existing methods. An application to large financial systems illustrates horizon-specific connectedness.

Sparse VARs Do Not Imply Sparse Local Projections: Robust Inference for High-Dimensional Granger Causality

Abstract

This paper studies multi-horizon Granger causality using high-dimensional local projections in sparse Vector Autoregressive (VAR) systems. Since local projection coefficients are nonlinear transformations of the underlying VAR parameters, existing approaches, such as de-biased least absolute shrinkage and selection operator (LASSO) and post-double-selection methods applied directly to local projections, lack a general justification, as sparsity of the VAR does not always propagate to higher horizons. We propose a two-step framework that avoids imposing sparsity at each horizon and delivers valid inference without relying on heteroskedasticityand autocorrelation-consistent (HAC) corrections. We establish large sample theory for the proposed estimators and develop feasible Wald tests. Monte Carlo experiments demonstrate improved size control across horizons relative to existing methods. An application to large financial systems illustrates horizon-specific connectedness.
Paper Structure (20 sections, 8 theorems, 31 equations, 6 figures)

This paper contains 20 sections, 8 theorems, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Econometric and financial motivations
  3. Local projections for multi-horizon Granger causality
  4. Sparsity does not propagate to local projections
  5. Connectedness measures
  6. Two-stage estimation
  7. Two-stage identification
  8. De-biased two-stage estimator
  9. Newey--West--type HAC estimator
  10. Robust inference without serial correlation correction
  11. Asymptotic properties of two-stage estimators
  12. Assumptions
  13. Theoretical results
  14. Monte Carlo simulations
  15. Empirical application
  16. ...and 5 more sections

Key Result

Proposition 3.1

Suppose Assumption cond_HC holds. Then $\mathbb E[s_t s_\tau']=0$ for all $t\neq\tau$.

Figures (6)

  • Figure 1: Empirical size of joint horizon-$h$ Granger-causality tests at the 5% nominal level under the tridiagonal root design.
  • Figure 2: Empirical size of joint horizon-$h$ Granger-causality tests at the 5% nominal level under the upper-triangular root design.
  • Figure 3: Rolling-window estimates (100 trading days) of total connectedness measured by DGC across horizons $h$ at $\alpha=0.99$.
  • Figure 4: Rolling-window estimates (100 trading days) of total connectedness measured by DGC across horizons $h$ at $\alpha=0.999$.
  • Figure 5: One-day Granger causality networks around major financial shocks.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Proposition 3.1
  • Lemma 4.1: Consistency results
  • Theorem 4.2: Inference for the de-2S estimator
  • Lemma A.1: Lemma A.2 of krampe2023structural
  • Lemma A.2
  • Lemma A.3
  • Lemma A.4
  • Lemma A.5