[Paper Review] High-Dimensional Granger Causality Tests with an Application to VIX and News

Andrii Babii, Éric Ghysels+1 more|arXiv (Cornell University)|Dec 12, 2019

Statistical Methods and Inference21 citations

TL;DR

This paper develops high-dimensional Granger causality tests using sparse-group LASSO regularization and heteroskedasticity and autocorrelation consistent (HAC) inference for time series. It establishes a debiased central limit theorem and a new Fuk-Nagaev inequality for τ-mixing processes with heavy tails, enabling valid inference on individual coefficients and groups, including Granger causality, in high-dimensional settings with dependent, non-Gaussian data. The method is applied to study causality between the VIX and financial news.

ABSTRACT

We study Granger causality testing for high-dimensional time series using regularized regressions. To perform proper inference, we rely on heteroskedasticity and autocorrelation consistent (HAC) estimation of the asymptotic variance and develop the inferential theory in the high-dimensional setting. To recognize the time series data structures we focus on the sparse-group LASSO estimator, which includes the LASSO and the group LASSO as special cases. We establish the debiased central limit theorem for low dimensional groups of regression coefficients and study the HAC estimator of the long-run variance based on the sparse-group LASSO residuals. This leads to valid time series inference for individual regression coefficients as well as groups, including Granger causality tests. The treatment relies on a new Fuk-Nagaev inequality for a class of $\ au$-mixing processes with heavier than Gaussian tails, which is of independent interest. In an empirical application, we study the Granger causal relationship between the VIX and financial news.

Motivation & Objective

• Address the challenge of performing valid Granger causality testing in high-dimensional time series with many predictors and complex dependence structures.
• Develop inferential theory for regularized regression coefficients in high-dimensional time series, particularly focusing on sparse-group LASSO estimators.
• Establish asymptotic normality and valid inference for individual coefficients and groups under weak dependence and heavy-tailed innovations.
• Provide a robust framework for causality testing that remains valid under model misspecification and heteroskedasticity.
• Apply the method to study the causal relationship between the VIX and financial news derived from high-dimensional text data.

Proposed method

• Utilizes the sparse-group LASSO estimator to simultaneously perform variable selection and group selection in high-dimensional time series regression.
• Employs a debiasing procedure to correct the bias of the sparse-group LASSO estimator, enabling asymptotic normality of the debiased coefficients.
• Derives a heteroskedasticity and autocorrelation consistent (HAC) estimator for the long-run variance of the debiased coefficients using residuals from the sparse-group LASSO.
• Establishes a new Fuk-Nagaev inequality for τ-mixing processes with heavier-than-Gaussian tails, which is critical for deriving the asymptotic distribution under weak dependence.
• Applies the HAC-based inference to test Granger causality at both individual and group levels, including testing whether news series Granger-cause the VIX.
• Uses high-frequency financial news data (180 topic series from Wall Street Journal) to estimate predictive relationships with the VIX.

Experimental results

Research questions

• RQ1 Can valid Granger causality testing be performed in high-dimensional time series with dependent, heavy-tailed innovations?
• RQ2 How can inference be conducted on individual regression coefficients and groups of coefficients when using regularized estimators like sparse-group LASSO?
• RQ3 What is the asymptotic distribution of the debiased sparse-group LASSO estimator under weak dependence and heavy tails?
• RQ4 Does financial news Granger-cause the VIX in high-dimensional settings, and which specific news topics are predictive?
• RQ5 How does the proposed HAC-based inference compare to standard asymptotic inference in finite samples under model misspecification?

Key findings

• The paper establishes a debiased central limit theorem for low-dimensional groups of coefficients in high-dimensional time series with τ-mixing, heavy-tailed errors.
• A new Fuk-Nagaev inequality for τ-mixing processes with heavier-than-Gaussian tails is derived, enabling the theoretical foundation for inference under weak dependence and heavy tails.
• The HAC estimator of the long-run variance based on sparse-group LASSO residuals is shown to be consistent and valid for inference, even under model misspecification.
• The proposed method enables valid Granger causality testing at both individual and group levels, with empirical evidence showing that financial news Granger-causes the VIX.
• The application to 180 news topic series from the Wall Street Journal reveals that topics related to economic uncertainty, financial markets, and political risk are significant predictors of the VIX.
• The method outperforms standard LASSO-based inference in terms of size and power in finite samples, particularly under heteroskedasticity and serial correlation.