Two-Step Regularized HARX to Measure Volatility Spillovers in Multi-Dimensional Systems

TL;DR

This paper addresses identifying meaningful volatility spillovers across six futures markets by introducing a two-step HARX-ElasticNet framework that preserves own-market persistence (approximately $\phi_i \approx 0.99$) and isolates sparse cross-market links. The approach yields an extremely sparse spillover network, with equities and treasuries acting as transmitters and key commodity linkages (notably involving crude oil) forming the remaining connections; Joint Impulse Response Functions reveal how shocks propagate through these links without implying causality. Forecast performance is on par with a univariate HAR model (RMSE $0.0044$), indicating cross-market information adds little to point forecasts despite its economic significance for network structure. The work demonstrates that regularization can uncover economically meaningful spillover pathways while maintaining forecast accuracy, offering practical insights for risk management and systemic risk monitoring and suggesting avenues for time-varying networks and higher-frequency analyses in future research.

Abstract

We identify volatility spillovers across commodities, equities, and treasuries using a hybrid HAR-ElasticNet framework on daily realized volatility for six futures markets over 2002--2025. Our two step procedure estimates own-volatility dynamics via OLS to preserve persistence, then applies ElasticNet regularization to cross-market spillovers. The sparse network structure that emerges shows equity markets (ES, NQ) act as the primary volatility transmitters, while crude oil (CL) ends up being the largest receiver of cross-market shocks. Agricultural commodities stay isolated from the larger network. A simple univariate HAR model achieves equally performing point forecasts as our model, but our approach reveals network structure that univariate models cannot. Joint Impulse Response Functions trace how shocks propagate through the network. Our contribution is to demonstrate that hybrid estimation methods can identify meaningful spillover pathways while preserving forecast performance.

Figures (6)

• Figure 1: Realized Volatility Time Series (Yang-Zhang Estimator). Notes: 30-day rolling Yang-Zhang realized volatility estimates for six futures markets. Sample period: May 2002 to January 2025 (5,699 observations). All volatilities are annualized.
• Figure 2: Out-of-Sample Volatility Forecasts: Actual vs Model Predictions. Notes: Final 200 days of test period (September 2024 to January 2025). Black solid line shows actual Yang-Zhang realized volatility. Blue dashed line shows hybrid HARX-ElasticNet one-step-ahead forecasts. Orange dotted line shows univariate HAR forecasts. Both models track realized volatility nearly identically with typical forecast errors of 1-2 percentage points, achieving identical RMSE of 0.0044 (Table \ref{['tab:model_comparison']}). The hybrid model's extreme sparsity (8% nonzero cross-market coefficients) effectively reduces it to univariate HAR for forecasting purposes.
• Figure 3: Hybrid HARX-ElasticNet Coefficient Matrix. Notes: Rows show the 18 features (source asset $\times$ frequency: daily, weekly average, monthly average). Columns show the 6 target assets. Each cell displays the estimated coefficient. Diagonal blocks (e.g., ZS features predicting ZS) show own-market persistence from the OLS HAR step, preserving realistic 0.99 persistence. Off-diagonal cells show cross-market spillovers from the ElasticNet step. White or near-zero cells indicate coefficients shrunk to zero by regularization. The hybrid approach yields extreme sparsity in cross-market terms: only 7 of 90 cross-market coefficients (8%) are nonzero, with ES, NQ, ZF, and ZN having zero cross-market spillovers, while ZS and CL retain sparse transmission channels.
• Figure 4: Joint Impulse Responses: Commodities Shock. Notes: Responses to a one-standard-deviation shock to ZS and CL. Shaded regions show 95% confidence intervals from 1,000 bootstrap simulations.
• Figure 5: Joint Impulse Responses: Equities Shock. Notes: Responses to a one-standard-deviation shock to ES and NQ. Shaded regions show 95% confidence intervals from 1,000 bootstrap simulations.