Hybrid HAR-ElasticNet Framework

• The paper introduces a hybrid HAR-ElasticNet framework that uses unpenalized OLS HAR to preserve own-market persistence while applying ElasticNet for sparse cross-market spillovers.
• The methodology reveals that commodity markets exhibit measurable spillovers, whereas equities and treasuries show negligible cross-market effects.
• The approach achieves forecasting accuracy comparable to univariate HAR models while offering a detailed, interpretable mapping of volatility transmission networks.

The hybrid HAR-ElasticNet framework is a two-step regularization-based econometric methodology designed for analyzing volatility spillovers across high-dimensional financial systems. Developed specifically for studying daily realized volatility interactions among commodities, equities, and treasuries, it uses an initial OLS-based estimation to preserve the high persistence of own-market volatility while employing ElasticNet regularization to induce sparsity and interpretability in cross-market spillover effects. By uniting strong time-domain persistence and contemporary machine learning tools, this approach enables detailed mapping of volatility transmission networks without sacrificing forecast accuracy (Mallory, 6 Jan 2026).

1. Heterogeneous Autoregressive (HAR) Model for Volatility Persistence

At its foundation, the framework builds upon the Heterogeneous Autoregressive (HAR) model, which captures the long-memory and high persistence of realized volatility observed in financial markets. For each market $i$, the volatility $RV_{i,t}$ is modeled as a function of its own daily, weekly, and monthly lagged volatilities:

$$RV_{i,t} = \alpha_i + \beta_i^{(d)}RV_{i,t-1} + \beta_i^{(w)}\frac{1}{5}\sum_{j=1}^5 RV_{i,t-j} + \beta_i^{(m)}\frac{1}{22}\sum_{j=1}^{22}RV_{i,t-j} + u_{i,t}$$

Persistence is quantitated by the sum $$\phi_i = \beta_i^{(d)} + \beta_i^{(w)} + \beta_i^{(m)}$$, which in empirical OLS estimation typically lies in the range $0.95$–$0.99$. This slow decay of volatility shocks underpins the need to avoid penalization or shrinkage of own-lag coefficients to preserve realistic market dynamics.

2. Hybrid HAR-ElasticNet Estimation Procedure

The prevention of overfitting and the encouragement of sparsity in cross-market volatility effects is achieved through a disciplined two-step process:

• Step 1 (Own-lag OLS HAR): Each market's univariate HAR model is fit by OLS, yielding unrestricted estimates $$\{\hat\alpha_i, \hat\beta_i^{(d)}, \hat\beta_i^{(w)}, \hat\beta_i^{(m)}\}$$ and ensuring near-unit own-lag persistence ($\hat\phi_i \approx 0.99$). The own-dynamics fit is

$$\widehat{RV}_{i,t}^{\rm own} = \hat\alpha_i + \hat\beta_i^{(d)}RV_{i,t-1} + \hat\beta_i^{(w)}RV_{i,t-1:t-5}^{(w)} + \hat\beta_i^{(m)}RV_{i,t-1:t-22}^{(m)}.$$

• Step 2 (Cross-market ElasticNet): Residuals $$\tilde{u}_{i,t} = RV_{i,t} - \widehat{RV}_{i,t}^{\rm own}$$ are regressed on lagged realized volatilities of all other markets. Sparse $\gamma$-vectors for cross-market coefficients are estimated by ElasticNet:

$$\hat\gamma_i = \arg\min_\gamma \Bigg\{\frac{1}{2} \sum_t (\tilde{u}_{i,t} - X_{i,t}\gamma)^2 + \lambda(\alpha \|\gamma\|_1 + \frac{1-\alpha}{2}\|\gamma\|_2^2) \Bigg\}$$

where $\lambda$ controls the penalty (set to $0.5$ for equal L1/L2 mixing), and $\alpha$ is market-specifically tuned by time-series cross-validation.

3. Model Integration and Volatility Spillover Networks

The full volatility dynamics composition for market $i$ is given as:

$RV_{i,t} = \widehat{RV}_{i,t}^{\rm own}_{\rm OLS\,HAR} + \sum_{j\neq i} [\hat\gamma_{ij}^{(d)} RV_{j,t-1} + \hat\gamma_{ij}^{(w)}RV_{j,t-1:t-5}^{(w)} + \hat\gamma_{ij}^{(m)}RV_{j,t-1:t-22}^{(m)} ] + \hat\xi_{i,t}$

This formulation strictly penalizes cross-market terms, yielding a highly sparse yet interpretable network of volatility spillovers. Empirical application to six U.S. futures markets retaining only $7$ out of $90$ possible cross-market coefficients ($\sim 8\%$), with the majority of markets either not transmitting or receiving any measurable volatility spillovers after regularization (Mallory, 6 Jan 2026).

Model Component	Estimator	Penalization
Own-market HAR terms	OLS	None
Cross-market spillovers	ElasticNet	L1 and L2 (CV)

4. Joint Impulse Response Functions (JIRFs) for Systemic Shock Propagation

To analyze the propagation of simultaneous shocks across markets, the hybrid HAR-ElasticNet approach incorporates Joint Impulse Response Functions (JIRFs), generalizing the standard IRF to the multivariate, sparse setting. For a set $\mathcal{S}$ of shocked markets, the JIRF at horizon $h$ for market $j$ is

$$JIRF_j^{\mathcal S}(h) = E[RV_{j,t+h} \mid \text{1-s.d. shock in } \mathcal{S} \text{ at } t ] - E[RV_{j,t+h} \mid \text{no shock}]$$

Shocks are drawn respecting the residual covariance $\hat\Sigma_u$ to reflect empirically observed contemporaneous correlations. Confidence bands are constructed via 1000 block-bootstrap samples of the full model.

5. Empirical Volatility Spillover Patterns

Application of the hybrid HAR-ElasticNet framework to six major U.S. futures markets from 2002–2025 reveals several salient empirical findings (Mallory, 6 Jan 2026):

• Network Sparsity: Only $7$ out of $90$ cross-market coefficients survive ElasticNet regularization. All equity and treasury assets (ES, NQ, ZF, ZN) exhibit zero cross-market coefficients, implying they neither transmit nor receive volatility shocks in this network.
• Commodity Linkages: Nonzero spillovers are confined to commodities. Crude oil (CL) receives minor daily and monthly spillover effects from equities (ES, NQ) and from 5-year treasuries (ZF). Soybeans (ZS) receive a very small daily effect from ZF and a minuscule daily CL$\rightarrow$ZS effect. All other cross-class spillovers are eliminated.
• Forecasting Performance: One-step-ahead RMSE (out-of-sample) matches that of a univariate HAR model: average RMSE = $0.0044$ (per-asset range: $0.0011$ for ZF to $0.0086$ for CL), with MAE and MAPE in agreement. Thus, cross-market information adds negligible predictive accuracy once own-market persistence is correctly specified.
• Impulse Response Evidence: JIRFs show commodity shocks (ZS, CL) do not propagate into equities or treasuries; equity or treasury shocks affect only own-markets, and cross-market mean responses are not statistically significant—though some volatility-of-volatility effects appear in peripheral markets during systemic shocks.

6. Methodological Significance and Interpretation

The hybrid HAR-ElasticNet approach demonstrates three key methodological principles (Mallory, 6 Jan 2026):

• Preservation of own-market HAR persistence via unpenalized OLS estimation is essential; shrinkage of these terms can induce spurious links in the spillover network.
• Application of ElasticNet solely to cross-market terms yields a highly interpretable and parsimonious network, facilitating identification of economically meaningful linkages without overfitting.
• The hybrid estimation delivers forecast accuracy on par with univariate HAR models, but uniquely enables system-wide network analysis and nonlinear joint impulse response investigation that standard univariate protocols cannot provide.

A plausible implication is that, in high-dimensional risk management contexts, meaningful network analysis of volatility spillovers must carefully distinguish own-dynamics from cross-linkages and employ penalization methods capable of uncovering sparse, interpretable structures.

7. Context Within Financial Econometrics

The methodology advances existing volatility modeling by combining machine learning regularization with classical time series models, addressing both the curse of dimensionality and interpretability requirements in financial systems. It further introduces a robust framework for studying multidimensional impulse responses under empirically-driven shock scenarios. This approach aligns with contemporary work seeking to make sense of complex, high-dimensional economic networks while maintaining statistical rigor and practical forecast relevance (Mallory, 6 Jan 2026).