Joint Impulse Response Functions

• Joint Impulse Response Functions are simulation-based tools that quantify the dynamic impact of joint shocks on interconnected markets by comparing forecast trajectories under shocked and baseline conditions.
• They employ a two-step hybrid HAR–ElasticNet procedure to separately estimate own-market persistence and sparse cross-market spillovers in high-dimensional volatility systems.
• Empirical results demonstrate that JIRFs effectively reveal directional spillovers and structural risk in financial networks while preserving forecast accuracy.

Joint Impulse Response Functions (JIRFs) provide a rigorous and interpretable mechanism for tracing the propagation of shocks across interconnected financial markets within a sparse multivariate network structure. In the context of high-dimensional volatility systems, JIRFs quantify the dynamic impact of exogenous disturbances—applied simultaneously to multiple series—on the realized volatility of each market, conditional on the network structure inferred from hybrid estimation procedures. Such functions are instrumental for empirical spillover analysis, particularly in revealing both the direction and magnitude of multivariate shock transmission, extending the impulse response paradigm from classic VAR settings into penalized, high-dimensional regimes (Mallory, 6 Jan 2026).

1. Formal Definition and Conceptualization

The JIRF quantifies, for each target market $i$ at forecast horizon $h$, the differential effect between two simulated multivariate trajectories: one under a prescribed joint shock to a subset of markets $\mathcal{S}$, and the other under a baseline (no shock) scenario. The JIRF is defined as the average across bootstrap realizations of the difference between these two paths, holding all data-generating processes—own- and cross-spillover coefficients—fixed (Mallory, 6 Jan 2026). This simulation-based approach accommodates nonlinearities and complex lag dependencies present in estimated networks, such as those generated by the hybrid HAR–ElasticNet framework.

2. Methodological Implementation in Hybrid HAR–ElasticNet Frameworks

JIRFs are computed atop a two-step estimation pipeline:

1. HAR Model Fitting: Own-market volatility dynamics for each asset $i$ are estimated by OLS via the univariate HAR specification:

$$RV_{i,t} = \alpha_i + \beta_i^{(d)}\,RV_{i,t-1} + \beta_i^{(w)}\,\overline{RV}^{(5)}_{i,t-1} + \beta_i^{(m)}\,\overline{RV}^{(22)}_{i,t-1} + u_{i,t}$$

with daily, weekly, and monthly aggregation operators, producing highly persistent coefficients ($\phi_i \approx 0.99$).

2. Cross-Market Regularization: Residuals $\tilde u_{i,t}$ are regressed on lagged volatilities of other assets $j \neq i$ via ElasticNet, producing a sparse estimate of cross-market spillover coefficients $\gamma_{ij}^{(\cdot)}$ for daily, weekly, and monthly lags. Hyperparameters are selected by time-series cross-validation, and the sparsity pattern is used to define the empirical network structure.

Given the estimated network, JIRFs are constructed following a generalized impulse response approach [Wiesen & Beaumont (2024); Pesaran & Shin (1998)]:

• Two forecast trajectories are recursively constructed: one applying a (possibly multi-asset, $\mathcal{S}$) shock at $t_0$, another running baseline dynamics.
• At each period $t_{0}+h$, the difference in realized volatility across these trajectories defines the JIRF at horizon $h$.
• Features, including daily/weekly/monthly aggregation, are updated from simulated histories, ensuring the full lag structure is respected (Mallory, 6 Jan 2026).

Block bootstrap (length 50) with 1,000 replications is used to generate 95% confidence bands. This accounts for serial dependence due to the rolling window construction of realized volatility measures.

3. Empirical Spillover Findings in High-Dimensional Volatility Networks

Application to a six-market system (soybeans [ZS], crude oil [CL], S&P 500 [ES], Nasdaq-100 [NQ], 5-year Treasury [ZF], 10-year Treasury [ZN]) reveals a highly sparse network:

• Only 7 of 90 possible cross-market coefficients are nonzero (≈8 %).
• Equities (ES, NQ) function as transmitters, emitting spillovers (nonzero $\gamma$'s into CL); they themselves receive no significant cross-market spillovers.
• Crude oil (CL) is the largest receiver, affected by small daily and monthly spillovers from equities and a minor daily effect from ZF.
• Treasuries and soybeans remain largely isolated; ZF has minimal outgoing links.

JIRFs computed within this system delineate the dynamic and horizon-specific propagation of shocks; e.g., a joint equity shock traces its effect on CL volatility across multiple days, with empirical variability quantified by bootstrap confidence intervals (Mallory, 6 Jan 2026).

4. Forecast Performance and Comparative Evaluation

The hybrid HAR–ElasticNet approach delivers one-step-ahead RMSE, MAE, and MAPE metrics that are indistinguishable from those of the univariate HAR model. For example, average out-of-sample RMSE over July 2020–January 2025 is $0.0044$ for both approaches; average MAE (≈0.0026) and average MAPE (≈1.6 %) are likewise identical. Thus, explicit modeling of the sparse spillover network underlying JIRFs does not compromise forecast accuracy but enables structural revelation unattainable by univariate modeling (Mallory, 6 Jan 2026).

5. Advantages, Practical Guidelines, and Limitations

Advantages:

• Separation of own-market and spillover dynamics preserves HAR persistence ($\phi\approx 0.99$) while isolating identification uncertainty to cross-market channels.
• Exact sparsity in the cross-market network supports valid interpretation of edges and IRF pathways.
• JIRFs specifically trace joint, multi-asset shocks, extending analytic capabilities beyond traditional single-shock IRF analysis.

Guidelines:

• Estimation should proceed with OLS for own-market HAR coefficients, followed by ElasticNet regularization exclusively on cross-market terms.
• Time-series cross-validation with nonoverlapping folds is recommended for penalty parameter selection.
• Use block bootstrap (block length $\geq$ rolling window) for inference within JIRF computation.
• Network matrix visualization is critical to verify sparsity prior to interpreting directional spillovers.

Limitations:

• The static network specification (2002–2025) precludes detection of time-varying spillovers or crisis regime shifts. Possible extensions include rolling ElasticNet and state-space priors.
• Adaptations such as group-LASSO or adaptive LASSO may respect asset-class structures, and higher-frequency data could facilitate finer-scale propagation analysis (Mallory, 6 Jan 2026).

6. Theoretical and Methodological Context

The JIRF methodology adapts the generalized impulse paradigm to modern, high-dimensional systems characterized by ultra-sparse network connectivity, as evidenced in the HAR–ElasticNet implementation. Compared to direct ElasticNet estimation on the full system (which yields overly shrunk own-lags and spurious links), the hybrid approach with JIRFs produces more plausible long-memory dynamics and interpretable shock trajectories. As such, JIRFs bridge the gap between structural network modeling and dynamic causal tracing in contemporary financial econometrics.

7. Broader Implications and Ongoing Research

JIRFs elucidate systemic risk channels and enable robust scenario analysis in high-dimensional settings where traditional VAR-based methods are intractable or non-interpretable. The hybrid HAR–ElasticNet with JIRFs offers a methodological blueprint for future work on regime-dependent, time-varying, or frequency-domain spillovers, as well as for applications beyond volatility to other domains such as macroeconomic or climate networks. A plausible implication is that as empirical networks grow in size and complexity, the interpretability and simulation-based nature of JIRFs will become increasingly central to multi-asset risk management and systemic stability analysis (Mallory, 6 Jan 2026).