Crisis-Regime Dynamic Volatility Spillovers in U.S. Commodity Markets: A Bayesian Mixture-Identified SVAR Approach

Abstract

Conventional VAR-based volatility spillover measures rely on homoskedasticity and single-Gaussian assumptions, limiting their ability to capture structural breaks and heterogeneous shocks during crises. This study develops a flexible framework to analyze volatility transmission in U.S. commodity markets under multiple crisis regimes. We propose a Bayesian Structural Vector Autoregressive Mixture Normal (BSVAR-MIX) model that embeds finite normal mixtures within a mixture-based heteroskedastic structural VAR framework. The model combines generalized forecast error variance decomposition with posterior-probability weighting. Daily data for eight U.S. benchmark commodities across food, energy, and precious metals markets are examined over the 2008–2016 global financial crisis and the 2017–2025 multi-crisis period, including COVID-19 and the Russia–Ukraine conflict. The BSVAR-MIX framework provides a flexible descriptive setting for capturing multimodal shocks, heteroskedastic volatility states, and regime-dependent spillover patterns in commodity markets. Empirically, Gold and oil dominate systemic volatility transmission, soybeans amplify food–energy spillovers, while coal and wheat exhibit rising fragility under policy and geopolitical shocks. Assets commonly viewed as safe havens may contribute to systemic stress during extreme events. Overall, the framework offers a robust tool for structural shock identification and cross-commodity risk monitoring relevant to U.S. macroprudential policy.

1. Introduction

Commodity markets provide a particularly informative environment for studying risk transmission. Energy, metals, and agricultural commodities jointly underpin global production systems and are interconnected through supply chains, input costs, and portfolio allocation channels. While cross-market volatility spillovers are widely studied, fewer studies explicitly compare spillover structures across major commodity groups under multiple crisis regimes using structural shock identification. This gap becomes especially relevant during periods of severe stress, such as the COVID-19 pandemic and the Russia–Ukraine conflict, when disruptions to energy supply, food security, and safe-haven demand interact and reinforce one another. Energy price shocks affect agricultural markets through production and transportation costs; food price fluctuations influence precious metals via inflation expectations and risk hedging behavior; and gold, in turn, shapes broader portfolio reallocation. These interactions form a risk transmission network whose structure may change substantially under crisis conditions.

Most empirical studies (Antonakakis et al. 2020; Hammoudeh et al. 2024) measure volatility spillovers using variance-decomposition techniques based on vector autoregressive (VAR) models (Diebold and Yilmaz 2009, 2012). Although these frameworks provide a convenient representation of interconnectedness, they rely on restrictive assumptions regarding the distribution and stability of model disturbances. In particular, shocks are typically assumed to be Gaussian and homoskedastic. Such assumptions limit the ability of standard models to capture sudden crisis shocks—large, unexpected disturbances associated with financial, geopolitical, or health-related crises—that play a central role in systemic risk amplification.

When extreme shocks occur, contemporaneous relationships among markets may shift and volatility transmission mechanisms may change. Historical episodes, including the oil crises of the 1970s and the global financial crisis, illustrate how abrupt disturbances can alter risk propagation and generate persistent instability. Structural VAR (SVAR) models partially address this issue by imposing identifying restrictions, yet their application to commodity price data remains challenging. Commodity markets are frequently exposed to crisis-related shocks that generate multimodal and heavy-tailed price distributions (Malsiner-Walli et al. 2016). At the same time, volatility in commodity prices is highly state dependent, varying systematically across business-cycle phases (Brunnermeier et al. 2021). These distributional and volatility features are often accompanied by shifts in transmission mechanisms between low- and high-volatility regimes (Lanne and Lütkepohl 2010). Together, these characteristics complicate the identification of structural shocks and hinder robustness measurement of regime-dependent risk spillovers.

To better capture these characteristics, this study adopts a risk-oriented empirical framework based on the Bayesian Structural VAR with Normal Mixtures (BSVAR-MIX). The mixture-based structure provides a flexible identification tool that accommodates distributional heterogeneity and regime-dependent volatility, allowing crisis-related structural shocks to be identified without imposing ex ante breakpoints or crisis dummy variables. Rather than emphasizing methodological novelty, the framework is employed to improve the empirical characterization of volatility spillovers under extreme events.

Although BSVAR-MIX has strong theoretical appeal, its application to volatility spillovers in commodity markets remains limited. This study addresses this gap by combining mixture-based heteroskedastic identification with variance decomposition, allowing regime-dependent spillovers associated with sudden crisis shocks to be summarized in a transparent manner.

Empirically, the study examines daily prices of eight U.S. benchmark commodities—rice, soybeans, wheat, crude oil (WTI), coal, natural gas, gold, and silver—over two economically distinct periods: the global financial crisis and post-crisis normalization phase (period 1: 14 January 2008–30 December 2016), and the subsequent multi-crisis period shaped by the COVID-19 pandemic and geopolitical disruptions (period 2: 3 January 2017–23 June 2025). This sample design facilitates a transparent comparison across two economically distinct stress environments without imposing observation-level crisis dummy variables within each estimation sample. While the subsample split is introduced ex ante for comparative purposes, the volatility states and spillover structures within each subsample are identified from the data through the estimated mixture-based heteroskedastic framework.

By documenting how the magnitude and direction of volatility spillovers vary across commodities and crisis regimes, this study contributes to the literature on systemic risk in commodity markets. The findings provide a data-driven assessment of how sudden crisis shocks reshape cross-market risk transmission, offering insights into the dynamics of volatility spillovers under extreme uncertainty.

This study is organized into five main sections. Section 2 provides a comprehensive review of the relevant literature. Section 3 presents the methodological framework and scenario-based applications. Section 4 reports the empirical results, validating the model’s robustness under variable reordering, quantifying its forecasting advantage in persistent volatility clustering, and uncovering time-varying asymmetric transmission mechanisms between the energy and equity markets. Section 5 concludes with a summary of three key contributions.

2. Literature Review

2.1. Volatility Spillovers in Commodity Markets

A growing body of empirical research documents volatility spillovers across commodity markets, reflecting the increasing financialization and interconnectedness of these assets (Chan et al. 2018; Mellouli et al. 2025). Early studies emphasize that shocks originating in one commodity market can transmit rapidly to others, particularly during periods of heightened uncertainty, thereby weakening diversification benefits. Empirical evidence shows that energy commodities often play a central role in volatility transmission, while metals and agricultural commodities may act as either transmitters or receivers depending on market conditions (Tessmann et al. 2024).

To quantify these transmission mechanisms, many studies employ vector autoregressive (VAR) models combined with forecast error variance decomposition techniques (Baruník and Křehlík 2018; Balli et al. 2023; Naeem et al. 2024). The connectedness framework proposed by Diebold and Yilmaz (2012) has become a standard tool in this literature, providing intuitive measures of total, directional, and net spillovers across markets. Applications of this framework to commodity markets have yielded robust evidence of time variation in spillover intensity, with pronounced increases during major crisis episodes.

More recent studies reinforce this view. For example, Kočenda and Moravcová (2024) show that connectedness among U.S. energy commodities rises over time and becomes especially pronounced during distress episodes such as the Russia–Ukraine war, the COVID-19 pandemic, and the global financial crisis. Similarly, Cunado et al. (2024) document stronger realized volatility spillovers between energy and metal markets in the post-COVID period, further supporting the view that commodity interconnectedness intensifies under major global shocks.

Despite these advances, much of the existing literature relies on constant-parameter models or full-sample estimation, implicitly assuming that spillover dynamics are sufficiently stable over time. While this approach is informative for long-run connectedness, it may mask important differences in volatility transmission across market states, particularly when stress originates from distinct economic sources.

This limitation is particularly relevant for the present study, because our results likewise suggest that connectedness is not constant across time: both the level and the structure of spillovers differ markedly across crisis environments, with stronger systemic transmission emerging in the later multi-crisis period.

2.2. Crisis Effects and the Use of Dummy Variables

A common strategy in empirical studies is to account for crisis effects through the inclusion of dummy variables that distinguish crisis from non-crisis periods (Al-Gamrh et al. 2018; Sapiri et al. 2026). This approach has been widely adopted in studies examining the impact of the Global Financial Crisis, the European sovereign debt crisis, and other episodes of market turmoil on volatility spillovers (Rodríguez et al. 2025).

While crisis dummy variables offer a convenient way to isolate average differences between turbulent and tranquil periods, they impose restrictive assumptions that may be problematic in the context of volatility spillovers. In particular, crisis periods must be dated ex ante, crisis effects are assumed to be homogeneous within each interval, and changes in spillover intensity are typically modeled as discrete shifts rather than gradual transitions. As a result, dummy-based specifications may overlook variations in crisis intensity and fail to capture changes in the structure of shock transmission across markets.

Several studies have noted that volatility dynamics often evolve continuously and that spillover patterns may differ not only between crisis and non-crisis periods but also across phases within a given crisis episode (Zhang et al. 2025b; Bethlendi and Szabó 2025). These observations suggest the need for empirical frameworks that allow spillover behavior to vary with market conditions rather than being imposed through binary indicators.

This concern is directly relevant to our setting, where the empirical results indicate not only stronger spillovers during crisis periods, but also shifts in the identities of net transmitters and receivers across different stress environments.

2.3. Time-Varying and State-Dependent Spillover Dynamics

In response to the limitations of constant-parameter and dummy-based approaches, a strand of the literature introduces time-varying (Antonakakis et al. 2020) or state-dependent models to study volatility spillovers (Zhu et al. 2025). These studies employ extensions of VAR models that allow parameters or volatility processes to evolve over time, capturing gradual changes in market conditions and shifts in interconnectedness.

Empirical findings from several studies (Masmoudi Kammoun 2025) indicate that spillovers tend to intensify during periods of elevated volatility and that the roles of individual markets as transmitters or receivers are not stable over time. Importantly, the structure of volatility transmission may differ across crises driven by financial system fragility versus those triggered by exogenous shocks, such as global health events or geopolitical tensions (Ben Salem and El Aoun 2025). This evidence highlights the relevance of distinguishing between different stress environments when analyzing spillover dynamics.

However, much of the existing work focuses on methodological extensions or emphasizes forecasting performance, sometimes at the expense of economic interpretation. In applied settings, there remains a need for empirical analyses that balance flexibility with transparency and focus on documenting economically meaningful patterns in volatility transmission across distinct crisis environments.

This is precisely where the present study contributes: rather than ranking models by predictive performance, it uses a mixture-based structural VAR framework to describe how crisis-dependent volatility states reshape connectedness across energy, agricultural, and precious-metal markets. In this respect, our findings that gold and oil dominate systemic transmission, soybeans strengthen food–energy spillovers, and coal and wheat become more fragile under policy and geopolitical stress provide an economically interpretable complement to the recent time-varying connectedness literature.

2.4. Positioning of the Present Study

The present study builds on the above literature by adopting an application-oriented perspective on volatility spillovers in commodity markets. Rather than relying on crisis dummy variables or full-sample estimation, the analysis allows volatility dynamics and spillover patterns to evolve with market conditions. This design reflects the view that crises are heterogeneous in origin and intensity and that their effects on volatility transmission cannot be adequately captured by binary indicators.

By distinguishing between two economically distinct crisis environments—the financial crisis period and the more recent pandemic–geopolitical episode—the study provides a transparent comparison of spillover patterns across stress regimes. The empirical framework combines standard connectedness measures with a multivariate setting that accommodates time-varying volatility, thereby documenting how the magnitude and direction of spillovers differ across market states without imposing strong a priori restrictions.

In doing so, the study complements existing research by emphasizing descriptive robustness and economic interpretation rather than methodological novelty or model ranking. The results contribute to a clearer understanding of how volatility spillovers in commodity markets depend on the nature of underlying stress environments.

More specifically, the study speaks directly to recent evidence that commodity connectedness rises under overlapping macro-financial, geopolitical, and pandemic-related shocks (Kočenda and Moravcová 2024; Cunado et al. 2024; Balcilar et al. 2024). By applying a BSVAR-MIX framework, we extend this literature by showing not only that connectedness intensifies under crisis conditions, but also that the composition of systemic transmitters and receivers changes across volatility regimes. This literature-to-results link is central to the paper: our later-period findings of stronger systemic spillovers, dominant transmission from gold and oil, and heightened fragility in coal and wheat provide concrete evidence that commodity connectedness is both crisis-sensitive and regime-dependent.

3. Methodology and Scenario Analysis

3.1. Empirical Framework

The BSVAR-MIX model extends the traditional Structural Vector Autoregression (SVAR) framework (Lütkepohl et al. 2024), with its main contribution being a Bayesian specification of the variance–covariance matrix of structural shocks, modeled as a finite mixture of normal distributions. This approach facilitates shock identification by exploiting heteroskedasticity (Woźniak and Droumaguet 2015) across the mixture components. Let ${\mathrm{y}}_{\mathrm{t}}$ denote an $\mathrm{N}\times 1$ vector of daily commodity prices returns. The empirical analysis is conducted within a structural vector autoregressive SVAR framework of order $\mathrm{p}$,

$${\mathrm{y}}_{\mathrm{t}}={\mathrm{A}}_{\mathrm{d}}{\mathrm{d}}_{\mathrm{d}}+{\sum }_{\mathrm{i}}^{\mathrm{p}}{\mathrm{A}}_{\mathrm{i}}{\mathrm{y}}_{\mathrm{t}-\mathrm{i}}+{\mathrm{B}}_0{\mathsf{\epsilon }}_{\mathrm{t}},{\mathrm{B}}_0{{\mathsf{\epsilon }}_{\mathrm{t}}}_{\mathrm{t}}={\mathrm{u}}_{\mathrm{t}}$$

(1)

where ${\mathrm{A}}_1$, $\dots$, ${\mathrm{A}}_{\mathrm{p}}$ denote the autoregressive coefficient matrices, ${\mathsf{\epsilon }}_{\mathrm{t}}$ the reduced-form residuals, and ${\mathrm{B}}_0$ the structural matrix captures contemporaneous relationships among variables. The structural matrix ${\mathrm{B}}_0$ transforms ${\mathsf{\epsilon }}_{\mathrm{t}}$ into structural shocks ${\mathrm{u}}_{\mathrm{t}}$, i.e., ${\mathrm{B}}_0{{\mathsf{\epsilon }}_{\mathrm{t}}}_{\mathrm{t}}={\mathrm{u}}_{\mathrm{t}}$. The term ${\mathrm{d}}_{\mathrm{d}}$ represents a D-dimensional vector of deterministic components, which invariably includes a constant term and may incorporate dummy variables and exogenous regressors, while ${\mathrm{A}}_{\mathrm{d}}$ denotes the corresponding $\mathrm{N}\times \mathrm{D}$ parameter matrix.

Unlike conventional VAR and TVP-VAR models, the structural form of the SVAR model establishes a linear relationship between reduced-form residuals ${\mathsf{\epsilon }}_{\mathrm{t}}$ and structural shocks ${\mathrm{u}}_{\mathrm{t}}$ via the $\mathrm{N}\times \mathrm{N}$ structural matrix ${\mathrm{B}}_0$. The BSVAR-MIX model further posits that structural shocks ${\mathrm{u}}_{\mathrm{t}}$ follow an M-component finite mixture of normal distributions with mixture-based heteroskedastic variances under high- and low-volatility states. Specifically, the data-generating process assumes draws from a finite set of exchangeable mixture components, each corresponding to a distinct data cluster, with variances exhibiting non-constancy across volatility regimes.

Formally, for $\mathrm{N}$ observed time series variables ${\mathrm{y}}_{\mathrm{t}}=({\mathrm{y}}_{1\mathrm{t}}$,$\dots ,{\mathrm{y}}_{\mathrm{Nt}})$, all models adhere to the property that structural shocks follow a zero-mean normal distribution conditional on past observations and volatility state $\mathrm{s}$, where ${\mathrm{y}}_{\mathrm{it}}\in {\mathrm{R}}^{\mathrm{r}}$. This is characterized by an M-component ($\mathrm{M}=2$) mixture normal distribution:

$${\mathrm{u}}_{\mathrm{Mt}}|{\mathrm{y}}_{\mathrm{it}},\mathrm{s}=\left\{\begin{array}{l}{\mathrm{u}}_{\mathrm{M}1\mathrm{t}}|{\mathrm{y}}_{\mathrm{it}},\mathrm{S}~{\mathrm{N}}_{\mathrm{N}}(0,{\mathrm{I}}_{\mathrm{N}})\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}{\mathsf{\pi }}_{\mathrm{S}}\\ {\mathrm{u}}_{\mathrm{M}2\mathrm{t}}|{\mathrm{y}}_{\mathrm{it}},\mathrm{S}~{\mathrm{N}}_{\mathrm{N}}(0,\mathsf{\Theta })\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}1-{\mathsf{\pi }}_{\mathrm{S}}\end{array}\right.$$

(2)

Here, structural shock states are classified into high-volatility (${\mathrm{S}}_{1\mathrm{t}}$) and low-volatility (${\mathrm{S}}_{2\mathrm{t}}$) regimes, denoted collectively as $\mathrm{S}=({\mathrm{S}}_{1\mathrm{t}},{\mathrm{S}}_{2\mathrm{t}})$. The regime-specific probabilities are $\mathrm{P}({\mathrm{u}}_{\mathrm{M}1\mathrm{t}}\mid \mathrm{S})={\mathsf{\pi }}_{\mathrm{s}}$ and $\mathrm{P}({\mathrm{u}}_{\mathrm{M}2\mathrm{t}}\mid \mathrm{S})=1-{\mathsf{\pi }}_{\mathrm{s}}$. The matrix ${\mathrm{I}}_{\mathrm{N}}$ is the identity matrix, and $\mathsf{\Theta }=\mathrm{diag}({\mathsf{\theta }}_1,\dots ,{\mathsf{\theta }}_{\mathrm{N}})$ is a diagonal parameter matrix with positive elements ${\mathsf{\theta }}_{\mathrm{j}}(\mathrm{j}=1,\dots ,\mathrm{N})$. Notably, when ${\mathsf{\theta }}_{\mathrm{j}}=1$, the j-th component of ${\mathsf{\epsilon }}_{\mathrm{t}}$ reduces to a standard normal distribution. Consequently, certain components of ${\mathsf{\epsilon }}_{\mathrm{t}}$ may not exhibit mixture properties. Indeed, when $\mathsf{\Theta }={\mathrm{I}}_{\mathrm{N}},{\mathrm{u}}_{\mathrm{Mt}}\sim {\mathrm{N}}_{\mathrm{N}}(0,{\mathrm{I}}_{\mathrm{N}})$. Such specifications with conventional errors constitute a special case within our framework.

Following Song and Woźniak (2021), the probabilities of the latent volatility regimes are obtained from the posterior distribution of the regime indicators. Let denote the unobserved regime indicator at time t, where ${\mathrm{S}}_{\mathrm{t}}\in \left\{\mathrm{1,2}\right\}$ represents the low-volatility and high-volatility states, respectively. For each posterior draw of the model parameters, the regime probability is computed as $\mathrm{P}({\mathrm{u}}_{\mathrm{M}1\mathrm{t}}\mid {\mathrm{S}}_{\mathrm{t}}=\mathrm{j},{\mathrm{y}}_{\mathrm{it}},{\mathsf{\Theta }}^{\left(\mathrm{m}\right)})$, where ${\mathrm{y}}_{\mathrm{it}}$ denotes the observed data up to time t, ${\mathsf{\Theta }}^{\left(\mathrm{m}\right)}$ represents the m-th posterior draw of the model parameters, and $\mathrm{j}\in \{1,2\}$ denotes the regime index. In this study, we report the filtered regime probabilities, which condition only on information available up to time t.

It is critical to note that ${\mathrm{u}}_{\mathrm{t}}$ has zero mean and a covariance matrix ${\mathsf{\pi }}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{N}}+(1-{\mathsf{\pi }}_{\mathrm{s}})\mathsf{\Theta }$. Within our research framework, the mixture probability ${\mathsf{\pi }}_{\mathrm{s}}(0<{\mathsf{\pi }}_{\mathrm{s}}<1)$ is also a model parameter. Thus, the mixture distribution of structural shocks is formally expressed as:

$${\mathrm{u}}_{\mathrm{Mt}}|{\mathrm{y}}_{\mathrm{it}},\mathrm{s}~{\mathrm{N}}_{\mathrm{N}}(0,{\mathsf{\pi }}_{\mathrm{S}}{\mathrm{I}}_{\mathrm{N}}+(1-{\mathsf{\pi }}_{\mathrm{S}})\mathsf{\Theta })$$

(3)

This specification enables the model to capture multimodal volatility patterns frequently observed in financial time series.

3.2. Measurement of Volatility Spillovers

Variance decomposition quantifies the proportional contribution of each structural shock to the forecast error variance. To simplify derivations, let ${\mathsf{\Phi }}_{\mathrm{s}}={\mathsf{\pi }}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{N}}+(1-{\mathsf{\pi }}_{\mathrm{s}})\mathsf{\Theta }$ and ${\mathrm{A}}_0={\mathrm{A}}_{\mathrm{d}}{\mathrm{d}}_{\mathrm{d}}$. In the BSVAR-MIX($\mathrm{p}$) model, the mean term $\mathsf{\mu }$ represents the long-run equilibrium level of variables, while the component-specific covariance matrix ${\mathsf{\Phi }}_{\mathrm{s}}=\mathrm{diag}({\mathsf{\sigma }}_{\mathrm{Ns}}^2)$ is positive definite and invertible. The variance decomposition proceeds as follows:

First, Equation (1) is transformed into its $\mathrm{VMA}(\infty )$ representation:

$${\mathrm{y}}_{\mathrm{t}}=\mathsf{\mu }+\sum _{\mathrm{s}=0}^{\infty }{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{u}}_{\mathrm{t}-\mathrm{s}}$$

(4)

whereas, the transformation in Equation (4) requires incorporating the model’s stationarity condition for $\mathsf{\mu }$ and expectation operator. Firstly, since ${\mathrm{B}}_0{\mathsf{\epsilon }}_{\mathrm{t}}={\mathrm{u}}_{\mathrm{t}},$ the original model is simplified to ${\mathrm{y}}_{\mathrm{t}}={\mathrm{A}}_0+{\sum _{\mathrm{i}=1}^{\mathrm{p}}\mathrm{A}}_{\mathrm{i}}{\mathrm{y}}_{\mathrm{t}-\mathrm{i}}+{\mathrm{u}}_{\mathrm{t}}$. Given that the model satisfies the second-order stationarity condition, assuming $\mathrm{E}[{\mathrm{y}}_{\mathrm{t}}]=\mathrm{E}[{\mathrm{y}}_{\mathrm{t}-\mathrm{i}}]=\mathsf{\mu }$, we obtain: $\mathsf{\mu }={\mathrm{A}}_0+\sum _{\mathrm{i}=1}^{\mathrm{p}}{\mathrm{A}}_{\mathrm{i}}\mathsf{\mu }+\mathrm{E}\left[{\mathrm{u}}_{\mathrm{t}}\right]$. Since the unconditional mean of the structural shock satisfies $\mathrm{E}[{\mathrm{u}}_{\mathrm{t}}]=0$, we derive: $\mathsf{\mu }={\left({\mathrm{I}}_{\mathrm{n}}-\sum _{\mathrm{i}=1}^{\mathrm{p}}{\mathrm{A}}_{\mathrm{i}}\right)}^{-1}{\mathrm{A}}_0$. Secondly, apply mean adjustment to the original model (2), defining ${\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}}={\mathrm{y}}_{\mathrm{t}}-\mathsf{\mu }$. Substituting into the model yields: ${\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}}+\mathsf{\mu }={\mathrm{A}}_0+\sum _{\mathrm{i}=1}^{\mathrm{p}}{\mathrm{A}}_{\mathrm{i}}({\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}-\mathrm{i}}+\mathsf{\mu })+{\mathrm{u}}_{\mathrm{t}}$. Rearranging gives: ${\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}}=\sum _{\mathrm{i}=1}^{\mathrm{p}}{\mathrm{A}}_{\mathrm{i}}{\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}-\mathrm{i}}+{\mathrm{u}}_{\mathrm{t}}$. Thirdly, applying the lag operator L and its inverse, we obtain the infinite vector moving average (VMA(∞)) representation: ${\stackrel{\mathrm{~}}{\mathrm{y}}}_{\mathrm{t}}=\mathrm{A}(\mathrm{L}{)}^{-1}{\mathrm{u}}_{\mathrm{t}}=\sum _{\mathrm{s}=0}^{\infty }{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{u}}_{\mathrm{t}-\mathrm{s}}$. where ${\mathsf{\Psi }}_{\mathrm{s}}$ represents the impulse response matrices, satisfying the recursive condition:${\mathsf{\Psi }}_{\mathrm{s}}=\sum _{\mathrm{i}=1}^{\min (\mathrm{p},\mathrm{s})}{\mathrm{A}}_{\mathrm{i}}{\mathsf{\Psi }}_{\mathrm{s}-\mathrm{i}}$ and ${\mathsf{\Psi }}_0={\mathrm{I}}_{\mathrm{n}}$. Under stationarity, the roots of the characteristic equation det[A(z)] = 0 satisfy ∣z∣ > 1, ensuring the invertibility of $\mathrm{A}(\mathrm{L}{)}^{-1}$, which admits the representation: $\mathrm{A}(\mathrm{L}{)}^{-1}=\sum _{\mathrm{s}=0}^{\infty }{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{L}}^{\mathrm{s}}$. By multiplying both sides by A(L), we obtain $\mathrm{A}(\mathrm{L}{)}^{-1}{\mathrm{u}}_{\mathrm{t}}=\sum _{\mathrm{s}=0}^{\infty }{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{L}}^{\mathrm{s}}{\mathrm{u}}_{\mathrm{t}}$. According to the definition of the lag operator, ${\mathrm{L}}^{\mathrm{k}}{\mathrm{u}}_{\mathrm{t}}$ = ${\mathrm{u}}_{\mathrm{t}-\mathrm{k}}$, which implies that the time series will lag k periods behind. Therefore, the s-lag term becomes ${\mathrm{L}}^{\mathrm{s}}{\mathsf{\epsilon }}_{\mathrm{t}}$ = ${\mathrm{u}}_{\mathrm{t}-\mathrm{s}}$. Next, substituting the redefined expression ${\tilde{\mathrm{y}}}_{\mathrm{t}}={\mathrm{y}}_{\mathrm{t}}-\mathsf{\mu }$ into the model, we get ${\tilde{\mathrm{y}}}_{\mathrm{t}}=\mathrm{A}(\mathrm{L}{)}^{-1}{\mathrm{u}}_{\mathrm{t}}=\sum _{\mathrm{s}=0}^{\mathrm{\infty }}{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{u}}_{\mathrm{t}-\mathrm{s}}$. Thus, the original model becomes ${\mathrm{y}}_{\mathrm{t}}-\mathsf{\mu }=\sum _{\mathrm{s}=0}^{\mathrm{\infty }}{\mathsf{\Psi }}_{\mathrm{s}}{\mathrm{u}}_{\mathrm{t}-\mathrm{s}}$. Finally, let $\mathsf{\mu }=\mathrm{E}\left[{\mathrm{y}}_{\mathrm{t}}\right]$ denote the unconditional mean of the return vector. Under covariance stationarity, the model can be written in mean-adjusted form as ${\mathrm{y}}_{\mathrm{t}}-\mathsf{\mu }$, which yields the corresponding $\mathrm{VMA}(\infty )$ representation for deviations from the unconditional mean. For daily commodity returns, $\mathsf{\mu }$ may be small in magnitude, but it is retained here for theoretical completeness.

Under the conditions imposed on the impulse response matrix in Equation (4), we follow the generalized forecast error variance decomposition (GFEVD) framework introduced by Diebold and Yilmaz (2012) to empirically analyze the strength and direction of volatility spillovers. Considering an H-step forecasting horizon, the proportion of forecast error variance in variable i that is attributable to shocks in variable j under the mixture distribution M is defined as:

$${\mathsf{\theta }}_{\mathrm{ij}|\mathrm{S}}\left(\mathrm{H}\right)={\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{jj}|\mathrm{S}}^{-1}{\mathsf{\Sigma }}_{\mathrm{h}=0}^{\mathrm{H}-1}({\mathrm{e}}_{\mathrm{i}}’{\mathsf{\Psi }}_{\mathrm{h}}{\mathsf{\Phi }}_{\mathrm{S}}{\mathrm{e}}_{\mathrm{j}}{)}^2}{{\mathsf{\Sigma }}_{\mathrm{h}=0}^{\mathrm{H}-1}{\mathrm{e}}_{\mathrm{i}}’{\mathsf{\Psi }}_{\mathrm{h}}{\mathsf{\Phi }}_{\mathrm{S}}{\mathsf{\Psi }}_{\mathrm{h}}’{\mathrm{e}}_{\mathrm{i}}}}$$

(5)

Here, the numerator captures the cumulative impact of shock $\mathrm{j}$ across all mixture components, while the denominator normalizes by the total forecast error variance. The selection vector ${\mathrm{e}}_{\mathrm{j}}$ ensures additive results, ${\mathsf{\sigma }}_{\mathrm{jj}}$ denotes the standard deviation of the error term in the j-th equation, and ${\mathsf{\Psi }}_{\mathrm{h}}$ is the moving average coefficient matrix computed recursively from structural parameters. To obtain holistic volatility spillover measures, we average across states:

$${\mathsf{\theta }}_{\mathrm{ij}}\left(\mathrm{H}\right)={\mathsf{\Sigma }}_{\mathrm{S}=1}^2{\mathsf{\pi }}_{\mathrm{S}}{\mathsf{\theta }}_{\mathrm{ij}|\mathrm{S}}\left(\mathrm{H}\right)$$

(6)

where ${\mathsf{\pi }}_{\mathrm{s}}$ denotes the posterior weight associated with mixture component M. In the present specification, these weights are obtained from the estimated finite-mixture distribution rather than from an explicit Markov transition system (Lütkepohl and Woźniak 2020). Based on this, we construct four core indices:

Total Volatility Spillover Index (TVSI): Measures systemic risk transmission intensity:

$$\mathrm{TVSI}\left(\mathrm{H}\right)=({\mathsf{\Sigma }}_{\mathrm{i}\ne \mathrm{j}}{\mathsf{\theta }}_{\mathrm{ij}}\left(\mathrm{H}\right))/\mathrm{n}\times 100$$

(7)

Directional Spillover Index (DSI): Identifies risk transmitter/receiver roles:

$$\mathrm{DS}{\mathrm{I}}_{\mathrm{i}\to \mathrm{j}}\left(\mathrm{H}\right)={\mathsf{\theta }}_{\mathrm{ji}}\left(\mathrm{H}\right)/{\mathsf{\Sigma }}_{\mathrm{k}=1}^{\mathrm{n}}{\mathsf{\theta }}_{\mathrm{jk}}\left(\mathrm{H}\right)\times 100$$

(8)

Net Spillover Index (NSI): Quantifies net risk transmission status:

$$\mathrm{NS}{\mathrm{I}}_{\mathrm{i}}\left(\mathrm{H}\right)=\mathrm{DS}{\mathrm{I}}_{\mathrm{i}\to ·}\left(\mathrm{H}\right)-\mathrm{DS}{\mathrm{I}}_{\text{·}\to \mathrm{i}}\left(\mathrm{H}\right)$$

(9)

Integrating via state-dependent weights, these indicators capture risk transmission characteristics across both normal and extreme regimes, providing a multidimensional perspective for financial stability assessment.

These spillover measures are constructed objects rather than directly observable economic quantities. Accordingly, they are interpreted as descriptive summaries of cross-market volatility transmission rather than as structural measures of systemic risk or forecasting performance. The analysis emphasizes differences in spillover magnitude and direction across market states and crisis environments.

3.3. Time-Varying Volatility Spillover and Crisis Dynamics

Consistent with the empirical design of this study, crisis dynamics are not imposed through exogenous dummy variables. Instead, time variation in volatility spillovers is captured using a rolling-window framework, allowing the intensity of cross-market risk transmission to evolve continuously over time.

The rolling-window length used to compute dynamic volatility spillovers is selected based on the RMSE-based Volatility Spillover efficiency Index (RMSE-VSEI = $\sqrt{{\displaystyle \frac{1}{\mathrm{N}}}{\sum _{\mathrm{j}=1}^{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{N}}\left({\widehat{\mathrm{TVSI}}}_{\mathrm{ijt}}-{\mathrm{TVSI}}_{\mathrm{ijt}}^{\mathrm{benchmark}}\right)}^2}$). Benchmark selection must balance theoretical rigor with empirical plausibility: historical extreme-event peaks (e.g., cross-market volatility transmission during the 2008 global financial crisis) or theoretical bounds (e.g., zero spillover under complete market segmentation) may be employed. Critically, benchmark choice significantly impacts metric validity—static historical benchmarks may overestimate current market segmentation during financial integration. Thus, dynamic rolling-window benchmarks (e.g., moving averages of recent spillover peaks) enhance temporal adaptability. Specifically, a set of candidate window lengths is evaluated, and the window that minimizes the RMSE-VSEI criterion is chosen as optimal. This data-driven procedure ensures that the window length balances estimation stability and responsiveness to changes in volatility dynamics.

3.4. Data Sources and Construction

Figure 1 plots the logarithmic price series of the eight commodity futures considered in this study. Because these markets are quoted in different units (e.g., dollars per barrel, dollars per ounce, or dollars per bushel), the logarithmic transformation allows the series to be presented on a comparable scale. The figure also highlights the baseline subsample split on 3 January 2017, which separates the relatively stable post-GFC period from the subsequent multi-crisis environment characterized by trade tensions, the COVID-19 pandemic, and geopolitical disruptions. As is standard in the spillover literature, volatility transmission is analyzed using log-returns rather than price levels in order to ensure stationarity and avoid spurious dynamic relationships. Unit root tests (available upon request) confirm that the log-return series are stationary.

Daily commodity future prices are obtained from the Markets Insider platform (INSIDER 2025): https://markets.businessinsider.com/commodities (accessed on 23 June 2025), which serves as a publicly accessible data aggregation portal rather than a proprietary database. The underlying prices reported on the platform are sourced from the major exchanges on which the respective futures contracts are traded.

Specifically, gold and silver futures prices are sourced from COMEX; crude oil (WTI) and natural gas futures prices are sourced from the Chicago Mercantile Exchange (CME); and agricultural commodities, including rice, wheat, and soybeans, are sourced from the Chicago Board of Trade (CBOT). Coal prices reflect benchmark futures or index-linked contracts reported through exchange-based trading venues. For each commodity, the platform provides continuous historical close price series corresponding to the most actively traded futures contracts.

The analysis employs front-month (nearest-to-maturity futures contracts, which typically exhibit the highest liquidity). Continuous price series are constructed using the rolling procedure applied by the data provider, whereby the active contract is switched to the next maturity as the current contract approaches expiration. This approach ensures continuity and liquidity in the reported series and is standard in empirical studies focusing on volatility dynamics rather than term-structure effects.

The reported prices correspond to settlement prices, or exchange-equivalent closing prices, as disseminated by the underlying exchanges and aggregated by Markets Insider. Settlement prices are used because they reflect official end-of-day valuations employed for margining and risk management, providing a consistent benchmark across markets.

Daily returns are constructed as log differences in prices: ${\mathrm{r}}_{\mathrm{i},\mathrm{t}}=100\times (\log {\mathrm{P}}_{\mathrm{i},\mathrm{t}}-\log {\mathrm{P}}_{\mathrm{i},\mathrm{t}-1}),$ where ${\mathrm{P}}_{\mathrm{i},\mathrm{t}}$ denotes the settlement price of the continuous front-month futures contract for commodity $\mathrm{i}$ on day $\mathrm{t}$.

3.5. Sample Period and Subsample Design

The full sample spans 14 January 2008 to 23 June 2025. The empirical analysis is conducted separately for two subsamples corresponding to distinct crisis environments, as discussed in the Introduction. The first subsample 2008–2016 covers the Global Financial Crisis, the European sovereign debt crisis, and the subsequent post-crisis normalization phase. The second subsample 2017–2025 reflects a different risk environment that precedes and includes the COVID-19 pandemic and subsequent geopolitical disruptions. This subsample design facilitates a transparent comparison of volatility spillover patterns across economically different stress regimes while avoiding the imposition of exogenous crisis indicators at the observation level.

The division into 2008–2016 and 2017–2025 is a deliberate empirical design choice intended to compare two broad and economically distinct stress environments: a financial-crisis/post-crisis environment and a pandemic–geopolitical environment. We do not interpret the 2016–2017 boundary as an endogenously estimated structural break. Rather, the split is used to improve interpretability and comparability across major stress episodes, while crisis-state heterogeneity within each subsample is still allowed to emerge from the estimated mixture-based volatility structure.

3.6. Scope and Interpretation

The SVAR is estimated in a Bayesian setting, which provides a stable empirical environment for multivariate return dynamics under volatility clustering and parameter uncertainty. The Bayesian framework is employed as an estimation tool rather than as a methodological contribution, and the analysis focuses on the implied patterns of volatility transmission across markets.

The empirical framework is designed to document state- and crisis-dependent patterns in volatility spillovers rather than to establish predictive dominance or causal policy effects. Spillover measures are interpreted as descriptive indicators of interconnectedness under different market conditions. Technical details regarding prior specification, estimation algorithms, and additional robustness checks are reported in the literature (Woźniak 2024).

4. Results Analysis

4.1. Stationarity Tests

Table 1. Unit root test.

Variables	Level Data (t-Value)			First Difference Data
	ADF	PP	KPSS	ADF	PP	KPSS
Rice	−2.85 (0.22)	−3.61 (0.03)	4.70 (0.01)	−18.96 (0.01)	−67.10 (0.01)	0.02 (0.10)
Soybeans	−2.71 (0.28)	−2.56 (0.34)	2.61 (0.01)	−15.56 (0.01)	−64.85 (0.01)	0.04 (0.10)
Wheat	−3.09 (0.12)	−3.02 (0.15)	7.92 (0.01)	−15.93 (0.01)	−63.83 (0.01)	0.06 (0.10)
Oil (WTI)	−2.83 (0.23)	−2.54 (0.35)	5.99 (0.01)	−14.27 (0.01)	−66.08 (0.01)	0.05 (0.10)
Coal	−2.62 (0.32)	−2.93 (0.19)	5.25 (0.01)	−15.41 (0.01)	−58.75 (0.01)	0.03 (0.10)
Natural Gas	−3.81 (0.02)	−3.41 (0.05)	5.28 (0.01)	−14.71 (0.01)	−70.11 (0.01)	0.10 (0.10)
Gold	1.05 (0.99)	0.78 (0.99)	20.55 (0.01)	−16.91 (0.01)	−66.35 (0.01)	0.56 (0.03)
Silver	−1.75 (0.68)	−2.02 (0.57)	3.38 (0.01)	−17.64 (0.01)	−65.72 (0.01)	0.09 (0.10)

Notes: Parentheses denote p-values. When the p-value is much smaller than 0.01, it is reported as 0.01. When the p-value is much larger than 0.1, it is reported as 0.1.

and

Table 2. Zivot–Andrews unit root test results with variance breaks.

Variable	Test_Statistic	CV_1	CV_5	CV_10	Break_Date	Decision_at_10 (Reject H0)
Rice	−5.3026	0.01	−5.57	0.05	11 July 2014	(stationary after break)
Soybeans	−3.9074	0.01	−5.57	0.05	24 August 2020	(stationary after break)
Wheat	−4.6029	0.01	−5.57	0.05	9 July 2021	(stationary after break)
Oil (WTI)	−4.3306	0.01	−5.57	0.05	29 September 2014	(stationary after break)
Coal	−5.8234	0.01	−5.57	0.05	31 December 2021	(stationary after break)
Natural Gas	−6.9813	0.01	−5.57	0.05	7 July 2008	(stationary after break)
Gold	−2.3842	0.01	−5.57	0.05	26 July 2023	(stationary after break)
Silver	−4.042	0.01	−5.57	0.05	29 November 2012	(stationary after break)

Notes: CV_1, CV_5, and CV_10 denote the 1%, 5%, and 10% critical values of the Zivot–Andrews unit root test, respectively. If the reported test statistic is smaller (more negative) than the corresponding critical value, the null hypothesis of a unit root is rejected at that significance level, indicating stationarity after accounting for the structural break.

report unit root test results for the eight commodity price series. Conventional ADF, PP, and KPSS tests (Table 1) consistently indicate that all series are integrated of order one. At levels, most ADF and PP p-values exceed 0.05 (e.g., soybean ADF p = 0.28; wheat PP p = 0.15), failing to reject the unit root null, while KPSS p-values are uniformly 0.01, rejecting stationarity. After first differencing, ADF and PP p-values fall to 0.01, and most KPSS p-values rise to 0.10 (e.g., natural gas and silver), with gold as the only exception (KPSS p = 0.03), though its stationarity is still supported by ADF and PP results. Overall, these findings classify all series as I(1).

However, standard unit root tests are known to have low power in the presence of structural breaks, which are common in commodity prices exposed to major economic and geopolitical shocks. To account for this possibility, Table 2 reports results from the Zivot and Andrews (2002) unit root test under the “both” specification, allowing for an endogenous break in both the intercept and trend. Using critical values of −5.57 (1%), −5.08 (5%), and −4.82 (10%), the results show that several commodities—including rice, WTI crude oil, and coal—exhibit stationarity once structural breaks are incorporated.

The estimated break dates are economically meaningful and align with major events such as the 2014 oil price collapse, the COVID-19 pandemic in 2020, and the Russia–Ukraine conflict in 2021. These findings suggest that part of the apparent persistence in commodity prices reflects break-induced dynamics rather than genuine unit roots, reinforcing the importance of accounting for structural changes when modeling volatility regime-switching behavior.

The Zivot and Andrews (2002) test is employed because it allows for an endogenous structural break within the unit root testing framework. While alternative approaches, such as the Bai–Perron multiple break test, can identify several structural changes in regression models, the Zivot–Andrews procedure is more appropriate here, as our objective is to test the stationarity of the series in the presence of a potential structural break rather than to estimate multiple breakpoints. In commodity markets, major global shocks (e.g., financial crises or geopolitical conflicts) often generate dominant structural changes, making a single-break framework a reasonable approximation for preliminary stationarity testing.

To examine whether the empirical findings depend on the specific subsample split, we conduct a robustness analysis using several alternative cutoff dates around the baseline break (January 2017). For each candidate cutoff date, the sample is divided into two subsamples and the total connectedness index (TCI) is recalculated using the VAR-based connectedness framework.

As shown in Figure 2, the level of connectedness in the later subsample consistently exceeds that in the earlier subsample across all alternative cutoff choices. Moreover, the magnitude of the TCI remains relatively stable when the cutoff is shifted from mid-2016 to mid-2017. This evidence indicates that the key conclusion of stronger volatility spillovers in the later period is not driven by the specific choice of the baseline breakpoint.

Table 3. Lag order selection and robustness check.

	Period 1				Period 2
Lag	TCI (%)	AIC	HQ	SC	TCI (%)	AIC	HQ	SC
1	18.60	9.65	9.72	9.84	12.41	10.72	10.80	10.92
2	19.23	9.64	9.77	10.00	12.72	10.74	10.88	11.11
3	19.43	9.65	9.85	10.18	13.05	10.76	10.96	11.31
4	19.79	9.67	9.93	10.37	13.34	10.77	11.03	11.49
5	20.07	9.69	10.01	10.56	13.51	10.80	11.13	11.70

reports the lag order selection results together with the sensitivity of the total connectedness index (TCI) to alternative lag specifications. The optimal lag length was initially evaluated using several standard information criteria, including the Akaike Information Criterion (AIC), the Hannan–Quinn Criterion (HQ), and the Schwarz Criterion (SC). For Period 1 (2008–2016), the AIC reaches its minimum at lag two, while HQ and SC favor a more parsimonious specification with one lag. For Period 2 (2017–2025), all criteria indicate a lag order of one. Considering the trade-off between model parsimony and the need to capture sufficiently rich dynamic interactions among commodity markets, a lag order of two is adopted as the baseline specification.

To further verify that the results are not driven by the choice of lag length, the VAR model is re-estimated using lag orders ranging from one to five. As shown in Table 3, the estimated TCI varies only slightly across alternative lag specifications. In Period 1, TCI ranges from 18.60% to 20.07%, while in Period 2, it varies between 12.41% and 13.51%. Importantly, the incremental change in TCI associated with each additional lag is consistently below one percentage point, indicating that the overall level of connectedness is largely insensitive to the lag specification. Therefore, the empirical findings are robust to alternative lag choices. The incremental change in TCI associated with each additional lag is consistently below one percentage point.

It is also worth noting that the lag order selection is based on the reduced-form VAR specification rather than the structural identification scheme. In Bayesian structural VAR models, the structural restrictions affect the identification of shocks but do not alter the underlying lag dynamics of the system. Consequently, the lag length determined from the reduced-form VAR model can be directly applied to the BSVAR framework. This practice is standard in the VAR literature, where the lag order is typically determined prior to structural identification. Given that (i) the AIC favors a lag order of two in the first subsample, (ii) the connectedness measures remain highly stable across alternative lag specifications, and (iii) the lag structure is independent of the Bayesian structural identification, we adopt p = 2 as the baseline lag order in the subsequent BSVAR-MIX estimations. The robustness analysis reported in Table 3 further confirms that the choice of lag length does not materially affect the estimated connectedness levels or the interpretation of the results.

As the baseline subsample split is motivated by comparative crisis-environment design rather than statistical break-date estimation, we further assess whether the main connectedness patterns are sensitive to nearby alternative cutoff dates and to modest changes in the VAR lag order. The results show that the identity of the major net transmitters and net receivers remains broadly stable, suggesting that the core conclusions are not driven by a single partition rule or lag choice.

4.2. Volatility Posterior Regime Probabilities Under Extreme Events

Figure 3 shows the smoothed posterior probabilities of two volatility states over the full sample period. The density of the colored vertical bars reflects the frequency of regime switching, with denser segments indicating rapid transitions and wider blank intervals indicating regime persistence. Two periods stand out with particularly frequent switching: early 2008–2009 and January 2020–late 2023.

The 2008–2009 episode corresponds to the global financial crisis, during which sharp contractions in demand, liquidity shortages, and heightened financial stress repeatedly altered volatility conditions across commodity markets, leading to frequent shifts between volatility states. A similar pattern is observed during 2020–2023, coinciding with the COVID-19 pandemic and the escalation of the Russia–Ukraine conflict. In this later period, sanctions on Russian energy exports and rising geopolitical tensions substantially increased energy policy uncertainty and global risk exposure. Supporting evidence is provided by Pan and Sun (2023), who document intensified volatility spillovers in global crude-oil futures following the 2022 conflict, and by Chen et al. (2023), who report a pronounced surge in energy-market volatility—particularly in natural gas—after the onset of the conflict.

These episodes illustrate that frequent volatility state transitions arise when markets are exposed to overlapping financial, geopolitical, and policy shocks, rather than isolated events, underscoring the relevance of a regime-switching heteroskedasticity framework for identifying structural shocks under evolving volatility conditions.

4.3. Analysis of Volatility Spillover Efficiency Index

Figure 4 reports the RMSE-VSEI criterion used to select the rolling-window length for the dynamic spillover analysis. It is used here as a window-selection device rather than as evidence of forecast superiority over competing models. RMSE-VSEI values are relatively large for window lengths below 200 and above 450; therefore, the analysis focuses on the range of 200–450 observations. Within this range, the BSVAR-MIX model selects optimal windows of 438 and 404 for the two subsamples, respectively. These RMSE-VSEI–minimizing windows are used for all subsequent dynamic spillover calculations.

4.4. Tests for Multi-Normality Distributions

Figure 5 and Figure 6 report density plots of rolling realized volatility for eight commodities (coal, gold, natural gas, crude oil, rice, silver, soybeans, and wheat) in Periods 1 and 2, respectively. Volatility is measured as the standard deviation of daily returns using window lengths of W = 438 and W = 404, and its distribution is approximated by a two-component normal mixture model estimated via the EM algorithm. In Period 1, volatility distributions display clear multimodality and skewness; for example, crude oil volatility shows multiple peaks in the range of 1.0–3.5, deviating from a single Gaussian shape. In Period 2, non-Gaussian features become more pronounced: gold and silver concentrate more mass in high-volatility regions, while soybeans and wheat exhibit left-skewed distributions, indicating clustering of extreme risks. These patterns provide visual evidence of shock heterogeneity across crisis regimes and motivate the use of a finite-mixture BSVAR framework to characterize complex volatility dynamics.

4.5. Evidence on Heteroskedasticity

Table 4. Tests for Heteroskedasticity.

	log(SDDR)	NSE	Pr [H0|Data]	Pr [H1|Data]
period 1
Shock1	−269.59	0.00	0.00	1.00
Shock2	−409.66	0.00	0.00	1.00
Shock3	−422.10	0.00	0.00	1.00
Shock4	−340.27	0.00	0.00	1.00
Shock5	−1667.08	0.00	0.00	1.00
Shock6	−76.45	0.00	0.00	1.00
Shock7	−291.09	0.00	0.00	1.00
Shock8	−226.60	0.00	0.00	1.00
period 2
Shock1	−287.19	0.00	0.00	1.00
Shock2	−401.43	0.00	0.00	1.00
Shock3	−411.51	0.00	0.00	1.00
Shock4	−332.17	0.00	0.00	1.00
Shock5	−1702.42	0.00	0.00	1.00
Shock6	−43.57	0.00	0.00	1.00
Shock7	−304.46	0.00	0.00	1.00
Shock8	−229.82	0.00	0.00	1.00

Note: All values are rounded to the nearest (specified) decimal. SDDR is computed following the formula proposed by Verdinelli and Wasserman (1995); see their paper for details.

reports the Bayesian factor test based on the Savage–Dickey Density Ratio (SDDR) (Verdinelli and Wasserman 1995), which examines the heteroskedasticity of structural shocks in the eight major commodity markets over the periods 1 and 2. The test compares the null hypothesis H₀ (δ = 1, homoskedasticity) against the alternative H₁ (δ ≠ 1, heteroskedasticity). The results show that the log(SDDR) values for all structural shocks are highly negative (e.g., for Shock 5 during the financial crisis, log(SDDR) = −1667.08), with numerical standard errors (NSE) approaching zero. Moreover, the posterior probabilities consistently yield Pr [H₀|data] = 0, and Pr [H₁|data] = 1.

This outcome far exceeds the conventional threshold of log(SDDR) < −5 for decisive evidence, indicating that the Bayes factor strongly supports the heteroskedastic specification. The findings confirm that the variances of shocks exhibit pronounced time-varying properties. From an economic perspective, such heteroskedasticity is highly consistent with the volatility clustering observed during extreme events—for instance, the liquidity crisis triggered by the collapse of Lehman Brothers in 2008 or the supply-chain disruptions caused by pandemic-related lockdowns in 2020, both of which substantially magnified the variance of shock distributions.

4.6. Evidence on Structural Shocks

Table 5. Structural Shock Tests.

	Mean	SD	5% Quantile	95% Quantile
14 January 2008–30 December 2016
shock1	−0.01	0.06	−0.10	0.08
shock2	−0.02	0.06	−0.11	0.07
shock3	0.02	0.06	−0.07	0.10
Shock4	0.00	0.07	−0.10	0.10
Shock5	0.00	0.04	−0.05	0.06
Shock6	0.00	0.09	−0.13	0.13
Shock7	0.00	0.08	−0.11	0.11
Shock8	−0.02	0.08	−0.14	0.10
3 January 2017–23 June 2025
shock1	−0.01	0.06	−0.10	0.08
shock2	−0.03	0.06	−0.12	0.06
shock3	0.01	0.06	−0.08	0.10
Shock4	−0.01	0.07	−0.11	0.10
Shock5	0.01	0.04	−0.05	0.06
Shock6	−0.01	0.09	−0.14	0.12
Shock7	−0.01	0.07	−0.12	0.09
Shock8	−0.02	0.08	−0.14	0.11

Note: All values are rounded to the nearest (specified) decimal.

reports Bayesian posterior interval estimates to assess the statistical significance of structural shocks. During both the 14 January 2008–30 December 2016 financial crisis and the 3 January 2017–23 June 2025 multi-crisis period, the posterior means of all shocks (e.g., Shock 3 with a mean of 0.02 during the financial crisis) fall strictly within the corresponding 95% credible intervals (e.g., Shock 3: [−0.07, 0.10]). This evidence probabilistically rejects the null hypothesis of zero-mean shocks, thereby confirming their statistical significance.

As an illustration, consider the crude oil market (Shock 4). During the financial crisis, the posterior mean is 0.00 (SD = 0.07), with a 95% credible interval of [−0.10, 0.10]. This interval fully encompasses economically meaningful scenarios of exogenous supply disruptions—such as OPEC production cuts or geopolitical conflicts—indicating that the model successfully identifies structural shocks with clear economic interpretation.

4.7. Volatility Spillover Analysis

4.7.1. Static Volatility Spillover

The static unnormalized volatility spillover index at horizon 10 is used to assess overall volatility transmission, with results reported in

Table 6. Static volatility spillover index during period 1.

To	From
	Rice	Soybeans	Wheat	Oil (WTI)	Coal	Natural Gas	Gold	Silver	From
Rice	92.77	0.92	1	0.51	1.42	0.31	1.67	1.39	7.23
Soybeans	6.96	85.66	1.32	0.77	1.38	0.3	1.62	1.99	14.34
Wheat	4.06	8.62	83.06	0.64	1.13	0.27	1.28	0.95	16.94
Oil (WTI)	3.71	7.08	2.26	80.42	1.62	0.34	3.04	1.54	19.58
Coal	1.6	0.97	1.84	1.64	91.56	0.28	1.06	1.05	8.44
Natural Gas	3.16	2.96	2.82	4.12	4.8	75.58	3.91	2.63	24.42
Gold	1.75	1.19	0.9	2.73	1.45	0.24	91.15	0.58	8.85
Silver	2.19	2.1	1.11	4.46	1.56	0.35	54.34	33.89	66.11
To	23.42	23.85	11.26	14.88	13.37	2.09	66.91	10.12	165.9
Net	16.19	9.51	−5.68	−4.71	4.93	−22.32	58.07	−55.99	20.74

Note: The “From” and “To” columns represent the unnormalized sums of variance contributions.

and

Table 7. Static volatility spillover index during period 2.

To	From
	Rice	Soybeans	Wheat	Oil (WTI)	Coal	NaturalGas	Gold	Silver	From
Rice	87.74	2.31	1.02	0.77	1.16	0.32	4.76	1.93	12.26
Soybeans	2.03	93.68	0.75	0.23	0.47	0.10	1.86	0.88	6.32
Wheat	3.49	10.66	79.38	0.60	0.44	0.16	4.31	0.96	20.62
Oil (WTI)	4.06	12.82	3.81	71.38	0.98	0.53	4.23	2.20	28.62
Coal	4.07	4.31	3.07	2.04	75.87	0.34	7.90	2.41	24.13
Natural Gas	4.55	6.87	3.70	3.03	2.99	67.68	8.64	2.53	32.32
Gold	0.50	0.96	0.44	0.50	0.30	0.11	96.75	0.45	3.25
Silver	0.63	1.38	0.66	0.71	0.30	0.12	65.59	30.61	69.39
To	19.32	39.31	13.46	7.87	6.64	1.68	97.29	11.35	196.92
Net	7.06	32.99	−7.16	−20.74	−17.50	−30.64	94.04	−58.04	24.62

Note: see Table 6.

. In energy markets, crude oil illustrates a clear shift in spillover roles. During the financial crisis, oil acted as a net receiver (Net = −4.71), whereas in the multi-crisis period, its net inflows increased markedly (Net = −20.74), alongside elevated volatility levels (e.g., exceeding 60% in 2022). Natural gas remained a net receiver across both periods (Net = −22.32 → −30.64), with incoming spillovers rising from 24.42 to 32.32. Coal shifted from a marginal net transmitter (Net = 4.93) to a net receiver (Net = −17.50).

Agricultural markets display heterogeneous patterns. Rice maintained a positive but declining net spillover role (Net = 16.19 → 7.06). Soybeans show a pronounced increase in net spillovers (Net = 9.51 → 32.99), indicating a strengthened transmission role, while wheat remained a net receiver with spillovers declining further to −7.16.

In precious metals, gold consistently acted as a dominant net transmitter, with net spillovers rising from 58.07 to 94.04. By contrast, silver remained a strong net receiver (Net = −55.99 → −58.04), with external spillovers increasing from 66.11 to 69.39.

Across all commodities, the unnormalized total spillover measure increases from 165.90 to 196.92. To facilitate comparison, the Total Connectedness Index (TCI) is computed as TCI = (Total/(N * 100)) * 100% = [Value]%, where N = 8 is the number of variables. Based on this normalization, the TCI rises from 20.74% to 24.62%, indicating a marked intensification of overall market connectedness during the multi-crisis period. Specifically, changes in key transmission roles—particularly the strengthened dominance of gold and crude oil and the increased passivity of natural gas, coal, wheat, and silver—highlight the evolving structure of volatility transmission under overlapping crisis conditions. To examine time-varying volatility spillovers, all subsequent dynamic results are computed at horizon 10 using rolling windows of 438 and 404 observations for Periods 1 and 2, respectively.

4.7.2. Dynamics of Total Volatility Spillovers

Figure 7a (left panel) shows the time-varying total volatility spillover index during the global financial crisis. Between 2008Q3 and 2009Q1, the index rose sharply from around 20% to a peak of 25%, reflecting intensified contagion following the collapse of Lehman Brothers. It then declined to about 22% by 2009Q4, indicating partial stabilization, although spillovers remained above the pre-crisis level of roughly 18%.

Figure 7b (right panel) displays a distinct dual-peak pattern in the multi-crisis period. The first peak occurs in 2020Q2, when the index surged to approximately 35% during the initial COVID-19 shock, followed by a second peak of about 37% in 2022Q1, coinciding with the Russia–Ukraine conflict. Between these peaks, spillovers remained elevated with pronounced fluctuations. Compared with the financial crisis, this period is characterized by higher levels and longer persistence of volatility spillovers, reflecting more complex and sustained risk transmission under overlapping shocks.

4.7.3. Net Volatility Spillover

The net volatility spillover index is used to examine the directional transmission of volatility across markets, with results shown in Figure 8 and Figure 9 for the financial crisis period and the post-pandemic period, respectively. The results highlight pronounced changes in market roles under different crisis environments.

During the financial crisis, rice evolved into a persistent net transmitter, peaking at around 40 in 2009–2011, while soybeans largely remained net receivers with only a brief positive phase. Wheat showed short-lived net transmission in 2008–2009 before shifting to sustained net reception, reaching −40 in 2010. Crude oil (WTI) exhibited a V-shaped pattern, moving from net reception below −20 in 2008 to net transmission above 20 after 2012, whereas coal followed a similar transition. Natural gas remained a deep net receiver throughout, falling to −50 in 2009–2010. Gold consistently acted as a strong net transmitter, peaking near 80 in 2011, while silver stayed predominantly in negative territory, fluctuating between −20 and −50.

In the post-pandemic period, rice reversed from an early net receiver (−20) to a strong net transmitter, exceeding 80 in 2021. Soybeans acted as net transmitters for most of the period, with values between 40 and 60 in 2021–2022, whereas wheat remained a net receiver and declined to −40 following the 2022 Russia–Ukraine conflict. Crude oil fluctuated around zero (−20 to 20) without a stable role. Coal and natural gas both remained net receivers, with natural gas reaching values as low as −60. Gold further strengthened its transmitter role, peaking near 140, while silver showed large negative swings between −20 and −70.

4.7.4. Pairwise Volatility Spillover

The pairwise volatility spillover index is used to examine bilateral spillover relationships between commodity markets, with results reported in Figure 10 and Figure 11 for the financial crisis and the post-pandemic period, respectively. The figures highlight pronounced time variation and heterogeneity in cross-market risk transmission.

During the financial crisis, gold initially acted as a net receiver of volatility from crude oil (WTI) in 2008–2009, before switching to a net transmitter after 2010, with spillovers peaking above 10. Silver exhibited a similar regime change, moving from negative spillovers (around −5) in the early crisis phase to positive values exceeding 10 during the recovery. Spillovers from gold and silver to rice and soybeans were predominantly negative in 2008–2010, while spillovers to wheat were mildly positive.

In the post-pandemic period, gold displayed persistently negative spillovers to crude oil, reaching a trough of about −15 in 2020. Silver’s spillover to crude oil remained negative between 2020 and 2021 (minimum around −8) before turning positive after 2023. Spillovers from precious metals to rice and soybeans were generally negative, whereas spillovers to wheat became positive, indicating a reversal in transmission direction under heightened geopolitical uncertainty.

4.8. Impulse Response Analysis

Figure 12 and Figure 13 report impulse response functions across commodity markets for the financial crisis and the post-pandemic period, highlighting period-specific heterogeneity in shock transmission.

During the financial crisis, gold shows a pronounced short-term positive response to crude oil, peaking above 0.4 at lags 0–4, before converging to zero after lag 10. Its response to rice is also positive in the short run (peak around 0.25 at lags 0–2) but fades by lag 8, while silver’s response to rice remains weak (peak about 0.15). Gold’s response to soybeans follows a non-monotonic pattern—positive in the short term (≈0.3), turning negative in mid to long horizons (around −0.2). Silver exhibits a similar but weaker pattern. Both gold and silver generate persistently negative responses to wheat, with troughs around −0.2 and −0.25, respectively.

In the post-pandemic period, gold continues to respond positively to crude oil in the short run (peak ≈ 0.4 at lags 0–2), but the effect dissipates faster, converging by lag 6. Silver’s response to oil turns from positive in 2020 (≈0.3) to negative in 2022 (≈−0.1). Gold’s responses to rice, soybeans, and wheat become persistently negative, reaching about −0.2 at lag 4, whereas silver’s responses remain weakly negative (around −0.05 to −0.1). Overall, impulse responses in the post-pandemic period are shorter-lived but more asymmetric, reflecting altered transmission mechanisms under overlapping shocks.

5. Discussion and Implications

This study applies the BSVAR-MIX framework to uncover regime-dependent volatility transmission in U.S. commodity markets under extreme events. By accommodating multimodal shock distributions and endogenous regime shifts, the analysis reveals several economically meaningful patterns that can be interpreted through the lens of financial risk management and portfolio theory.

More broadly, these results contribute to the growing literature on volatility connectedness and systemic risk transmission across commodity markets (Cunado et al. 2024; Balcilar et al. 2024), while extending this literature by showing how spillover structures evolve across crisis-driven volatility regimes. Recent studies increasingly emphasize that commodity connectedness intensifies under global shocks and macro-financial stress (e.g., Balcilar et al. 2024; Charteris et al. 2025), suggesting that regime-dependent models may provide additional insights into systemic risk propagation.

5.1. Systemic Transmitters and the Repricing of Portfolio Risk

A first key finding is that safe-haven assets and strategically central commodities dominate systemic volatility transmission during crisis regimes. Gold consistently emerges as a major net transmitter, with its net spillover rising to 94.04 in the multi-crisis period and short-term impulse responses to crude oil peaking at approximately 0.4. From a portfolio risk management perspective, this reflects large-scale capital reallocation toward perceived safe assets under heightened uncertainty. When systemic risk rises, investors often rebalance portfolios toward assets historically considered safe havens, such as gold. However, when such reallocations occur simultaneously across global markets, these assets may become central nodes through which volatility is transmitted rather than absorbed. In this sense, safe-haven demand can paradoxically increase systemic connectedness when capital flows concentrate in a small set of assets.

This behavior aligns with connectedness-based evidence in Diebold and Yilmaz (2012) and Antonakakis et al. (2020), who document that core assets increasingly drive system-wide risk during turbulent periods. At the same time, it contrasts with findings such as Gabauer and Gupta (2018), who emphasize gold’s shock-absorbing role in standard network models. The divergence is economically intuitive: under stress regimes characterized by heteroskedasticity and non-Gaussian shocks, correlation-based hedging breaks down, and assets that dominate portfolio reallocations can amplify, rather than dampen, volatility transmission. Recent connectedness studies also document that precious metals may shift from hedging assets to systemic transmitters during periods of extreme financial uncertainty (e.g., Balcilar et al. 2024; Charteris et al. 2025), reinforcing the view that safe-haven behavior is regime-dependent rather than constant over time. The BSVAR-MIX framework explicitly captures these stress-regime asymmetries that conventional homoskedastic models overlook.

5.2. Hybrid Commodities and the Amplification Channel

A second pattern concerns commodities with hybrid financial–real characteristics. Soybeans transition from a moderate net transmitter (9.51) to a dominant spillover source (32.99) in the multi-crisis period. From a risk management standpoint, this reflects the tightening linkage between food markets, energy markets, and policy-driven demand. Biofuel mandates, climate-related supply risks, and trade disruptions increase the covariance between agricultural and energy assets, weakening diversification benefits in multi-asset portfolios.

This finding is consistent with Abdi et al. (2020), who highlight the food–energy nexus as a key channel of systemic transmission. More broadly, it echoes portfolio theory insights that assets subject to policy-induced demand shocks can become endogenous risk amplifiers. Recent empirical studies similarly document that agricultural commodities increasingly interact with energy markets through biofuel policies and climate-related policy shocks (Declerck et al. 2023; Zhang et al. 2025a). These linkages strengthen cross-market volatility spillovers, particularly when macroeconomic shocks simultaneously affect energy demand, food security concerns, and global trade conditions. In such settings, volatility spillovers arise not merely from price comovement but from synchronized exposure to regulatory and macroeconomic risk factors, which are particularly salient during crisis regimes.

5.3. Policy-Sensitive Assets and Rising Fragility

Third, policy-sensitive markets display increasing vulnerability rather than transmission power. Coal shifts from a marginal net transmitter (4.93) to a net receiver (−17.50), while wheat’s net spillover declines to −7.16. These results indicate that assets exposed to structural transitions—such as decarbonization policies or agricultural supply constraints—are less able to propagate risk and instead absorb shocks originating elsewhere in the system.

This pattern is consistent with Broadstock et al. (2022) and Hammoudeh et al. (2024), who show that environmental regulation and supply-side disruptions weaken the stabilizing role of certain commodity markets. From a financial risk perspective, such assets behave as residual risk holders in stressed portfolios, becoming more sensitive to external volatility while offering limited diversification benefits.

5.4. Implications for Risk Monitoring and Portfolio Management

These findings have direct implications for risk-oriented monitoring and portfolio management. First, systemic risk surveillance should distinguish between core transmitters (e.g., gold, crude oil, soybeans) and peripheral receivers (e.g., coal, wheat) under crisis regimes. Monitoring frameworks that ignore directional spillovers risk underestimating systemic exposure concentrated in a small set of dominant assets. Recent financial network studies similarly emphasize that monitoring directional spillovers can improve early detection of systemic commodity risk and enhance portfolio stress-testing frameworks (e.g., Kočenda and Moravcová 2024).

Second, for institutional investors, the results highlight the limits of static diversification strategies. Negative or rapidly changing impulse responses—such as the gold–oil spillover reversal observed in 2022—indicate periods when traditional hedging relationships weaken. Incorporating dynamic spillover measures into portfolio rebalancing and tail-risk hedging strategies can improve resilience under extreme market conditions. This suggests that regime-aware portfolio allocation strategies may outperform static diversification approaches when systemic commodity risk intensifies.

Overall, the evidence suggests that crisis-driven volatility transmission is shaped not only by market fundamentals but also by portfolio reallocation behavior and policy-induced risk channels. By capturing these dynamics, the BSVAR-MIX framework provides a risk-focused perspective on how systemic shocks propagate through interconnected commodity markets and highlights the importance of regime-aware econometric models for monitoring systemic commodity risk.

6. Conclusions

This study examines volatility spillovers across major U.S. commodity markets under extreme events using a Bayesian Structural VAR with Normal Mixtures (BSVAR-MIX). By allowing for multimodal shock distributions and regime-dependent volatility dynamics, the proposed framework captures time-varying and asymmetric risk transmission without relying on exogenous crisis dummy variables.

Empirical results reveal three central findings. First, systemic volatility during crisis regimes is concentrated in a small set of core assets. Gold and crude oil consistently emerge as dominant transmitters, with connectedness intensifying markedly during overlapping shocks. This indicates that assets typically viewed as hedges can become channels of risk propagation when portfolio rebalancing and safe-haven demand dominate market behavior. Second, commodities with hybrid real–financial characteristics, such as soybeans, exhibit a pronounced shift toward active volatility transmission under policy-driven and energy-related shocks, weakening diversification benefits across asset classes. Third, policy-sensitive and supply-constrained markets—including coal and wheat—increasingly act as net receivers of volatility, highlighting their vulnerability to external shocks and limited capacity to stabilize the system.

From a risk management perspective, these findings underscore the importance of distinguishing between volatility transmitters and receivers when monitoring systemic risk. Static correlation-based diversification strategies may fail during crisis regimes, as spillover directions and intensities change rapidly across states. Incorporating dynamic spillover measures into portfolio allocation, stress testing, and tail-risk hedging can improve resilience under extreme uncertainty.

Overall, the study contributes to the risk literature by providing a regime-sensitive view of volatility transmission in commodity markets. The BSVAR-MIX framework offers a practical tool for identifying evolving sources of systemic risk and supports more adaptive approaches to financial risk surveillance and portfolio management in environments characterized by recurring and overlapping crises.

Future research may extend the framework to broader multi-asset settings, incorporate higher-frequency or real-time risk monitoring, and compare mixture-based structural spillover measures with alternative state-dependent connectedness models.