Bayesian Spillover Graphs (BSG)

• Bayesian Spillover Graphs (BSG) are a framework that combines Bayesian VAR modeling with forecast error variance decomposition to quantify and visualize dynamic spillover effects.
• The methodology employs conjugate priors and Gibbs sampling to generate horizon-dependent, directed, and weighted network graphs, enabling systemic risk and influence assessment.
• BSG has practical applications in environmental risk and financial systems, offering clear insights with posterior uncertainty quantification to support public health and economic policy decisions.

Bayesian Spillover Graphs (BSG) are a principled framework for learning, quantifying, and visualizing dynamic spillover effects within multivariate time series and panel data. By integrating Bayesian time series modeling with forecast error variance decomposition (FEVD) or similar structural tools, BSG provides interpretable, horizon-dependent network representations of how shocks to one subsystem propagate across others, while yielding posterior-credible quantification of systemic risk, node-level influence, and transient versus equilibrium network behavior. In contrast with classical Granger-causality graphs or static structure learning, BSG uniquely tailors edge weights to explain multi-horizon prediction variability and supports rigorous uncertainty assessment (Deng et al., 2022, Costola et al., 2023).

1. Bayesian Backbone and Modeling of Dynamic Systems

The BSG framework for time series originates from the Bayesian vector autoregression (VAR) model. For a zero-mean, stationary $d$-dimensional time series $\mathbf{z}_t \in \mathbb{R}^d$, a VAR($p$) is defined as

$$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$

where $\phi_j$ are autoregressive matrices and $\Sigma_a$ is the error covariance. Estimation employs conjugate Normal–Inverse–Wishart priors for $(\beta, \Sigma_a)$ with fully tractable posterior updates. This Bayesian paradigm enables the sampling of model parameters, providing the foundation for spillover quantification with posterior-credible intervals (Deng et al., 2022).

In the spatial autoregressive (SAR–SV) extension, panel data $y_t \in \mathbb{R}^n$ are modeled with multi-layer time-varying network weights $W_{k,t}$, individual exposure coefficients $\rho_j$, and stochastic volatility in structural shocks. The observed process can be written as

$\mathbf{z}_t \in \mathbb{R}^d$0

where $\mathbf{z}_t \in \mathbb{R}^d$1, with $\mathbf{z}_t \in \mathbb{R}^d$2 a convex combination of $\mathbf{z}_t \in \mathbb{R}^d$3 network layers and $\mathbf{z}_t \in \mathbb{R}^d$4 (Costola et al., 2023).

2. Forecast-Error Variance Decomposition and Graph Construction

In BSG for time series, the forecasting power of each node $\mathbf{z}_t \in \mathbb{R}^d$5 on another node $\mathbf{z}_t \in \mathbb{R}^d$6 after $\mathbf{z}_t \in \mathbb{R}^d$7 steps is determined via FEVD. Under stationarity, the VAR is written as a moving-average: $\mathbf{z}_t \in \mathbb{R}^d$8 The $\mathbf{z}_t \in \mathbb{R}^d$9-step FEVD for node $p$0's variance from shocks to node $p$1 is

$p$2

normalized as

$p$3

The BSG at horizon $p$4 is the directed, weighted graph with edges $p$5, interpreted as the percentage of node $p$6's forecast error variance due to shocks from node $p$7 at that horizon.

For SAR-based BSG, graph construction derives from the spatial multiplier $p$8. Directed edge weights at time $p$9 are

$$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$0

optionally normalized by the jth-column sum (Costola et al., 2023). Both models compute edge weights for each posterior sample, yielding credible intervals for spillover strengths.

3. Horizon Hyperparameter, Spillover Equilibrium, and Temporal Resolution

The forecast-horizon parameter $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$1 provides interpretive control over temporal relationships. Small $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$2 recapitulates short-term (e.g., one-step Granger-like) effects; large $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$3 or $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$4 reflects long-run or equilibrium spillover structure. In practice, $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$5 is increased until edge weights $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$6 stabilize within a numeric tolerance $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$7, yielding a data-driven definition of the system's equilibrium spillover network (Deng et al., 2022).

In multilayer, time-varying BSG, the graph may also vary through time $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$8 relative to exogenous covariates or evolving network topologies, yielding dynamic families of BSGs indexed by both $$\mathbf{z}_t = \phi_1 \mathbf{z}_{t-1} + \cdots + \phi_p \mathbf{z}_{t-p} + \mathbf{a}_t, \quad \mathbf{a}_t \stackrel{\mathrm{iid}}{\sim} \mathcal{N}(\mathbf{0}, \Sigma_a),$$9 and $\phi_j$0 (Costola et al., 2023).

4. Inference Algorithms and Uncertainty Quantification

Parameter inference in BSG is Gibbs-style MCMC. For the VAR approach:

1. Sample posterior draws $\phi_j$1 alternately.
2. For each draw and candidate $\phi_j$2, compute $\phi_j$3, $\phi_j$4, and $\phi_j$5.
3. Average over $\phi_j$6 for posterior means and quantify credible intervals (e.g., highest posterior density) for each edge (Deng et al., 2022).

In SAR–SV BSG, block-Gibbs or slice sampling is used for $\phi_j$7, latent volatilities $\phi_j$8, network weights $\phi_j$9, and exposures $\Sigma_a$0. The edge weights $\Sigma_a$1 are computed at each MCMC iteration, and their empirical distribution defines posterior mean and uncertainty at all time points (Costola et al., 2023).

This comprehensive propagation of posterior uncertainty through all graph weights and node scores enables credible interval estimation for all systemic risk and influence metrics.

5. Systemic Risk, Influence, and Vulnerability Metrics

The BSG induces natural systemic risk and node importance measures:

• Total $\Sigma_a$2-spillover index: $\Sigma_a$3, summarizing overall network connectedness.
• Vulnerability (sink) score for node $\Sigma_a$4: $\Sigma_a$5, representing the fraction of $\Sigma_a$6's forecast error variance due to others.
• Influence (source) score for node $\Sigma_a$7: $\Sigma_a$8, quantifying the share of systemic spillover originating from $\Sigma_a$9.

Analogous decompositions for direct, indirect, and total spillover effects are specified in dynamic SAR-based BSG via the spatial multiplier $(\beta, \Sigma_a)$0, separating contributions from own shocks vs. propagated (network) shocks (Costola et al., 2023). These metrics inherit full Bayesian uncertainty quantification.

6. Empirical Comparisons and Application Domains

Synthetic studies for VAR-based BSG (e.g., $(\beta, \Sigma_a)$1, various network topologies and noise configurations) show that BSG yields near-perfect normalized discounted cumulative gain (NDCG@20 $(\beta, \Sigma_a)$2–$(\beta, \Sigma_a)$3) in ranking true source or sink nodes, significantly outperforming dynamic Bayesian network (DBN) and graphical VAR (GVAR) baselines (typical NDCG 0.71–0.90), particularly under correlated errors and complex network structure (Deng et al., 2022). BSG accurately recovers influencer and sink structure in both linear and nonlinear (e.g., Lotka–Volterra) data.

As a real-world demonstration, VAR-based BSG mapped PM2.5 spillovers during the 2019 Kincade wildfire across Northern California, delineating sources (Sonoma County; influence $(\beta, \Sigma_a)$441% with 95% HPDI [17.9%, 62.7%]) and vulnerable sinks (Alameda, Contra Costa, etc.), clarifying both direct and indirect transmission pathways for environmental risk and supporting the design of public-health interventions (Deng et al., 2022).

Multilayer SAR–SV BSG has been applied to international stock returns and volatility for the G7, revealing distinct roles of cooperative versus conflictual network ties, exposure heterogeneity (e.g., U.S. vs. EU members), and dynamic patterns of direct/indirect spillovers across time and economic shocks (Costola et al., 2023).

7. Extensions and Scope of the BSG Framework

BSG encompasses both stationary vector autoregressive and panel spatial autoregressive settings, supporting a range of structural assumptions:

• Multi-horizon, multi-layer, and time-varying graphs: BSG can model evolving network dependencies, allowing for covariate-driven or exogenously observed edge evolution (Costola et al., 2023).
• Structural and observation noise sophistication: Inclusion of stochastic volatility or non-Gaussian error models supports robust inference in volatile environments.
• Multimodal uncertainty quantification: BSG's use of posterior sampling provides distributional—rather than pointwise—systemic risk estimands, supporting decision-making under uncertainty.

A distinctive property is interpretability: edge weights in BSG are directly tied to the FeVD or SAR structure, providing a natural causal and predictive decomposition. Current implementations, sample code, and applications can be found at https://github.com/gdeng96/bsg (Deng et al., 2022).

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Key references:

• Bayesian Spillover Graphs for Dynamic Networks (Deng et al., 2022)
• Bayesian SAR model with stochastic volatility and multiple time-varying weights (Costola et al., 2023)