Network spillover Model

Financial crises reveal linkages that appear invisible in calm times. Before 2008, individual bank risk metrics looked fine; the problem was the network of exposures linking them. Diebold and Yilmaz (2009) proposed a framework that measures interconnectedness directly from observable time series, without requiring bilateral exposure data. The idea: estimate a VAR on $N$ variables (country equity returns, sector volatilities, sovereign CDS spreads), compute the generalized forecast-error variance decomposition (GFEVD), and read the off-diagonal shares as spillover intensities. The total spillover index aggregates these into a single number that tracks system-wide connectedness over time.

The mechanics work in two steps. First, estimate a VAR($p$) on $N$ stationary series and compute the $H$-step-ahead generalized forecast-error variance decomposition (Pesaran and Shin, 1998). Unlike Cholesky-based decompositions, the generalized version does not depend on variable ordering---critical when there is no theoretical ordering among, say, 20 country equity markets. The GFEVD produces an $N \times N$ matrix $\theta^H$ where $\theta^H_{ij}$ is the share of variable $i$'s $H$-step forecast-error variance attributable to shocks in variable $j$. Row-normalize so each row sums to 1. The off-diagonal elements are spillovers; the diagonal elements are own-variance shares.

The total spillover index is the average off-diagonal share: $S = 100 \cdot N^{-1} \sum_{i \neq j} \theta^H_{ij}$. This single number ranges from 0 (all variables are isolated---own shocks explain everything) to 100 (all forecast uncertainty comes from cross-variable shocks). Rolling-window estimation produces a time series of $S_t$ that spikes during crises: the 2008 Global Financial Crisis, the 2010 - 12 Euro-area debt crisis, the 2020 COVID crash. Directional spillovers---how much variable $i$ transmits to versus receives from others---identify net transmitters (systemically important nodes) and net receivers (vulnerable nodes).

The framework is used at the IMF for financial surveillance (Global Financial Stability Report), at the BIS for systemic-risk monitoring, at the Federal Reserve for tracking cross-asset contagion, and in academic research on international business-cycle transmission, oil-price spillovers, and climate-risk propagation. Barunik and Krehlik (2018) extended it to frequency-domain spillovers, decomposing connectedness by short-run versus long-run horizons. Demirer, Diebold, Liu, and Yilmaz (2018) scaled it to networks of 100+ global banks.

Inputs are $N$ stationary time series observed at the same frequency over $T$ periods. In the original application: realized volatilities of equity indices for major economies. In other applications: credit spreads, bilateral trade flows (differenced), sector-level output growth, or bank-level CDS spreads. The series must be stationary; log-differencing, realized volatility, or HP-filter deviations are common transformations. The data enter a VAR($p$) where $p$ is selected by AIC/BIC.

The model estimates the VAR $y_t = c + A_1 y_{t-1} + \cdots + A_p y_{t-p} + e_t$, then inverts it into its moving-average representation $y_t = \sum_{h=0}^\infty \Phi_h e_{t-h}$ where $\Phi_h$ are $N \times N$ MA coefficient matrices. The generalized FEVD decomposes the $H$-step forecast error $y_{t+H} - \hat{y}_{t+H|t}$ into contributions from each variable's innovations. The $(i,j)$ element of the unnormalized GFEVD is $\tilde{\theta}^H_{ij} = \sigma_{jj}^{-1} \sum_{h=0}^{H-1} (e_i' \Phi_h \Sigma e_j)^2 / \sum_{h=0}^{H-1} e_i' \Phi_h \Sigma \Phi_h' e_i$. Row-normalizing gives $\theta^H_{ij} = \tilde{\theta}^H_{ij} / \sum_k \tilde{\theta}^H_{ik}$.

Rolling-window estimation repeats this for overlapping subsamples of width $W$ (typically 100 - 200 observations for daily data, 40 - 60 for quarterly). Each window produces a spillover matrix $\theta^H_t$, a total spillover index $S_t$, and directional spillover profiles for each variable. The resulting time series of $S_t$ is the primary output: a real-time connectedness tracker. Alternative approaches---Bayesian TVP-VARs or DCC-GARCH-based spillovers---avoid the window-width choice but add computational complexity.

The IMF's Global Financial Stability Report uses the Diebold-Yilmaz framework to track cross-country financial connectedness. A rolling-window spillover index on sovereign CDS spreads of 20+ countries reveals when contagion risk is elevated. During the 2010 - 12 euro-area crisis, the total spillover index spiked from around 40 to over 70, driven by directional spillovers from Greece, Ireland, and Portugal to Spain and Italy. This time-varying measure informs the IMF's surveillance discussions about systemic-risk buildup.

Central banks monitor cross-asset connectedness for macro-prudential policy. The Federal Reserve Bank of New York computes spillover indices across US equity sectors, Treasury volatility, credit spreads, and currency markets. A rising spillover index signals that asset classes are moving together more than usual---a precursor to financial stress. This complements traditional financial-conditions indices by providing directionality: which markets are transmitting stress and which are receiving it.

Academic research on oil-price spillovers uses the framework to measure how crude-oil shocks propagate to equity markets, exchange rates, and real economic activity across countries. Diebold and Yilmaz (2012) showed that oil-price volatility spillovers to US equities increased dramatically after 2003, coinciding with the financialization of commodity markets. This finding influenced the debate about whether financial speculation in commodities amplifies real-economy volatility.

The framework breaks down when $N$ is large relative to $T$ (or relative to the rolling window $W$). A VAR with 50 variables and 2 lags has $50 \times (50 \times 2 + 1) = 5050$ parameters. With a 200-observation rolling window, the VAR is severely overparameterized. Solutions include LASSO-VAR (penalized estimation), factor-augmented approaches (compress $N$ variables into fewer factors before running the VAR), or the elastic-net VAR of Demirer et al. (2018). The framework also does not identify causal mechanisms: a high spillover from variable $i$ to $j$ means $i$'s innovations help forecast $j$'s future, but this could reflect common exposure to a third factor rather than direct transmission.