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How much of the forecast uncertainty in one market, sector, or country is caused by shocks originating elsewhere, and how does that connectedness change over time?
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 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() on stationary series and compute the -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 matrix where is the share of variable 's -step forecast-error variance attributable to shocks in variable . 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: . 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 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 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 stationary time series observed at the same frequency over 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() where is selected by AIC/BIC.
The model estimates the VAR , then inverts it into its moving-average representation where are MA coefficient matrices. The generalized FEVD decomposes the -step forecast error into contributions from each variable's innovations. The element of the unnormalized GFEVD is . Row-normalizing gives .
Rolling-window estimation repeats this for overlapping subsamples of width (typically 100--200 observations for daily data, 40--60 for quarterly). Each window produces a spillover matrix , a total spillover index , and directional spillover profiles for each variable. The resulting time series of 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 is large relative to (or relative to the rolling window ). A VAR with 50 variables and 2 lags has parameters. With a 200-observation rolling window, the VAR is severely overparameterized. Solutions include LASSO-VAR (penalized estimation), factor-augmented approaches (compress 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 to means 's innovations help forecast 's future, but this could reflect common exposure to a third factor rather than direct transmission.
The share of variable 's -step forecast-error variance attributable to shocks in variable . Row-normalized to sum to 1.
System-wide connectedness: . Ranges from 0 (isolated) to 100 (fully interconnected).
How much variable contributes to the forecast-error variance of all other variables: .
How much of variable 's forecast-error variance comes from shocks in other variables: .
Transmitter minus receiver: . Positive means is a net transmitter; negative means net receiver.
The moving-average coefficient at horizon , derived from the VAR. Maps period- innovations to their contribution to the forecast at .
All series are covariance-stationary, or transformations have been applied to make them stationary.
If violated: Non-stationary series produce spurious VAR estimates and meaningless FEVD shares. Unit-root tests (ADF, KPSS) should precede estimation.
The VAR has lag order chosen by AIC/BIC, and this correctly captures the dynamic structure.
If violated: Misspecified lag order distorts the MA coefficients , which are the basis for the GFEVD. Residual autocorrelation tests should be run.
The generalized FEVD is exact under Gaussian innovations. For non-Gaussian data, it provides an approximation.
If violated: With heavy-tailed or skewed innovations (common in financial data), the GFEVD is approximate. Bootstrapped confidence intervals mitigate this.
The window width is large enough to estimate the VAR reliably but short enough to capture time variation.
If violated: Too-short windows produce noisy spillover time series with spurious spikes. Too-long windows smooth out genuine crisis-period jumps. Sensitivity to should be reported.
Within each rolling window, the VAR parameters are approximately constant.
If violated: A structural break inside a rolling window (e.g., a crisis onset) mixes pre- and post-break data, producing a blended VAR that represents neither regime accurately.
The GFEVD (Pesaran-Shin) does not depend on the ordering of variables in the VAR, unlike the Cholesky FEVD.
If violated: Not a failure per se, but the GFEVD rows do not sum to exactly 1 before normalization. The normalization is a convenience, not a structural result. This means spillover shares are relative, not absolute.