Volatility Spillover (Diebold–Yilmaz)

When one market catches fire, you want to know who else is standing close enough to burn.

That is the whole question of this chapter, and it is a different question from correlation. Correlation tells you that two assets tend to move together. It does not tell you who is moving whom, it does not tell you how much of one asset's turbulence is being imported from elsewhere, and it collapses everything into a single symmetric number that cannot distinguish a sender of stress from a receiver of it. What you actually want, when you are watching a regime turn, is a directed, quantitative map of who is exporting volatility into whom. That map is what the Diebold–Yilmaz framework gives you, and it is exactly what sits behind SVQ's vol-contagion network.

Let us build the intuition before the algebra.

Imagine you are forecasting tomorrow's volatility for, say, crude oil. You will be wrong by some amount — there is forecast error, and that error has a variance. Now ask a sharper question. Of that uncertainty, how much comes from oil's own surprises, and how much comes from shocks that started somewhere else — in the dollar, in equities, in rates — and propagated into oil? If almost all of oil's forecast-error variance traces back to its own shocks, oil is an island; it is volatile on its own terms and largely deaf to the rest of the system. But if a large slice of that variance is explained by shocks that originated in equities, then equities are spilling over into oil. That share — the fraction of one asset's forecast-error variance attributable to another asset's shocks — is the atom of connectedness. Everything else is bookkeeping on top of it.

Now the machinery. You start by fitting a vector autoregression, a VAR, to the volatilities of the assets in your panel — each asset's volatility regressed on recent lags of every asset's volatility, so that cross-asset dynamics are captured. From that VAR you compute the forecast-error variance decomposition at some horizon, typically around ten days. The decomposition answers our atom question for every ordered pair: what share of asset i's H-step forecast-error variance is due to shocks in asset j. Diebold and Yilmaz use the generalized variance decomposition of Pesaran and Shin, and the word "generalized" matters. The older Cholesky decomposition forces you to pick an ordering of the assets, and your answers change if you reshuffle that order — an unacceptable property when no asset has a natural claim to go first. The generalized version is invariant to ordering. The price you pay is that its rows do not sum to one, so in practice each entry is normalized by its row sum so the contributions are comparable.

Collect those shares into a square table. The entry in row i, column j is the percent of i's forecast-error variance coming from j. The diagonal is the "own" share — how much of i is just i. The off-diagonal entries are the spillovers, and now the table does real work.

Sum across a row, excluding the diagonal, and you get how much volatility asset i receives from all others. Sum down a column, excluding the diagonal, and you get how much asset j transmits to all others. The difference between what an asset sends and what it receives is its net spillover: a positive net marks a source, a propagator of stress; a negative net marks a sink, an absorber. And the sum of all the off-diagonal entries, divided by the grand total of the row-normalized table — which is just the number of assets, since every row sums to one hundred — is the total connectedness index. Equivalently, it is the average across assets of the share of forecast-error variance each one imports from the rest. It is one number, between zero and one hundred percent, summarizing how tightly the whole system is wired together right now. In calm regimes that index sits low; as a crisis builds, it ratchets up, often sharply, well before the headlines.

This is where the throughline of the institute returns. A single asset printing a big volatility number is not, by itself, evidence that the system is about to move. It might be an island having a bad day. What turns a number into genuine evidence of contagion is direction and share — seeing that a specific asset's shocks are claiming a rising fraction of everyone else's forecast-error variance. The connectedness table is precisely the instrument that separates "this one asset is loud" from "this asset is now driving the room."