Forecast Error Variance Decomposition

Fit a vector autoregression on, say, oil prices and equity returns, and you can forecast several periods ahead. But forecasts are never exact — there's error, and that error grows the further out you look. Forecast error variance decomposition (FEVD) asks: of the total forecast-error variance for equity returns at some horizon, how much comes from oil-price shocks versus equity's own shocks? It turns a fitted VAR into a statement about which variable's surprises actually matter for another variable's uncertainty.

The idea

At each forecast horizon, the total forecast-error variance for a variable can be split into pieces attributable to each shock in the system, using the model's moving-average representation. These shares sum to 100% at every horizon and typically shift over time — a variable often explains most of its own short-horizon variance but cedes a growing share to other variables at longer horizons, as their effects propagate through the system's dynamics.

Worked example

A two-variable VAR of oil returns and airline-stock returns shows that at a one-day horizon, 92% of airline-return forecast error variance comes from airline shocks and 8% from oil shocks. By 20 days out, that splits closer to 60/40 — oil shocks have had time to feed through into airline profitability expectations, so they explain a much larger share of the uncertainty about airline returns further into the future.