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.