Variance decomposition of forecast errors


In econometrics and other applications of multivariate time series analysis, a variance decomposition or forecast error variance decomposition is used to aid in the interpretation of a vector autoregression model once it has been fitted. The variance decomposition indicates the amount of information each variable contributes to the other variables in the autoregression. It determines how much of the forecast error variance of each of the variables can be explained by exogenous shocks to the other variables.

Calculating the forecast error variance

For the VAR of form
This can be changed to a VAR structure by writing it in companion form
where, and are dimensional column vectors, is by dimensional matrix and, and are dimensional column vectors.
The mean squared error of the h-step forecast of variable is
and where
The amount of forecast error variance of variable accounted for by exogenous shocks to variable is given by