In econometrics and other applications of multivariate time series analysis, a variance decomposition or forecast error variance decomposition (FEVD) is used to aid in the interpretation of a vector autoregression (VAR) 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 (p) of form

1. .

This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))

1. where

dimensional column vectors.

The mean squared error is

and where

• is the jth refers to that element of the matrix

  is a lower triangular matrix obtained by a Cholesky decomposition is the covariance matrix

  dimensional matrix.

See also

• Analysis of variance

Notes