Variance decomposition of forecast errors
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
This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))
- , , and
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 j is
- is the jth column of and the subscript refers to that element of the matrix
- where is a lower triangular matrix obtained by a Cholesky decomposition of such that , where is the covariance matrix of the errors
- where so that is a by dimensional matrix.
The amount of forecast error variance of variable accounted for by exogenous shocks to variable is given by
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- Lütkepohl, H. (2007) New Introduction to Multiple Time Series Analysis, Springer. p. 63.