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.[1] 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

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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))

  where
  ,  ,   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   is

 

and where

  •   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  

 

See also

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Notes

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  1. ^ Lütkepohl, H. (2007) New Introduction to Multiple Time Series Analysis, Springer. p. 63.