Variance decomposition
Encyclopedia
Variance decomposition or forecast error variance decomposition indicates the amount of information each variable contributes to the other variables in a vector autoregression
(VAR) models. Variance decomposition determines how much of the forecast error variance of each of the variable can be explained by exogenous shocks to the other variables.
.
Change this to a VAR (1) by writing it in companion form (see general matrix notation of a VAR(p))
where
Vector autoregression
Vector autoregression is a statistical model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregression models. All the variables in a VAR are treated symmetrically; each variable has an equation explaining its evolution based on...
(VAR) models. Variance decomposition determines how much of the forecast error variance of each of the variable can be explained by exogenous shocks to the other variables.
Calculating the forecast error variance
For the VAR (p) of form .Change this to a VAR (1) 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.
Calculate the mean squared error of the h-step forecast of variable j,
,
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 decompositionCholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
of
such that 
. 
where 
so 
is 
by 
dimensional matrix. 
is the covariance matrix of the errors 
.
The amount of forecast error variance of variable
accounted for by exogenous shocks to variable 
is given by 
