Variance decomposition of forecast errors

For the VAR (p) of form

${\displaystyle y_{t}=\nu +A_{1}y_{t-1}+\dots +A_{p}y_{t-p}+u_{t}}$ .

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

${\displaystyle Y_{t}=V+AY_{t-1}+U_{t}}$ where

${\displaystyle A={\begin{bmatrix}A_{1}&A_{2}&\dots &A_{p-1}&A_{p}\\\mathbf {I} _{k}&0&\dots &0&0\\0&\mathbf {I} _{k}&&0&0\\\vdots &&\ddots &\vdots &\vdots \\0&0&\dots &\mathbf {I} _{k}&0\\\end{bmatrix}}}$ , ${\displaystyle Y={\begin{bmatrix}y_{1}\\\vdots \\y_{p}\end{bmatrix}}}$, ${\displaystyle V={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}}$ and ${\displaystyle U_{t}={\begin{bmatrix}u_{t}\\0\\\vdots \\0\end{bmatrix}}}$

where ${\displaystyle y_{t}}$, ${\displaystyle \nu }$ and ${\displaystyle u}$ are ${\displaystyle k}$ dimensional column vectors, ${\displaystyle A}$ is ${\displaystyle kp}$ by ${\displaystyle kp}$ dimensional matrix and ${\displaystyle Y}$, ${\displaystyle V}$ and ${\displaystyle U}$ are ${\displaystyle kp}$ dimensional column vectors.

The mean squared error of the h-step forecast of variable ${\displaystyle j}$ is

${\displaystyle \mathbf {MSE} [y_{j,t}(h)]=\sum _{i=0}^{h-1}\sum _{l=1}^{k}(e_{j}'\Theta _{i}e_{l})^{2}={\bigg (}\sum _{i=0}^{h-1}\Theta _{i}\Theta _{i}'{\bigg )}_{jj}={\bigg (}\sum _{i=0}^{h-1}\Phi _{i}\Sigma _{u}\Phi _{i}'{\bigg )}_{jj},}$

and where

• ${\displaystyle e_{j}}$ is the j^th column of ${\displaystyle I_{k}}$ and the subscript ${\displaystyle jj}$ refers to that element of the matrix

• ${\displaystyle \Theta _{i}=\Phi _{i}P,}$ where ${\displaystyle P}$ is a lower triangular matrix obtained by a Cholesky decomposition of ${\displaystyle \Sigma _{u}}$ such that ${\displaystyle \Sigma _{u}=PP'}$, where ${\displaystyle \Sigma _{u}}$ is the covariance matrix of the errors ${\displaystyle u_{t}}$

• ${\displaystyle \Phi _{i}=JA^{i}J',}$ where ${\displaystyle J={\begin{bmatrix}\mathbf {I} _{k}&0&\dots &0\end{bmatrix}},}$ so that ${\displaystyle J}$ is a ${\displaystyle k}$ by ${\displaystyle kp}$ dimensional matrix.

The amount of forecast error variance of variable ${\displaystyle j}$ accounted for by exogenous shocks to variable ${\displaystyle l}$ is given by ${\displaystyle \omega _{jl,h},}$

${\displaystyle \omega _{jl,h}=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{l})^{2}/MSE[y_{j,t}(h)].}$

• Analysis of variance