Variance decomposition of forecast errors

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. ^[1] 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

y_{t}=\nu +A_{1}y_{t-1}+\dots +A_{p}y_{t-p}+u

Change this to a VAR (1) by writing it in companion form (see General matrix notation of a VAR(p))

Y_{t}=\mathbf {\nu } +AY_{t-1}+U

where

A={\begin{bmatrix}A_{1}&A_{1}&\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}}

,

Y={\begin{bmatrix}y_{1}\\\vdots \\y_{p}\end{bmatrix}}

,

\mathbf {\nu } ={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}

and

\mathbf {\nu } ={\begin{bmatrix}\nu \\0\\\vdots \\0\end{bmatrix}}

,

U={\begin{bmatrix}u\\0\\\vdots \\0\end{bmatrix}}

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

Calculate the mean squared error of the variables, $\mathbf {MSE} [y_{j,t}(h)]$ , which is given by the diagonal elements of the mean squared error matrix $\Sigma _{y}(h)$ . $\mathbf {MSE} [y_{j,t}(h)]=\operatorname {diag} (\Sigma _{y}(h))$ .

\Sigma _{y}(h)=\sum _{i=0}^{h-1}(e_{j}'\Theta _{i}e_{k})^{2}

where $e_{j}$ is the ${\text{j}}^{\text{th}}$ column of $I_{k}$ and $\Theta _{i}=\Phi _{i}P$ . $P$ is a lower triangular matrix obtain by a Cholesky decomposition of $\Sigma _{u}$ such that $\Sigma _{u}=PP'$

where $\Phi _{i}=JA^{i}J'$ where $J={\begin{bmatrix}\mathbf {I} _{k}&0&\dots &0\end{bmatrix}}$ so $J$ is $k$ by $kp$ dimensional matrix. $\Sigma _{u}$ is the covariance matric of the errors $u$ .