Variance decomposition

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.

Calculating the forecast error variance

For the VAR (p) of form

$y_t=\nu +A_1y_{t-1}+\dots+A_p y_{t-p}+u$ .

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

$Y_t=\mathbf{\nu} +A Y_{t-1}+U$ where

$A=\begin{bmatrix} A_1 & A_2 & \dots & A_{p-1} & A_p \\ \mathbf{I}_k & 0 & \dots & 0 & 0 \\ 0 & \mathbf{I}_k & & 0 & 0 \\ \vdots & & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & \mathbf{I}_k & 0 \\ \end{bmatrix}$ , $Y=\begin{bmatrix} y_1 \\ \vdots \\ y_p \end{bmatrix}$, $\mathbf{\nu}=\begin{bmatrix} \nu \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ and $U=\begin{bmatrix} u \\ 0 \\ \vdots \\ 0 \end{bmatrix}$

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 h-step forecast of variable j, $\mathbf{MSE}[y_{j,t}(h)]$,

$\mathbf{MSE}[y_{j,t}(h)]=\sum_{i=0}^{h-1}\sum_{k=1}^{K}(e_j'\Theta_ie_k)^2=\bigg(\sum_{i=0}^{h-1}\Theta_i\Theta_i'\bigg)_{jj}=\bigg(\sum_{i=0}^{h-1}\Phi_i\Sigma_u\Phi_i'\bigg)_{jj},$

where $e_j$ is the j^th column of $I_K$ and the subscript $jj$ refers to that element of the matrix. $\Theta_i=\Phi_i P$ where $P$ is a lower triangular matrix obtained by a Cholesky decomposition of $\Sigma_u$ such that $\Sigma_u = PP'$. $\Phi_i=J A^i J'$ 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 matrix of the errors $u$.

The amount of forecast error variance of variable $j$ accounted for by exogenous shocks to variable $k$ is given by $\omega_{jk,h} .$

$\omega_{jk,h}=\sum_{i=0}^{h-1}(e_j'\Theta_ie_k)^2/MSE[y_{j,t}(h)] .$