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.
For the VAR (p) of form
yt=\nu+A1yt-1+...+Apyt-p+ut
This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))
Yt=V+AYt-1+Ut
A=\begin{bmatrix} A1&A2&...&Ap-1&Ap\\ Ik&0&...&0&0\\ 0&Ik&&0&0\\ \vdots&&\ddots&\vdots&\vdots\\ 0&0&...&Ik&0\\ \end{bmatrix}
Y=\begin{bmatrix} y1\ \vdots\ yp\end{bmatrix}
V=\begin{bmatrix} \nu\ 0\ \vdots\ 0\end{bmatrix}
Ut=\begin{bmatrix} ut\ 0\ \vdots\ 0\end{bmatrix}
where
yt
\nu
u
k
A
kp
kp
Y
V
U
kp
The mean squared error of the h-step forecast of variable
j
MSE[yj,t
h-1 | |
(h)]=\sum | |
i=0 |
k | |
\sum | |
l=1 |
(ej'\Thetaie
h-1 | |
i=0 |
\Thetai\Thetai')jj
h-1 | |
=(\sum | |
i=0 |
\Phii\Sigmau\Phii')jj,
ej
Ik
jj
\Thetai=\PhiiP,
P
\Sigmau
\Sigmau=PP'
\Sigmau
ut
\Phii=JAiJ',
J=\begin{bmatrix} Ik&0&...&0\end{bmatrix},
J
k
kp
The amount of forecast error variance of variable
j
l
\omegajl,h,
\omegajl,h
h-1 | |
=\sum | |
i=0 |
(ej'\Thetaie
2/MSE[y | |
j,t |
(h)].