Variance decomposition of forecast errors explained

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.

Calculating the forecast error variance

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

where

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}

and

Ut=\begin{bmatrix} ut\ 0\\vdots\ 0\end{bmatrix}

where

yt

,

\nu

and

u

are

k

dimensional column vectors,

A

is

kp

by

kp

dimensional matrix and

Y

,

V

and

U

are

kp

dimensional column vectors.

The mean squared error of the h-step forecast of variable

j

is

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,

and where

ej

is the jth column of

Ik

and the subscript

jj

refers to that element of the matrix

\Thetai=\PhiiP,

where

P

is a lower triangular matrix obtained by a Cholesky decomposition of

\Sigmau

such that

\Sigmau=PP'

, where

\Sigmau

is the covariance matrix of the errors

ut

\Phii=JAiJ',

where

J=\begin{bmatrix} Ik&0&...&0\end{bmatrix},

so that

J

is a

k

by

kp

dimensional matrix.

The amount of forecast error variance of variable

j

accounted for by exogenous shocks to variable

l

is given by

\omegajl,h,

\omegajl,h

h-1
=\sum
i=0

(ej'\Thetaie

2/MSE[y
j,t

(h)].

See also

Notes and References

  1. Lütkepohl, H. (2007) New Introduction to Multiple Time Series Analysis, Springer. p. 63.