Variance decomposition of forecast errors

For the VAR (p) of form

:$$

y_t=\nu +A_1y_{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))

:$$

Y_t=V+A Y_{t-1}+U_t

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}

, $V=\backslash begin\{bmatrix\}$

\nu \\ 0 \\ \vdots \\ 0 \end{bmatrix}

and $$

U_t=\begin{bmatrix}

u_t \\ 0 \\ \vdots \\ 0 \end{bmatrix}

where $y\_t$, $\backslash nu$ 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

:$$

\mathbf{MSE}[y_{j,t}(h)]=\sum_{i=0}^{h-1}\sum_{l=1}^{k}(e_j'\Theta_ie_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

:*$e\_j$ is the j^th column of $I\_k$ and the subscript $jj$ refers to that element of the matrix

:*$\backslash Theta\_i=\backslash Phi\_i\; P\; ,$ where $P$ is a lower triangular matrix obtained by a Cholesky decomposition of $\backslash Sigma\_u$ such that $\backslash Sigma\_u\; =\; PP\text{'}$, where $\backslash Sigma\_u$ is the covariance matrix of the errors $u\_t$

:* $\backslash Phi\_i=J\; A^\{i\}\; J\text{'},$ where $$

J=\begin{bmatrix}

\mathbf{I}_k &0 & \dots & 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 $\backslash omega\_\{jl,h\}\; ,$

:$$

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

{{Refimprove|date=March 2011}}

• Analysis of variance

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Category:Multivariate time series