Variance Decomposition - Calculating The Forecast Error Variance

Calculating The Forecast Error Variance

For the VAR (p) of form

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

This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))

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

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

where, and are dimensional column vectors, is by dimensional matrix and, and are dimensional column vectors.

The mean squared error of the h-step forecast of variable j is, where

$\mathbf{MSE}=\sum_{i=0}^{h-1}\sum_{k=1}^{K}(e_j'\Theta_ie_k)^2=\bigg(\sum_{i=0}^{h-1}\Theta_i\Theta_i'\bigg)_{jj}=\bigg(\sum_{i=0}^{h-1}\Phi_i\Sigma_u\Phi_i'\bigg)_{jj},$

and where

is the jth column of and the subscript refers to that element of the matrix

where is a lower triangular matrix obtained by a Cholesky decomposition of such that, where is the covariance matrix of the errors

where $J=\begin{bmatrix} \mathbf{I}_k &0 & \dots & 0\end{bmatrix} ,$ so that is a by dimensional matrix.

The amount of forecast error variance of variable accounted for by exogenous shocks to variable is given by

$\omega_{jk,h}=\sum_{i=0}^{h-1}(e_j'\Theta_ie_k)^2/MSE .$

Read more about this topic: Variance Decomposition

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