10.2 Orthogonal IRF

• The impulse response function (IRF) in VAR models is interpreted as the reaction of each endogenous variable to a unit shock caused by a change in one of the error terms $u_{1,t}$ or $u_{2,t}$

• In the case where the error terms $u_{1,t}$ are $u_{2,t}$ are correlated, an orthogonalization procedure is applied, i.e. non-diagonal matrix $\Sigma$ is transformed into diagonal one by Cholesky decomposition

• Cholesky decomposition transforms a non-diagonal matrix $\Sigma$ into the product of a lower triangular matrix $L$ with positive elements on the main diagonal, and its transpose $L^T$

$$\begin{equation} \Sigma=L L^T \tag{10.5} \end{equation}$$

• When the lower triangular matrix $L$ is found, its inverse is used to transform correlated error terms into uncorrelated ones

$$\begin{equation} \begin{aligned} \varepsilon_t&=L^{-1} u_t \\ \Omega&=L^{-1} \Sigma {(L^{-1})}^{T} \end{aligned} \tag{10.6} \end{equation}$$

• Finally, the orthogonal impulse response function (OIRF) is formed, generating response matrices in every period

TABLE 10.1: Orthogonal response matrices

Period	Response
$~~~~~~~0~~~~~~~$	$0$
$1$	$L$
$2$	$A_1 L$
$3$	$A_1^2 L$
$4$	$A_1^3 L$
$5$	$A_1^4 L$
$\cdots$	$~~~~~~~~~~\cdots~~~~~~~~~~$
$k$	$A_1^{k-1} L$