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