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Background:

I am working on orthogonal impuls response functions (OIRFs) for vector error correction models (VECMs). Its an exercise to develop understanding. I am given a bivariate VECM: $$ \Delta y_t = \begin{bmatrix}-0.15 \\0.3\end{bmatrix} \begin{bmatrix} 1&1.15 \end{bmatrix} y_{t-1}+ \begin{bmatrix}-0.2&0.15\\0.15&-0.2\end{bmatrix}\Delta y_{t-1} + \begin{bmatrix}-0.05 & -0.05 \\ 0 &-0.2\end{bmatrix}\Delta y_{t-2} + v_t $$ with $$ \Sigma_{vv}=\begin{bmatrix}0.2&0.3\\0.3&0.7\end{bmatrix}$$

To compute OIRFs I first rephrased the vecm as a VAR(3). It now looks like: $$ y_t=\begin{bmatrix}0.65&-0.0225 \\0.45&1.1450\end{bmatrix}y_{t-1}+\begin{bmatrix}0.15&-0.2\\-0.15&0.0\end{bmatrix}y_{t-2}+\begin{bmatrix}0.05 &0.05\\0.00&0.20\end{bmatrix}y_{t-3}+v_t$$ Then I transformed it to a VMA process: $$y_t=v_t +\begin{bmatrix}0.65&-0.0225 \\0.45&1.1450\end{bmatrix}v_{t-1}+\begin{bmatrix}0.5725&-0.1994938\\0.0525&1.3110250\end{bmatrix}v_{t-2}+...+\begin{bmatrix}0.2416860048&0.0004147842\\-0.0001924644&9.0691554099\end{bmatrix}v_{t-10}$$ using $$\psi_i=\sum_{j=1}^i\phi_j \psi_{i-j}$$ and $$\phi_j=0 \text{ for } j>p$$ The lower triangle of the cholesky decomposed variance-covarinace matrix is $$\begin{bmatrix}0.4472136 &0.0\\0.6708204&0.5\end{bmatrix}$$

Question 1: Do I calculate the OIRFs $\theta$ for horizon $h=1,\dots,10$ correcty if I calculate: $$\theta_h=\begin{bmatrix}0.4472136 &0.0\\0.6708204&0.5\end{bmatrix}*\psi_h$$ So for example: $$\theta_1=\begin{bmatrix}0.4472136 &0.0\\0.6708204&0.5\end{bmatrix}*\begin{bmatrix}0.65&-0.0225 \\0.45&1.1450\end{bmatrix}$$ or do I need to switch them?

Question 2: How do I calculate the forecast error variance decomposition a given $h$?

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  • $\begingroup$ I thought vmatrix looked too much like determinants, so I replaced them by bmatrix. Hope you do not mind. $\endgroup$ – Richard Hardy Sep 8 '17 at 14:19

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