# Generalized Impulse responses VAR(2)

I have a VAR(2) model:

$$\textbf{y}_t=\textbf{A}_1\textbf{y}_{t-1}+\textbf{A}_2\textbf{y}_{t-2}+\textbf{u}_t$$

where $$\textbf{y}_t$$ is a 2x1 vector, $$\textbf{A}_1, \textbf{A}_2$$ are two 2x2 matrices and $$\textbf{u}_t$$ is a 2x1 vector of white noise distrubuted as $$N(0,\Sigma)$$. Notice that $$\Sigma$$ is not diagonal, i.e. the noise are contemporaneous correlated. Moreover: the two matrices $$\textbf{A}_1, \textbf{A}_2$$ are respectively:

$$\begin{array}{cc} 0.6 & 0.2 \\ 0.2 & 0.4 \\ \end{array}$$

and

$$\begin{array}{cc} -0.2 & 0.2 \\ 0.4 & -0.4 \\ \end{array}$$

and $$\Sigma$$ $$\begin{array}{cc} -0.2 & 0.3 \\ 0.1 & -0.4 \\ \end{array}$$

Can someone help me in writing $$\textbf{y}_{t+h}$$ foh $$h=1,2$$ as a function of $$\textbf{y}_0$$ and $$\textbf{y}_{-1}$$ only? Notice that I have that $$\textbf{y}_0=(1,1)$$ and $$\textbf{y}_{-1}=(0,0)$$.

This will be needed to calculat the generalized impulse response function:

$$IR_{it}(h)=E_t[\textbf{y}_{t+h}|u_{it=k},\textbf{y}_{t-1}.. ]-E_t[\textbf{y}_{t+h}|\textbf{y}_{t-1}.. ]$$

You only need to substitute t+h into your original equation:

$$๐ฒ_๐ก=๐_1๐ฒ_{๐กโ1}+๐_2๐ฒ_{๐กโ2}+๐ฎ_๐ก$$

So you would have:

$$๐ฒ_{๐ก+h}=๐_1๐ฒ_{๐ก+hโ1}+๐_2๐ฒ_{๐ก+hโ2}+๐ฎ_{๐ก+h}$$

When h=1,2 respectievly

• $$๐ฒ_{๐ก+1}=๐_1๐ฒ_{๐ก}+๐_2๐ฒ_{๐ก-1}+๐ฎ_{๐ก+1}$$
• $$๐ฒ_{๐ก+2}=๐_1๐ฒ_{๐ก+1}+๐_2๐ฒ_{๐ก}+๐ฎ_{๐ก+2}$$

When t=-1 and h=1,2 respectievly

• $$๐ฒ_{0}=๐_1๐ฒ_{-1}+๐_2๐ฒ_{-2}+๐ฎ_{0}$$
• $$๐ฒ_{1}=๐_1๐ฒ_{0}+๐_2๐ฒ_{-1}+๐ฎ_{1}$$

When t=0 and h=1,2 respectievly

• $$๐ฒ_{1}=๐_1๐ฒ_{0}+๐_2๐ฒ_{-1}+๐ฎ_{1}$$
• $$๐ฒ_{2}=๐_1๐ฒ_{1}+๐_2๐ฒ_{0}+๐ฎ_{2}$$

Now you have your ingredeints for you complete solution.

$$๐ฒ_{1}$$ is a function of $$๐ฒ_{0}$$ and $$๐ฒ_{-1}$$ and $$๐ฒ_{2}$$ is a function of $$๐ฒ_{1}$$ and $$๐ฒ_{0}$$

• Unofortunately this is not what I meant. My aim is to express $\textbf{y}_{t+1}$ and $\textbf{y}_{t+1}$ as functions of $\textbf{y}_{-1}$ and $\textbf{y}_{0}$ only leaving the $t$ index generic, i.e. I am not allowed to arbitrary fix $t$. Essentially, my problem is to derive a \enquote{nice} representation of $\textbf{y}_{t+1}$ by iteratively substituting backwards until time 0 and 1, but I have troubles in seeing the right pattern – giorgio Jul 14 '19 at 17:05
• You should look carefully at the answer. What you seek is there. Even though I didn't spell it out for you. – grldsndrs Jul 14 '19 at 22:00