Consider the VAR lag-p model: $$Bx_t = \Gamma_0 + \sum_{i=1}^p\Gamma_i x_{t-i} + \epsilon_t,\quad x_t\in\Bbb R^n,\,\forall t\in\Bbb Z$$ Setting $B$ to be upper-triangular and $A_0:=B^{-1}\Gamma_0,\,A_i:=B^{-1}\Gamma_i \,\,($i$=1,\cdots,p),\,e_t:=B^{-1}\epsilon_t$, we may write $$x_t = A_0 + \sum_{i=1}^p A_ix_{t-i} + e_t,\quad \forall t\in\Bbb Z$$ Then what's the impulse response formula for this model?

The case where $p=1$ is well-known: $$\begin{align} x_t &= (\sum_{i=0}^n A_1^i)A_0+A_1^{n+1}x_{t-n-1}+\sum_{i=0}^n A_1^ie_{t-i}\\ &=(\sum_{i=0}^\infty A_1^i)A_0+\sum_{i=0}^\infty A_1^ie_{t-i}\\ &=\Bbb E(x_t) + \sum_{i=0}^\infty A_1^ie_{t-i} \end{align}$$ which clearly converges provided that $\|A_1\| < 1$.

The case $p>1$ is a bit complicated but still algebraically tractable. What I got is $$\begin{align} x_t &= (I+\sum_{i_1}A_{i_1}+\sum_{i_1,i_2}A_{i_1}A_{i_2}+\cdots+\sum_{i_1,i_2,\cdots,i_n}A_{i_1}A_{i_2}\cdots A_{i_n})A_0+A_1^{n+1}x_{t-n-1}+\sum_{i_1,i_2,\cdots,i_n,i_{n+1}}A_{i_1}A_{i_2}\cdots A_{i_n}A_{i_{n+1}}e_{t-i_1-\cdots-i_{n+1}}+(e_t+\sum_{i_1}A_{i_1}e_{t-i_1}+\sum_{i_1,i_2}A_{i_1}A_{i_2}e_{t-i_1-i_2}+\cdots+\sum_{i_1,i_2,\cdots,i_n}A_{i_1}A_{i_2}\cdots A_{i_n}e_{t-i_1-\cdots-i_n})\\ &=(\sum_{k=1}^\infty\sum_{i_1,\cdots,i_k}\prod_{s=1}^k A_{i_s})A_0 + \sum_{k=1}^\infty(\sum_{i_1,\cdots,i_k}(\prod_{s=1}^k A_{i_s})e_{t-i_1-\cdots-i_k}) \end{align}$$ where the last equality is only heuristical but the convergence isn't actually guaranteed. One sufficent but conceivably too strong condition I see for the convergence to hold is that $p\|A_i\|<1,\,i=1,\cdots,p$, from which we immediately have $$\|\sum_{i_1,\cdots,i_k}\prod_{s=1}^k A_{i_s}\|\le p^k \max \|\prod_{s=1}^k A_{i_s}\| \le p^k\max_i \|A_i\|^k\le (\max_i p\|A_i\|)^k \le (1-\delta)^k$$ for some $\delta>0$, and the series becomes dominated by a geometric series. However, do we really need such a strong condition? That seems pretty unreasonable. But I really can't come up with anything that can strengthen the current result.

  • $\begingroup$ You may want to add a tag related to the algebraic aspect of the problem. You seem to have formulated it well enough so that special knowledge of VAR is at this point less crucial than good skills in inequalities. $\endgroup$ – Richard Hardy Oct 23 '18 at 14:51
  • $\begingroup$ @RichardHardy thanks, I added the "convergence" tag. But I do think the VAR tag is still worth keeping because this topic may have been well studied in VAR research but is relatively unfamiliar outside researcher circle and perhaps somebody with expertise in VAR can point me to some helpful reference materials or give a nice explanation on their own. $\endgroup$ – Vim Oct 23 '18 at 14:55
  • $\begingroup$ Your are right. I did not mean that the var tag should be removed, just that another one should be added. But for example the autoregressive tag could perhaps be replaced by probability-inequalities or something like that. Just an idea. $\endgroup$ – Richard Hardy Oct 23 '18 at 15:10

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