I wonder if there is a proof that estimates of impulse response functions for VAR model are consistent given that an estimator for autoregressive matrices is consistent? Should be true, but I have never encountered formal prove.

Consider zero-mean VAR(1) $$y_t = Ay_{t-1} + u_t,$$ with $y_t \in \mathbb{R}^K$, $A \in \mathbb{R}^{K\times K}$ and $u_t \in \mathbb{R}^K$.

Let $\zeta_h \in \mathbb{R}^K$ be impulse responses of $K$ variables to a shock in variable $k$ at step $h$. Then, it can be shown, that $$\zeta_h = A^he_k,$$ where $e_k$ is a vector with $1$ at $k^{th}$ position and zeros elsewhere.

Then, estimate of $\zeta_h$ is given by $\hat\zeta_h = \hat A^he_k$. Given that $plim(\hat A) = A$, we want to prove that $plim(\hat\zeta_h) = \zeta_h$. Applying continuous mapping theorem (CMP) we are immediately done. However, what bothers me is that CMP requires function $g(\hat A)$ to be continuous. In our case $g(\hat A) = \hat A^h$ and I have not come up so far with the way to show this.

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