IRF for VAR GARCH (Impulse Response Function)

Question

When there is ARCH effects on VAR residuals $\varepsilon_t$, we can use a GARCH model to remove them : $\zeta_t = \Sigma_{t|t-1}^{-\frac{1}{2}} \varepsilon_t$. Following [Lutkepohl, New Introduction to Multiple Time Series Analysis], coefficients of the IRF correspond to the coefficients of the well-known Moving-Average representation :

$\Delta P_{t} = J \mathbf{\Delta P}_{t} = \sum_{i=0}^{\infty} J \mathbf{\Phi}^i J^{\tau} J \mathbf{E}_{t-i} = \sum_{i=0}^{\infty} J \mathbf{\Phi}^i J^{\tau} \Sigma_{t-i|t-i-1}^{\frac{1}{2}} \Sigma_{t-i|t-i-1}^{-\frac{1}{2}} \varepsilon_{t-i} = \sum_{i=0}^{\infty} \mathbf{\Psi}_{i,t} \zeta_{t-i}$

And so the impulse response for a given $t$ after $N$ periods is the corresponding MA coefficient : $\mathbf{\Psi}_{N,t} = J \mathbf{\Phi}^N J^{\tau} \Sigma_{t-N|t-N-1}^{\frac{1}{2}}$.

I am not sure about the indices of $\Sigma$, should I use forecast of it ? Thanks for your help.

EDIT : Maybe I could directly apply IRF on VAR residuals, is it a necessary condition to remove ARCH effects to use IRF ? I don't think so, and so I could use : $\Delta P_{t} = \sum_{i=0}^{\infty} J \mathbf{\Phi}^i J^{\tau} \varepsilon_{t-i} = \sum_{i=0}^{\infty} \mathbf{\Psi}_{i,t} \varepsilon_{t-i}$.

And so the impulse response after N periods is the corresponding MA coefficient : $\mathbf{\Psi}_{N} = J \mathbf{\Phi}^N J^{\tau}$, which not depends on t anymore.

Answer 1

Since in the GARCH model the variance covariance matrix of the residuals $\Sigma_t$ is time varying you would have $t = 1, \ldots, T$ covariance matrices, each of which is associated to an impulse response. That's to say that the impulse responses will be time varying as well i.e for $t = 1980$ the impulse response function of output to a monetary policy shock is allowed to have a different shape with respect to impulse response for $t = 2000$.

This is completely unrelated to the time span you want to observe the effect of a shock on a given variable $t - N$.

That's to say you don't need to forecast anything, you just need to obtain the $\Sigma_t$ for $t=1, \ldots, T$, fix a time span and calculate the impulse responses.

Answering the second part, if you have GARCH effects in the true DGP and you estimate an homoskedastic VAR by OLS, you would get inefficient estimates of the coefficients (higher variance) and of the impulse responses as well.