# IRF for VAR GARCH (Impulse Response Function)

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.

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.

• Thanks @Giorgetto, that's exactly what I had in mind. But one thing I'm wondering, : maybe I could directly apply IRF on VAR residuals, is it a necessary conditions 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. Aug 13, 2021 at 10:12
• I have tried to answer also this bit Aug 13, 2021 at 11:47
• Thanks @Giorgetto, your answer was very clear. But using VAR-GARCH residuals instead of VAR residuals won't remove the serial correlation, it will juste remove ARCH effects, and so coefficients estimates will still be biased and inconsistent. Is their a way to remove the serial correlation ? Papers don't mention this serial correlation issue before using IRF. Aug 13, 2021 at 12:31
• Ok I understand, thank you very much for your time, I accepted your answer :) Aug 13, 2021 at 13:52
• Will autocorrelation yield inconsistent estimates? Biased for sure (even without autocorrelation) but inconsistent? Also, ignoring GARCH will not yield inconsistent, only inefficient estimates. Therefore I think at least your last comment is wrong. Aug 14, 2021 at 11:10