I am currently working on a model where I am trying to compute the response of macroeconomic variables like gdp and CPI as well as Gini Koefficient to monetary policy shocks. My problem now is I have non stationary data and would like to generate Impuls response functions. I thought I might do a Var Model on first differences for short run and vecm to look in to the long run relationships, so that I could get my IRF from the Var of first differences...is that logical ? If not can anyone advise me on what I should do ? Thanks !!!
I would advice to only estimate a VECM if your variables are cointegrated. In this case there are long run relationships. If they are not cointegrated, you may want to estimate a VAR on your differenced data, you are only concerned with the short run effects in this case. Macroeconomic variables are typically integrated of order 1 or 2, if they are integrated of order 2 you may need to difference 2x.
Impulse response analysis is only concerned with your estimated $\hat A_i$. I see no problem in analyzing IRF when your estimator is obtained through a regression on differenced data. It should be a valid estimate of the population parameter $A_i$ given that the usual assumptions are satisfied (well specified model).