I am currently studying the monetary mechanism transmission in Brazil, and using 4 variables into the analysis(interest, exchange rate, output and inflation). The goal is to study the impulse response functions.
I've made the 4 basic tests for the presence of the unit roots in these series, and found the results
Inflation being I(0)
Exchange Rate and Interest being I(1)
The output variable has been tough. ADF and Phillips Perron tests indicate that the variable is I(1), but ADF GLS and the KPSS test point to it being a I(2) variable. When the second difference is applied, all tests agrees in stationarity at this level. So far, I'm leaning into treating output as I(2).
So, can I use the VAR to estimate this equation and the impulse response function? Seeing some people suggesting ARDL but also read about how ARDL cannot be used in the case of I(2) and I(1) variables combined.
Use the first differences of variables, which are I(1) and levels for variables of I(0)! Good Luck!
The VAR model usefulness stems from its ability to fit data. So yes, you can model variables with different orders of integration. Additionally, VAR model is frequently used to forecast short term fluctuations in inflation, interest and exchange rates. You just need to make sure that there is no time trends in your data.
I(2) for output is unusual. Be aware that standard tests can be fooled by breaks in the trend or level component of a series. Maybe try some tests that allow for structural break(s).
I would avoid using all the non-stationary variables in differences, since then would be loss of information and also mix-up of levels with rates of growth..
If you 'ignore' ADF etc tests, then the most appropriate model would be VAR, given that the Johansen tests indicate co-integration - which is possible if output is I(1).
A meet-point would be to use all variables in levels and output in first difference=rate of growth. Therefore, according to ADF, you have one I(0) variable and three I(1) variables, and again, you check for co-integration and accordingly use the impulse-response application...
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