My final goal is to generate Impulse Response Functions in R.
I have variables that are non stationary when I set k = 5 in a Unit Root test, and they are cointegrated which to my understanding prompts the use of the VECM, from which the Vec2Var argument is used to then generate IRFs. However, my response functions from this methodology do not decay over time and mostly do not revert to the zero line.
Furthermore, I noticed that when I input first differenced variables into the VECM as opposed to level data above, response function do revert to the zero line. Therefore my first question is: 1) is it appropriate to use differenced variables in a VECM model?
Secondly, as an alternative I am considering restricting the number of lags used in the Unit Root tests so that the variables are not all non-stationary; in this case 2) would the use of VECM be void and the VAR with first differences be a more appropriate model? Again, using differenced data is giving me better response functions that revert to zero in the long run. However, 3) is it ok to use the VAR for stationary/non-stationary data in levels where these variables are still cointegrated?
Also, I am using growth rate variables, 4) should I still long transform all variables and use logs in all tests?
Thank you for the help!
Best Answer
I assume you use ADF test for stationarity check and that
kis the number of lags (in my example, $p$ is the number of lags).A VECM model $$\Delta Q_t = \Gamma_0 + \Gamma_1Q_{t-1} + \sum_{i=1}^p\Lambda_i\Delta Q_{t-i} + e_t$$ where $Q'_t = (Y_t \quad X_t \quad Z_t)$
is comprised of a "VAR" part (which is the differences part within the summation) and additional levels part, which $\Gamma_1$ is its coefficient (matrix notation). Thus, when you ask if it is appropriate to use differenced variables in a VECM model then the answer is its already being used anyway. What is usually of particular interest is hypothesis testing regarding $\Gamma_1$ which represents the long-run relationships.
When dealing with non-stationary variables and applying the VECM model, you want to see what is the % off the long-term relationship that is being corrected each period.
It sounds like your series are not stationary thus you cannot use VAR (unless you use differences to eliminate non-stationarity, but then everything you'll learn will be correct only for the $i$th difference! and not for the levels. You didn't say what is the scenario and the use-case so it's hard to tell what is more appropriate).
So coming back to your first problem of non-decaying IRFs - I would guess that the error correction term for your model is positive, which means that the process is not converging in the long run. It probably means that something is wrong with the specification of your model, or the presence of structural breaks which are not accounted for.