Vector autoregression model with unit root in the exogenous variable and endogenous variables

Question

I was wondering whether I should worry about the fact that I have one unit root in my exogenous variable. I think based on what I understand that I should first difference the variable with unit root, before using it in my VAR. Can someone verify whether this is correct?

Second, I was also wondering, suppose I have multiple unit roots, but they are all in my exogenous variables, I guess I should then test for cointegration. However if I find them to be cointegrated, should I then switch to a vector error correction model? Or do I just difference them and then add them to my VAR?

In addition, I was also wondering, suppose I have one exogenous variable that has a unit root and an endogenous variable that has a unit root, should I then test both of them for cointegration? If yes, and I find cointegration, what should I do?

Hopefully someone can shine some light upon this, because I am having difficulties figuring this out.

Answer 1

I am not an expert of this topic, but here is what seems reasonable to me.

Case 1: all endogenous variables are stationary, one exogenous variable is integrated (has a unit root).
I would take first differences of the integrated exogenous variable and include that in the VAR model. You cannot include levels of an integrated variable because you would end up with a stationary variable on the left hand side and a non-stationary combination (made up of some stationary variables and one non-stationary variable) on the right hand side, which is a contradiction.

Case 2: all endogenous variables are stationary, some exogenous variables are integrated.
A: If the exogenous variables are not cointegrated, include their first differences just as in Case 1.
B: If the exogenous variables are cointegrated, include both their first differences and the stationary combinations (error correction terms).
In both A and B cases the left-hand-side variables will be the regular endogenous variables. My answer is about what to include on the right hand side.

Case 3: one endogenous variable is integrated, one exogenous variables is integrated.
A: If the two variables are not cointegrated, include their first differences just as in Case 2A.
B: If the two variables are cointegrated, include both their first differences and their stationary combination (the error correction term).
In both A and B cases the left-hand-side variables will be the regular endogenous variables except for the integrated one which is replaced by its own first differences. My answer is about what to include on the right hand side.