# Vector autoregression for mix of stationary and nonstationary variables

I am currently investigating the impact of certain indicators such as GDP and inflation on the stock market. However some of my variables are non-stationary and some stationary in levels. All variables are stationary in first differences.

My question:

1. Since I have a mix of I(0) and I(1), I need to take the first difference of the I(1) variables and then use VAR. Say that I have a dependent variable $y$ which is I(0) and independent variables $x_1$ which is I(1) and $x_2$ which is I(0). Do I take the first differences of only $x_1$ and then apply the VAR model? Is that correct?

2. If I take the first differences, then might I lose the long-run relationship between the variables?

3. Is ARDL the better method in this case?

If you have any references regarding the above I would be grateful.

• I have answered a related question here. – Richard Hardy Jan 31 '16 at 13:28

## 1 Answer

1. In your example there is only one integrated variable, $x_1$; hence, taking the first differences of $x_1$ and leaving the other variables $y$ and $x_2$ in levels before estimating a VAR model is fine. But if you had more than one integrated variable, you should consider cointegration.

2. If there is more than one integrated variable and some of the integrated variables are cointegrated, then taking the first differences of the integrated variables and ignoring the cointegrating relationship would indeed be suboptimal. You could say that then you lose the long-run relationship.

3. I don't know.

Note: differencing variables that are I(0) in levels should be avoided. It induces extra structure that need not be there. The term is "overdifferencing". See this, this or this.

• Thanks. Regarding 1) So if I have 8 variables, where four of them are I(0), and the rest are I(1) then I should use maybe Johansen test disregarding the fact that I have a mix of stationarity? – Parash Dejmar Jan 31 '16 at 13:09
• I don't remember this right away, but there should be some guidelines for what to do with mixes of I(0) and I(1) variables. You would have to do a search on that. If I had to answer immediately, I would say do the Johansen procedure on the I(1) variables only and then add the I(0) ones when building your model. But that might not be optimal. – Richard Hardy Jan 31 '16 at 13:20