# Time series: one I(1) and one I(0) variable, should I use VAR/VEC, test for cointegration?

Like the title says, I've got two time series, one is stationary to begin with and thus has no unit root, the other time serie is stationary after one-time differencing.

I want to create a model out of this and I know that when unit roots are present, I should test for cointegration. But I've read in Engle & Granger (1987) that cointegration tests are only to be done when you have two or more I(1) variables, is that correct?

So I cannot find in literature if I should now use a VAR model on differences or test for cointegration and perhaps do a Vector Error Correction model.

Can anyone help me? I would be very thankful!

• Please give full references. Minimal name (date) references may be widely familiar in your field, but the complete reference may help many people. – Nick Cox May 29 '13 at 9:38
• My apoligies, this is the paper. I am unable to cite the passage on which I concluded that cointegration is not possible to be tested with a I(1) and I(0) variable, so perhaps I'm incorrect. I hope some one can enlighten me on this process. – Robin Papa May 29 '13 at 9:49

A $I(0)$ and a $I(1)$ timeseries can not be cointegrated. There is no linear combination of the timeseries that is stationary. And the definition of cointegration is if there is a combination of them that is stationary, they're cointegrated.

I think you should fit a VAR with the stationary variable in levels and the non-stationary variable in first difference.

Good luck!

• That is one way to look at it. Johansen (and others) do however also allow for $I(0)$ variables, arguing that such a variable is a (trivial) linear combination of variables that leads to stationary series, viz. just the variable itself. – Christoph Hanck Feb 24 '15 at 12:42

If johansen test result is not significant, meaning no cointegration, then take the 1st difference of the other variable to ensure stationarity. In some cases you may need to take the ln of the 1 st difference.

In case you have a mix of I(0) and I(1) variables, you can apply the tests proposed by Pesaran et al (2001), where you can test for cointegration. The link to the paper is:

http://onlinelibrary.wiley.com/doi/10.1002/jae.616/pdf

All the best

• Please provide a complete citation & a precis of the information in the article so future readers can decide if it is something they want to pursue. – gung - Reinstate Monica Jul 1 '15 at 20:43

ARDL model approach described by Pesaran is the only way to find the cointegration among the variables having different orders I(0) and I(1) but keeping in mind none of the variable should stationery at I(2)