# How to deal with a mix of I(1) and I(0) variables?

It seems that choosing the appropriate model for a mix of I(1) and I(0) variables is an hot topic on Stack Exchange but I was not able to find the solution I am looking for :

Considering a TS model with an I(1) dependent variable (y) and an I(0) explanatory variable (x),

• a model of VAR cannot be selected because y is non-stationary.
• a model of VECM is not appropriate since x is stationary. (*I have seen that I can add an explanatory variable which is non stationary I(1) in order to compute a VECM but this not possible in my case).

Is an ARDL model the best way to deal with my data ?

Is it possible to differentiate only the dependent variable y to compute a model of VAR?

• I have been asking and reading for years if one can mix I(1) and I(0) variables in ARDL or VAR and have never gotten an answer. – user54285 Apr 19 '20 at 22:48
• This paper might be useful: onlinelibrary.wiley.com/doi/abs/10.1002/jae.616 – Christoph Hanck Apr 23 '20 at 11:46
• A few related questions that I have answered can be found here. – Richard Hardy Apr 24 '20 at 5:35

## 2 Answers

Standard VAR model is applicable to integrated variables, after adding more lags to accommodate the degree of integration. For example, if maximum degree of integration among the variables is 2, one adds 2 lags to the model, in addition to chosen by, say AIC. This is standard practice in analysis of macroeconomic variables.

Estimation of single equation ARDL models require stationarity. If you believe your variables in an ARDL model are non-stationary but the error term is stationary, this means cointegration among your variables, which brings you back to VAR.

• Both parahraphs sound a little unusual to me. Could you offer any references? – Richard Hardy Apr 23 '20 at 16:52
• VAR with integrated variables was first considered in Toda and Yamamoto 1995, I believe. – Michael Apr 23 '20 at 21:53
• Note that the OP's situation is where $y$ is integrated but $x$s are not. A VAR on such data is certainly not appropriate; the left hand side of the model will diverge from the right hand side. Thus your first paragraph might be correct but irrelevant/misleading for the OP. Now what about the second paragraph? Does it not contradict Giles "ARDL Models - Part II - Bounds Tests"? – Richard Hardy Apr 24 '20 at 5:43

I agree with your logic. I think it is safe to difference $$y$$ and run an ARDL model. Then you will have a model with both the left and the right hand side being $$I(0)$$. The OLS estimators will have standard distributions, making inference straightforward.