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My dependent variable, Y, is I(1) and is co-integrated with an I(1) independent variable X1. I understand I can estimate the regression Y~ X1 with OLS. Now can I include another I(0) independent variable X2 in the Engle-Granger test? If I find co-integration in the EG test, can I further estimate the regression, Y~X1+X2, with OLS?

I found conflicting ideas on Cross Validated for the similar questions where response variable is I(1) and covariates are mixed of I(1) and I(0).

Good to have covariates of both I(1) and I(0) in co-integration: I(1) and I(0) variables and cointegration in a trivariate system

Cannot involve covariate of I(0) in co-integration: Cointegration of I(0) and I(1)

Can anyone help to address my questions or comment on the conflicting ideas on co-integration? Thank you in advance.

RMIHANH
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  • I think you fliipped the links but the answer is no, you can't include the I(0) variable in the FINAL regression if the other two are I(1) and cointegrated. But, of course, if, in the general case, if you have three variables and two out of the three variables are I(1), then it's possible that those two can be cointegrated. Just don't include the third I(0) variable in the final model. – mlofton Sep 03 '19 at 03:15
  • Hi @mlofton, I revised the links. I would like to understand the consequence of including the I(0), and how it would violate the assumptions of co-integration or linear regression. My guess is that: there are only two I(1) variables in this case, so the stationary combination of the two I(1) variables must be unique. Now if I add the I(0) variable to the regression, the OLS estimates of the I(1) variables would be different from the stationary combination coefficients. Please let me know your thoughts. – RMIHANH Sep 03 '19 at 13:04
  • Hi: I think we're on the same page. I'm not very experienced in this but suppose the I(1) variables are cointegrated. So, there's a long run relationship between them and a short run relationship, (i.e: an ECM ). Now, if you then add an I(0) variable to the relationship, it's going to "mess up" the current model relationship. So, that's why you leave it out. – mlofton Sep 03 '19 at 13:21
  • Good question. I think you can add I(0). It would just take a form of VECM, I think. Maybe just skim through the Johansson test procedure in some textbook. You may get an idea there. In any case what you can do alternatively is extract error terms from cointegration regression and regress them on your I(0) variable separately. However, if the former is allowed then that would be a superior method because the estimation would be simultaneous. – Dayne Sep 04 '19 at 01:22
  • Hi @Dayne, I would like try to clarify your comments. I agree that I can always add the I(0) term in the VECM or ECM model, since everything in the function is stationary. But under what condition do you think I can add it to the long-run (level) regression. Thank you in advance. – RMIHANH Sep 05 '19 at 14:09
  • I think it wouldn't change things much. As I understand, cointegration looks for common stochastic trend in two variables. In VECM, first the error from long term relationship is extracted (which is stationary if there is cointegration) and then this error (call it $\epsilon_t$)is modeled by a VAR (or ARMA). Adding a new I(0) variable (call it X2) to long term relationship is just like adding a new variable to the VAR of $\epsilon_t$. Only thing is that in this VAR you are considering only one way relationship, i.e. you are not adding X2 in second relationship. – Dayne Sep 06 '19 at 04:24
  • (cont...) This is as good as fitting an ARMAX on $\epsilon_t$ in place of ARMA. The reason I talked about Johansson test is that in Multivariate time series with say p time series also, it is possible to have cointegration among q

     – Dayne Sep 06 '19 at 04:27

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