I assume that you test for cointegration by applying the ADF for the regression residuals. Probably, the exogenous variables might enter (in levels) in the cointegration relationship. In that case, you need to adjust the ADF critical values to account for the additional regressor in the long run. Even if they are not part of the cointegrating relationship, you can include them in the ECM to  enrich the short run dynamics. Here, they enter in differenced form to ensure their stationarity.   

 Yes, surely you have to include the exogenous variables. To test for cointegration between two or more non-stationary time series, it simply requires running an OLS regression by including the exogenous variables in the regression equation, saving the residuals and then running the ADF test on the residual to determine if it is stationary. The time series are said to be cointegrated if the residual is itself stationary. In effect the non-stationary I(1) series have cancelled each other out to produce a stationary I(0) residual.

Interesting, I have used Eviews a litte, and it says that including exogenous variables may distort critical values, which is also what Christian Deger says. I take it that this is the same in VECM, as well as in Engle-Granger cointegration? That is however contradictory to what you say Mounir Belloumi, or? And another question, how do I add exogenous variables (in first differences) in VECM in Stata?

Note that the critical value of the ADF applied ro a series or to regression residuals differ. The latter depend, among others, on the number of exogenous regressors in the cointegration relationship. The more you include, the easier it is to produce stationary residuals.

If the aditional exogeneous variables are not in the cointegrated relationship they might still enter the ecm,  but in differenced form, if they are integrated in levels

Thanks, in that case it feels more safe to include them in the ECM (in differenced form). Seems hard to do in VECM in Stata however.

I do not think so there is any option to add any exogenous variable in E-G test

But in Johansen cointegration test, there is an option

In Stata? I get an error message trying to use  "si(variable variable)" in VEC in Stata. For E-G test I don't even find a command.

Hi Christian: You clearly know this material so excuse the dumn question but I never knew that exogenous variables could be thrown into an ECM that was derived from a cointegrating relationship. Is there material on this anywhere ? Wouldn't that mean that you couldn't just the cointegration estimates in the ECM and would have to re-estimate the whole thing from scratch ? Essentially, what I'm asking is: What role is the cointegrating relationship playing if you also include include exogenous variables in the ECM that were not involved in the cointegrating relation in the first place ? Thanks.

If the variables are cointegrated an ECM exists between precisely these variables and vice versa. So the (ordinary) ECM will include variables from the cointegrating relationship, i.e. the lagged deviations from the long run relationship and first differences of the variables in question, depending on their significance. However, as you already found cointegration, you can think to extend the (ordinary) ECM by adding further stationary variables. So I see no problem with that  

Hi Christian: I agree, except that a lot of times, the regression coefficient estimate of the cointegration relation is used in the ensuing estimation of the ecm ( i.e: the beta is used to calculate the short run component and then the ecm is estimated ) .. In this case, where you add in other exogenous variables,  I'm not sure but I don't think that trick of using the beta from the cointegrating relation  still holds so you would probably need to re-estimate the ecm from scratch.  but I agree that they can be added in. Anyway, I never thought about adding other exogenous variables so thanks for that wisdom.

Yes. Suppose there is cointegration between y and x, and the countegration residual is u=y-beta*x (we are in the EG approach). Then u(t-1) enters the ECM with a coefficient of alpha, which should be negative. Suppose there is another variable z that has nothing to do with your cointegrating relation, but is I(1). Then, you  can include dz, dz(-1), ... in the ECM. This does not affect the EG-cointegration result, which has been determined before. Probably it affects the alpha, but this is not a serious drawback.

Thanks Christian. But I would think that a wrong parameter estimate is not a good thing. It's  probably best to re-restimate the ECM from scratch and I think it then becomes non-linear ( unless I'm missing somethjing ) so all the new problems of local minimums then come into the story. that's a serious drawback IMHO.

Engle -Granger tests if a linear combination of a set of I(1) variables is stationary. There is no assumption that these variables are endogenous or exogenous and both may be included in the E-G regression.  The stationary linear combination (suitable normalised) is the equilibrium error.  The E-G residual based test for cointegration is a test that this is not-stationary. If this is rejected one proceeds on the basis that this equilibrium error is stationary.

Adding any non-stationary variable to the equilibrium error will make it non-stationary.

The ECM is a regression which measures the effect of departures from equilibrium and other short term effects (stationary) on  one of the variables.  The ECM should not include any endogenous variables either in the equilibrium error or added in first differences. If there are endogenous variables you should use a Johansen VECM.  This allows you not only test for weal exogeneity but also to allow for the extra feedbacks caused by the endogeneity. 

Hi John: Yes,  that's all true. But Christian is referrring to adding an exogenous variable to the ecm after the eg regression ( that doesn't include it ) has been estimated. I'm not clear what this has to do with what you said. But I agree with what you said.

Some variables might have only a temporary impact. For example, think of explaining inflation (I assume that this is I(0)). Then, price levels, money stocks and unit labour costs (or whatever) enter the cointegrration relationship and the cointegrating residual is calculated from them. But in the ECM, i.e. the inflation equation, other (stationary) variables will enter, such as unemployment gaps, oil price changes etc. But these variables will not  enter the cointegrating relationship

Hi Christaian: I see what you're saying  and I agree. I'm just saying that, in this case, the whole ECM neecs to be estimated from scratch.

Hi Mark, Agreed. For some reason I was reading something in the question or one of the earlier answers that simply was not there.

Mark, what do you mean by "the whole ECM needs to be estimated from scratch"? I mean, first you look for cointegration of y and x in level, and then insert the coefficient (if there is cointegration) into the ecm, along with y and x in first differences, and perhaps some stationary exogenous variables. What part needs to be made from scratch?

Anyway, what I can't wrap my head around, is if there is any difference between E-G ECM and VECM in all of this. STATA tells me I can't insert time series code (lags or first differences) of exogenous variables in VECM...

Hi Lars: What I mean is,  how can you assume that the insertion of the cointegration coeffficient is okay, since you're adding exogenous variables also. I would think the coefficient wouldn't necessarily stay the same since you are introducing new variables in the ecm.  Maybe one assumes that the exogenous variables don't effect the short term ( pullback ) component of the ecm ? I would find it unlikely that the beta coefficient wouldn't change, once the exogenous variable(s) is ( are) included.  Thanks for clarifying my confusion.

Hehe, no, I think I'm the most confused one. You might very well be right, but from what the others here write it seems as if the cointegration coeffficient will remain the same, even in the face of exogenous variables. The exogenous variables will most certainly effect the short term component of the ecm however. In fact, this was my original question. 

Lars and Mark: The VECM would be a different approach, as it allows for more than one cointegrating vector, allows all variables to adjust to short run deviations from the steady state (depending on weak exogeneity) etc. Exogenous variables can be included in the analysis, but do not necesssarily enter the cointegration vector, see the Johansen article on international parity conditions, where he introduced the oil price, probably in the Journal of Econometrics, 1992. Regarding the inclusion of variables for the short run: I would prefer to estimate the coinntegration relationship only between nonstationary variables. If you try to add stationary variables, you will end up with a cointegration rank exceeding 1, simply due to the presence of trivial cointegration vectors (they only include the stationary variable). Therefore, your "beta" should be be treated as fixed. Besides that, the coefficient of stationary variables in a relationship between nonstationary variables should eventually converge to 0.   

Ok. Christian, so it is ok to include exogenous variables in the VECM, which are not present in the Johansen cointegration test?

According to the Johansen article yes. But I would be cautios since I(1) "exogeneous" variables can affect the cointegration finding. The argument for the oil price in the analysis of international parity conditions (PPP, UIP) is that it can smooth out  "irregular" fluctuations or to achieve normality in the residuals. So, it  you detect cointegration only after including I(1) exogeneous variables, they should be definitely part of the cointegration vector. I do not expect serious problems if you include additional stationary variables in the ECM.

Hi Christian: So, are you saying that, in order to not have to worry about the exogenous variables effecting the short component, you can use a VECM to deal with that ?

This discussion is getting beyond me because I currently only grasp cointegration in the engle-granger sense ( if anyone knows a of a  good  intuitive explanation on Johahsen cointegration testing, it's appreciated ). But all the comments are appreciated and it's good to meet everyone.

VECM is only an alternative strategy to E-G, but superior,, accordiing to my view, as E-G has particular deficits (only oone cointegrating vector, de facto distinction between endogenous and exogenous variables etc). The inclusiion of further stationary variables in the equations describing the short run dynamics should be OK, whether or not you have E-G or a VECM. Despite that, economic theory could restrict an impact to the short run. For the Johansen approach, the best explanation might be the book of Katarina Juselius on the Cointegrated VAR model 

Hi Christian. I have that book so I'll check it out when I have time. ( hopefully in this lifetime ). Thanks for your beautiful explanations.

Engel Granger cointegration test is used to check for causality test

Hi all, thank you for this interesting and very informative discussion. I would like to just clarify one specific question I have with regards to my own thesis.

When adding exogenous variables to the ECM (second stage), these should be differenced if non-stationary I(1) and should be in levels if they are stationary I(0)?

Many thanks in advance.

I presume that you propose adding your exogenous variable to the short-run part of your equation ( along with lagged variables). If so your proposal is correct.

To validate this procedure you need two assumptions

I do not know of any way to validate these assumptions using the EG single equation system. I would think that a VECM analysis is more appropriate. Have you looked at the references given you by Christian Dreger ? Section 5.1 of Hunter, Burke and Canapa (2017), Multivariate Modelling of Non-Stationary Economic Time Series contains a good summary of this material. This material is not easy but it does provide an answer to the questions that you have asked.

Econometrics is the application of statistical/mathematical methods to an economic analysis. It may be possible that if one had an understanding of your economic analysis and data that one might be able to give a more appropriate answer or provide an alternative solution.

Thank you for your response John, I will read the material you have provided.

My thesis is to test whether positive and negative changes in the oil price impact Norway's trade balance symmetrically (analysing the trade balance in three ways, trade balance (TB), oil trade balance (OTB) and non-oil trade balance (NOTB)).

I have quarterly averages fro Brent Crude Oil I(1), quarterly trade balance data and quarterly GDP I(1) and USDNOK I(1) data to use as control variables.

I have found that TB and OTB are both I(1). The residuals from the first stage for TB are I(0), therefore long-run cointegration exists and I will use an ECM to assess, adding in differenced control variables in the second stage to help explain short-run dynamics. I am comfortable with this but the next two throw me...

2. The residuals from OTB's first stage are I(1), therefore no long-run cointegration and only short-run dynamics can be modeled with all variables in differences along with the differenced control variables?

1. NOTB is I(0). How do I analyse this relationship with the oil price and the control variables given they are I(1)?

I presume that your trade balances are Exports - Imports (at current prices?). Would you be better modelling the ratio (or the log of the ratio) of Exports to Imports? Using logs your model will have interpretation in terms of percentage changes or growth rates.

I would also consider using the logs of the Brent Crude price (?), GDP (at current prices ?) and the exchange rate rather than their levels.

Consider using Volume data rather than value data. Using the log of the export/ import ratio may also mitigate the effects of prices.

I do not have any knowledge of the Norwegian economy but I would be concerned about missing variables and your model specification. You may, therefore, encounter problems with specification tests on your model.

John, thank you for your input it was much appreciated. I was originally using logs but analysing the trade balances as ratios helped with the significance and interpretation of my results!