# Should multicollinearity problem be looked into while doing cointegration?

Multicollinearity and cointegration is not the same thing; however, if the series actually move together in the long-run i.e. are cointegrated, won't they also be collinear, making e.g. autoregressive models (for cointegration) and Johansen's test biased?

• A related question here. – Richard Hardy Mar 14 '16 at 14:59

Multicollinearity doesn't make estimators biased, rather it increases their variances. You could also think of it as reducing the effective sample size.

Also, in presence of cointegrated time series the models are often formulated in first differences and stationary combinations of variables. E.g. the vector error correction model has first differences of the dependent variables on the left hand side and both stationary combinations (the error correction terms) and first differences (lags of the dependent variables) on the right hand side. This makes multicollinearity in levels redundant.

Also, if cointegrated series are used in levels, superconsistency of estimators acts in opposite to multicollinearity, the former effectively increasing the sample size and the latter effectively reducing it. I am not sure of the relative effects of the two (which is stronger and which is weaker), but if the effect of superconsistency is at least as strong as that of multicollinearity, then the latter should be of limited importance in practice.

I don't have an answer regarding the Johansen test.

Edit: a good point raised by @Sympa is that multicollinearity regards the regressors as a group but not the relationship between the regressors and the regressand. So, for example, "multicollinearity" between $y$ and $x$ in a model $y=\beta_0+\beta_1 x+\varepsilon$ is not really an example of multicollinearity and is by no means harmful. Also, multicollinearity can be encountered simultaneously with cointegration in cases where several regressors are cointegrated and multicollinear. (For completeness, multicollinearity would be irrelevant in case there is only one regressor that is cointegrated with the regressand, just as in the equation above.)

• Is it any way to measure super-consistency? If the series are cointegrated they should be super-consistent so I guess thats an arguement for reduced multicolliniearity. Further if there is no serial correlation, heteroskedasticity, non-normality, misspecification etc then maybe its fine to have a fairly high multicollinearity ? – Parash Dejmar Mar 14 '16 at 15:33
• 1. Superconsistency as a term applies to OLS estimators, not to the time series themselves. 2. Superconsistency will not reduce multicollinearity (these are two different things), but it could compensate its effect. 3. Under no serial correlation, homoskedasticity, normality etc. multicollinearity will still have its effect. How bad it is depends on the data set at hand. – Richard Hardy Mar 14 '16 at 15:37
• Regarding your Edit. This implies that multicollinear is indeed a problem for a model such as: $$\Delta y_{t} = \beta_{0} + \beta_{1} \Delta x_{t-1} + \beta_{2}\Delta x2_{t-1}+ \beta_{3}x_{t-1} +\beta_{4}x2_{t-1}+\beta_{5}y_{t-1} + u_{t}$$? Because here $x,x2,y$ can be cointegrated but if $x,x2$ are excellent explanatories for $y$ then there will be multicollinearity. This is somewhat confusing. – Parash Dejmar Mar 15 '16 at 9:06
• Once you separate multicollinearity and cointegration in the thought process, this need not be confusing anymore; or is it still? – Richard Hardy Mar 15 '16 at 9:36
• Sorry, the confusing part is related to how to assess both the collinearity and cointegration, and when (or if it is) its fine to have collinearity when cointegration is present. Im running my model now according to the one above, and find cointegration. However the VIF is high around 20, im not sure how to proceed. – Parash Dejmar Mar 15 '16 at 9:47

Multicollinearity is an issue between independent variables. Very high or excessive correlations between independent variables can be problematic for several reasons. When correlation approaches 1, the Tolerance approaches 0, and the closed form Matrix algebra used to solve regression equations collapses.

Cointegration is not an issue, but rather a remedy whereby your regression that has variables with unit roots on either side of the regression equation is still ok (it is not spurious). And, that is because even though those variables have unit roots, they do move somewhat together.

In summary, multicollinearity is an issue solely between independent variables. And, should always be looked at as such. It does not always need to be resolved. Cointegration is a property between the dependent variable and at least one independent variable. And, that is a good thing.

When you model a dependent variable it makes sense that you want a close relationship between it and an independent variable (cointegration). But, that is completely different from independent variables having very close relationships between themselves (multicollinearity) which can at times be a problem.

• Yes I do agree however if we consider cointegration from an autoregressive distribution lags (ARDL) perspective $$\Delta y_{t} = \beta_{0} + \beta_{1} \Delta x_{t-1} + \beta_{2}\Delta x_{t-2} + \beta_{3}y_{t-1}+\beta_{4}x_{t-1}+\Delta u_{t}$$. Where the cointegration tests, tests if $\beta_{3}=\beta_{4}=0$. If they are cointegrated then they may also be colliniear. – Parash Dejmar Mar 14 '16 at 15:24
• Good point that multicollinearity only applies to the right-hand-side variables. However, this does not preclude it happening under cointegration where several right-hand-side variables may be cointegrated among themselves. Thus your statement Cointegration is a property between the dependent variable and at least one independent variable is not generally true. (Cointegration considers groups of variables in general and does not only apply to cases where a regression model has already been formulated.) – Richard Hardy Mar 14 '16 at 15:41
• Parish, when you deal with autoregressive distribution lags, does'nt it warrant special consideration related to multicollinearity? In other words, if you use one or more lags of the same variable, it is more likely that they may hit multicollinearity thresholds. But, in such situations multicollinearity may be ok? – Sympa Mar 14 '16 at 16:02
• yea exactly Sympa, thats what Im wondering :) It feels like atleast some multicollinearity should be ok... – Parash Dejmar Mar 14 '16 at 17:05
• I would like to encourage you to revise the statement where you associate cointegration with dependent vs. independent variables, as I noted above. Or if you maintain that I am mistaken, I would be interested to hear your arguments. – Richard Hardy Mar 14 '16 at 18:47